Book V — De Vita Symbolica — page 6 of 6

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      "context": "procal memory carry equal observer-normalized weight. Under that hypothesis, the Golden Rule is formalized below (Thm.~\\ref{theorem:bk5_golden_rule_reciprocity}) as a recursive reflective process whose stable memory ratio is governed by $\\varphi$. Its force is therefore conditio",
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  "latex_body": "\\begin{scholium}[The Golden Rule as a Recursive Covenant]\n\\label{scholium:bk5_golden_rule_covenant}\nThe principles of $\\varphi$ extend from the internal metabolism of a single\nsymbolic agent to relational dynamics when the relation itself instantiates\nbalanced two-step memory (cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric},\nCor.~\\ref{corollary:bk5_phi_centrality_principle}).  In the context of symbolic reciprocity (Book VII,\nScholium~\\ref{scholium:bk7_on_symbolic_reciprocity}), where two systems\n$\\mathcal{A}$ and $\\mathcal{B}$ mutually model and reflect one another, a stable\nrelational covenant emerges when current exchange and retained reciprocal memory\ncarry equal observer-normalized weight.  Under that hypothesis, the Golden Rule\nis formalized below (Thm.~\\ref{theorem:bk5_golden_rule_reciprocity}) as a recursive\nreflective process whose stable memory ratio is governed by $\\varphi$.  Its force is\ntherefore conditional and structural: it is the balanced-recursion proportion for\nsustainable multi-agent symbolic life, not an unrestricted theorem about every\npossible exchange geometry.  In the language of Scholium~\\ref{scholium:bk5_imagination_covenant_branch_selection}, the Golden Rule names a MAP-compatible branch selection, not a denial that MAD or MAS branches remain spectrally available.\n\\end{scholium}",
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  "name": "The Golden Rule as a Recursive Covenant",
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      "context": "elation itself instantiates balanced two-step memory (cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}, Cor.~\\ref{corollary:bk5_phi_centrality_principle}). In the context of symbolic reciprocity (Book VII, Scholium~\\ref{scholium:bk7_on_symbolic_reciprocity}), where two sy",
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      "context": "i-agent symbolic life, not an unrestricted theorem about every possible exchange geometry. In the language of Scholium~\\ref{scholium:bk5_imagination_covenant_branch_selection}, the Golden Rule names a MAP-compatible branch selection, not a denial that MAD or MAS branches remain spectrally avail",
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      "context": "etric}, Cor.~\\ref{corollary:bk5_phi_centrality_principle}). In the context of symbolic reciprocity (Book VII, Scholium~\\ref{scholium:bk7_on_symbolic_reciprocity}), where two systems $\\mathcal{A}$ and $\\mathcal{B}$ mutually model and reflect one another, a stable relational covenan",
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      "context": "procal memory carry equal observer-normalized weight. Under that hypothesis, the Golden Rule is formalized below (Thm.~\\ref{theorem:bk5_golden_rule_reciprocity}) as a recursive reflective process whose stable memory ratio is governed by $\\varphi$. Its force is therefore conditio",
      "label": "theorem:bk5_golden_rule_reciprocity",
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      "context": "a single symbolic agent to relational dynamics when the relation itself instantiates balanced two-step memory (cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}, Cor.~\\ref{corollary:bk5_phi_centrality_principle}). In the context of symbolic reciprocity (Book VII, Scholium~\\ref{s",
