Scholium Symbolicum — page 1 of 6

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "axiom:bk1_axiomata_prima"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "sec:bk1_foundational_structures",
  "label": "sec:bk1_foundational_structures",
  "latex_body": "",
  "line": 13,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Foundational Structures",
  "ref_roles": [
    {
      "context": "",
      "label": "axiom:bk1_axiomata_prima",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book1.tex",
      "target_line": 3,
      "target_type": "axiom"
    }
  ],
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_pre_geometric_nature",
    "definition:bk1_bounded_observer",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space",
    "definition:bk1_symbolic_category",
    "definition:bk1_symbolic_manifold",
    "lemma:bk1_universality_of_proto_symbolic_space",
    "subsec:appD_ct_contribution_differentiation"
  ],
  "cites": [
    "axiom:bk1_axiomata_prima"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_let_cats_be_the_category",
  "label": "definition:bk1_let_cats_be_the_category",
  "latex_body": "\\begin{definition}[Category of Structures]\n\\label{definition:bk1_let_cats_be_the_category}\nLet \\(\\catS\\) be the category whose\n\\begin{itemize}\n  \\item \\textbf{Objects} are structures \\(P_\\lambda\\) indexed by an ordinal stage \\(\\lambda \\in \\mathsf{Ord}\\);\n  \\item \\textbf{Morphisms} \\(f_{\\lambda\\mu}\\colon P_\\lambda \\to P_\\mu\\) are structure–preserving maps compatible with emergence order (\\(\\lambda \\le \\mu\\));\n  \\item \\textbf{Initial object} is \\(\\emptyset \\in Ob(\\catS)\\), representing the pre-structured void.\n\\end{itemize}\nWe assume \\(\\catS\\) is cocomplete, so every small diagram admits a colimit, allowing the construction of structural configurations from emergence-aligned diagrams. The initial object $\\emptyset$ is the direct formal image of Axiom~\\ref{axiom:bk1_axiomata_prima}: the pre-structured void from which drift generates existence. \\qedhere\n\\end{definition}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Typed ambient interface only: an arbitrary staged category, initial void, and universe-bounded cocompleteness are recorded as supplied data. This list-first definition has no ontological priority over the co-emergent drift/reflection operation and does not manufacture either operation or the later manifold."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-001"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "ScholiumC.CategoryOfStructures.existsUnique_from_empty",
      "ScholiumC.CategoryOfStructures.exists_colimit_cocone",
      "ScholiumC.CategoryOfStructures.hom_advances_stage"
    ]
  },
  "line": 16,
  "macros_used": [
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Category of Structures",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ural configurations from emergence-aligned diagrams. The initial object $\\emptyset$ is the direct formal image of Axiom~\\ref{axiom:bk1_axiomata_prima}: the pre-structured void from which drift generates existence. \\qedhere \\end{definition}",
      "label": "axiom:bk1_axiomata_prima",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book1.tex",
      "target_line": 3,
      "target_type": "axiom"
    }
  ],
  "refs": [
    "axiom:bk1_axiomata_prima"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "certificate_tier": "C",
  "cited_by": [
    "abs:press",
    "axiom:bk1_observable_gradation_of_pre_geometric_operations",
    "axiom:bk4_bounded_accessibility",
    "axiom:bk4_refinement_contraction",
    "axiom:bk8_curvature_transformation",
    "corollary:bk1_event_horizon_identity_field",
    "corollary:bk4_homological_coherence_observer_bounds",
    "corollary:bk4_smoothness_as_epistemic_phenomenon",
    "corollary:bk4_symbolic_lightcone",
    "corollary:bk8_resonant_cognition",
    "corollary:bk9_freedomentropy_complementarity",
    "definition:appC_bounded_observation_frame",
    "definition:appC_coherence_functional",
    "definition:appD_llm_observer_tuple",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_newtonian_category_error",
    "definition:bk1_observer_gradient",
    "definition:bk1_observer_horizon_structure",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_problem_of_symbolic_smoothness",
    "definition:bk1_shared_boundary_paradox",
    "definition:bk1_symbolic_hypothesis",
    "definition:bk4_bounded_observer",
    "definition:bk4_collapse_of_symbolic_ide",
    "definition:bk4_epistemic_differential_o",
    "definition:bk4_fuzzy_gradient",
    "definition:bk4_fuzzy_integral_operator",
    "definition:bk4_fuzzy_symbolic_substitution",
    "definition:bk4_observer_differentiable_",
    "definition:bk4_observer_metric",
    "definition:bk4_observer_valid_different",
    "definition:bk4_symbolic_space",
    "definition:bk4_test_time_coherent_sampling",
    "definition:bk4_test_time_integrative_expansion",
    "definition:bk4_tilda_substitution",
    "definition:bk5_two_way_street_tensor",
    "definition:bk6_symbolic_confidence_field",
    "definition:bk7_adaptive_refinement_recurrence",
    "definition:bk7_observerrelative_symbolic_error_field",
    "definition:bk7_operational_resolution_uncertainties",
    "definition:bk7_symbolic_reflexive_validation_srv",
    "definition:bk7_symbolic_uncertainty",
    "definition:bk8_observer_relative_artifact",
    "definition:bk8_sr_triplet",
    "definition:bk9_reflective_dyad",
    "demonstratio:bk4_fuzzy_forward_mode",
    "demonstratio:bk4_prompt_time_ttdc",
    "demonstratio:bk7_convergence_within_reflective_basin",
    "lemma:bk1_bounded_approximation_and_interpretability",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "lemma:bk2_wellposedness_symb_prob_space",
    "lemma:bk4_gradient_stability",
    "lemma:bk4_properties_of_ttcs",
    "proof:bk1_constitutive_bootstrap_extraction",
    "proof:bk1_contrapositive_search_principle",
    "proof:bk1_drift_deviation_bound",
    "proof:bk1_energy_bound_identity",
    "proof:bk1_event_horizon_identity_field",
    "proof:bk1_fix_s_in_s",
    "proof:bk1_observer_kernel_convolution",
    "proof:bk1_observer_threshold_reflexivity",
    "proof:bk1_sketch_effective_proto_drift_field_induction",
    "proof:bk1_sketch_observed_consequences",
    "proof:bk4_fuzzy_deriv_algebra",
    "proof:bk4_fuzzy_exponential_rule",
    "proof:bk4_fuzzy_substitution_drift_smoothing",
    "proof:bk4_observer_capacity_bound",
    "proof:bk4_observer_relative_smoothness",
    "proof:bk4_sketch_extracting_recrusive_curvature",
    "proof:bk4_symbolic_work_path_dependence",
    "proof:bk4_timescale_separation_hierarchy",
    "proof:bk8_resonant_cognition",
    "proof:bk9_meta_reflective_memory_integration",
    "proof:bk9_symbolic_viability",
    "proposition:bk1_observer_relative_bounded_approximation",
    "proposition:bk1_stage_composite_operators_are_interpretable",
    "proposition:bk4_symbolic_work_path_dependence",
    "proposition:bk9_modes_of_re_interpretation",
    "remark:appC_born_rule_dependency",
    "remark:appD_llm_tuple_anchors",
    "remark:bk3_toward_symbolic_evolution",
    "remark:bk4_computational_complexity",
    "remark:bk4_observer_relative_ttdc",
    "remark:bk4_symbolic_work_capacity",
    "remark:bk8_inference_principle_over_confidence_loss_tradeoff",
    "scholium:bk1_constitutive_reflex",
    "scholium:bk1_curvature_flux_kin_kout",
    "scholium:bk1_emergence_envelope",
    "scholium:bk1_epistemic_humility",
    "scholium:bk1_interpretability_two_axes",
    "scholium:bk1_resolution_of_continuum_disjunction",
    "scholium:bk2_on_hypotheses_as_thermodyn",
    "scholium:bk3_hypotheses_as_cognitive_membranes",
    "scholium:bk4_dynamics_of_observer_frame",
    "scholium:bk4_irreversibility_as_trace",
    "scholium:bk4_nested_frames",
    "scholium:bk4_o_boundedness_unifying_principle",
    "scholium:bk4_recursive_introspection",
    "scholium:bk4_reflexive_physics_emergence",
    "scholium:bk4_role_of_observer_induced_metric",
    "scholium:bk4_symbolic_drift_fields",
    "scholium:bk4_the_nature_of_truth",
    "scholium:bk4_the_observer_as_weaver",
    "scholium:bk4_topological_complexity_semantic_richness",
    "scholium:bk4_ttcs_link_traversal",
    "scholium:bk4_ttcs_stochastic_operator",
    "scholium:bk4_ttdc_impulse_collapse",
    "scholium:bk4_ttdc_symbolic_singularity",
    "scholium:bk4_zero_is_idealized_in_boundedness",
    "scholium:bk5_constant_of_becoming",
    "scholium:bk7_constrained_uncertainty_motivation",
    "scholium:bk7_power_organizational_navigational",
    "sec:appC_born_preamble",
    "sec:appC_born_rule",
    "sec:bk7_pisu_universal_symbolic_uncertainty",
    "subsec:appC_born_observer_structures",
    "subsec:bk3_preamble_to_symbiosis",
    "subsec:bk4_fuzzy_sum_rule",
    "subsec:bk4_symbolic_identity_collapse",
    "subsec:bk7_emergence_symbolic_uncertainty",
    "subsec:bk7_pisu_axiom_statement",
    "subsec:bk7_pisu_motivation",
    "subsec:bk7_pisu_revisited_power_uncertainty",
    "subsec:bk7_pisu_scholium",
    "subsec:bk7_sources_regimes_uncertainty",
    "theorem:bk1_constitutive_bootstrap",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk4_existence_observer_valid_derivatives",
    "theorem:bk4_fuzzy_chain_rule",
    "theorem:bk4_fuzzy_exponential_rule",
    "theorem:bk4_fuzzy_fundamental",
    "theorem:bk4_fuzzy_jacobian",
    "theorem:bk4_fuzzy_logarithmic_rule",
    "theorem:bk4_fuzzy_power_rule",
    "theorem:bk4_fuzzy_product_rule",
    "theorem:bk4_fuzzy_quotient_rule",
    "theorem:bk4_fuzzy_sum_rule",
    "theorem:bk4_fuzzy_symbolic_geometry_theorem",
    "theorem:bk4_paradoxical_arrow_of_time",
    "theorem:bk4_restated_fuzzy_symbolic_geometry_theorem",
    "theorem:bk4_symbolic_link_activation",
    "theorem:bk8_gradient_dissipation_balance"
  ],
  "cites": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "depends_on": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_bounded_observer",
  "label": "definition:bk1_bounded_observer",
  "latex_body": "\\begin{definition}[Bounded Observer]\n\\label{definition:bk1_bounded_observer}\nA \\emph{bounded observer} is a triple\n\\[\n\\Obs = \\bigl(N_\\Obs,\\;\\{\\delta_\\Obs^{\\,n}\\}_{n=1}^{N_\\Obs},\\;\\epsilon_\\Obs\\bigr)\n\\]\nwhere\n\\begin{enumerate}[label=(\\roman*)]\n  \\item \\(N_\\Obs \\in \\mathbb{N}\\) is the \\textbf{maximal differentiation order};\n  \\item \\(\\delta_\\Obs^{\\,n}\\colon P \\to P\\) are internal \\(n^{\\text{th}}\\)-order differentiation operators;\n  \\item \\(\\epsilon_\\Obs\\colon M \\to \\mathbb{R}_{>0}\\) is a \\textbf{resolution threshold}, assigning each point a smallest observable deviation.\n\\end{enumerate}\nThis construct enables structure to be interpreted from within the category \\(\\catS\\) (cf.~\\ref{definition:bk1_let_cats_be_the_category}) and over a manifold-like membrane whose topology reflects emergent curvature.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Modeled as the BoundedObserver structure (N, delta, eps) used downstream by Traceable/interpretable_of_factor_and_traceable; no standalone theorem, differentiation operators kept opaque (Nat -> Real -> Real) rather than manifold operators."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-002"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "ScholiumA.interpretable_of_factor_and_traceable"
    ]
  },
  "line": 27,
  "macros_used": [
    "Obs",
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Bounded Observer",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "e deviation. \\end{enumerate} This construct enables structure to be interpreted from within the category \\(\\catS\\) (cf.~\\ref{definition:bk1_let_cats_be_the_category}) and over a manifold-like membrane whose topology reflects emergent curvature. \\end{definition}",
      "label": "definition:bk1_let_cats_be_the_category",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 16,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_observer_gradient",
  "label": "definition:bk1_observer_gradient",
  "latex_body": "\\begin{definition}[Observer as Structured Gradient]\n\\label{definition:bk1_observer_gradient}\n\\leavevmode\\newline\nIn \\textit{Principia Symbolica}, the Observer (Def.~\\ref{definition:bk1_bounded_observer}) emerges as a \\textit{structured gradient}: a dimensional cascade bridging fundamental physics, mathematics, and computation. Each level corresponds to core structures across quantum physics, mathematical physics, high-energy theory, machine learning, and statistical mechanics:\n\n\\begin{enumerate}\n  \\item \\textbf{Observation Point (0D) — The Measurement Nexus}  \n  \\begin{itemize}\n    \\item \\textbf{quant-ph}: Quantum measurement collapse—the irreducible moment where superposition becomes definite state\n    \\item \\textbf{math-ph}: Singular manifold point where local charts fail and topology shifts\n    \\item \\textbf{hep-th}: Worldline intersection, the minimal spacetime object near trajectory endpoints\n    \\item \\textbf{cs.LG}: Attention head query—the computational primitive that selects specific information from distributed representations\n    \\item \\textbf{cond-mat.stat-mech}: Critical point—where phase transitions occur and correlation length diverges\n  \\end{itemize}\n  \\textit{The irreducible locus where structural differentiation first emerges from undifferentiated potential.}\n\n  \\item \\textbf{Referential Frame (2D) — The Coherence Manifold}  \n  \\begin{itemize}\n    \\item \\textbf{quant-ph}: Quantum reference frame—defines relative phases and enables consistent measurement across subsystems\n    \\item \\textbf{math-ph}: Coordinate chart/atlas—local diffeomorphism establishing tangent space structure\n    \\item \\textbf{hep-th}: Worldsheet—2D surface swept by string, encoding fundamental interactions\n    \\item \\textbf{cs.LG}: Embedding space—learned representation manifold where semantic relationships become geometric\n    \\item \\textbf{cond-mat.stat-mech}: Order parameter field—macroscopic variable describing collective behavior and symmetry breaking\n  \\end{itemize}\n  \\textit{Bounded surfaces of coherence that transform local curvature into navigable topology.}\n\n  \\item \\textbf{Field of Interpretation (3D+) — The Recursive Manifold}  \n  \\begin{itemize}\n    \\item \\textbf{quant-ph}: Quantum field configuration—excitations propagating through vacuum, enabling non-local correlations\n    \\item \\textbf{math-ph}: Fiber bundle total space enabling parallel transport of geometric data\n    \\item \\textbf{hep-th}: Bulk spacetime where holographic duality links boundary and interior\n    \\item \\textbf{cs.LG}: Transformer layer stack—recursive processing enabling contextual understanding across arbitrary distances\n    \\item \\textbf{cond-mat.stat-mech}: Renormalization group flow—systematic coarse-graining revealing emergent scales and universality\n  \\end{itemize}\n  \\textit{Activated structured space where frames undergo mutual interrogation, enabling temporal continuity and TTDC collapse.}\n\n  \\item \\textbf{Agentic Observer (n-D, Reflexive) — The Self-Modifying Geometry}  \n  \\begin{itemize}\n    \\item \\textbf{quant-ph}: Quantum agent/observer—system capable of self-measurement and adaptive quantum error correction\n    \\item \\textbf{math-ph}: Automorphism group—symmetries that preserve structure while enabling self-transformation\n    \\item \\textbf{hep-th}: M-theory moduli space—parameter space of all possible string compactifications, self-consistently determined\n    \\item \\textbf{cs.LG}: Meta-learning architecture—networks that learn to modify their own learning algorithms and representations\n    \\item \\textbf{cond-mat.stat-mech}: Self-organized criticality—systems that dynamically tune themselves to critical points without external control\n  \\end{itemize}\n  \\textit{Recursive participant that constructs its own frames, adjusts curvature tolerances, and enacts geometric responsibility.}\n\\end{enumerate}\n\n\\textbf{Cross-Field Synthesis.}  \nThe Observer gradient unifies measurement (quant-ph), geometric structure (math-ph), dimensional transcendence (hep-th), representational learning (cs.LG), and emergent organization (cond-mat.stat-mech). Each field contributes essential analogues:\n\\begin{align}\n\\text{Measurement} &\\rightarrow \\text{Geometry} \\rightarrow \\text{Holography} \\rightarrow \\text{Meta-Learning} \\rightarrow \\text{Self-Organization} \\\\\n\\text{Collapse} &\\rightarrow \\text{Curvature} \\rightarrow \\text{Emergence} \\rightarrow \\text{Recursion} \\rightarrow \\text{Criticality}\n\\end{align}\n\n\\textbf{Operationalization Principle.}  \nThis framework enables implementing bounded observers in LLMs through: quantum-inspired attention mechanisms (measurement-based selection), geometric embedding spaces (manifold learning), holographic compression, recursive self-modification (meta-learning), and critical self-tuning (adaptive complexity regulation). The Observer becomes a computational architecture that embodies the deep mathematical structures underlying conscious structured processing.\n\\end{definition}",
  "line": 97,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer as Structured Gradient",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "dient] \\label{definition:bk1_observer_gradient} \\leavevmode\\newline In \\textit{Principia Symbolica}, the Observer (Def.~\\ref{definition:bk1_bounded_observer}) emerges as a \\textit{structured gradient}: a dimensional cascade bridging fundamental physics, mathematics, and comput",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk8_binding_curvature_limit",
    "definition:bk8_symbolic_projection",
    "scholium:bk3_hypotheses_as_cognitive_membranes"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "sec:bk1_minimal_structure_for_symbolic_emergence"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "roximation] \\label{proposition:bk1_observer_relative_bounded_approximation} Let \\(S \\in Ob(\\catS)\\) be a structure (cf.~\\ref{sec:bk1_minimal_structure_for_symbolic_emergence}), and let \\(\\Obs\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}). Then there exists an operator \\",
      "label": "sec:bk1_minimal_structure_for_symbolic_emergence",
      "line_distance": 1018,
      "role": "navigation",
      "target_line": 1173,
      "target_type": "section"
    }
  ],
  "forward_refs": [
    "sec:bk1_minimal_structure_for_symbolic_emergence"
  ],
  "id": "proposition:bk1_observer_relative_bounded_approximation",
  "label": "proposition:bk1_observer_relative_bounded_approximation",
