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The Wicked Prior as a Bounded-Observer Manifold

Atlas Fracture, Stackelberg Parentage, and Grace-Flow Repair

Wicked Geometry is the working name for this line of research on bounded-observer manifolds and atlas fracture.

Paul Tiffany | November 2025

Read the Paper (LaTeX source) Download PDF View Repository V-Baum Protocol
LLM-ready signals

Abstract & Core Hypothesis

We formalize the Wicked Prior not as a literary metaphor, but as a rigorous instantiation of bounded symbolic geometry under dual-horizon constraints. Oz is modeled as a bounded symbolic manifold \mathcal{M} equipped with:

We demonstrate that the unmasking of the Wizard constitutes an atlas fracture - a measurable discontinuity in semantic entropy where the outer chart fails to map the inner territory. Part II repairs the manifold via a Ricci-type grace flow with adaptive cadence, enabling reconciliation while preserving identity.

TL;DR (alignment/econ): Treat narrative as a curved manifold; the Wizard unmasking yields the dominant curvature spike. That same atlas-fracture geometry explains mis-specified Stackelberg games in institutions; a grace-flow schedule reconciles outer/inner metrics without erasing identity, leading to testable predictions (P1–P6) and training levers (A1–A3).

Line graph 'V-Baum Validation' plotting ||Ric|| curvature against Narrative Time. A distinct 4-sigma spike occurs at t=0.78 (Chapter 15), confirming the Atlas Fracture hypothesis.
Figure 1: Validation Protocol V-Baum. Embedding curvature ||Ric|| over the narrative arc of Baum's 1900 text. The dashed line marks the precise moment of "The Screen Falls" (Chapter 15), correlating with a spike in semantic volatility consistent with atlas fracture.

Key Theoretical Definitions

To facilitate rigorous discussion, we define the following terms from the paper:

Atlas Fracture
A failure to transition smoothly between inner and outer horizons (gin and gout) at bounded resolution. Mathematically, this manifests as a blow-up in extrinsic curvature where the institutional map can no longer represent the emergent territory. As resolution floor εres -> 0, curvature concentrates: ||Ricout|| ≳ K/εres2.
Stackelberg Parentage Paradox
A game-theoretic condition where a Leader (the Wizard) generates a world whose contradictions necessarily produce a corrective truth-seeker (Elphaba), then misclassifies that corrective as adversarial noise. The paradox resolves only when the leader admits generative parentage: πL => E => πF, recognizing the follower as an ancestor of the leader's own latent objective.
Grace Flow φ(τ)
A geometric repair process modeled as a Ricci-type flow with adaptive cadence that restores the manifold: &partial;g/&partial;τ = φ(τ)[Ric⊥(g(τ)) + ∇∇εres(τ)]. This allows reconciliation by transforming rigid categories into a shared basin of attraction, with boundary conditions g(0) = gout, g(&infty;) = ggrace.
Resolution Floor εres
The observer's representational bandwidth constraint. Identity features below scale εres are truncated in outer projection. As εres -> 0, identities are forced into low-dimensional categories, producing geometric stress and curvature concentration.

Falsifiable Predictions

The framework generates six independently testable predictions (P1-P6):

P1: Defying Gravity Curvature Peak

Prediction: The dominant curvature peak aligns with the climactic unmasking moment.

Falsification: If the maximum curvature occurs >2 scenes from the climax, or if peaks drift substantially across embedding models, then curvature is not tracking a narrative invariant.

P2: Animal Transport Failure

Prediction: Animal-scene embeddings persist while their transport fidelity to outer decision nodes decreases through Part I (connection annihilation).

Falsification: If Animal scenes remain fully connected to outer-horizon decision embeddings, the annihilation model is incorrect.

P3: Grace Flow Graduality

Prediction: Character arcs show increasing cadence φ(τ) and monotone decrease in curvature proxies through Part II.

Falsification: If repair occurs as a single discontinuous jump (no gradual flow), the grace-flow model fails.