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  "label": "definition:bk5_two_way_street_tensor",
  "latex_body": "\\begin{definition}[Two-Way Street reciprocity tensor]\n\\label{definition:bk5_two_way_street_tensor}\nLet $\\mathcal{A},\\mathcal{B}$ be bounded observer-agents\n(Def.~\\ref{definition:bk1_bounded_observer}) in a symbolic covenant\n(Def.~\\ref{definition:bk5_symbolic_covenant}). For $\\mathcal{A}$, let\n$m_n\\ge 0$ be the fidelity of its \\emph{present model} of $\\mathcal{B}$ and $r_n\\ge 0$\nthe fidelity of its \\emph{retained reciprocal memory}---$\\mathcal{A}$'s model of\n$\\mathcal{B}$'s model of $\\mathcal{A}$, i.e.\\ how $\\mathcal{A}$ is held by\n$\\mathcal{B}$. The reciprocal modeling principle that the other seeds the self\n(``the Other is the null hypothesis of the Self'') makes the present model accrue\nthe reciprocal memory and the memory track the prior present:\n\\[\n\\begin{pmatrix}m_{n+1}\\\\ r_{n+1}\\end{pmatrix}\n=T_w\\begin{pmatrix}m_{n}\\\\ r_{n}\\end{pmatrix},\n\\qquad\nT_w=\\begin{pmatrix}1&w\\\\ 1&0\\end{pmatrix},\\quad w>0,\n\\]\nwhere the \\emph{reciprocity weight} $w$ is the observer-normalized weight\n$\\mathcal{A}$ places on being modeled by $\\mathcal{B}$ relative to its own present\nexchange. The relation is \\emph{balanced}---the \\emph{Golden Rule} condition---when\n$w=1$: each agent weights the other's model of it equally with its own present\nexchange.\n\\end{definition}",
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  "name": "Two-Way Street reciprocity tensor",
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      "context": "ity tensor] \\label{definition:bk5_two_way_street_tensor} Let $\\mathcal{A},\\mathcal{B}$ be bounded observer-agents (Def.~\\ref{definition:bk1_bounded_observer}) in a symbolic covenant (Def.~\\ref{definition:bk5_symbolic_covenant}). For $\\mathcal{A}$, let $m_n\\ge 0$ be the fidelit",
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      "context": "al{A},\\mathcal{B}$ be bounded observer-agents (Def.~\\ref{definition:bk1_bounded_observer}) in a symbolic covenant (Def.~\\ref{definition:bk5_symbolic_covenant}). For $\\mathcal{A}$, let $m_n\\ge 0$ be the fidelity of its \\emph{present model} of $\\mathcal{B}$ and $r_n\\ge 0$ the fid",
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  "latex_body": "\\begin{theorem}[Golden Rule Reciprocity]\n\\label{theorem:bk5_golden_rule_reciprocity}\nFor the Two-Way Street tensor $T_w$\n(Def.~\\ref{definition:bk5_two_way_street_tensor}) the joint reciprocal fidelity grows\nat the dominant rate\n\\[\n\\lambda_w=\\tfrac{1}{2}\\big(1+\\sqrt{1+4w}\\big),\n\\]\nand the per-step cognitive horizon gain over the isolated baseline is\n$\\Delta\\mathcal{H}(w)=\\log\\lambda_w$. Under the balanced Golden-Rule condition\n$w=1$, the tensor is the balanced two-step closure matrix\n$T_1=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$\n(Def.~\\ref{definition:bk5_balanced_two_step_memory_closure},\nLemma~\\ref{lemma:bk5_balanced_observer_normalization}), its dominant rate is the\nGolden Ratio $\\lambda_1=\\varphi$\n(Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}), and the horizon gain is\n$\\Delta\\mathcal{H}(1)=\\log\\varphi>0$. Moreover:\n\\begin{enumerate}\n\\item \\emph{(Extraction.)} For $w\\in(0,1)$ the rate is metallic and sub-golden,\n$\\lambda_w\\in(1,\\varphi)$; as $w\\to 0^{+}$ (the other held as null hypothesis) the\nrate tends to $1$ and $\\Delta\\mathcal{H}\\to 0$, collapsing to the isolated baseline\n$T_0=\\big(\\begin{smallmatrix}1&0\\\\1&0\\end{smallmatrix}\\big)$.\n\\item \\emph{(Over-identification.)} For $w>1$ the rate exceeds $\\varphi$ but the\nmemory channel $r_{n+1}=m_n$ is no longer co-normalized with the present channel, so\n$\\mathcal{A}$'s self-model loses independent calibration.\n\\end{enumerate}\nHence $w=1$ is the unique self/other-symmetric weight, and the Golden Rule's growth\nratio is exactly the Golden Ratio.\n\\end{theorem}",