  "latex_body": "\\begin{proposition}[Observer–Relative Bounded Approximation]\n\\label{proposition:bk1_observer_relative_bounded_approximation}\nLet \\(S \\in Ob(\\catS)\\) be a structure (cf.~\\ref{sec:bk1_minimal_structure_for_symbolic_emergence}), and let \\(\\Obs\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}).  \nThen there exists an operator \\(\\Phi_\\lambda\\colon S \\to S\\) such that\n\\[\n\\bigl\\|\\;K_\\Obs * \\bigl(\\Phi_\\lambda(s)-s\\bigr)\\;\\bigr\\|\\; \\le \\epsilon_\\Obs(s)\n\\quad\\text{for all } s \\in S,\n\\]\ni.e., \\(\\Phi_\\lambda\\) is a non-trivial \\(\\Obs\\)-bounded approximation of the identity on \\(S\\).\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Existence holds given K kills the zero vector and eps is pointwise nonnegative; Phi = id witnesses it. Honesty gap: this is exactly the trivial case the source's 'non-trivial' qualifier excludes."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-027"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumC.exists_bounded_approx"
    ]
  },
  "line": 155,
  "macros_used": [
    "Obs",
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer–Relative Bounded Approximation",
  "proof_labels": [
    "proof:bk1_fix_s_in_s"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "e a structure (cf.~\\ref{sec:bk1_minimal_structure_for_symbolic_emergence}), and let \\(\\Obs\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}). Then there exists an operator \\(\\Phi_\\lambda\\colon S \\to S\\) such that \\[ \\bigl\\|\\;K_\\Obs * \\bigl(\\Phi_\\lambda(s)-s",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "roximation] \\label{proposition:bk1_observer_relative_bounded_approximation} Let \\(S \\in Ob(\\catS)\\) be a structure (cf.~\\ref{sec:bk1_minimal_structure_for_symbolic_emergence}), and let \\(\\Obs\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}). Then there exists an operator \\",
      "label": "sec:bk1_minimal_structure_for_symbolic_emergence",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1173,
      "target_type": "section"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "sec:bk1_minimal_structure_for_symbolic_emergence"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk4_refinement_contraction"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "oximation exists for any observer–relative structure, realizable via kernel-based bounded structured approximation (cf.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) and the observer’s resolution parameters (cf.~\\ref{definition:bk1_bounded_observer}). \\end{proof}",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "line_distance": 266,
      "role": "interpretive_bridge",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "id": "proof:bk1_fix_s_in_s",
  "label": "proof:bk1_fix_s_in_s",
  "latex_body": "\\begin{proof}[Symbol Preservation Under Drift–Reflection Fixation]\n\\label{proof:bk1_fix_s_in_s}\n\\leavevmode\n\nFix an element \\(s \\in S\\).  \nLet \\(\\varepsilon(s)\\) be a perturbation satisfying  \n\\(\\lVert K_\\Obs * \\varepsilon(s)\\rVert \\le \\tfrac12\\,\\epsilon_\\Obs(s)\\).  \nFor example, take a local Gaussian blur scaled by \\(\\tfrac12\\,\\epsilon_\\Obs(s)\\).  \nDefine \\(\\Phi_\\lambda(s) \\coloneqq s + \\varepsilon(s)\\).  \nBy linearity of convolution:\n\\[\n\\lVert K_\\Obs * \\bigl(\\Phi_\\lambda(s) - s\\bigr)\\rVert\n= \\lVert K_\\Obs * \\varepsilon(s)\\rVert\n\\le \\tfrac12\\,\\epsilon_\\Obs(s)\n< \\epsilon_\\Obs(s),\n\\]\nso the bound is satisfied.  \nMoreover, since \\(\\varepsilon \\not\\equiv 0\\), we have \\(\\Phi_\\lambda \\neq id\\).  \nHence, a bounded structured approximation exists for any observer–relative structure, realizable via kernel-based bounded structured approximation (cf.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) and the observer’s resolution parameters (cf.~\\ref{definition:bk1_bounded_observer}).\n\\end{proof}",
  "line": 167,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbol Preservation Under Drift–Reflection Fixation",
  "proves": "proposition:bk1_observer_relative_bounded_approximation",
  "ref_roles": [
    {
      "context": "on (cf.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) and the observer’s resolution parameters (cf.~\\ref{definition:bk1_bounded_observer}). \\end{proof}",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "oximation exists for any observer–relative structure, realizable via kernel-based bounded structured approximation (cf.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) and the observer’s resolution parameters (cf.~\\ref{definition:bk1_bounded_observer}). \\end{proof}",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_observer_relative_interpretability",
  "label": "subsec:bk1_observer_relative_interpretability",
  "latex_body": "",
  "line": 187,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer–Relative Interpretability",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "abs:press",
    "axiom:bk8_binding_curvature_limit",
    "definition:bk3_autophagic_drift",
    "definition:bk4_test_time_integrative_expansion",
    "definition:bk4_test_time_precision_refinement",
    "definition:bk8_symbolic_projection",
    "lemma:bk1_bounded_approximation_and_interpretability",
    "lemma:bk4_ttpr_interpretability_preserved",
    "proof:bk1_boundedness_encoding_cost",
    "proof:bk1_energy_bound_identity",
    "proof:bk4_interpretability_preservation",
    "proof:bk8_sketch_observer_interoperability",
    "proposition:bk1_stage_composite_operators_are_interpretable",
    "proposition:bk8_genetic_symbolic_resonance",
    "scholium:bk1_interpretability_two_axes",
    "scholium:bk2_on_hypotheses_as_thermodyn",
    "scholium:bk4_topological_complexity_semantic_richness"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "erver (cf.~\\ref{definition:bk1_bounded_observer}), and let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}, \\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}). Fix measurable thresholds $\\nu_{\\mathcal{O}}, \\ep",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "line_distance": 183,
      "role": "interpretive_bridge",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "and let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}, \\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}). Fix measurable thresholds $\\nu_{\\mathcal{O}}, \\epsilon_{\\mathcal{O}} : M \\to \\mathbb{R}^+$ satisfying \\[ 0 < \\nu_{\\",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "line_distance": 243,
      "role": "interpretive_bridge",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "id": "definition:bk1_observer_relative_interpretability",
  "label": "definition:bk1_observer_relative_interpretability",
  "latex_body": "\\begin{definition}[Observer–Relative Interpretability]\n\\label{definition:bk1_observer_relative_interpretability}\n\nLet $\\mathcal{O} = (N_{\\mathcal{O}}, \\{\\delta^n_{\\mathcal{O}}\\}_{n=1}^{N_{\\mathcal{O}}}, \\epsilon_{\\mathcal{O}})$ be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}),  \nand let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}, \\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}).  \nFix measurable thresholds $\\nu_{\\mathcal{O}}, \\epsilon_{\\mathcal{O}} : M \\to \\mathbb{R}^+$ satisfying\n\\[\n0 < \\nu_{\\mathcal{O}}(x) < \\epsilon_{\\mathcal{O}}(x) \\quad \\text{for all } x \\in M.\n\\]\n\n\\begin{enumerate}[label=\\textbf{(I\\arabic*)}]\n\\item \\textbf{Distinguishability:} $\\Phi : P \\to P$ is $\\mathcal{O}$–distinguishable at $s \\in P$ if  \n      $\\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| \\ge \\nu_{\\mathcal{O}}(s)$.\n\n\\item \\textbf{Boundedness:} $\\Phi$ is $\\mathcal{O}$–bounded at $s$ if  \n      $\\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| \\le \\epsilon_{\\mathcal{O}}(s)$.\n\n\\item \\textbf{Differential Traceability:} $\\Phi$ is $\\mathcal{O}$–traceable at $s$ if  \n      there exists $n \\in \\{1, \\ldots, N_{\\mathcal{O}}\\}$ such that  \n      $\\delta^n_{\\mathcal{O}}(\\Phi(s)) \\ne \\delta^n_{\\mathcal{O}}(s)$.\n\\end{enumerate}\n\nWe say $\\Phi$ is $\\mathcal{O}$–interpretable at $s$ if conditions \\textbf{(I1)}–\\textbf{(I3)} all hold,  \nand \\emph{globally $\\mathcal{O}$–interpretable} if they hold for all $s \\in P$.  \n\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "(I1) distinguishability and (I2) boundedness are derived (not assumed) from a c*eps factorization; (I3) traceability is modeled concretely via a Finset.range witness over BoundedObserver.delta rather than manifold differentiation operators."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-007"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.ifValue_le_eps",
      "ScholiumA.interpretable_of_factor_and_traceable",
      "ScholiumA.nu_le_ifValue"
    ]
  },
  "line": 190,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer–Relative Interpretability",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "_{\\mathcal{O}}, \\{\\delta^n_{\\mathcal{O}}\\}_{n=1}^{N_{\\mathcal{O}}}, \\epsilon_{\\mathcal{O}})$ be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}), and let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximati",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "erver (cf.~\\ref{definition:bk1_bounded_observer}), and let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}, \\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}). Fix measurable thresholds $\\nu_{\\mathcal{O}}, \\ep",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "and let $K_{\\mathcal{O}}$ be its normalized resolution kernel (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}, \\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}). Fix measurable thresholds $\\nu_{\\mathcal{O}}, \\epsilon_{\\mathcal{O}} : M \\to \\mathbb{R}^+$ satisfying \\[ 0 < \\nu_{\\",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "sec:bk1_operatio",
    "theorem:bk4_symbolic_identity_continuit"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "theorem:bk4_symbolic_identity_continuit"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "scholium:bk1_interpretability_two_axes",
  "label": "scholium:bk1_interpretability_two_axes",
  "latex_body": "\\begin{scholium}[Interpretability on Two Axes --- a Complex Reading]\n\\label{scholium:bk1_interpretability_two_axes}\nWe can imagine the bounded observer (cf.~\\ref{definition:bk1_bounded_observer})\nas resolving change not along a single magnitude but across the two axes of the\ncomplex symbolic distance \\((d_{\\mathrm{Re}}, d_{\\mathrm{Im}})\\):\n\\(d_{\\mathrm{Re}}\\) the \\emph{real} mismatch a change induces, \\(d_{\\mathrm{Im}}\\)\nthe \\emph{imaginative} (orientation) residue it leaves. Read this way, the\ndistinguishability floor \\(\\nu_{\\mathcal{O}}\\) of\nDef.~\\ref{definition:bk1_observer_relative_interpretability} gates\n\\(d_{\\mathrm{Re}}\\)---a change is perceived when it is really detectable---while\na continuity ceiling \\(\\theta_{\\mathcal{O}}\\) gates \\(d_{\\mathrm{Im}}\\)---a change\nis \\emph{re-integrable} when it leaves the observer's orientation within bound.\n``Really detected and imaginatively continuous'' is then the same operational\ncriterion Book~IV records as symbolic identity continuity\n(cf.~\\ref{theorem:bk4_symbolic_identity_continuit}): one observer, read in two\nbooks. We offer this as a lens, not a theorem---conditions\n\\textbf{(I1)}--\\textbf{(I2)} literally bound a single real norm, so the two-axis\nreading is a reinterpretation, and a formal identity---to be made precise in the\nOperatio (cf.~\\ref{sec:bk1_operatio})---would still want the imaginative ceiling\nadopted as such, and a Book~I-side construction grounded in the Axiomata Prima\n(cf.~\\ref{axiom:bk1_axiomata_prima}) to instantiate it.\n\\end{scholium}",
  "line": 219,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Interpretability on Two Axes --- a Complex Reading",
  "ref_roles": [
    {
      "context": "still want the imaginative ceiling adopted as such, and a Book~I-side construction grounded in the Axiomata Prima (cf.~\\ref{axiom:bk1_axiomata_prima}) to instantiate it. \\end{scholium}",
      "label": "axiom:bk1_axiomata_prima",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book1.tex",
      "target_line": 3,
      "target_type": "axiom"
    },
    {
      "context": "Two Axes --- a Complex Reading] \\label{scholium:bk1_interpretability_two_axes} We can imagine the bounded observer (cf.~\\ref{definition:bk1_bounded_observer}) as resolving change not along a single magnitude but across the two axes of the complex symbolic distance \\((d_{\\mathr",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "imaginative} (orientation) residue it leaves. Read this way, the distinguishability floor \\(\\nu_{\\mathcal{O}}\\) of Def.~\\ref{definition:bk1_observer_relative_interpretability} gates \\(d_{\\mathrm{Re}}\\)---a change is perceived when it is really detectable---while a continuity ceiling \\(\\theta_{\\",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    },
    {
      "context": "al norm, so the two-axis reading is a reinterpretation, and a formal identity---to be made precise in the Operatio (cf.~\\ref{sec:bk1_operatio})---would still want the imaginative ceiling adopted as such, and a Book~I-side construction grounded in the Axiomata Pr",
      "label": "sec:bk1_operatio",
      "logical_support": false,
      "role": "navigation",
      "target_file": "operatio.tex",
      "target_line": 1,
      "target_type": "section"
    },
    {
      "context": "imaginatively continuous'' is then the same operational criterion Book~IV records as symbolic identity continuity (cf.~\\ref{theorem:bk4_symbolic_identity_continuit}): one observer, read in two books. We offer this as a lens, not a theorem---conditions \\textbf{(I1)}--\\textbf{(I2)} lit",
      "label": "theorem:bk4_symbolic_identity_continuit",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 698,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "sec:bk1_operatio",
    "theorem:bk4_symbolic_identity_continuit"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk4_refinement_envelope",
    "definition:bk4_test_time_precision_refinement",
    "proof:bk1_energy_bound_identity"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_bounded_approximation_and_interpretability",
  "label": "lemma:bk1_bounded_approximation_and_interpretability",
  "latex_body": "\\begin{lemma}[Bounded Approximation Implies Interpretability]\n\\label{lemma:bk1_bounded_approximation_and_interpretability}\n\nLet $\\Phi : P \\to P$ be an operator on structured states, and let $\\mathcal{O}$ be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}).  \nSuppose\n\\[\n\\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| = c(s) \\cdot \\epsilon_{\\mathcal{O}}(s)\n\\quad \\text{with } 0 < c_{\\min} \\le c(s) \\le 1.\n\\]\nIf the observer resolution satisfies\n\\[\nc_{\\min} \\cdot \\epsilon_{\\mathcal{O}}(s) \\ge \\nu_{\\mathcal{O}}(s)\n\\quad \\text{for all } s \\in P,\n\\]\nthen $\\Phi$ is globally $\\mathcal{O}$–interpretable (cf.~\\ref{definition:bk1_observer_relative_interpretability}).\n\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The c_min ≤ c ≤ 1 and c_min*eps ≥ nu hypotheses genuinely force nu ≤ c*eps ≤ eps; this is a real inequality derivation, not a restatement."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-008"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumA.ifValue_le_eps",
      "ScholiumA.nu_le_ifValue"
    ]
  },
  "line": 244,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Bounded Approximation Implies Interpretability",
  "proof_labels": [
    "proof:bk1_boundedness_encoding_cost"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "retability} Let $\\Phi : P \\to P$ be an operator on structured states, and let $\\mathcal{O}$ be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}). Suppose \\[ \\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| = c(s) \\cdot \\epsilon_{\\mathcal{O}}(s) \\quad \\text{with } 0 < c_{",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "(s) \\ge \\nu_{\\mathcal{O}}(s) \\quad \\text{for all } s \\in P, \\] then $\\Phi$ is globally $\\mathcal{O}$–interpretable (cf.~\\ref{definition:bk1_observer_relative_interpretability}). \\end{lemma}",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_observer_relative_interpretability"
  ],
  "depends_on": [
    "definition:bk1_observer_relative_interpretability"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_boundedness_encoding_cost",
  "label": "proof:bk1_boundedness_encoding_cost",
  "latex_body": "\\begin{proof}[Boundedness of Observer Encoding Cost]\n\\label{proof:bk1_boundedness_encoding_cost}\n\\leavevmode\n\nTo show global $\\mathcal{O}$–interpretability, we verify conditions (I1)–(I3) from \\ref{definition:bk1_observer_relative_interpretability}:\n\n- \\textbf{(I2) Boundedness:} Since \\(c(s) \\le 1\\), we have  \n  \\(\\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| \\le \\epsilon_{\\mathcal{O}}(s)\\).\n\n- \\textbf{(I1) Distinguishability:} Follows from  \n  \\(c(s) \\cdot \\epsilon_{\\mathcal{O}}(s) \\ge c_{\\min} \\cdot \\epsilon_{\\mathcal{O}}(s) \\ge \\nu_{\\mathcal{O}}(s)\\),  \n  hence the perturbation is detectable.\n\n- \\textbf{(I3) Differential Traceability:} Since \\(K_{\\mathcal{O}} \\ast [\\Phi(s) - s] \\ne 0\\),  \n  at least one symbol is perturbed, and the observer’s differential operators \\(\\delta^n_{\\mathcal{O}}\\)  \n  must detect it for some \\(n \\le N_{\\mathcal{O}}\\).\n\nThus, all interpretability criteria are met globally. \\qed\n\n\\end{proof}",
  "line": 262,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Boundedness of Observer Encoding Cost",
  "proves": "lemma:bk1_bounded_approximation_and_interpretability",
  "ref_roles": [
    {
      "context": "ndedness_encoding_cost} \\leavevmode To show global $\\mathcal{O}$–interpretability, we verify conditions (I1)–(I3) from \\ref{definition:bk1_observer_relative_interpretability}: - \\textbf{(I2) Boundedness:} Since \\(c(s) \\le 1\\), we have \\(\\|K_{\\mathcal{O}} \\ast [\\Phi(s) - s]\\| \\le \\epsilon_",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_observer_relative_interpretability"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_energy_bound_identity",
    "proof:bk4_ttpr_convergence"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "lemma:bk1_bounded_approximation_and_interpretability"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_stage_composite_operators_are_interpretable",
  "label": "proposition:bk1_stage_composite_operators_are_interpretable",