P4: Coupling-Dependent Convergence

Prediction: Character pairs with strong coupling converge to a stable Lyapunov basin; weakly coupled pairs diverge.

Falsification: If convergence occurs independently of coupling strength, basin emergence is not the operative mechanism.

P5: Regulatory Capture Concentration

Prediction: Policies exhibiting Stackelberg misclassification show sustained increases in market concentration relative to adjacent sectors.

Falsification: If concentration changes are statistically indistinguishable from control sectors, the capture model doesn't predict industrial outcomes.

P6: Safety-Innovation Decoupling

Prediction: True risk reduction decouples from innovation suppression; captured regimes show apparent safety improvements without systemic risk reduction.

Falsification: If innovation suppression always correlates with genuine risk reduction, the Stackelberg framework doesn't distinguish capture from regulation.

Validation Protocol V-Baum

We validate the geometric thesis empirically by applying oz_fracture.py to Baum's 1900 public-domain text. The protocol uses sentence-transformers (all-MiniLM-L6-v2) to compute embedding curvature as a proxy for semantic entropy.

Hypothesis

Let t* denote the window containing Chapter 15 ("The Discovery of Oz, the Terrible"), where the curtain falls. Under the dual-horizon hypothesis, we predict a measurable discontinuity in semantic entropy around t*.

Method

# Uses sentence-transformers to map narrative topology
from sentence_transformers import SentenceTransformer
from scipy.spatial.distance import cosine
import numpy as np

def compute_curvature(embeddings):
    """
    Estimates extrinsic curvature in the embedding ambient space.
    Peaks indicate rapid semantic acceleration consistent with atlas fracture.
    """
    curvature = []
    for i in range(1, len(embeddings)):
        # Cosine distance as proxy for geodesic deviation
        k = cosine(embeddings[i-1], embeddings[i])
        curvature.append(k)
    return np.array(curvature)

Running the Code

git clone https://github.com/PaulTiffany/wicked-geometry.git
cd wicked-geometry
pip install -r requirements.txt
python oz_fracture.py

This generates atlas_fracture_evidence.png showing semantic volatility over the narrative arc. The code is intentionally minimal and reproducible - anyone can run it and verify (or falsify) the claimed peak alignment.

-> View full source code

Applications Beyond Narrative

AI Alignment Corollaries

Economic & Regulatory Applications

The Isomorphism Hypothesis:

We posit that the geometric stress of Atlas Fracture is scale-invariant. The same curvature concentration observed in the narrative micro-scale (Oz) appears in the institutional macro-scale (Regulatory Capture) because both are bounded systems attempting to map infinite complexity onto finite categories.

Historical Substrate: Baum's Monetary Allegory

Baum wrote The Wonderful Wizard of Oz during the 1890s monetary politics debates - gold standard orthodoxy, free silver, and the perceived "greenback illusion" of paper claims detaching from productive value. In the standard Populist reading, the Yellow Brick Road encodes a hard gold path, Dorothy's silver slippers symbolize free silver coinage, and the Emerald City renders value literally through a green optical filter.

Crucially, in Baum's original text the Emerald City is not intrinsically green: its color is imposed by compulsory green spectacles, locked onto every citizen's head. The spectacles implement an outer metric gout on an underlying manifold with its own inner structure gin. The moment the spectacles are removed corresponds to driving the resolution floor εres -> 0, revealing a non-isometry between the two metrics.

This historical substrate functions as an anchored prior over the space of institutional geometries, rather than as an arbitrary narrative toy model.

Citation

If you use this framework or code, please cite:

@article{tiffany2025wicked,
  title={The Wicked Prior as a Bounded-Observer Manifold: 
         Atlas Fracture, Stackelberg Parentage, and Grace-Flow Repair},
  author={Tiffany, Paul},
  year={2025},
  month={11},
  url={https://paultiffany.github.io/wicked-geometry/}
}