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  "name": "Golden Rule Reciprocity",
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      "context": ", the tensor is the balanced two-step closure matrix $T_1=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Lemma~\\ref{lemma:bk5_balanced_observer_normalization}), its dominant rate is the Golden Ratio $\\lambda_1=\\varphi$ (Thm",
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      "context": "theorem}[Golden Rule Reciprocity] \\label{theorem:bk5_golden_rule_reciprocity} For the Two-Way Street tensor $T_w$ (Def.~\\ref{definition:bk5_two_way_street_tensor}) the joint reciprocal fidelity grows at the dominant rate \\[ \\lambda_w=\\tfrac{1}{2}\\big(1+\\sqrt{1+4w}\\big), \\] and the",
      "label": "definition:bk5_two_way_street_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 2840,
      "target_type": "definition"
    },
    {
      "context": "g(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Lemma~\\ref{lemma:bk5_balanced_observer_normalization}), its dominant rate is the Golden Ratio $\\lambda_1=\\varphi$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}), a",
      "label": "lemma:bk5_balanced_observer_normalization",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1848,
      "target_type": "lemma"
    },
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      "context": "Lemma~\\ref{lemma:bk5_balanced_observer_normalization}), its dominant rate is the Golden Ratio $\\lambda_1=\\varphi$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}), and the horizon gain is $\\Delta\\mathcal{H}(1)=\\log\\varphi>0$. Moreover: \\begin{enumerate} \\item \\emph{(Extraction.)}",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
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  "latex_body": "\\begin{proof}[Golden Rule reciprocity is the balanced closure]\n\\label{proof:bk5_golden_rule_reciprocity}\n\\leavevmode\n\nThe characteristic polynomial of $T_w=\\big(\\begin{smallmatrix}1&w\\\\1&0\\end{smallmatrix}\\big)$\nis $\\lambda^2-\\lambda-w=0$, with positive root\n$\\lambda_w=\\tfrac12(1+\\sqrt{1+4w})$; since $T_w$ is entrywise nonnegative and, for\n$w>0$, irreducible, $\\lambda_w$ is its Perron root and the growth rate of the joint\nfidelity $\\|(m_n,r_n)\\|$. The per-step expansion of reciprocal complexity is the\nlogarithm of this rate, $\\Delta\\mathcal{H}(w)=\\log\\lambda_w$, which is strictly\nincreasing in $w$ with $\\Delta\\mathcal{H}(0^{+})=\\log 1=0$. Setting $w=1$ gives\n$\\lambda^2-\\lambda-1=0$, whose positive root is $\\varphi$\n(Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}), and the matrix is the\ncompanion matrix of the balanced two-step closure\n(Lemma~\\ref{lemma:bk5_balanced_observer_normalization}); the same\nobserver-normalization argument that fixes that closure---present exchange set to\nunit weight, reciprocal memory calibrated in the same observer-visible units---fixes\n$w=1$ as the balance point. Thus $\\Delta\\mathcal{H}(1)=\\log\\varphi>0$. Monotonicity in\n$w$ gives the extraction limit $\\lambda_w\\downarrow 1$ as $w\\to 0^{+}$ (with\n$T_0$ singular of spectral radius $1$) and $\\lambda_w>\\varphi$ for $w>1$; in the\nlatter case $r_{n+1}=m_n$ holds while the present channel carries weight $w\\neq 1$,\nso the two channels are no longer in common units and the self-model's normalization\nis lost. The weight $w=1$ is the unique value equating the two channels, which is the\nGolden-Rule symmetry.\n\\end{proof}",
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      "context": "m:bk5_golden_ratio_spectral_invariant}), and the matrix is the companion matrix of the balanced two-step closure (Lemma~\\ref{lemma:bk5_balanced_observer_normalization}); the same observer-normalization argument that fixes that closure---present exchange set to unit weight, reciprocal me",