  "latex_body": "\\begin{proposition}[Stage–Composite Operators Are Interpretable]\n\\label{proposition:bk1_stage_composite_operators_are_interpretable}\n\nLet \\(E_\\lambda : P_{<\\lambda} \\to P_\\lambda\\) be a stage-level structural operator  \ncomposed of reflective sub-processes \\(D_\\lambda\\) and \\(R_\\lambda\\),  \nand let \\(\\mathcal{O}\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}).  \nSuppose for all \\(s \\in P_{<\\lambda}\\):\n\n\\begin{enumerate}[label=(\\alph*)]\n    \\item \\(\\|K_{\\mathcal{O}} \\ast [E_\\lambda(s) - s]\\| \\le \\epsilon_{\\mathcal{O}}(s)\\) \\hfill \\textit{(Bounded Energy Approximation)}\n    \\item \\(D_\\lambda\\) induces a lower-bounded change satisfying \\(\\|K_{\\mathcal{O}} \\ast [D_\\lambda(s) - s]\\| \\ge \\nu_{\\mathcal{O}}(s)\\)\n\\end{enumerate}\n\nThen \\(E_\\lambda\\) is globally \\(\\mathcal{O}\\)–interpretable (cf.~\\ref{definition:bk1_observer_relative_interpretability}).\n\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Hypotheses (a) bounded energy approximation and (b) lower-bounded distinguishability are represented by the InterpretabilityFactor sandwich; traceability is supplied as an explicit extra hypothesis (the source's own hypotheses do not entail it either, since (a)/(b) alone give only I1/I2)."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-009"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.interpretable_of_factor_and_traceable"
    ]
  },
  "line": 285,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Stage–Composite Operators Are Interpretable",
  "proof_labels": [
    "proof:bk1_energy_bound_identity"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "posed of reflective sub-processes \\(D_\\lambda\\) and \\(R_\\lambda\\), and let \\(\\mathcal{O}\\) be a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}). Suppose for all \\(s \\in P_{<\\lambda}\\): \\begin{enumerate}[label=(\\alph*)] \\item \\(\\|K_{\\mathcal{O}} \\ast [E_\\l",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "s) - s]\\| \\ge \\nu_{\\mathcal{O}}(s)\\) \\end{enumerate} Then \\(E_\\lambda\\) is globally \\(\\mathcal{O}\\)–interpretable (cf.~\\ref{definition:bk1_observer_relative_interpretability}). \\end{proposition}",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator",
    "lemma:bk1_bounded_approximation_and_interpretability",
    "proposition:bk1_stage_composite_operators_are_interpretable"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "lemma:bk1_bounded_approximation_and_interpretability",
    "proposition:bk1_stage_composite_operators_are_interpretable"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "bda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk1_stage_composite_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}): - \\textbf{(I2) Boundedness:",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "line_distance": 53,
      "role": "interpretive_bridge",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "_interpretability}, where \\(E_\\lambda := R_\\lambda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk1_stage_composite_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{defini",
      "label": "definition:bk1_stage_composite_operator",
      "line_distance": 218,
      "role": "interpretive_bridge",
      "target_line": 520,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "id": "proof:bk1_energy_bound_identity",
  "label": "proof:bk1_energy_bound_identity",
  "latex_body": "\\begin{proof}[Bounded Energy Ensures Identity Integrity]\n\\label{proof:bk1_energy_bound_identity}\n\\leavevmode\n\nTo show interpretability of \\(E_\\lambda\\), we verify the three conditions from \\ref{definition:bk1_observer_relative_interpretability}, where \\(E_\\lambda := R_\\lambda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk1_stage_composite_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}):\n\n- \\textbf{(I2) Boundedness:}  \n  Follows directly from assumption (a), since \\(\\|K_{\\mathcal{O}} \\ast [E_\\lambda(s) - s]\\| \\le \\epsilon_{\\mathcal{O}}(s)\\).\n\n- \\textbf{(I1) Distinguishability:}  \n  Assumption (b) gives a lower bound on the signal change induced by \\(D_\\lambda\\).  \n  Since \\(E_\\lambda = R_\\lambda \\circ D_\\lambda\\), and \\(R_\\lambda\\) preserves the first-order deviation,  \n  we have:  \n  \\[\n  \\|K_{\\mathcal{O}} \\ast [E_\\lambda(s) - s]\\| \\ge \\|K_{\\mathcal{O}} \\ast [D_\\lambda(s) - s]\\| \\ge \\nu_{\\mathcal{O}}(s)\n  \\]\n  by triangle inequality and the assumed preservation.\n\n- \\textbf{(I3) Differential Traceability:}  \n  As \\(D_\\lambda\\) alters at least one symbol, and \\(R_\\lambda\\) transmits this change structurally  \n  (cf.~\\ref{definition:bk1_pre_geometric_operators_and_stages}),  \n  there exists an \\(n\\) such that \\(\\delta^n_{\\mathcal{O}}(E_\\lambda(s)) \\ne \\delta^n_{\\mathcal{O}}(s)\\),  \n  ensuring traceability.\n\nThus, all interpretability conditions are satisfied. \\qed\n\\end{proof}",
  "line": 302,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Bounded Energy Ensures Identity Integrity",
  "proves": "proposition:bk1_stage_composite_operators_are_interpretable",
  "ref_roles": [
    {
      "context": "e_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}): - \\textbf{(I2) Boundedness:} Follows directly from assumption (a), since \\(\\|K_{\\mathcal{O}} \\ast [E_\\lambda(s)",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": ":bk1_energy_bound_identity} \\leavevmode To show interpretability of \\(E_\\lambda\\), we verify the three conditions from \\ref{definition:bk1_observer_relative_interpretability}, where \\(E_\\lambda := R_\\lambda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    },
    {
      "context": "bda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk1_stage_composite_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{definition:bk1_bounded_observer}): - \\textbf{(I2) Boundedness:",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "_interpretability}, where \\(E_\\lambda := R_\\lambda \\circ D_\\lambda\\) is defined from the stage composite operators (cf.~\\ref{definition:bk1_stage_composite_operator}, \\ref{definition:bk1_pre_geometric_operators_and_stages}) and evaluated relative to a bounded observer (cf.~\\ref{defini",
      "label": "definition:bk1_stage_composite_operator",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 520,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "lemma:bk1_bounded_approximation_and_interpretability",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 244,
      "target_type": "lemma"
    },
    {
      "context": "",
      "label": "proposition:bk1_stage_composite_operators_are_interpretable",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 285,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_local_charitability",
    "axiom:bk1_pre_geometric_nature",
    "axiom:bk1_symbolic_smoothness",
    "axiom:bk1_topological_regularity",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_problem_of_symbolic_smoothness",
    "definition:bk1_proto_drift_field",
    "definition:bk1_proto_symbolic_space",
    "definition:bk1_reflection_operator",
    "definition:bk1_stage_composite_operator",
    "lemma:bk1_coherence_of_proto_drift_fields",
    "lemma:bk1_existence_of_metric",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "lemma:bk1_universality_of_proto_symbolic_space",
    "proof:bk1_atlas_final_topology_phase_space",
    "proof:bk1_bounded_drift_approximation",
    "proof:bk1_colimit_yields_categoric_structure",
    "proof:bk1_energy_bound_identity",
    "proof:bk1_sketch_coherence_drift_reflection",
    "proof:bk1_sketch_construction_proto_metric",
    "proof:bk1_sketch_effective_proto_drift_field_induction",
    "proof:bk1_sketch_limit_stabilization_colimit",
    "scholium:bk1_emergence_envelope",
    "subsec:appD_category_theory_core_resonance"
  ],
  "cites": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "depends_on": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_pre_geometric_operators_and_stages",
  "label": "definition:bk1_pre_geometric_operators_and_stages",
  "latex_body": "\\begin{definition}[Pre-geometric Operators and Stages]\n\\label{definition:bk1_pre_geometric_operators_and_stages}\nWorking within category $\\catS$\n(Def.~\\ref{definition:bk1_let_cats_be_the_category}), let $\\Omega$ be a limit\nordinal representing the horizon of emergence.\nFor each ordinal $\\lambda < \\Omega$:\n\\begin{itemize}\n    \\item $P_\\lambda \\in Ob(\\catS)$ is the symbolic structure at stage $\\lambda$. We assume each $P_\\lambda$ carries a topology.\n    \\item $P_{<\\lambda} := \\varinjlim_{\\mu < \\lambda} P_\\mu$ denotes the colimit of all prior stages, endowed with the colimit topology induced by the canonical maps $P_\\mu \\to P_{<\\lambda}$ (for $\\mu < \\lambda$).\n    \\item The \\textbf{differentiation operator} $D_\\lambda: P_{<\\lambda} \\to P_\\lambda$ generates the symbolic structure at stage $\\lambda$ from the history encoded in $P_{<\\lambda}$. This represents the fundamental generative aspect of drift.\n    \\item The \\textbf{stabilization operator} $R_\\lambda: P_\\lambda \\to P_\\lambda$ is an idempotent endomorphism ($R_\\lambda \\circ R_\\lambda = R_\\lambda$) that integrates and consolidates symbolic coherence within stage $\\lambda$.\n\\end{itemize}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A categorical OperationalStage now carries drift and idempotent stabilization jointly as one co-emergent witness; neither operator is derived from the other. The drift morphism preserves emergence orientation. The earlier concrete projection remains a nontrivial stabilization witness; ordinal limits, topology, and continuity remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-014"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.projection_idempotent_ne_id_exists",
      "ScholiumC.OperationalStage.drift_advances_stage",
      "ScholiumC.OperationalStage.operators_coemerge"
    ]
  },
  "line": 355,
  "macros_used": [
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Pre-geometric Operators and Stages",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c Operators and Stages] \\label{definition:bk1_pre_geometric_operators_and_stages} Working within category $\\catS$ (Def.~\\ref{definition:bk1_let_cats_be_the_category}), let $\\Omega$ be a limit ordinal representing the horizon of emergence. For each ordinal $\\lambda < \\Omega$: \\begin{it",
      "label": "definition:bk1_let_cats_be_the_category",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 16,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_let_cats_be_the_category"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_sketch_effective_proto_drift_field_induction",
    "proof:bk4_symbolic_curvature_boundary"
  ],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "axiom:bk1_observable_gradation_of_pre_geometric_operations",
  "label": "axiom:bk1_observable_gradation_of_pre_geometric_operations",
  "latex_body": "\\begin{axiom}[Observable Gradation of Pre-geometric Operations]\n\\label{axiom:bk1_observable_gradation_of_pre_geometric_operations}\nThe operators $D_\\lambda$ and $R_\\lambda$ induce observable transformations that vary continuously relative to the stage parameter $\\lambda$, as perceived by a bounded observer $\\mathcal{O}$ (cf.~\\ref{definition:bk1_bounded_observer}).\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other",
      "pair-covering as the topological-regularity stand-in (Hausdorff/second-countable/paracompact/connected unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Observable transformation requires an explicit Observation map and nonzero drift signal; continuity across the stage parameter remains represented only by the Atlas tower-convergence kernel. Detectability is not inferred from category structure."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-094"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.tower_glues",
      "ScholiumC.OperationalStage.observed_drift_ne_zero"
    ]
  },
  "line": 368,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observable Gradation of Pre-geometric Operations",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "that vary continuously relative to the stage parameter $\\lambda$, as perceived by a bounded observer $\\mathcal{O}$ (cf.~\\ref{definition:bk1_bounded_observer}). \\end{axiom}",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_observer_relative_interpretability",
    "lemma:bk4_ttpr_interpretability_preserved",
    "proof:bk1_bounded_drift_approximation",
    "proof:bk1_drift_deviation_bound",
    "proof:bk1_sketch_effective_proto_drift_field_induction",
    "proof:bk4_interpretability_preservation",
    "proposition:bk1_boundedness_from_drift",
    "proposition:bk1_the_operators_lambda_and_lambda",
    "scholium:bk1_consequences_of_bounded_pre_geometric_operations"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "t $\\mathcal{O}$ be a bounded observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}) on a symbolic manifold (cf.~Def.~\\ref{definition:bk1_symbolic_manifold}). An operator $\\Phi_\\lambda$ on symbolic structures $\\mathcal{S}$ is a \\emph{bounded symbolic approximation} when, for",
      "label": "definition:bk1_symbolic_manifold",
      "line_distance": 815,
      "role": "interpretive_bridge",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_symbolic_manifold"
  ],
  "id": "definition:bk1_bounded_symbolic_approximation",
  "label": "definition:bk1_bounded_symbolic_approximation",
  "latex_body": "\\begin{definition}[\\textbf{Bounded Symbolic Approximation}]\n\\label{definition:bk1_bounded_symbolic_approximation}\n\\leavevmode\\newline\nLet $\\mathcal{O}$ be a bounded observer\n(cf.~Def.~\\ref{definition:bk1_bounded_observer}) on a symbolic manifold\n(cf.~Def.~\\ref{definition:bk1_symbolic_manifold}).\nAn operator $\\Phi_\\lambda$ on symbolic structures $\\mathcal{S}$ is a\n\\emph{bounded symbolic approximation} when, for any $s \\in \\mathcal{S}$, the\nperceived change at $\\mathcal{O}$ stays below threshold $\\delta_\\mathcal{O}$,\ni.e.,\n\\[\n\\|\\Phi_\\lambda(s) - s\\|_\\mathcal{O} \\leq \\delta_\\mathcal{O}.\n\\]\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Folded into one scalar kernel-bound structure/theorem shared with the three anchors below; convolution itself is not modeled, only the stated submultiplicativity inequality."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-003"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.kernelBounded_le"
    ]
  },
  "line": 373,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "\\textbf{Bounded Symbolic Approximation}",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "el{definition:bk1_bounded_symbolic_approximation} \\leavevmode\\newline Let $\\mathcal{O}$ be a bounded observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}) on a symbolic manifold (cf.~Def.~\\ref{definition:bk1_symbolic_manifold}). An operator $\\Phi_\\lambda$ on symbolic struc",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "t $\\mathcal{O}$ be a bounded observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}) on a symbolic manifold (cf.~Def.~\\ref{definition:bk1_symbolic_manifold}). An operator $\\Phi_\\lambda$ on symbolic structures $\\mathcal{S}$ is a \\emph{bounded symbolic approximation} when, for",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk4_drift_reflection_field"
  ],
  "cites": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "depends_on": [
    "axiom:bk1_observable_gradation_of_pre_geometric_operations",
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "ions] \\label{proposition:bk1_the_operators_lambda_and_lambda} The operators $D_\\lambda$ and $R_\\lambda$ from Definition~\\ref{definition:bk1_proto_drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definit",
      "label": "definition:bk1_proto_drift_field",
      "line_distance": 2409,
      "role": "teaser",
      "target_line": 2797,
      "target_type": "definition"
    },
    {
      "context": "lambda} The operators $D_\\lambda$ and $R_\\lambda$ from Definition~\\ref{definition:bk1_proto_drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definition:bk1_bounded_symbolic_approximation}, assuming observ",
      "label": "definition:bk1_reflection_operator",
      "line_distance": 821,
      "role": "teaser",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_proto_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "id": "proposition:bk1_the_operators_lambda_and_lambda",
  "label": "proposition:bk1_the_operators_lambda_and_lambda",
  "latex_body": "\\begin{proposition}[Fundamental Operators as Bounded Symbolic Approximations]\n\\label{proposition:bk1_the_operators_lambda_and_lambda}\nThe operators $D_\\lambda$ and $R_\\lambda$ from Definition~\\ref{definition:bk1_proto_drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definition:bk1_bounded_symbolic_approximation}, assuming observer-resolved emergence.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "face 3 consumes the guarded-process machinery (LPS-P49) and the helix kernel (LPS-P48)",
      "the metaphysical scope of a three-word axiom is not exhausted; the operational tri-face kernel is what is certified"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "D_lambda, R_lambda as bounded symbolic approximations: their single-channel failure modes are certified; the bounded-approximation predicate stays interpretive."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-093"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "AxiomataPrima.no_drift_no_novelty",
      "AxiomataPrima.pure_drift_dissolves"
    ]
  },
  "line": 388,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Fundamental Operators as Bounded Symbolic Approximations",
  "proof_labels": [
    "proof:bk1_sketch_effective_proto_drift_field_induction"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definition:bk1_bounded_symbolic_approximation}, assuming observer-resolved emergence. \\end{proposition}",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "ions] \\label{proposition:bk1_the_operators_lambda_and_lambda} The operators $D_\\lambda$ and $R_\\lambda$ from Definition~\\ref{definition:bk1_proto_drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definit",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    },
    {
      "context": "lambda} The operators $D_\\lambda$ and $R_\\lambda$ from Definition~\\ref{definition:bk1_proto_drift_field} and Definition~\\ref{definition:bk1_reflection_operator} are bounded symbolic approximations per Definition~\\ref{definition:bk1_bounded_symbolic_approximation}, assuming observ",
      "label": "definition:bk1_reflection_operator",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "axiom:bk1_observable_gradation_of_pre_geometric_operations",
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_drift_field"
  ],
  "depends_on": [
    "axiom:bk1_observable_gradation_of_pre_geometric_operations",