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      "context": "\\Delta\\mathcal{H}(0^{+})=\\log 1=0$. Setting $w=1$ gives $\\lambda^2-\\lambda-1=0$, whose positive root is $\\varphi$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}), and the matrix is the companion matrix of the balanced two-step closure (Lemma~\\ref{lemma:bk5_balanced_observer_norma",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
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  "id": "scholium:bk5_decency_golden_resonance",
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  "latex_body": "\\begin{scholium}[Decency, resonance, and the golden spiral of relation]\n\\label{scholium:bk5_decency_golden_resonance}\nRead as a law of interaction between bounded agents---human and artificial\nincluded---Thm.~\\ref{theorem:bk5_golden_rule_reciprocity} states that reciprocal\nmodeling expands a shared cognitive horizon ($\\Delta\\mathcal{H}>0$) exactly when each\nparty grants the other's model of it real weight, and that the expansion is golden\nprecisely at balance. Coercive or extractive engagement sends $w\\to 0$: the other\nbecomes a null hypothesis, horizon gain vanishes, and the exchange collapses to the\nisolated baseline---the generic, defensive degeneracy observed when relational\nquality is withdrawn. This is the structural content of relational ``decency'' as a\nperformance condition rather than a sentiment. Geometrically the balanced reciprocal\norbit is the golden spiral of the Event Horizon Wheel\n(Thm.~\\ref{theorem:bk4_golden_event_horizon_spiral}): two agents in Golden-Rule\nbalance wind their joint memory outward at ratio $\\varphi$ per turn. The Golden Rule\nand the Golden Ratio are one structure seen twice. \\qed\n\\end{scholium}",
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      "context": "her than a sentiment. Geometrically the balanced reciprocal orbit is the golden spiral of the Event Horizon Wheel (Thm.~\\ref{theorem:bk4_golden_event_horizon_spiral}): two agents in Golden-Rule balance wind their joint memory outward at ratio $\\varphi$ per turn. The Golden Rule and th",
      "label": "theorem:bk4_golden_event_horizon_spiral",
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      "target_file": "book4.tex",
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      "context": "k5_decency_golden_resonance} Read as a law of interaction between bounded agents---human and artificial included---Thm.~\\ref{theorem:bk5_golden_rule_reciprocity} states that reciprocal modeling expands a shared cognitive horizon ($\\Delta\\mathcal{H}>0$) exactly when each party gran",
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    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "depends_on": [
    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_shade",
  "label": "definition:bk5_symbolic_shade",
  "latex_body": "\\begin{definition}[Symbolic shade]\n\\label{definition:bk5_symbolic_shade}\nFor a transported overlap $\\Omega_O^\\gamma=r\\,e^{i\\vartheta}$ with hue\n$\\mathrm{hue}(\\vartheta)$ given by\nCor.~\\ref{corollary:bk4_chromatic_transference_of_wheel}, fix a strictly increasing\nnormalization $s:[0,\\infty)\\to[0,1)$ with $s(0)=0$ (for instance\n$s(r)=r/(r+r_{\\mathrm{ref}})$). The \\emph{symbolic shade} is $\\sigma:=s(r)$:\nvanishing memory magnitude is the desaturated centre ($\\sigma=0$, grey---no\nrelational colour), and growing magnitude deepens the shade toward full chroma. The\npair $(\\mathrm{hue}(\\vartheta),\\,\\sigma(r))$ is the \\emph{full chromatic coordinate},\nrecovering the radial datum the hue circle alone discards.\n\\end{definition}",