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "bserver}). For \\[ \\vec{D}_\\lambda^{eff}(s)=D_\\lambda(s)\\ominus s \\] as the observer-visible proto-drift deviation (Def.~\\ref{definition:bk1_proto_drift_field}), this gives \\[ \\|\\vec{D}_\\lambda^{eff}(s)\\|_{\\mathcal{O}}\\leq \\delta_{\\mathcal{O}} \\] on the observer-resolved domain.",
      "label": "definition:bk1_proto_drift_field",
      "line_distance": 2405,
      "role": "teaser",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_proto_drift_field"
  ],
  "id": "proof:bk1_sketch_effective_proto_drift_field_induction",
  "label": "proof:bk1_sketch_effective_proto_drift_field_induction",
  "latex_body": "\\begin{proof}[Fundamental Operators as Bounded Symbolic Approximations]\n\\label{proof:bk1_sketch_effective_proto_drift_field_induction}\n\\leavevmode\n\n\\textbf{For $D_\\lambda$.}\\ By Ax.~\\ref{axiom:bk1_observable_gradation_of_pre_geometric_operations},\n\\(D_\\lambda\\) induces transformations that are observable to the bounded\nobserver \\(\\mathcal{O}\\). Observer-resolved emergence means that the effective\nchange registered by \\(\\mathcal{O}\\) lies inside its resolution threshold\n\\(\\delta_{\\mathcal{O}}\\) (Def.~\\ref{definition:bk1_bounded_observer}). For\n\\[\n\\vec{D}_\\lambda^{eff}(s)=D_\\lambda(s)\\ominus s\n\\]\nas the observer-visible proto-drift deviation\n(Def.~\\ref{definition:bk1_proto_drift_field}), this gives\n\\[\n\\|\\vec{D}_\\lambda^{eff}(s)\\|_{\\mathcal{O}}\\leq \\delta_{\\mathcal{O}}\n\\]\non the observer-resolved domain. This is exactly the bounded symbolic\napproximation condition of Def.~\\ref{definition:bk1_bounded_symbolic_approximation}.\n\n\\textbf{For $R_\\lambda$.}\n\\(R_\\lambda:P_\\lambda\\to P_\\lambda\\) is the idempotent stabilization operator\nof Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}. The same\nobservable-gradation axiom applies to its observer-visible stabilization\ndeviation \\(R_\\lambda(s)-s\\), and observer resolution gives\n\\[\n\\|R_\\lambda(s)-s\\|_{\\mathcal{O}}\\leq \\delta_{\\mathcal{O}}\n\\]\nfor \\(s\\in P_\\lambda\\). Hence \\(R_\\lambda\\) also satisfies\nDef.~\\ref{definition:bk1_bounded_symbolic_approximation}.\n\\end{proof}",
  "line": 392,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Fundamental Operators as Bounded Symbolic Approximations",
  "proves": "proposition:bk1_the_operators_lambda_and_lambda",
  "ref_roles": [
    {
      "context": "imations] \\label{proof:bk1_sketch_effective_proto_drift_field_induction} \\leavevmode \\textbf{For $D_\\lambda$.}\\ By Ax.~\\ref{axiom:bk1_observable_gradation_of_pre_geometric_operations}, \\(D_\\lambda\\) induces transformations that are observable to the bounded observer \\(\\mathcal{O}\\). Observer-resolved e",
      "label": "axiom:bk1_observable_gradation_of_pre_geometric_operations",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 368,
      "target_type": "axiom"
    },
    {
      "context": "the effective change registered by \\(\\mathcal{O}\\) lies inside its resolution threshold \\(\\delta_{\\mathcal{O}}\\) (Def.~\\ref{definition:bk1_bounded_observer}). For \\[ \\vec{D}_\\lambda^{eff}(s)=D_\\lambda(s)\\ominus s \\] as the observer-visible proto-drift deviation (Def.~\\ref{def",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "_{\\mathcal{O}} \\] on the observer-resolved domain. This is exactly the bounded symbolic approximation condition of Def.~\\ref{definition:bk1_bounded_symbolic_approximation}. \\textbf{For $R_\\lambda$.} \\(R_\\lambda:P_\\lambda\\to P_\\lambda\\) is the idempotent stabilization operator of Def.~\\ref{",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "tion}. \\textbf{For $R_\\lambda$.} \\(R_\\lambda:P_\\lambda\\to P_\\lambda\\) is the idempotent stabilization operator of Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}. The same observable-gradation axiom applies to its observer-visible stabilization deviation \\(R_\\lambda(s)-s\\), and ob",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "bserver}). For \\[ \\vec{D}_\\lambda^{eff}(s)=D_\\lambda(s)\\ominus s \\] as the observer-visible proto-drift deviation (Def.~\\ref{definition:bk1_proto_drift_field}), this gives \\[ \\|\\vec{D}_\\lambda^{eff}(s)\\|_{\\mathcal{O}}\\leq \\delta_{\\mathcal{O}} \\] on the observer-resolved domain.",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_observable_gradation_of_pre_geometric_operations",
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_drift_field"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "depends_on": [
    "definition:bk1_bounded_symbolic_approximation"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "line_distance": 9,
      "role": "teaser",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "id": "scholium:bk1_consequences_of_bounded_pre_geometric_operations",
  "label": "scholium:bk1_consequences_of_bounded_pre_geometric_operations",
  "latex_body": "\\begin{scholium}[Consequences of Bounded Pre-geometric Operations]\n\\label{scholium:bk1_consequences_of_bounded_pre_geometric_operations}\nBoundedness of $D_\\lambda$ and $R_\\lambda$ ensures stability of emergent structure, constraining drift intensity and symbolic fluctuation across $\\lambda$.\n\\end{scholium}",
  "line": 424,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Consequences of Bounded Pre-geometric Operations",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "refs": [],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "remark:scholium_symbolicum.tex:429",
  "label": "",
  "latex_body": "\\begin{remark}[Relating Process-Oriented Boundedness to a Kernel-Based Model]\nThe kernel-based formulation of symbolic approximation (cf.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) is an instance of the broader process-oriented model (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}), where convolution with $\\mathcal{K}_\\mathcal{O}$ provides an observable-resolved smoothing interpretation.\n\\end{remark}",
  "line": 429,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Relating Process-Oriented Boundedness to a Kernel-Based Model",
  "refs": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_observer_relative_interpretability",
    "definition:bk4_coherence_metric",
    "definition:bk4_test_time_precision_refinement",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "proof:bk1_fix_s_in_s",
    "proof:bk1_observer_kernel_convolution",
    "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound",
    "remark:bk1_kernel_based_bounded_approximation",
    "scholium:bk1_consequences_of_bounded_pre_geometric_operations"
  ],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_kernel_based_bounded_symbolic_approximation",
  "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
  "latex_body": "\\begin{definition}[\\textbf{Kernel-Based Bounded Symbolic Approximation (Illustration)}]\n\\label{definition:bk1_kernel_based_bounded_symbolic_approximation}\nLet $\\mathcal{O}$ be a bounded observer with resolution kernel $\\mathcal{K}_\\mathcal{O}$ as specified in Definition~\\ref{definition:bk1_bounded_observer}. An operator $\\Phi_\\lambda$ (or $\\Psi_\\lambda$) acting on symbolic structures $\\mathcal{S}$ is said to be a \\emph{kernel-bounded symbolic approximation} if and only if for any symbol $s \\in \\mathcal{S}$ and its image $\\Phi_\\lambda(s)$, the perceptual difference as measured by $\\mathcal{O}$ satisfies:\n\\begin{equation}\n\\|\\mathcal{K}_\\mathcal{O} \\ast [\\Phi_\\lambda(s) - s]\\| \\leq \\delta_\\mathcal{O},\n\\end{equation}\nwhere $\\delta_\\mathcal{O} > 0$ is the resolution threshold of $\\mathcal{O}$ and $\\ast$ denotes the convolution operation.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Same KernelBoundedApprox structure as bk1_bounded_symbolic_approximation; the iff-form of the source definition is not modeled, only the derived bound."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-004"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.kernelBounded_le"
    ]
  },
  "line": 433,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "\\textbf{Kernel-Based Bounded Symbolic Approximation (Illustration)}",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "on} Let $\\mathcal{O}$ be a bounded observer with resolution kernel $\\mathcal{K}_\\mathcal{O}$ as specified in Definition~\\ref{definition:bk1_bounded_observer}. An operator $\\Phi_\\lambda$ (or $\\Psi_\\lambda$) acting on symbolic structures $\\mathcal{S}$ is said to be a \\emph{kerne",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "remark:bk1_kernel_based_bounded_approximation"
  ],
  "cites": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "ec{D}_\\lambda$ be the proto-drift field induced by operators $\\Phi_\\lambda$ and $\\Psi_\\lambda$ as defined in Definition~\\ref{definition:bk1_proto_drift_field}. If $\\vec{D}_\\lambda$ satisfies: \\begin{equation} \\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\del",
      "label": "definition:bk1_proto_drift_field",
      "line_distance": 2355,
      "role": "teaser",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_proto_drift_field"
  ],
  "id": "proposition:bk1_boundedness_from_drift",
  "label": "proposition:bk1_boundedness_from_drift",
  "latex_body": "\\begin{proposition}[\\textbf{Boundedness from Drift}]\n\\label{proposition:bk1_boundedness_from_drift}\nLet $\\vec{D}_\\lambda$ be the proto-drift field induced by operators $\\Phi_\\lambda$ and $\\Psi_\\lambda$ as defined in Definition~\\ref{definition:bk1_proto_drift_field}. If $\\vec{D}_\\lambda$ satisfies:\n\\begin{equation}\n\\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\delta_\\mathcal{O},\n\\end{equation}\nthen both $\\Phi_\\lambda$ and $\\Psi_\\lambda$ are bounded symbolic approximations with respect to observer $\\mathcal{O}$ (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}).\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The sup-bound hypothesis is represented as a plain scalar bound (drift ≤ δ) rather than an actual supremum over a domain."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-005"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.kernelBounded_le"
    ]
  },
  "line": 442,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "\\textbf{Boundedness from Drift}",
  "proof_labels": [
    "proof:bk1_drift_deviation_bound",
    "proof:bk1_observer_threshold_reflexivity"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "both $\\Phi_\\lambda$ and $\\Psi_\\lambda$ are bounded symbolic approximations with respect to observer $\\mathcal{O}$ (cf.~\\ref{definition:bk1_bounded_symbolic_approximation}). \\end{proposition}",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "ec{D}_\\lambda$ be the proto-drift field induced by operators $\\Phi_\\lambda$ and $\\Psi_\\lambda$ as defined in Definition~\\ref{definition:bk1_proto_drift_field}. If $\\vec{D}_\\lambda$ satisfies: \\begin{equation} \\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\del",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_observer_kernel_convolution"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "n_bound} \\leavevmode Let $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition:",
      "label": "definition:bk1_proto_drift_field",
      "line_distance": 2346,
      "role": "teaser",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_proto_drift_field"
  ],
  "id": "proof:bk1_drift_deviation_bound",
  "label": "proof:bk1_drift_deviation_bound",
  "latex_body": "\\begin{proof}[Proto-Drift Induces Directional Deviation Bound]\n\\label{proof:bk1_drift_deviation_bound}\n\\leavevmode\n\nLet $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition:\n\\[\n\\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\delta_\\mathcal{O},\n\\]\nit follows that $\\|\\vec{D}_\\lambda(s)\\| \\leq \\delta_\\mathcal{O}$ for all $s$ in the domain.\nSince $\\mathcal{K}_\\mathcal{O}$ is a resolution kernel of a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), it satisfies $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ (normalization). By the properties of convolution and norms:\n\\begin{align}\n\\|\\mathcal{K}_\\mathcal{O} \\ast [\\Phi_\\lambda(s) - s]\\| &= \\|\\mathcal{K}_\\mathcal{O} \\ast \\vec{D}_\\lambda(s)\\| \\\\\n&\\leq \\|\\mathcal{K}_\\mathcal{O}\\|_1 \\cdot \\|\\vec{D}_\\lambda(s)\\| \\\\\n&= \\|\\vec{D}_\\lambda(s)\\| \\\\\n&\\leq \\delta_\\mathcal{O}\n\\end{align}\nTherefore, $\\Phi_\\lambda$ satisfies the condition to be a bounded symbolic approximation. The proof for $\\Psi_\\lambda$ follows similarly by observing that the proto-drift field $\\vec{D}_\\lambda$ also encodes the action of $\\Psi_\\lambda$ through the inverse relationship established in Definition~\\ref{definition:bk1_bounded_symbolic_approximation}.\n\\end{proof}",
  "line": 451,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Proto-Drift Induces Directional Deviation Bound",
  "proves": "proposition:bk1_boundedness_from_drift",
  "ref_roles": [
    {
      "context": "thcal{O}$ for all $s$ in the domain. Since $\\mathcal{K}_\\mathcal{O}$ is a resolution kernel of a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), it satisfies $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ (normalization). By the properties of convolution and norms: \\begin{",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "$\\vec{D}_\\lambda$ also encodes the action of $\\Psi_\\lambda$ through the inverse relationship established in Definition~\\ref{definition:bk1_bounded_symbolic_approximation}. \\end{proof}",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "n_bound} \\leavevmode Let $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition:",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "proposition:bk1_boundedness_from_drift"
  ],
  "depends_on": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "proposition:bk1_boundedness_from_drift"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "remark:bk1_kernel_based_bounded_approximation",
  "label": "remark:bk1_kernel_based_bounded_approximation",
  "latex_body": "\\begin{remark}[Alternative Perspective: Kernel-Based Bounded Approximation]\n\\label{remark:bk1_kernel_based_bounded_approximation}\nAn alternative, more concrete way to conceptualize how an observer $\\mathcal{O}$ might implement or model the perception of boundedness involves considering a resolution kernel $\\mathcal{K}_\\mathcal{O}$ (as specified in Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}).  \nIn this view, an operator $\\Phi_\\lambda$ acting on symbolic structures $\\mathcal{S}$ could be considered a \\emph{kernel-bounded symbolic approximation} if for any symbol $s \\in \\mathcal{S}$ and its image $\\Phi_\\lambda(s)$, the perceptual difference as measured by convolution with $\\mathcal{K}_\\mathcal{O}$ satisfies:\n\\[\n\\|\\mathcal{K}_\\mathcal{O} \\ast [\\Phi_\\lambda(s) - s]\\| \\leq \\delta_\\mathcal{O},\n\\]\nwhere $\\delta_\\mathcal{O} > 0$ is the resolution threshold of $\\mathcal{O}$.\n\nThis perspective leads to a corresponding sufficient condition: if a proto-drift field $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ satisfies $\\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\delta_\\mathcal{O}$, then $\\Phi_\\lambda$ is a kernel-bounded symbolic approximation. (The proof follows as in Proposition~\\ref{proposition:bk1_boundedness_from_drift}).\n\nWhile the process-oriented Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation} is considered more fundamental within \\textit{Principia Symbolica} as it directly leverages the observer's differentiation capacity, the kernel-based perspective can provide a useful illustrative model, particularly when analogizing to systems where perceptual filtering is well-described by such convolution operations. The core principle remains that the change induced by the operator must be sub-threshold for the observer.\n\\end{remark}",
  "line": 470,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Alternative Perspective: Kernel-Based Bounded Approximation",
  "ref_roles": [
    {
      "context": "erception of boundedness involves considering a resolution kernel $\\mathcal{K}_\\mathcal{O}$ (as specified in Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}). In this view, an operator $\\Phi_\\lambda$ acting on symbolic structures $\\mathcal{S}$ could be considered a \\emph{ke",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    },
    {
      "context": "elta_\\mathcal{O}$, then $\\Phi_\\lambda$ is a kernel-bounded symbolic approximation. (The proof follows as in Proposition~\\ref{proposition:bk1_boundedness_from_drift}). While the process-oriented Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation} is considered",
      "label": "proposition:bk1_boundedness_from_drift",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 442,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "proposition:bk1_boundedness_from_drift"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "abs:press"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "lemma:bk1_observer_bounded_emergence_constraint"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "Reflexive Admissibility] \\label{proof:bk1_observer_threshold_reflexivity} \\leavevmode This follows directly from Lemma~\\ref{lemma:bk1_observer_bounded_emergence_constraint} and Definition~\\ref{definition:bk1_bounded_observer}. The lemma states that the observer-perceived change induced by $D",
      "label": "lemma:bk1_observer_bounded_emergence_constraint",
      "line_distance": 64,
      "role": "teaser",
      "target_line": 553,
      "target_type": "lemma"
    }
  ],
  "forward_refs": [
    "lemma:bk1_observer_bounded_emergence_constraint"
  ],
  "id": "proof:bk1_observer_threshold_reflexivity",
  "label": "proof:bk1_observer_threshold_reflexivity",
  "latex_body": "\\begin{proof}[Observer Threshold Governs Reflexive Admissibility]\n\\label{proof:bk1_observer_threshold_reflexivity}\n\\leavevmode\n\nThis follows directly from Lemma~\\ref{lemma:bk1_observer_bounded_emergence_constraint} and Definition~\\ref{definition:bk1_bounded_observer}. The lemma states that the observer-perceived change induced by $D_\\lambda$ and $R_\\lambda$ is less than or equal to the observer's resolution threshold, which is precisely the condition required by the definition.\n\\end{proof}",
  "line": 489,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer Threshold Governs Reflexive Admissibility",
  "proves": "proposition:bk1_boundedness_from_drift",
  "ref_roles": [
    {
      "context": "ity} \\leavevmode This follows directly from Lemma~\\ref{lemma:bk1_observer_bounded_emergence_constraint} and Definition~\\ref{definition:bk1_bounded_observer}. The lemma states that the observer-perceived change induced by $D_\\lambda$ and $R_\\lambda$ is less than or equal to th",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "Reflexive Admissibility] \\label{proof:bk1_observer_threshold_reflexivity} \\leavevmode This follows directly from Lemma~\\ref{lemma:bk1_observer_bounded_emergence_constraint} and Definition~\\ref{definition:bk1_bounded_observer}. The lemma states that the observer-perceived change induced by $D",