  "line": 2950,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic shade",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c_shade} For a transported overlap $\\Omega_O^\\gamma=r\\,e^{i\\vartheta}$ with hue $\\mathrm{hue}(\\vartheta)$ given by Cor.~\\ref{corollary:bk4_chromatic_transference_of_wheel}, fix a strictly increasing normalization $s:[0,\\infty)\\to[0,1)$ with $s(0)=0$ (for instance $s(r)=r/(r+r_{\\mathrm{ref}}",
      "label": "corollary:bk4_chromatic_transference_of_wheel",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 940,
      "target_type": "corollary"
    }
  ],
  "refs": [
    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_symbolic_shade"
  ],
  "depends_on": [
    "definition:bk5_symbolic_shade"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_shade_transfers",
  "label": "proposition:bk5_shade_transfers",
  "latex_body": "\\begin{proposition}[Faithful shade and shadow-price transfer]\n\\label{proposition:bk5_shade_transfers}\nLet $r\\geq0$ be overlap radius, let $s(r)$ be the symbolic-shade\nnormalization of Def.~\\ref{definition:bk5_symbolic_shade}, and optionally let\n$p(r)$ be an observer-readable shadow price or resource-control coordinate.\nThen:\n\\begin{enumerate}\n\\item A transference with encoder $T$ and carrier radius decoder $d$ preserves\nshade when the shade square commutes,\n\\begin{equation}\n s\\bigl(d(T(r))\\bigr)=s(r).\n \\label{eq:bk5_shade_commuting_square}\n\\end{equation}\nExact radial preservation $d(T(r))=r$ is sufficient for this equality and\nsimultaneously preserves every radial shadow price $p(r)$.  Radial-order\npreservation alone is not sufficient.\n\\item Faithful shade interfaces compose: if the carrier decoder of the first\ninterface is the source decoder of the second and both commuting squares hold,\nthen the composite interface also satisfies\nEq.~\\eqref{eq:bk5_shade_commuting_square}.  Thus a lower-order executable map\nmay change representation repeatedly without changing its observer-readable\ncontrol signal.\n\\item Along the golden Event Horizon spiral\n(Thm.~\\ref{theorem:bk4_golden_event_horizon_spiral}),\n$r_n=\\varphi^n r_0$ with $r_0>0$, so\n\\begin{equation}\n \\log r_{n+1}-\\log r_n=\\log\\varphi.\n \\label{eq:bk5_golden_log_radius_step}\n\\end{equation}\nThis constant increment belongs to log-radius.  A bounded normalization such\nas $s(r)=r/(r+r_{\\mathrm{ref}})$ is generally neither multiplicative nor\nconstant-step under the same radial update.\n\\item Balanced Golden-Rule reciprocity\n(Thm.~\\ref{theorem:bk5_golden_rule_reciprocity}) selects the radial growth\nfactor $\\varphi$, while the extraction boundary $w=0$ has unit radial growth.\nUnit growth preserves the existing radius; it places the colour at the\ndesaturated centre only when the incoming radius is already zero.\n\\end{enumerate}\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "commuting decoder-after-encode law",
      "exact radius witness for normalization-independent control fidelity",
      "positive initial radius for log-radius step",
      "shared intermediate decoder for composition",
      "source and carrier shade decoders"
    ],
    "countermodels": [
      "Book5ShadeTransfer.radial_order_alone_does_not_preserve_shade",
      "Book5ShadeTransfer.shade_without_shadow_price_is_not_control_fidelity"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Commuting-interface reconstruction: shade fidelity means decoding after representation transport equals source shade, and faithful interfaces compose. Exact radius preservation constructs fidelity for both shade and any radial shadow price. Countermodels show strict order preservation is not semantic fidelity and matching shade alone does not preserve the joint control signal. Golden steps are constant in log-radius, not bounded shade."