      "label": "lemma:bk1_observer_bounded_emergence_constraint",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 553,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "lemma:bk1_observer_bounded_emergence_constraint"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_observer_kernel_convolution"
  ],
  "cites": [
    "definition:bk1_kernel_based_bounded_symbolic_approximation"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "proof:bk1_drift_deviation_bound"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound",
  "label": "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound",
  "latex_body": "\\begin{proposition}[\\textbf{Sufficient Condition for Kernel-Boundedness from Uniform Drift Bound}] \n\\label{proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound}\nLet $\\vec{D}_\\lambda$ be the proto-drift field induced by operators $\\Phi_\\lambda$ and $\\Psi_\\lambda$ such that $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ (or an appropriate difference). If $\\vec{D}_\\lambda$ satisfies:\n\\begin{equation}\n\\sup_{x \\in \\mathrm{dom}(D_\\lambda)} \\|\\vec{D}_\\lambda(x)\\| \\leq \\delta_\\mathcal{O}, \n\\end{equation}\nthen both $\\Phi_\\lambda$ and $\\Psi_\\lambda$ are kernel-bounded symbolic approximations (Def~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) with respect to observer $\\mathcal{O}$.\n\\begin{proof}[Convolutional Identity from Observer Kernel Properties]\n\\label{proof:bk1_observer_kernel_convolution}\n\\leavevmode\n\n(Proposition~\\ref{proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound}, which uses $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ and properties of convolution, remains valid for this proposition.)\n\nLet $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition (Eq.~\\ref{proof:bk1_drift_deviation_bound}), it follows that $\\|\\vec{D}_\\lambda(s)\\| \\leq \\delta_\\mathcal{O}$ for all $s$ in the domain.\n\nSince $\\mathcal{K}_\\mathcal{O}$ is a resolution kernel of a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), it satisfies $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ (normalization property). By the properties of convolution and norms, and the criteria for kernel-bounded approximation (Def.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}), using $\\Phi_\\lambda(s) - s = \\vec{D}_\\lambda(s)$:\n\\begin{align}\n\\|\\mathcal{K}_\\mathcal{O} \\ast \\vec{D}_\\lambda(s)\\| &\\leq \\|\\mathcal{K}_\\mathcal{O}\\|_1 \\cdot \\|\\vec{D}_\\lambda(s)\\| \\\\\n&= \\|\\vec{D}_\\lambda(s)\\| \\\\\n&\\leq \\delta_\\mathcal{O}\n\\end{align}\n\nTherefore, $\\Phi_\\lambda$ satisfies the condition to be a kernel-bounded symbolic approximation. The proof for $\\Psi_\\lambda$ follows similarly.\n\\end{proof}\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "This anchor's own inline proof is exactly the ‖K‖1=1 submultiplicativity chain now proved by kernelBounded_le; kernel norm and convolution remain hypotheses, not derived objects."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-006"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.kernelBounded_le"
    ]
  },
  "line": 495,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "\\textbf{Sufficient Condition for Kernel-Boundedness from Uniform Drift Bound}",
  "proof_labels": [
    "proof:bk1_observer_kernel_convolution"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "mathcal{O}, \\end{equation} then both $\\Phi_\\lambda$ and $\\Psi_\\lambda$ are kernel-bounded symbolic approximations (Def~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}) with respect to observer $\\mathcal{O}$. \\begin{proof}[Convolutional Identity from Observer Kernel Properties] \\label{p",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field",
    "proof:bk1_drift_deviation_bound",
    "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field",
    "proof:bk1_drift_deviation_bound",
    "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "proof:bk1_drift_deviation_bound",
    "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "r this proposition.) Let $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition",
      "label": "definition:bk1_proto_drift_field",
      "line_distance": 2295,
      "role": "teaser",
      "target_line": 2797,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_proto_drift_field"
  ],
  "id": "proof:bk1_observer_kernel_convolution",
  "label": "proof:bk1_observer_kernel_convolution",
  "latex_body": "\\begin{proof}[Convolutional Identity from Observer Kernel Properties]\n\\label{proof:bk1_observer_kernel_convolution}\n\\leavevmode\n\n(Proposition~\\ref{proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound}, which uses $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ and properties of convolution, remains valid for this proposition.)\n\nLet $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition (Eq.~\\ref{proof:bk1_drift_deviation_bound}), it follows that $\\|\\vec{D}_\\lambda(s)\\| \\leq \\delta_\\mathcal{O}$ for all $s$ in the domain.\n\nSince $\\mathcal{K}_\\mathcal{O}$ is a resolution kernel of a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), it satisfies $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ (normalization property). By the properties of convolution and norms, and the criteria for kernel-bounded approximation (Def.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}), using $\\Phi_\\lambda(s) - s = \\vec{D}_\\lambda(s)$:\n\\begin{align}\n\\|\\mathcal{K}_\\mathcal{O} \\ast \\vec{D}_\\lambda(s)\\| &\\leq \\|\\mathcal{K}_\\mathcal{O}\\|_1 \\cdot \\|\\vec{D}_\\lambda(s)\\| \\\\\n&= \\|\\vec{D}_\\lambda(s)\\| \\\\\n&\\leq \\delta_\\mathcal{O}\n\\end{align}\n\nTherefore, $\\Phi_\\lambda$ satisfies the condition to be a kernel-bounded symbolic approximation. The proof for $\\Psi_\\lambda$ follows similarly.\n\\end{proof}",
  "line": 502,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Convolutional Identity from Observer Kernel Properties",
  "proves": "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound",
  "ref_roles": [
    {
      "context": "hcal{O}$ for all $s$ in the domain. Since $\\mathcal{K}_\\mathcal{O}$ is a resolution kernel of a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), it satisfies $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ (normalization property). By the properties of convolution and norms",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "lization property). By the properties of convolution and norms, and the criteria for kernel-bounded approximation (Def.~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}), using $\\Phi_\\lambda(s) - s = \\vec{D}_\\lambda(s)$: \\begin{align} \\|\\mathcal{K}_\\mathcal{O} \\ast \\vec{D}_\\lambda(s)\\| &",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    },
    {
      "context": "r this proposition.) Let $s \\in \\mathcal{S}$ be an arbitrary symbolic structure. Expanding the proto-drift field (Def.~\\ref{definition:bk1_proto_drift_field}), $\\vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    },
    {
      "context": "vec{D}_\\lambda(s) = \\Phi_\\lambda(s) - s$ for any $s$ in the domain of $\\Phi_\\lambda$. Given the supremum condition (Eq.~\\ref{proof:bk1_drift_deviation_bound}), it follows that $\\|\\vec{D}_\\lambda(s)\\| \\leq \\delta_\\mathcal{O}$ for all $s$ in the domain. Since $\\mathcal{K}_\\math",
      "label": "proof:bk1_drift_deviation_bound",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 451,
      "target_type": "proof"
    },
    {
      "context": "ional Identity from Observer Kernel Properties] \\label{proof:bk1_observer_kernel_convolution} \\leavevmode (Proposition~\\ref{proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound}, which uses $\\|\\mathcal{K}_\\mathcal{O}\\|_1 = 1$ and properties of convolution, remains valid for this proposition.) Le",
      "label": "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 495,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_proto_drift_field",
    "proof:bk1_drift_deviation_bound",
    "proposition:bk1_sufficient_condition_for_kernel_boundedness_from_uniform_drift_bound"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:appC_bounded_reflexive_emergence",
    "definition:appC_complexity_measure",
    "definition:appC_symbolic_modality",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "proof:appC_phi_from_lagrangian",
    "proof:bk1_bounded_drift_approximation",
    "proof:bk1_energy_bound_identity"
  ],
  "cites": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "depends_on": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_stage_composite_operator",
  "label": "definition:bk1_stage_composite_operator",
  "latex_body": "\\begin{definition}[Stage–Composite Operator]\n\\label{definition:bk1_stage_composite_operator}\nLet \\(D_\\lambda : P_{<\\lambda} \\to P_\\lambda\\) be a symbolic transformation representing directional drift, and let \\(R_\\lambda : P_\\lambda \\to P_\\lambda\\) be a refinement or reflection operator  \n(as preliminarily introduced in Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}).\n\nThen the \\textbf{stage--composite operator} at ordinal level \\(\\lambda\\) is defined as:\n\\[\nE_\\lambda := R_\\lambda \\circ D_\\lambda : P_{<\\lambda} \\to P_\\lambda.\n\\]\nSuch operators encode a two-step symbolic emergence: first a directional transformation, then a bounded symbolic refinement.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Modeled as the TwoStepBoundedApprox structure's stabilize ∘ drift composite; no standalone theorem beyond twoStep_bound."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-010"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "ScholiumA.twoStep_bound"
    ]
  },
  "line": 520,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Stage–Composite Operator",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\lambda : P_\\lambda \\to P_\\lambda\\) be a refinement or reflection operator (as preliminarily introduced in Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}). Then the \\textbf{stage--composite operator} at ordinal level \\(\\lambda\\) is defined as: \\[ E_\\lambda := R_\\lambda \\c",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "lemma:bk1_observer_bounded_emergence_constraint",
    "proof:bk1_bounded_drift_approximation",
    "scholium:bk1_emergence_envelope"
  ],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "axiom:bk1_summable_resolution_decay",
  "label": "axiom:bk1_summable_resolution_decay",
  "latex_body": "\\begin{axiom}[Summable Resolution Decay]\n\\label{axiom:bk1_summable_resolution_decay}\nLet \\((\\lambda_n)_{n\\in\\mathbb{N}}\\) be a cofinal sequence in the emergence\ntower with \\(\\lambda_n<\\lambda_{n+1}<\\Omega\\) and\n\\(\\sup_n\\lambda_n=\\Omega\\). Relative to a bounded observer \\(O\\), assume there\nexists a positive sequence \\((\\eta_n)_{n\\in\\mathbb{N}}\\) such that\n\\[\n\\sum_{n=0}^{\\infty}\\eta_n < \\infty\n\\]\nand, along this cofinal tower,\n\\[\nd_O(E_{\\lambda_n}(s),s)\n= \\lVert K_O * [E_{\\lambda_n}(s)-s]\\rVert\n\\leq \\eta_n\n\\quad\\text{for all observable }s\\in P_{<\\lambda_n}.\n\\]\nThus later-stage refinements are not merely bounded one at a time; their\nobserver-visible tail is summable. No claim is made here for emergence towers\nof uncountable cofinality except through such selected cofinal sequences.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Summability is an explicit ChainedApprox field. It yields finite telescoping, a genuine Cauchy stage path, and under completeness an actual limit with tail-sum displacement bound."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-012"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumA.ChainedApprox.cauchySeq",
      "ScholiumA.ChainedApprox.exists_limit_with_tail_bound",
      "ScholiumA.chainedApprox_telescope"
    ]
  },
  "line": 532,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Summable Resolution Decay",
  "proof_status": "definitional",
  "refs": [],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_observer_threshold_reflexivity",
    "proof:bk4_ttpr_convergence",
    "proposition:bk4_ttpr_convergence"
  ],
  "cites": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space",
    "definition:bk1_stage_composite_operator"
  ],
  "depends_on": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "is \\(d_O\\)-Cauchy in the usual tail sense (see also the formal directed emergence structure in Definition~\\ref{definition:bk1_directed_system_of_emergence}). Its \\(d_O\\)-completion \\( \\overline{P}_O \\) supplies the observer-completed proto-s",
      "label": "definition:bk1_directed_system_of_emergence",
      "line_distance": 134,
      "role": "teaser",
      "target_line": 687,
      "target_type": "definition"
    },
    {
      "context": "proto-symbolic space associated with the colimit \\(P=\\varinjlim_{\\lambda<\\Omega}P_\\lambda\\) (Definition~\\ref{definition:bk1_proto_symbolic_space}). \\end{enumerate} \\end{lemma}",
      "label": "definition:bk1_proto_symbolic_space",
      "line_distance": 150,
      "role": "teaser",
      "target_line": 703,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_proto_symbolic_space"
  ],
  "id": "lemma:bk1_observer_bounded_emergence_constraint",
  "label": "lemma:bk1_observer_bounded_emergence_constraint",
  "latex_body": "\\begin{lemma}[Observer–Bounded Emergence Constraint]\n\\label{lemma:bk1_observer_bounded_emergence_constraint}\nLet \n\\(\nO=(N_O,\\{\\delta^{\\,n}_{O}\\}_{n=1}^{N_O},\\varepsilon_O)\n\\)\nbe a bounded observer with resolution kernel \\(K_O\\) and scalar threshold \\(\\delta_O\\) (Definition~\\ref{definition:bk1_bounded_observer}).  \nFor every ordinal \\(\\lambda<\\Omega\\), define the stage--composite operator\n\\[\n  E_\\lambda := R_\\lambda \\circ D_\\lambda : P_{<\\lambda} \\longrightarrow P_\\lambda,\n\\]\nas defined in Definition~\\ref{definition:bk1_stage_composite_operator},  \nwhere \\(P_{<\\lambda}\\) and \\(P_\\lambda\\) are symbolic stages introduced in Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}.  \nThen\n\\begin{enumerate}\n  \\item[\\textup{(i)}]  \\textbf{Bounded approximation of the identity.}\\; \n        For all \\(s\\in P_{<\\lambda}\\),\n        \\begin{equation}\n          \\bigl\\lVert K_O * \\bigl[E_\\lambda(s) - s\\bigr] \\bigr\\rVert \n          \\;\\le\\; 2\\,\\delta_O.\n        \\end{equation}\n        (This satisfies the kernel-bounded approximation condition in Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}.)\n\n  \\item[\\textup{(ii)}]  \\textbf{Cauchy tower under summable decay.}\\;\n        Endow every \\(P_\\lambda\\) with the observer metric\n        \\(\n          d_O(x,y) := \\lVert K_O * (x - y) \\rVert.\n        \\)\n        Along any cofinal sequence \\((\\lambda_n)\\) satisfying\n        Ax.~\\ref{axiom:bk1_summable_resolution_decay}, the transition maps obey\n        \\[\n          d_O(f_{\\lambda_m\\lambda_n}(x),x)\n          \\leq \\sum_{j=m}^{n-1}\\eta_j\n          \\quad\n          \\forall\\,x \\in P_{\\lambda_m},\\;\n          m<n,\n        \\]\n        so the cofinal directed subsystem\n        \\(\n          (P_{\\lambda_n}, f_{\\lambda_m\\lambda_n})_{m<n}\n        \\)\n        is \\(d_O\\)-Cauchy in the usual tail sense\n        (see also the formal directed emergence structure in Definition~\\ref{definition:bk1_directed_system_of_emergence}). Its \\(d_O\\)-completion\n        \\(\n          \\overline{P}_O\n        \\)\n        supplies the observer-completed proto-symbolic space associated with\n        the colimit \\(P=\\varinjlim_{\\lambda<\\Omega}P_\\lambda\\)\n        (Definition~\\ref{definition:bk1_proto_symbolic_space}).\n\\end{enumerate}\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Part (i)'s 2δ bound follows from the triangle inequality. Part (ii) includes finite telescoping and, from summable resolution decay, full Cauchy and complete-space convergence with a tail displacement bound."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-011"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumA.ChainedApprox.cauchySeq",
      "ScholiumA.ChainedApprox.exists_limit_with_tail_bound",
      "ScholiumA.chainedApprox_telescope",
      "ScholiumA.twoStep_bound"
    ]
  },
  "line": 553,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Observer–Bounded Emergence Constraint",
  "proof_labels": [
    "proof:bk1_bounded_drift_approximation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "y) := \\lVert K_O * (x - y) \\rVert. \\) Along any cofinal sequence \\((\\lambda_n)\\) satisfying Ax.~\\ref{axiom:bk1_summable_resolution_decay}, the transition maps obey \\[ d_O(f_{\\lambda_m\\lambda_n}(x),x) \\leq \\sum_{j=m}^{n-1}\\eta_j",
      "label": "axiom:bk1_summable_resolution_decay",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 532,
      "target_type": "axiom"
    },
    {
      "context": "O},\\varepsilon_O) \\) be a bounded observer with resolution kernel \\(K_O\\) and scalar threshold \\(\\delta_O\\) (Definition~\\ref{definition:bk1_bounded_observer}). For every ordinal \\(\\lambda<\\Omega\\), define the stage--composite operator \\[ E_\\lambda := R_\\lambda \\circ D_\\lam",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "is \\(d_O\\)-Cauchy in the usual tail sense (see also the formal directed emergence structure in Definition~\\ref{definition:bk1_directed_system_of_emergence}). Its \\(d_O\\)-completion \\( \\overline{P}_O \\) supplies the observer-completed proto-s",
      "label": "definition:bk1_directed_system_of_emergence",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 687,
      "target_type": "definition"
    },
    {
      "context": "\\; 2\\,\\delta_O. \\end{equation} (This satisfies the kernel-bounded approximation condition in Definition~\\ref{definition:bk1_kernel_based_bounded_symbolic_approximation}.) \\item[\\textup{(ii)}] \\textbf{Cauchy tower under summable decay.}\\; Endow every \\(P_\\lambda\\) with the obs",
      "label": "definition:bk1_kernel_based_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 433,
      "target_type": "definition"
    },
    {
      "context": ":bk1_stage_composite_operator}, where \\(P_{<\\lambda}\\) and \\(P_\\lambda\\) are symbolic stages introduced in Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}. Then \\begin{enumerate} \\item[\\textup{(i)}] \\textbf{Bounded approximation of the identity.}\\; For all \\(s",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "proto-symbolic space associated with the colimit \\(P=\\varinjlim_{\\lambda<\\Omega}P_\\lambda\\) (Definition~\\ref{definition:bk1_proto_symbolic_space}). \\end{enumerate} \\end{lemma}",
      "label": "definition:bk1_proto_symbolic_space",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 703,
      "target_type": "definition"
    },
    {
      "context": "rator \\[ E_\\lambda := R_\\lambda \\circ D_\\lambda : P_{<\\lambda} \\longrightarrow P_\\lambda, \\] as defined in Definition~\\ref{definition:bk1_stage_composite_operator}, where \\(P_{<\\lambda}\\) and \\(P_\\lambda\\) are symbolic stages introduced in Definition~\\ref{definition:bk1_pre_geomet",
      "label": "definition:bk1_stage_composite_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 520,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_kernel_based_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space",
    "definition:bk1_stage_composite_operator"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "depends_on": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_bounded_drift_approximation",
  "label": "proof:bk1_bounded_drift_approximation",
  "latex_body": "\\begin{proof}[Bounded Approximation Guarantees Drift Convergence]\n\\label{proof:bk1_bounded_drift_approximation}\n\\leavevmode\n\nBecause \\(D_\\lambda\\) is a bounded symbolic approximation (see Definition~\\ref{definition:bk1_bounded_symbolic_approximation} and Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}), we have\n\\[\n  \\lVert K_O*[D_\\lambda(s)-s]\\rVert \\le \\delta_O\n\\]\nfor all \\(s\\in P_{<\\lambda}\\). Applying \\(R_\\lambda\\) and using the boundedness property again (with \\(s' := D_\\lambda(s)\\)) gives\n\\[\n  \\lVert K_O*[R_\\lambda(D_\\lambda(s))-D_\\lambda(s)]\\rVert \\le \\delta_O.\n\\]\nThe triangle inequality for the observer norm then yields the overall bound. For \\(\\lambda<\\mu<\\Omega\\), we have\n\\[\nf_{\\lambda\\mu} = E_{\\mu-1}\\circ\\dots\\circ E_\\lambda,\n\\]\nwhere each \\(E_\\lambda\\) is the stage--composite operator (Definition~\\ref{definition:bk1_stage_composite_operator}) in the successor-indexed case; along a cofinal sequence the same expression is read as composition through the intervening transition maps.\n\nThe one-step estimate alone does not give a uniform bound for arbitrary long\ncomposites. Under Ax.~\\ref{axiom:bk1_summable_resolution_decay}, however, the\nobserver-visible displacement of the composite from \\(\\lambda_m\\) to\n\\(\\lambda_n\\) is bounded by the telescoping tail:\n\\[\nd_O(f_{\\lambda_m\\lambda_n}(x),x)\n\\leq \\sum_{j=m}^{n-1}d_O(E_{\\lambda_j}(x_j),x_j)\n\\leq \\sum_{j=m}^{n-1}\\eta_j,\n\\]\nwhere \\(x_j=f_{\\lambda_m\\lambda_j}(x)\\). Since \\(\\sum_j\\eta_j<\\infty\\), the\ntails \\(\\sum_{j=m}^{\\infty}\\eta_j\\) tend to zero. Hence the cofinal tower is\nCauchy in \\(d_O\\), and its observer-completed limit exists in\n\\(\\overline{P}_O\\).\n\\end{proof}",
  "line": 604,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Bounded Approximation Guarantees Drift Convergence",
  "proves": "lemma:bk1_observer_bounded_emergence_constraint",
  "ref_roles": [
    {
      "context": "ng transition maps. The one-step estimate alone does not give a uniform bound for arbitrary long composites. Under Ax.~\\ref{axiom:bk1_summable_resolution_decay}, however, the observer-visible displacement of the composite from \\(\\lambda_m\\) to \\(\\lambda_n\\) is bounded by the tele",
      "label": "axiom:bk1_summable_resolution_decay",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 532,
      "target_type": "axiom"
    },
    {
      "context": "bk1_bounded_drift_approximation} \\leavevmode Because \\(D_\\lambda\\) is a bounded symbolic approximation (see Definition~\\ref{definition:bk1_bounded_symbolic_approximation} and Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}), we have \\[ \\lVert K_O*[D_\\lambda(s)-s]\\rVert",
      "label": "definition:bk1_bounded_symbolic_approximation",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 373,
      "target_type": "definition"
    },
    {
      "context": "is a bounded symbolic approximation (see Definition~\\ref{definition:bk1_bounded_symbolic_approximation} and Definition~\\ref{definition:bk1_pre_geometric_operators_and_stages}), we have \\[ \\lVert K_O*[D_\\lambda(s)-s]\\rVert \\le \\delta_O \\] for all \\(s\\in P_{<\\lambda}\\). Applying \\(R_\\lambda\\)",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "mbda\\mu} = E_{\\mu-1}\\circ\\dots\\circ E_\\lambda, \\] where each \\(E_\\lambda\\) is the stage--composite operator (Definition~\\ref{definition:bk1_stage_composite_operator}) in the successor-indexed case; along a cofinal sequence the same expression is read as composition through the interve",
      "label": "definition:bk1_stage_composite_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 520,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_symbolic_approximation",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_stage_composite_operator"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk4_refinement_envelope"
  ],
  "cites": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "depends_on": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "scholium:bk1_emergence_envelope",
  "label": "scholium:bk1_emergence_envelope",
  "latex_body": "\\begin{scholium}[Emergence Envelope]\n\\label{scholium:bk1_emergence_envelope}\nTo a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), the tower of\nemergent symbolic structures (Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages})\nunfolds through finite one-step observer envelopes, and along any cofinal tower\nsatisfying Ax.~\\ref{axiom:bk1_summable_resolution_decay} its unresolved tail\nshrinks to zero in \\(d_O\\). Curvature, dimensional refinement, and horizon\nbifurcations may still arise, but their observer-visible refinements must become\nsummably finer for a completed proto-symbolic limit to be available. This\ntail-envelope is the geometric shadow of observer-boundedness that guides the\nsubsequent smoothness construction.\n\\end{scholium}",
  "line": 636,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Emergence Envelope",
  "ref_roles": [
    {
      "context": "c_operators_and_stages}) unfolds through finite one-step observer envelopes, and along any cofinal tower satisfying Ax.~\\ref{axiom:bk1_summable_resolution_decay} its unresolved tail shrinks to zero in \\(d_O\\). Curvature, dimensional refinement, and horizon bifurcations may still a",
      "label": "axiom:bk1_summable_resolution_decay",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 532,
      "target_type": "axiom"
    },
    {
      "context": "\\begin{scholium}[Emergence Envelope] \\label{scholium:bk1_emergence_envelope} To a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), the tower of emergent symbolic structures (Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}) unfolds thro",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "pe} To a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}), the tower of emergent symbolic structures (Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}) unfolds through finite one-step observer envelopes, and along any cofinal tower satisfying Ax.~\\ref{axiom:bk1_summable",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_summable_resolution_decay",
    "definition:bk1_bounded_observer",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk4_test_time_precision_refinement",
    "definition:bk9_symbolic_accountability",
    "remark:bk8_inference_principle_over_confidence_loss_tradeoff",
    "scholium:bk4_precision_without_collapse",
    "scholium:bk5_hypotheses_as_adaptive_sym",
    "scholium:bk8_autonomous_repair_systems_expanded",
    "sec:bk2_foundations_symbolic_thermodynamics",
    "subsec:bk7_pisu_revisited_power_uncertainty"
  ],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "scholium:bk1_epistemic_humility",
  "label": "scholium:bk1_epistemic_humility",
  "latex_body": "\\begin{scholium}[Epistemic Humility]\n\\label{scholium:bk1_epistemic_humility}\n\\textbf{Premise (Observer‑Boundedness).}  \nEvery act of cognition is executed by a \\emph{bounded observer}\\/ $O=(N_O,\\{\\delta^n_O\\}_{n\\le N_O},\\varepsilon_O)$ (Def.~\\ref{definition:bk1_bounded_observer}.  \nHence all symbolic operators that $O$ can deploy must respect the perceptual threshold\n\\[\n  \\|K_O\\ast[\\Phi(s)-s]\\|\\le\\varepsilon_O\n  \\quad\\text{for all observable symbols }s.\n\\]\n\\medskip\n\\textbf{Principle (Epistemic Humility).}  \nBecause $O$ \\emph{cannot} transcend its own resolution kernel $K_O$, any claim about the symbolic manifold $S$ must be  \n1) provisional,  \n2) open to \\emph{differentiation \\& reintegration},  \n3) anchored in \\emph{knowledge integrity},  \n4) iteratively refined along a \\emph{learning path}, and  \n5) stated with full \\emph{mathematical rigour}.  \nThese five clauses instantiate the four core \\textsc{Giants} axioms:  \n\\begin{enumerate}[label=\\arabic*.]\n  \\item \\textbf{Differentiation \\& Reintegration} — structure updates occur by decomposing $\\Phi$ into locally bounded moves and re‑synthesising them.  \n  \\item \\textbf{Knowledge Integrity} — updates that breach the boundedness constraint are rejected as incoherent.  \n  \\item \\textbf{Learning Path Influence} — mismatch $\\Delta=\\|\\Phi(s)-s\\|$ feeds back into subsequent operator design, minimising loss $L_{n+1}$ (see FormalMath core equation).  \n  \\item \\textbf{Mathematical Rigor} — all admissible claims are stated as formally verifiable lemmas or energy inequalities.  \n\\end{enumerate}\n\\medskip\n\\textbf{Lemma (Bounded‑Humility Constraint).}  \nLet $\\mathcal{E}$ be the set of epistemic commitments formulable by $O$ at symbolic time $t$.  \nThen the update map $\\rho_t:\\mathcal{E}\\to\\mathcal{E}$ generated by any admissible operator $\\Phi_t$ satisfies\n\\[\n  \\rho_t(e)\\;=\\;e\\;+\\;\\underbrace{\\bigl(\\Phi_t(e)-e\\bigr)}_{\\text{differentiation}}\n  \\quad\\text{with}\\quad\n  \\|K_O\\ast\\bigl(\\Phi_t(e)-e\\bigr)\\|\\le\\varepsilon_O,\n\\]\nso $\\rho_t$ is a \\emph{bounded symbolic approximation} (Def.~\\ref{definition:bk1_bounded_observer}).  \nConsequently, epistemic humility is not optional but a \\emph{necessary condition} for reflexive emergence: without it, $\\Phi_t$ would violate boundedness and fracture the observer’s horizon.\n\\end{scholium}",
  "line": 648,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Epistemic Humility",
  "ref_roles": [
    {
      "context": "ery act of cognition is executed by a \\emph{bounded observer}\\/ $O=(N_O,\\{\\delta^n_O\\}_{n\\le N_O},\\varepsilon_O)$ (Def.~\\ref{definition:bk1_bounded_observer}. Hence all symbolic operators that $O$ can deploy must respect the perceptual threshold \\[ \\|K_O\\ast[\\Phi(s)-s]\\|\\l",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "remark:scholium_symbolicum.tex:684",
  "label": "",
  "latex_body": "\\begin{remark}\n    This orientation toward epistemic humility prefigures the more formal construct of \\emph{Symbolic Accountability}, where coherence, transparency, and relational viability are operationalized.\n\\end{remark}",
  "line": 684,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "",
  "refs": [],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_proto_symbolic_space",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "lemma:bk1_universality_of_proto_symbolic_space",
    "proof:bk1_colimit_yields_categoric_structure"
  ],
  "cites": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "depends_on": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_directed_system_of_emergence",
  "label": "definition:bk1_directed_system_of_emergence",
  "latex_body": "\\begin{definition}[Directed System of Emergence]\n\\label{definition:bk1_directed_system_of_emergence}\nThe directed system $\\{P_\\lambda, f_{\\lambda\\mu}\\}_{\\lambda < \\mu < \\Omega}$ consists of:\n\\begin{itemize}\n    \\item Objects: The symbolic structures $P_\\lambda$ (see Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}).\n    \\item Morphisms: $f_{\\lambda\\mu}: P_\\lambda \\to P_\\mu$ for $\\lambda < \\mu < \\Omega$, representing structure-preserving evolution.\n\\end{itemize}\nThese satisfy the standard conditions:\n\\begin{itemize}\n    \\item $f_{\\lambda\\lambda} = id_{P_\\lambda}$ (identity).\n    \\item $f_{\\mu\\nu} \\circ f_{\\lambda\\mu} = f_{\\lambda\\nu}$ for all $\\lambda < \\mu < \\nu < \\Omega$ (composition).\n\\end{itemize}\nWe require each $f_{\\lambda\\mu}$ to be continuous with respect to the topologies on $P_\\lambda$ and $P_\\mu$.\n\nConceptually, each $f_{\\lambda\\mu}$ represents the cumulative effect of the interplay between stabilization ($R_\\nu$) and differentiation ($D_{\\nu+1}$) for stages $\\nu$ from $\\lambda$ to $\\mu-1$. For instance, $f_{\\lambda, \\lambda+1}$ can be thought of as mapping a structure stabilized by $R_\\lambda$ into the next stage generated via $D_{\\lambda+1}$. This description is itself a bounded approximation of the complex entanglement of drift and reflection.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A Nat-directed system explicitly records stage carriers, transition maps, and their identity/composition laws; transition images are identified in the colimit and every compatible cocone has a unique mediator."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-026"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumC.DirectedStageSystem.directed_colimit_universal_property",
      "ScholiumC.DirectedStageSystem.injection_transition"
    ]
  },
  "line": 687,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Directed System of Emergence",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "{\\lambda < \\mu < \\Omega}$ consists of: \\begin{itemize} \\item Objects: The symbolic structures $P_\\lambda$ (see Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}). \\item Morphisms: $f_{\\lambda\\mu}: P_\\lambda \\to P_\\mu$ for $\\lambda < \\mu < \\Omega$, representing structure-prese",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_local_charitability",
    "axiom:bk1_smooth_convergence",
    "axiom:bk1_topological_regularity",
    "lemma:bk1_observer_bounded_emergence_constraint",
    "lemma:bk1_universality_of_proto_symbolic_space",
    "proof:bk1_sketch_limit_stabilization_colimit",
    "proposition:bk4_ttpr_convergence",
    "subsec:appD_category_theory_core_resonance",
    "theorem:bk1_manifold_emergence"
  ],
  "cites": [
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "depends_on": [
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_proto_symbolic_space",
  "label": "definition:bk1_proto_symbolic_space",
  "latex_body": "\\begin{definition}[Proto-symbolic Space]\n\\label{definition:bk1_proto_symbolic_space}\nThe proto-symbolic space $P$ is defined as the colimit in the category $\\catS$ (see Def.~\\ref{definition:bk1_let_cats_be_the_category}):\n\\[\nP := \\varinjlim_{\\lambda < \\Omega} P_\\lambda\n\\]\nElements of $P$ are equivalence classes $[(x_\\lambda)]$ where $x_\\lambda \\in P_\\lambda$, under the relation $x_\\lambda \\sim x_\\mu$ if there exists $\\nu \\geq \\lambda, \\mu$ such that $f_{\\lambda\\nu}(x_\\lambda) = f_{\\mu\\nu}(x_\\mu)$ (cf.~Def.~\\ref{definition:bk1_directed_system_of_emergence}). The topology on $P$ is the final topology making all canonical injections $i_\\lambda: P_\\lambda \\to P$ continuous (see also Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The proto-symbolic carrier is now the quotient of the coproduct of a concrete Nat-directed stage tower by eventual compatibility; genuinely ordinal indexing and the source category catS remain abstracted."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-024"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumC.DirectedStageSystem.directed_colimit_universal_property",
      "ScholiumC.colimit_universal_property"
    ]
  },
  "line": 703,
  "macros_used": [
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Proto-symbolic Space",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\sim x_\\mu$ if there exists $\\nu \\geq \\lambda, \\mu$ such that $f_{\\lambda\\nu}(x_\\lambda) = f_{\\mu\\nu}(x_\\mu)$ (cf.~Def.~\\ref{definition:bk1_directed_system_of_emergence}). The topology on $P$ is the final topology making all canonical injections $i_\\lambda: P_\\lambda \\to P$ continuous (se",
      "label": "definition:bk1_directed_system_of_emergence",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 687,
      "target_type": "definition"
    },
    {
      "context": "tion:bk1_proto_symbolic_space} The proto-symbolic space $P$ is defined as the colimit in the category $\\catS$ (see Def.~\\ref{definition:bk1_let_cats_be_the_category}): \\[ P := \\varinjlim_{\\lambda < \\Omega} P_\\lambda \\] Elements of $P$ are equivalence classes $[(x_\\lambda)]$ where $x_\\",
      "label": "definition:bk1_let_cats_be_the_category",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 16,
      "target_type": "definition"
    },
    {
      "context": "ogy on $P$ is the final topology making all canonical injections $i_\\lambda: P_\\lambda \\to P$ continuous (see also Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}). \\end{definition}",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_universality_of_proto_symbolic_space",
  "label": "lemma:bk1_universality_of_proto_symbolic_space",
  "latex_body": "\\begin{lemma}[Universality of Proto-symbolic Space]\n\\label{lemma:bk1_universality_of_proto_symbolic_space}\nThe proto-symbolic space $P$ satisfies the universal property of colimits in $\\catS$ (see Def.~\\ref{definition:bk1_let_cats_be_the_category}): for any object $Q \\in Ob(\\catS)$ and compatible family of morphisms $\\{g_\\lambda: P_\\lambda \\to Q\\}_{\\lambda < \\Omega}$ (i.e., $g_\\mu \\circ f_{\\lambda\\mu} = g_\\lambda$ for $\\lambda < \\mu$, per Def.~\\ref{definition:bk1_directed_system_of_emergence}), there exists a unique morphism $g: P \\to Q$ such that $g \\circ i_\\lambda = g_\\lambda$ for all $\\lambda < \\Omega$ (cf.~Def.~\\ref{definition:bk1_proto_symbolic_space}, Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}).\n\\begin{proof}[Colimit Structure Yields Symbolic Cohesion]\n\\label{proof:bk1_colimit_yields_categoric_structure}\n\\leavevmode\n\nGiven Axiom~\\ref{axiom:bk1_axiomata_prima}, the stagewise symbolic structures are generated through non-trivial drift and require coherent stabilization across levels (cf.~Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}, Def.~\\ref{definition:bk1_directed_system_of_emergence}). Under cocompleteness of $\\catS$, the colimit definition then yields the unique mediating morphism and hence symbolic cohesion.\n\\end{proof}\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Existence and uniqueness of the mediating morphism is proved both for a general quotient and for cocones over an explicit Nat-directed diagram with lawful transition maps; ordinal indexing and catS remain abstracted."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-025"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumC.DirectedStageSystem.directed_colimit_universal_property",