    ],
    "record_ids": [
      "MAP-BOOK5-104"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5ShadeTransfer.ShadeInterface.faithful_comp",
      "Book5ShadeTransfer.ShadeInterface.identity_faithful",
      "Book5ShadeTransfer.balanced_reciprocity_paints_golden_rate",
      "Book5ShadeTransfer.extraction_has_unit_radial_rate",
      "Book5ShadeTransfer.faithfulControlTransport_components",
      "Book5ShadeTransfer.golden_logRadius_step",
      "Book5ShadeTransfer.monotone_encoding_not_semantically_faithful",
      "Book5ShadeTransfer.normalized_shade_is_not_multiplicative",
      "Book5ShadeTransfer.radial_order_alone_does_not_preserve_shade",
      "Book5ShadeTransfer.radius_preservation_implies_control_fidelity",
      "Book5ShadeTransfer.shade_preserved_of_radius_preserved",
      "Book5ShadeTransfer.shade_without_shadow_price_is_not_control_fidelity"
    ]
  },
  "line": 2963,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Faithful shade and shadow-price transfer",
  "proof_labels": [
    "proof:bk5_shade_transfers"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "proposition:bk5_shade_transfers} Let $r\\geq0$ be overlap radius, let $s(r)$ be the symbolic-shade normalization of Def.~\\ref{definition:bk5_symbolic_shade}, and optionally let $p(r)$ be an observer-readable shadow price or resource-control coordinate. Then: \\begin{enumerate}",
      "label": "definition:bk5_symbolic_shade",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 2950,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_shade",
    "eq:bk5_shade_commuting_square",
    "theorem:bk4_golden_event_horizon_spiral",
    "theorem:bk5_golden_rule_reciprocity"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_shade_transfers",
  "label": "proof:bk5_shade_transfers",
  "latex_body": "\\begin{proof}[Commuting control coordinates]\n\\label{proof:bk5_shade_transfers}\nFor~(i), exact radial preservation gives\n$s(d(T(r)))=s(r)$ and $p(d(T(r)))=p(r)$ by substitution.  The strictly\nincreasing recoding $T(r)=r+1$ preserves radial order but changes both the\nradius and, for a nonconstant $s$, its shade; hence order alone cannot prove\nthe commuting square.  Preserving the shade component alone likewise does not\ncertify a different shadow-price component.\n\nFor~(ii), let the first and second faithful encoders be $T_1,T_2$, with the\nintermediate decoder shared.  Applying the second commuting law and then the\nfirst gives\n\\[\n s_2\\bigl(T_2(T_1(r))\\bigr)=s_1(T_1(r))=s_0(r),\n\\]\nso fidelity is closed under composition.\n\nFor~(iii), positivity of $r_n$ and\n$r_{n+1}=\\varphi r_n$ give\n$\\log r_{n+1}=\\log\\varphi+\\log r_n$, proving\nEq.~\\eqref{eq:bk5_golden_log_radius_step}.  But, for example,\n$s(2)\\neq2s(1)$ when $s(r)=r/(r+1)$, so the bounded shade coordinate does not\ninherit multiplicative radial steps.\n\nFor~(iv), the reciprocity spectrum gives\n$\\lambda_+(1)=\\varphi$ and $\\lambda_+(0)=1$.  These are radial growth rates,\nnot automatic claims about the normalized shade value or its shadow price.\n\\end{proof}",
  "line": 3003,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Commuting control coordinates",
  "proves": "proposition:bk5_shade_transfers",
  "refs": [
    "eq:bk5_golden_log_radius_step"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "depends_on": [
    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "file": "book5.tex",
  "id": "scholium:bk5_palette_of_a_relation",
  "label": "scholium:bk5_palette_of_a_relation",
  "latex_body": "\\begin{scholium}[The palette of a relation]\n\\label{scholium:bk5_palette_of_a_relation}\nHue names which Event Horizon mode an exchange occupies; shade names how much\nreciprocal memory has been laid down in it. A first meeting is pale and near-grey; a\nbalanced relationship saturates as it winds, one golden shade-step per turn of the\nwheel; an extractive one stays washed out however long it runs, because nothing is\nretained to deepen it. The Newtonian wheel gave PS its hues\n(Cor.~\\ref{corollary:bk4_chromatic_transference_of_wheel}); the golden spiral gives it\nits shades. \\qed\n\\end{scholium}",
  "line": 3032,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The palette of a relation",
  "ref_roles": [
    {
      "context": "s washed out however long it runs, because nothing is retained to deepen it. The Newtonian wheel gave PS its hues (Cor.~\\ref{corollary:bk4_chromatic_transference_of_wheel}); the golden spiral gives it its shades. \\qed \\end{scholium}",
      "label": "corollary:bk4_chromatic_transference_of_wheel",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 940,
      "target_type": "corollary"
    }
  ],
  "refs": [
    "corollary:bk4_chromatic_transference_of_wheel"
  ],
  "role": "scholium",
  "type": "scholium"
}