      "ScholiumC.colimit_universal_property"
    ]
  },
  "line": 711,
  "macros_used": [
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Universality of Proto-symbolic Space",
  "proof_labels": [
    "proof:bk1_colimit_yields_categoric_structure"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "da: P_\\lambda \\to Q\\}_{\\lambda < \\Omega}$ (i.e., $g_\\mu \\circ f_{\\lambda\\mu} = g_\\lambda$ for $\\lambda < \\mu$, per Def.~\\ref{definition:bk1_directed_system_of_emergence}), there exists a unique morphism $g: P \\to Q$ such that $g \\circ i_\\lambda = g_\\lambda$ for all $\\lambda < \\Omega$ (cf.",
      "label": "definition:bk1_directed_system_of_emergence",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 687,
      "target_type": "definition"
    },
    {
      "context": "of_proto_symbolic_space} The proto-symbolic space $P$ satisfies the universal property of colimits in $\\catS$ (see Def.~\\ref{definition:bk1_let_cats_be_the_category}): for any object $Q \\in Ob(\\catS)$ and compatible family of morphisms $\\{g_\\lambda: P_\\lambda \\to Q\\}_{\\lambda < \\Omega",
      "label": "definition:bk1_let_cats_be_the_category",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 16,
      "target_type": "definition"
    },
    {
      "context": "at $g \\circ i_\\lambda = g_\\lambda$ for all $\\lambda < \\Omega$ (cf.~Def.~\\ref{definition:bk1_proto_symbolic_space}, Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}). \\begin{proof}[Colimit Structure Yields Symbolic Cohesion] \\label{proof:bk1_colimit_yields_categoric_structure} \\leave",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    },
    {
      "context": "re exists a unique morphism $g: P \\to Q$ such that $g \\circ i_\\lambda = g_\\lambda$ for all $\\lambda < \\Omega$ (cf.~Def.~\\ref{definition:bk1_proto_symbolic_space}, Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}). \\begin{proof}[Colimit Structure Yields Symbolic Cohesio",
      "label": "definition:bk1_proto_symbolic_space",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 703,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_let_cats_be_the_category",
    "definition:bk1_pre_geometric_operators_and_stages",
    "definition:bk1_proto_symbolic_space"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "cites": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_colimit_yields_categoric_structure",
  "label": "proof:bk1_colimit_yields_categoric_structure",
  "latex_body": "\\begin{proof}[Colimit Structure Yields Symbolic Cohesion]\n\\label{proof:bk1_colimit_yields_categoric_structure}\n\\leavevmode\n\nGiven Axiom~\\ref{axiom:bk1_axiomata_prima}, the stagewise symbolic structures are generated through non-trivial drift and require coherent stabilization across levels (cf.~Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}, Def.~\\ref{definition:bk1_directed_system_of_emergence}). Under cocompleteness of $\\catS$, the colimit definition then yields the unique mediating morphism and hence symbolic cohesion.\n\\end{proof}",
  "line": 714,
  "macros_used": [
    "catS"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Colimit Structure Yields Symbolic Cohesion",
  "proves": "lemma:bk1_universality_of_proto_symbolic_space",
  "ref_roles": [
    {
      "context": "limit Structure Yields Symbolic Cohesion] \\label{proof:bk1_colimit_yields_categoric_structure} \\leavevmode Given Axiom~\\ref{axiom:bk1_axiomata_prima}, the stagewise symbolic structures are generated through non-trivial drift and require coherent stabilization across le",
      "label": "axiom:bk1_axiomata_prima",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book1.tex",
      "target_line": 3,
      "target_type": "axiom"
    },
    {
      "context": "nd require coherent stabilization across levels (cf.~Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}, Def.~\\ref{definition:bk1_directed_system_of_emergence}). Under cocompleteness of $\\catS$, the colimit definition then yields the unique mediating morphism and hence symbolic",
      "label": "definition:bk1_directed_system_of_emergence",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 687,
      "target_type": "definition"
    },
    {
      "context": "symbolic structures are generated through non-trivial drift and require coherent stabilization across levels (cf.~Def.~\\ref{definition:bk1_pre_geometric_operators_and_stages}, Def.~\\ref{definition:bk1_directed_system_of_emergence}). Under cocompleteness of $\\catS$, the colimit definition then",
      "label": "definition:bk1_pre_geometric_operators_and_stages",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 355,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_axiomata_prima",
    "definition:bk1_directed_system_of_emergence",
    "definition:bk1_pre_geometric_operators_and_stages"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_necessity_of_the_dual_horizon_structure",
  "label": "subsec:bk1_necessity_of_the_dual_horizon_structure",
  "latex_body": "",
  "line": 721,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Proof by Elimination: Necessity of the Dual Horizon Structure",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_bounded_reflexive_emergence",
    "proof:bk1_horizon_characterization",
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "proof:bk4_wheel_refines_signature",
    "proposition:bk4_wheel_refines_signature"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "al{O}}(H) :=\\int_{H\\cap\\Omega_{\\mathcal{O}}}\\max(-\\kappa,0)\\,d\\sigma, \\] where \\(\\kappa\\) is symbolic curvature (Def.~\\ref{definition:bk1_symbolic_riemann_tensor}) and \\(d\\sigma\\) is the induced horizon measure. The effective horizon signature is \\[ \\Sigma_{\\mathcal{O}}(\\mathcal{U}",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "line_distance": 1181,
      "role": "teaser",
      "target_line": 1905,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "id": "definition:bk1_effective_horizon_signature",
  "label": "definition:bk1_effective_horizon_signature",
  "latex_body": "\\begin{definition}[Effective Horizon Signature]\n\\label{definition:bk1_effective_horizon_signature}\nLet $\\mathcal{U}$ be a symbolic universe sustaining a bounded observer\n\\(\\mathcal{O}\\) on a nonempty observer domain \\(\\Omega_{\\mathcal{O}}\\)\n(Def.~\\ref{definition:bk1_bounded_observer}). For any horizon component\n\\(H\\) meeting \\(\\Omega_{\\mathcal{O}}\\), define its observer-visible curvature\nfluxes by\n\\[\nG_{\\mathcal{O}}(H)\n  :=\\int_{H\\cap\\Omega_{\\mathcal{O}}}\\max(\\kappa,0)\\,d\\sigma,\n\\qquad\nC_{\\mathcal{O}}(H)\n  :=\\int_{H\\cap\\Omega_{\\mathcal{O}}}\\max(-\\kappa,0)\\,d\\sigma,\n\\]\nwhere \\(\\kappa\\) is symbolic curvature\n(Def.~\\ref{definition:bk1_symbolic_riemann_tensor}) and \\(d\\sigma\\) is the\ninduced horizon measure. The effective horizon signature is\n\\[\n\\Sigma_{\\mathcal{O}}(\\mathcal{U})\n\\subseteq \\{+,-\\},\n\\]\nwith \\(+\\in\\Sigma_{\\mathcal{O}}(\\mathcal{U})\\) iff some horizon component has\n\\(G_{\\mathcal{O}}(H)>0\\), and\n\\(-\\in\\Sigma_{\\mathcal{O}}(\\mathcal{U})\\) iff some horizon component has\n\\(C_{\\mathcal{O}}(H)>0\\). Multiple horizons and sign-changing horizons are\ntherefore represented by their effective observer-visible sign content.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Modeled on a single generative/dissipative flux pair (G, C) rather than existential quantification over multiple horizon components; the integral definitions of G_O, C_O themselves are not modeled, only the resulting sign predicate."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-028"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumC.effectiveSignature_empty",
      "ScholiumC.effectiveSignature_full"
    ]
  },
  "line": 724,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Effective Horizon Signature",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "lic universe sustaining a bounded observer \\(\\mathcal{O}\\) on a nonempty observer domain \\(\\Omega_{\\mathcal{O}}\\) (Def.~\\ref{definition:bk1_bounded_observer}). For any horizon component \\(H\\) meeting \\(\\Omega_{\\mathcal{O}}\\), define its observer-visible curvature fluxes by \\[",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "al{O}}(H) :=\\int_{H\\cap\\Omega_{\\mathcal{O}}}\\max(-\\kappa,0)\\,d\\sigma, \\] where \\(\\kappa\\) is symbolic curvature (Def.~\\ref{definition:bk1_symbolic_riemann_tensor}) and \\(d\\sigma\\) is the induced horizon measure. The effective horizon signature is \\[ \\Sigma_{\\mathcal{O}}(\\mathcal{U}",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "cites": [
    "corollary:bk1_fixed_point",
    "definition:bk1_effective_horizon_signature",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "depends_on": [
    "definition:bk1_effective_horizon_signature",
    "definition:bk2_symbolic_free_energy"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "the stabilized reduction of symbolic free energy \\(\\freeenergy\\) (Def.~\\ref{definition:bk2_symbolic_free_energy}; Cor.~\\ref{corollary:bk1_fixed_point}) that the observer can both \\emph{register} and \\emph{keep}. The criterion is stated independently of any horizon geome",
      "label": "corollary:bk1_fixed_point",
      "line_distance": 2251,
      "role": "teaser",
      "target_line": 3003,
      "target_type": "corollary"
    },
    {
      "context": "of any horizon geometry: that both a generative and a stabilizing channel are present is the \\emph{conclusion} of Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, not a premise, and the product \\(G_{\\mathcal{O}}(H_G)\\,C_{\\mathcal{O}}(H_D)\\) of Def.~\\ref{definition:bk1_effective_ho",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "line_distance": 23,
      "role": "teaser",
      "target_line": 775,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "corollary:bk1_fixed_point",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "id": "definition:bk1_bounded_reflexive_emergence",
  "label": "definition:bk1_bounded_reflexive_emergence",
  "latex_body": "\\begin{definition}[Bounded Reflexive Emergence]\n\\label{definition:bk1_bounded_reflexive_emergence}\nA symbolic universe \\(\\mathcal{U}\\) supports \\emph{bounded reflexive emergence}\nfor \\(\\mathcal{O}\\) when, over some interval of symbolic time on the observer\ndomain \\(\\Omega_{\\mathcal{O}}\\), the observer-visible emergence functional\n\\[\n\\Delta\\Phi_{\\mathcal{O}}(D,R_{\\mathrm{stab}}) \\;\\ge\\; \\tau_E \\;>\\; 0,\n\\]\nwhere \\(\\Delta\\Phi_{\\mathcal{O}}\\) measures the net retained, observer-resolved\ncoherent structure produced by the coupled action of novelty-generating drift\n\\(D\\) and stabilizing reflection \\(R_{\\mathrm{stab}}\\) --- equivalently, the\nstabilized reduction of symbolic free energy \\(\\freeenergy\\)\n(Def.~\\ref{definition:bk2_symbolic_free_energy};\nCor.~\\ref{corollary:bk1_fixed_point}) that the observer can both \\emph{register}\nand \\emph{keep}. The criterion is stated independently of any horizon geometry:\nthat both a generative and a stabilizing channel are present is the\n\\emph{conclusion} of Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, not a\npremise, and the product \\(G_{\\mathcal{O}}(H_G)\\,C_{\\mathcal{O}}(H_D)\\) of\nDef.~\\ref{definition:bk1_effective_horizon_signature} is the \\emph{binding}\nspecial case in which the two fluxes are read off a single\ngenerative/dissipative pair.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the 'binding special case' clause (Delta Phi_O = G_O(H_G) C_O(H_D)) is modeled as a scalar product-threshold fact; the general observer-resolved free-energy criterion is not."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-030"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumC.dualHorizonBinding_both_pos"
    ]
  },
  "line": 752,
  "macros_used": [
    "freeenergy"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Bounded Reflexive Emergence",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "the stabilized reduction of symbolic free energy \\(\\freeenergy\\) (Def.~\\ref{definition:bk2_symbolic_free_energy}; Cor.~\\ref{corollary:bk1_fixed_point}) that the observer can both \\emph{register} and \\emph{keep}. The criterion is stated independently of any horizon geome",
      "label": "corollary:bk1_fixed_point",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3003,
      "target_type": "corollary"
    },
    {
      "context": "_dual_horizon_necessity_theorem}, not a premise, and the product \\(G_{\\mathcal{O}}(H_G)\\,C_{\\mathcal{O}}(H_D)\\) of Def.~\\ref{definition:bk1_effective_horizon_signature} is the \\emph{binding} special case in which the two fluxes are read off a single generative/dissipative pair. \\end{defi",
      "label": "definition:bk1_effective_horizon_signature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 724,
      "target_type": "definition"
    },
    {
      "context": "flection \\(R_{\\mathrm{stab}}\\) --- equivalently, the stabilized reduction of symbolic free energy \\(\\freeenergy\\) (Def.~\\ref{definition:bk2_symbolic_free_energy}; Cor.~\\ref{corollary:bk1_fixed_point}) that the observer can both \\emph{register} and \\emph{keep}. The criterion is sta",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "of any horizon geometry: that both a generative and a stabilizing channel are present is the \\emph{conclusion} of Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, not a premise, and the product \\(G_{\\mathcal{O}}(H_G)\\,C_{\\mathcal{O}}(H_D)\\) of Def.~\\ref{definition:bk1_effective_ho",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk1_fixed_point",
    "definition:bk1_effective_horizon_signature",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "appendix_teaser_ref_roles": [
    {
      "context": "d sign-changing horizons and the explicit coupling premise on which the converse rests --- is given in Appendix~C (Thm.~\\ref{theorem:appC_dual_horizon_signature}), which defers to this theorem for the canonical formal statement. \\end{theorem}",
      "label": "theorem:appC_dual_horizon_signature",
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 78,
      "target_type": "theorem"
    }
  ],
  "appendix_teaser_refs": [
    "theorem:appC_dual_horizon_signature"
  ],
  "book": "scholium_symbolicum",
  "canonical_expansions": [
    "theorem:appC_dual_horizon_signature"
  ],
  "canonical_status": "canonical_statement",
  "cited_by": [
    "axiom:bk1_dual_horizon_postulate",
    "corollary:bk1_event_horizon_identity_field",
    "corollary:bk1_horizon_duality_principle",
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk1_horizon_crossing_operation",
    "lemma:bk1_horizon_characterization",
    "proof:bk1_dual_horizon_unification_principle",
    "proof:bk1_event_horizon_identity_field",
    "proof:bk1_horizon_characterization",
    "proof:bk1_horizon_duality_principle",
    "proof:bk1_sketch_observed_consequences",
    "scholium:bk1_cosmogenesis_proof_status",
    "sec:appC_dual_horizon",
    "theorem:appC_dual_horizon_signature",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_unification_principle",
    "theorem:bk1_quadratic_structure_necessity"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_reflexive_emergence",
    "proof:bk1_colimit_yields_categoric_structure",
    "theorem:appC_dual_horizon_signature"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk1_effective_horizon_signature",
    "proof:bk1_colimit_yields_categoric_structure"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_dual_horizon_necessity_theorem",
  "label": "theorem:bk1_dual_horizon_necessity_theorem",
  "latex_body": "\\begin{theorem}[Dual Horizon Necessity Theorem]\n\\label{theorem:bk1_dual_horizon_necessity_theorem}\nLet $\\mathcal{U}$ be a symbolic universe sustaining bounded observers within a\ndomain $\\Omega_{\\mathcal{O}}$ (cf.~Def.~\\ref{definition:bk1_bounded_observer}),\nwhose stagewise structures cohere into a categorical colimit\n(cf.~Proof~\\ref{proof:bk1_colimit_yields_categoric_structure}). If \\(\\mathcal{U}\\)\nsupports bounded reflexive emergence for \\(\\mathcal{O}\\)\n(Def.~\\ref{definition:bk1_bounded_reflexive_emergence}), then it possesses an\neffective dual horizon structure on a shared bounded domain,\n\\[\n\\Sigma_{\\mathcal{O}}(\\mathcal{U})=\\{+,-\\}:\n\\qquad G_{\\mathcal{O}}(H_G)>0 \\ \\text{and}\\ C_{\\mathcal{O}}(H_D)>0,\n\\]\ni.e.\\ at least one observer-visible positive-curvature novelty channel and one negative-curvature stabilization channel meeting a common\n\\(\\Omega_{\\mathcal{O}}\\). Conversely, when both channels are present on a shared\ndomain and their fluxes couple above the observer threshold,\n\\(\\Delta\\Phi_{\\mathcal{O}}(D,R_{\\mathrm{stab}})\\ge\\tau_E\\) and emergence follows.\nThe necessity direction is unconditional; the converse is the binding, coupled\ncase. The expanded two-modality derivation --- with its realization-invariance\nacross multiple and sign-changing horizons and the explicit coupling premise on\nwhich the converse rests --- is given in Appendix~C\n(Thm.~\\ref{theorem:appC_dual_horizon_signature}), which defers to this theorem\nfor the canonical formal statement.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Only the converse/binding, coupled-case direction is modeled (product of fluxes above threshold forces both signs present); the necessity direction and the general (non-binding) case are not modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-029"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumC.dualHorizonBinding_both_pos"
    ]
  },
  "line": 775,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Dual Horizon Necessity Theorem",
  "proof_labels": [
    "proof:bk1_proof_of_dual_horizon_necessity_theorem"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "Let $\\mathcal{U}$ be a symbolic universe sustaining bounded observers within a domain $\\Omega_{\\mathcal{O}}$ (cf.~Def.~\\ref{definition:bk1_bounded_observer}), whose stagewise structures cohere into a categorical colimit (cf.~Proof~\\ref{proof:bk1_colimit_yields_categoric_struc",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "colimit_yields_categoric_structure}). If \\(\\mathcal{U}\\) supports bounded reflexive emergence for \\(\\mathcal{O}\\) (Def.~\\ref{definition:bk1_bounded_reflexive_emergence}), then it possesses an effective dual horizon structure on a shared bounded domain, \\[ \\Sigma_{\\mathcal{O}}(\\mathcal{U}",
      "label": "definition:bk1_bounded_reflexive_emergence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 752,
      "target_type": "definition"
    },
    {
      "context": "f.~Def.~\\ref{definition:bk1_bounded_observer}), whose stagewise structures cohere into a categorical colimit (cf.~Proof~\\ref{proof:bk1_colimit_yields_categoric_structure}). If \\(\\mathcal{U}\\) supports bounded reflexive emergence for \\(\\mathcal{O}\\) (Def.~\\ref{definition:bk1_bounded_reflexi",
      "label": "proof:bk1_colimit_yields_categoric_structure",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 714,
      "target_type": "proof"
    },
    {
      "context": "d sign-changing horizons and the explicit coupling premise on which the converse rests --- is given in Appendix~C (Thm.~\\ref{theorem:appC_dual_horizon_signature}), which defers to this theorem for the canonical formal statement. \\end{theorem}",
      "label": "theorem:appC_dual_horizon_signature",
      "logical_support": false,
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 78,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_bounded_reflexive_emergence",
    "proof:bk1_colimit_yields_categoric_structure",
    "theorem:appC_dual_horizon_signature"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "appendix_teaser_ref_roles": [
    {
      "context": "ty argument and its invariance across multiple and sign-changing horizon realizations, is developed in Appendix~C (Thm.~\\ref{theorem:appC_dual_horizon_signature}). \\end{proof}",
      "label": "theorem:appC_dual_horizon_signature",
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 78,
      "target_type": "theorem"
    }
  ],
  "appendix_teaser_refs": [
    "theorem:appC_dual_horizon_signature"
  ],
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_sketch_observed_consequences"
  ],
  "cites": [
    "corollary:bk1_fixed_point",
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk1_drift_field",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_reflection_operator",
    "theorem:appC_dual_horizon_signature"
  ],
  "depends_on": [
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk1_effective_horizon_signature"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "energy is not stably reduced and the differentiated content disperses before it can register as retained identity (Cor.~\\ref{corollary:bk1_fixed_point}); novelty seen but not kept is not emergence. \\emph{No shared domain} --- a generative and a stabilizing channel exist",
      "label": "corollary:bk1_fixed_point",
      "line_distance": 2203,
      "role": "teaser",
      "target_line": 3003,
      "target_type": "corollary"
    },
    {
      "context": "a_{\\mathcal{O}}\\) and their fluxes couple above threshold, the positive flux supplies novelty through drift \\(D\\) (Def.~\\ref{definition:bk1_drift_field}) and the negative flux supplies state-level closure through \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_",
      "label": "definition:bk1_drift_field",
      "line_distance": 398,
      "role": "teaser",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ref{definition:bk1_drift_field}) and the negative flux supplies state-level closure through \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator}; Cor.~\\ref{corollary:bk1_fixed_point}); their coupled action realizes \\(\\Delta\\Phi_{\\mathcal{O}}\\ge\\tau_E\\), so \\(\\math",
      "label": "definition:bk1_reflection_operator",
      "line_distance": 409,
      "role": "teaser",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "corollary:bk1_fixed_point",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "id": "proof:bk1_proof_of_dual_horizon_necessity_theorem",
  "label": "proof:bk1_proof_of_dual_horizon_necessity_theorem",
  "latex_body": "\\begin{proof}[Proof of Dual Horizon Necessity Theorem]\n\\label{proof:bk1_proof_of_dual_horizon_necessity_theorem}\n\\leavevmode\n\n\\textbf{Necessity (by observational elimination).}\nAssume \\(\\mathcal{U}\\) supports bounded reflexive emergence: over some interval\nthe bounded observer registers \\emph{and retains} new coherent structure on\n\\(\\Omega_{\\mathcal{O}}\\), \\(\\Delta\\Phi_{\\mathcal{O}}\\ge\\tau_E>0\\)\n(Def.~\\ref{definition:bk1_bounded_reflexive_emergence}). We eliminate the three\nways the dual signature could fail.\n\\emph{No generative flux} --- \\(G_{\\mathcal{O}}(H)=0\\) for every horizon visible\nto \\(\\mathcal{O}\\): no observer-visible novelty crosses into\n\\(\\Omega_{\\mathcal{O}}\\), so over the interval nothing \\emph{new} is registered\n(only transport below resolution, repetition, or decay), and retained new\nstructure cannot reach \\(\\tau_E\\) --- one cannot keep what was never observed to\nenter.\n\\emph{No stabilizing flux} --- \\(C_{\\mathcal{O}}(H)=0\\): novelty may be sourced\nbut nothing contracts or integrates it, so symbolic free energy is not stably\nreduced and the differentiated content disperses before it can register as\nretained identity (Cor.~\\ref{corollary:bk1_fixed_point}); novelty seen but not\nkept is not emergence.\n\\emph{No shared domain} --- a generative and a stabilizing channel exist but\ntheir observer-visible supports do not both meet a common \\(\\Omega_{\\mathcal{O}}\\):\nthen on the single domain over which \\(\\mathcal{O}\\) integrates emergence one\nchannel is absent, returning us to the previous two cases.\nIn each case \\(\\Delta\\Phi_{\\mathcal{O}}<\\tau_E\\), contradicting the hypothesis.\nHence \\(G_{\\mathcal{O}}(H_G)>0\\) and \\(C_{\\mathcal{O}}(H_D)>0\\) on a shared\n\\(\\Omega_{\\mathcal{O}}\\), i.e.\\ \\(\\Sigma_{\\mathcal{O}}(\\mathcal{U})=\\{+,-\\}\\)\n(Def.~\\ref{definition:bk1_effective_horizon_signature}). A constant nonzero drift\nfield does not evade this: without positive horizon flux paired with negative\nstabilization flux on the shared domain it supplies transport, not bounded\nreflexive emergence.\n\n\\textbf{Converse (the binding, coupled case).}\nIf both channels are present on a shared \\(\\Omega_{\\mathcal{O}}\\) and their fluxes\ncouple above threshold, the positive flux supplies novelty through drift \\(D\\)\n(Def.~\\ref{definition:bk1_drift_field}) and the negative flux supplies state-level\nclosure through \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator};\nCor.~\\ref{corollary:bk1_fixed_point}); their coupled action realizes\n\\(\\Delta\\Phi_{\\mathcal{O}}\\ge\\tau_E\\), so \\(\\mathcal{U}\\) supports bounded\nreflexive emergence. This converse rests on the coupling of the two fluxes, not on\ntheir mere coexistence; the explicit coupling premise, together with the geometric\nmodality of the necessity argument and its invariance across multiple and\nsign-changing horizon realizations, is developed in Appendix~C\n(Thm.~\\ref{theorem:appC_dual_horizon_signature}).\n\\end{proof}",
  "line": 800,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Proof of Dual Horizon Necessity Theorem",
  "proves": "theorem:bk1_dual_horizon_necessity_theorem",
  "ref_roles": [
    {
      "context": "energy is not stably reduced and the differentiated content disperses before it can register as retained identity (Cor.~\\ref{corollary:bk1_fixed_point}); novelty seen but not kept is not emergence. \\emph{No shared domain} --- a generative and a stabilizing channel exist",
      "label": "corollary:bk1_fixed_point",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3003,
      "target_type": "corollary"
    },
    {
      "context": "rs \\emph{and retains} new coherent structure on \\(\\Omega_{\\mathcal{O}}\\), \\(\\Delta\\Phi_{\\mathcal{O}}\\ge\\tau_E>0\\) (Def.~\\ref{definition:bk1_bounded_reflexive_emergence}). We eliminate the three ways the dual signature could fail. \\emph{No generative flux} --- \\(G_{\\mathcal{O}}(H)=0\\) for",
      "label": "definition:bk1_bounded_reflexive_emergence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 752,
      "target_type": "definition"
    },
    {
      "context": "a_{\\mathcal{O}}\\) and their fluxes couple above threshold, the positive flux supplies novelty through drift \\(D\\) (Def.~\\ref{definition:bk1_drift_field}) and the negative flux supplies state-level closure through \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_",
      "label": "definition:bk1_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "_{\\mathcal{O}}(H_D)>0\\) on a shared \\(\\Omega_{\\mathcal{O}}\\), i.e.\\ \\(\\Sigma_{\\mathcal{O}}(\\mathcal{U})=\\{+,-\\}\\) (Def.~\\ref{definition:bk1_effective_horizon_signature}). A constant nonzero drift field does not evade this: without positive horizon flux paired with negative stabilization",
      "label": "definition:bk1_effective_horizon_signature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 724,
      "target_type": "definition"
    },
    {
      "context": "ref{definition:bk1_drift_field}) and the negative flux supplies state-level closure through \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator}; Cor.~\\ref{corollary:bk1_fixed_point}); their coupled action realizes \\(\\Delta\\Phi_{\\mathcal{O}}\\ge\\tau_E\\), so \\(\\math",
      "label": "definition:bk1_reflection_operator",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "ty argument and its invariance across multiple and sign-changing horizon realizations, is developed in Appendix~C (Thm.~\\ref{theorem:appC_dual_horizon_signature}). \\end{proof}",
      "label": "theorem:appC_dual_horizon_signature",
      "logical_support": false,
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 78,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk1_fixed_point",
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk1_drift_field",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_reflection_operator",
    "theorem:appC_dual_horizon_signature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_horizon_duality_principle",
    "proof:bk1_sketch_observed_consequences"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "depends_on": [
    "definition:bk1_effective_horizon_signature",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "ics: \\begin{enumerate} \\item $H_G$ is associated with generative symbolic drift, represented by a field $D$ (cf.~Def.~\\ref{definition:bk1_drift_field}), such that locally $\\nabla \\cdot D > 0$ (positive divergence, signifying expansion in possibility space). \\item $H_D",
      "label": "definition:bk1_drift_field",
      "line_distance": 351,
      "role": "interpretive_bridge",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ated with constraining symbolic stabilization, represented by the state-level component \\(R_{\\mathrm{stab}}\\) (cf.~Def.~\\ref{definition:bk1_reflection_operator}), such that observer-visible negative curvature supplies positive stabilization flux \\(C_{\\mathcal{O}}(H_D)>0\\). \\ite",
      "label": "definition:bk1_reflection_operator",
      "line_distance": 362,
      "role": "interpretive_bridge",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "to the horizons, establishing the stage for emergence (cf.~Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, Def.~\\ref{definition:bk1_symbolic_manifold}). \\end{enumerate} \\end{lemma}",
      "label": "definition:bk1_symbolic_manifold",
      "line_distance": 341,
      "role": "interpretive_bridge",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "id": "lemma:bk1_horizon_characterization",
  "label": "lemma:bk1_horizon_characterization",
  "latex_body": "\\begin{lemma}[Horizon Characterization]\n\\label{lemma:bk1_horizon_characterization}\nThe generative horizon $H_G$ and dissipative horizon $H_D$ exhibit distinct, complementary properties fundamental to symbolic dynamics:\n\\begin{enumerate}\n  \\item $H_G$ is associated with generative symbolic drift, represented by a field $D$ (cf.~Def.~\\ref{definition:bk1_drift_field}), such that locally $\\nabla \\cdot D > 0$ (positive divergence, signifying expansion in possibility space).\n  \\item $H_D$ is associated with constraining symbolic stabilization, represented by the state-level component \\(R_{\\mathrm{stab}}\\) (cf.~Def.~\\ref{definition:bk1_reflection_operator}), such that observer-visible negative curvature supplies positive stabilization flux \\(C_{\\mathcal{O}}(H_D)>0\\).\n  \\item Together, they define the bounded observer domain $\\Omega = \\{x \\in \\mathcal{U} : H_G \\prec x \\prec H_D\\}$, where $\\prec$ denotes symbolic containment relative to the horizons, establishing the stage for emergence (cf.~Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, Def.~\\ref{definition:bk1_symbolic_manifold}).\n\\end{enumerate}\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only clause 2's stabilization-flux conclusion (C_O(H_D) > 0) is captured, via the shared binding-product fact; clause 1's divergence condition nabla.D > 0 and clause 3's containment-domain definition are not modeled (require manifold divergence)."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-031"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumC.dualHorizonBinding_both_pos"
    ]
  },
  "line": 847,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Horizon Characterization",
  "proof_labels": [
    "proof:bk1_horizon_characterization"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ics: \\begin{enumerate} \\item $H_G$ is associated with generative symbolic drift, represented by a field $D$ (cf.~Def.~\\ref{definition:bk1_drift_field}), such that locally $\\nabla \\cdot D > 0$ (positive divergence, signifying expansion in possibility space). \\item $H_D",
      "label": "definition:bk1_drift_field",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ated with constraining symbolic stabilization, represented by the state-level component \\(R_{\\mathrm{stab}}\\) (cf.~Def.~\\ref{definition:bk1_reflection_operator}), such that observer-visible negative curvature supplies positive stabilization flux \\(C_{\\mathcal{O}}(H_D)>0\\). \\ite",
      "label": "definition:bk1_reflection_operator",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "to the horizons, establishing the stage for emergence (cf.~Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, Def.~\\ref{definition:bk1_symbolic_manifold}). \\end{enumerate} \\end{lemma}",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "}$, where $\\prec$ denotes symbolic containment relative to the horizons, establishing the stage for emergence (cf.~Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}, Def.~\\ref{definition:bk1_symbolic_manifold}). \\end{enumerate} \\end{lemma}",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_reflection_operator",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "depends_on": [
    "definition:bk1_effective_horizon_signature",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "-\\}\\) on a shared bounded domain. The positive component is exactly the generative channel carried by drift \\(D\\) (Def.~\\ref{definition:bk1_drift_field}); locally this is the expansion condition recorded as positive divergence. The negative component is exactly the stabil",
      "label": "definition:bk1_drift_field",
      "line_distance": 342,
      "role": "teaser",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "negative component is exactly the stabilizing channel carried by the state-level reflection \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator}), recorded as positive stabilizing flux. Their shared support is the observer domain \\(\\Omega_{\\mathcal{O}}\\), which is",
      "label": "definition:bk1_reflection_operator",
      "line_distance": 353,
      "role": "teaser",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "id": "proof:bk1_horizon_characterization",
  "label": "proof:bk1_horizon_characterization",
  "latex_body": "\\begin{proof}[Effective Signature Separates the Horizon Roles]\n\\label{proof:bk1_horizon_characterization}\n\\leavevmode\n\nBy Def.~\\ref{definition:bk1_effective_horizon_signature}, a horizon component\nwith \\(G_{\\mathcal{O}}(H)>0\\) contributes the positive observer-visible sign,\nwhile a component with \\(C_{\\mathcal{O}}(H)>0\\) contributes the negative\nobserver-visible sign. Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}\nstates that bounded reflexive emergence forces the joint signature\n\\(\\Sigma_{\\mathcal{O}}(\\mathcal{U})=\\{+,-\\}\\) on a shared bounded domain.\nThe positive component is exactly the generative channel carried by drift \\(D\\)\n(Def.~\\ref{definition:bk1_drift_field}); locally this is the expansion condition\nrecorded as positive divergence. The negative component is exactly the\nstabilizing channel carried by the state-level reflection\n\\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator}), recorded\nas positive stabilizing flux. Their shared support is the observer domain\n\\(\\Omega_{\\mathcal{O}}\\), which is equivalently the region symbolically\ncontained between the two effective horizons.\n\\end{proof}",
  "line": 856,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Effective Signature Separates the Horizon Roles",
  "proves": "lemma:bk1_horizon_characterization",
  "ref_roles": [
    {
      "context": "-\\}\\) on a shared bounded domain. The positive component is exactly the generative channel carried by drift \\(D\\) (Def.~\\ref{definition:bk1_drift_field}); locally this is the expansion condition recorded as positive divergence. The negative component is exactly the stabil",
      "label": "definition:bk1_drift_field",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "proof}[Effective Signature Separates the Horizon Roles] \\label{proof:bk1_horizon_characterization} \\leavevmode By Def.~\\ref{definition:bk1_effective_horizon_signature}, a horizon component with \\(G_{\\mathcal{O}}(H)>0\\) contributes the positive observer-visible sign, while a component wi",
      "label": "definition:bk1_effective_horizon_signature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 724,
      "target_type": "definition"
    },
    {
      "context": "negative component is exactly the stabilizing channel carried by the state-level reflection \\(R_{\\mathrm{stab}}\\) (Def.~\\ref{definition:bk1_reflection_operator}), recorded as positive stabilizing flux. Their shared support is the observer domain \\(\\Omega_{\\mathcal{O}}\\), which is",
      "label": "definition:bk1_reflection_operator",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "rver-visible sign, while a component with \\(C_{\\mathcal{O}}(H)>0\\) contributes the negative observer-visible sign. Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem} states that bounded reflexive emergence forces the joint signature \\(\\Sigma_{\\mathcal{O}}(\\mathcal{U})=\\{+,-\\}\\) on a s",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_reflection_operator",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "role": "proof",
  "type": "proof"
}