Cognitive Model: Curvature Diagnostics and Traversal-Rate Recovery for Number-Theoretic Distortion

Overview

This repository presents a theoretical and computational framework for analyzing discrete integer sequences through a geometry-inspired curvature model. By drawing a pedagogical analogy to relativistic distortions, it defines a forward diagnostic map that highlights structural irregularities—especially those arising from divisor density—and a calibrated recovery path that infers traversal rate v from observed Z sequences under an explicit support prior. The model remains pointwise forward in n and does not perform blind inversion of unknown integers, but it is now distributionally inverse in v when the source regime is calibrated.

🆕 The Cognitive Distortion Layer (CDL) standardizes κ(n) as the shared curvature signal across the Z Framework, providing unified primitives for prime diagnostics, QMC sampling, and signal normalization. Source modules now live under src/python/, with specifications in docs/specification/CDL_SPECIFICATION.md and docs/specification/INTEGRATION.md.

Key Concepts

1. Curvature Function: $\kappa(n) = \frac{d(n) \cdot \ln(n)}{e^2}$

   • d(n): Divisor count of $n$ (i.e., $\sigma_0(n)$).
   • ln(n): Natural logarithm of $n$.
   • Normalization: Constant $e^2$ determined empirically.
   • Interpretation: Higher divisor counts and larger values yield greater local "curvature".

2. Distortion Mapping (Forward Model): $\Delta_n = v \cdot \kappa(n)$

   • v: A traversal-rate parameter. In forward use it may be user-supplied. In recovery use it is inferred from observed Z values relative to a calibrated support prior.
   • $\Delta_n$: Modeled distortion at $n$.
   • Purpose: Encodes how rapid progression through integers skews apparent structure.

3. Perceived Value: $n_{\text{perceived}} = n \times \exp\bigl(\Delta_n\bigr)$

   • Applies exponential scaling to the true integer based on $\Delta_n$.
   • Emphasizes how distortion amplifies structural irregularities in composites.

4. Z-Transformation (Context-Dependent Normalization): $Z(n) ;=; \frac{n}{\exp\bigl(v \cdot \kappa(n)\bigr)}$

   • Pointwise forward in $n$: Assumes knowledge of $n$ and $v$ to normalize a specific integer.
   • Distributionally inverse in $v$: Under a calibrated support or sequence prior, observed Z values recover traversal rate through moment matching, MLE, or distributional fingerprints.
   • Outcome: Reveals underlying structural stability and exposes the process regime that generated the observed distortion pattern.

Empirical Validation

• Prime vs. Composite Curvature (n = 2–49)

  • Prime average curvature: ~0.739
  • Composite average curvature: ~2.252
  • Ratio: Composites ≈3.05× higher curvature

• Classification Test

  • Simple threshold on $\kappa(n)$ yields ~83% accuracy distinguishing primes from composites.

These results demonstrate that primes appear as "minimal-curvature geodesics" within the discrete sequence, providing a quantitative diagnostic measure of number complexity.

Implementation

• Language: Python 3

• Source Layout:

  • src/python/cdl.py: Canonical CDL primitives
  • src/python/cdl_continuous.py: Continuous-domain CDL extensions
  • src/python/v_recovery.py: Traversal-rate inference
  • src/python/cognitive_pilot.py: Sprint 6 cognitive pilot pipeline

• Repository Layout:

  • docs/: Specifications, summaries, roadmap, and concept notes
  • data/: Reference data and generated simulation traces
  • artifacts/: Generated reports and figures
  • experiments/: Research sprint outputs and benchmarks
  • scripts/: Demo, dashboard, report, and reproduction utilities
  • tests/: Reorganized pytest suite

Quick Start with CDL

The Cognitive Distortion Layer provides production-ready primitives:

import cdl

# Core primitive 1: Curvature signal
kappa_value = cdl.kappa(17)  # Returns κ(n)

# Core primitive 2: Threshold classifier
classification = cdl.classify(17, threshold=1.5)  # "prime" or "composite"

# Core primitive 3: Z-normalization
z_value = cdl.z_normalize(17, v=1.0)  # Returns Z(n)

# Integration helpers
likely_primes, likely_composites = cdl.prime_diagnostic_prefilter(candidates)
biased_candidates = cdl.qmc_sampling_bias(candidates, bias_strength=0.8)
normalized_signals = cdl.signal_normalize_pipeline(raw_signals, v=1.0)

See docs/specification/INTEGRATION.md for complete integration examples.

Quick Start with Traversal-Rate Recovery

The distributional recovery path treats v as an observable once the source regime is calibrated:

import numpy as np
from v_recovery import VRecovery, generate_z_sequence

rng = np.random.default_rng(321)
recovery = VRecovery(
    calibration_n_max=3000,
    sample_size=500,
    sequence_type="random",
    reference_trials=10,
    random_seed=321,
)

z_sequence, _ = generate_z_sequence(
    numbers=recovery.numbers,
    kappas=recovery.kappas,
    rng=rng,
    v=1.5,
    sample_size=500,
    sequence_type="random",
    prime_mask=recovery.prime_mask,
)

estimate = recovery.infer_v(z_sequence, method="fingerprint")
print(estimate.v_estimate)

The same recovery path is used in src/python/cognitive_pilot.py to recover participant-level v values from observed response traces.

Deterministic Cryptographic Prime Generation

The geodesic prime prefilter has moved to the standalone repository geodesic-prime-prefilter.

CDL remains the research home for the core curvature primitives and the broader Z Framework. The standalone repository now carries the production Python package, golden vectors, manual validation flow, and the multi-language porting surface for the prefilter.

Quick Start with Self-Contained Gist

The standalone gist lives at scripts/demos/curvature_gist.py and has only numpy as a dependency:

# Basic usage (n = 2-50, default parameters)
python scripts/demos/curvature_gist.py

# Extended analysis with 10,000 numbers
python scripts/demos/curvature_gist.py --max-n 10000

# Custom v-parameter for Z-transformation
python scripts/demos/curvature_gist.py --max-n 1000 --v-param 0.5

# Fewer bootstrap samples for faster execution
python scripts/demos/curvature_gist.py --max-n 100 --bootstrap-samples 500

Key Features:

• Instant computation for custom n ranges
• Built-in primality checks and bootstrap CI reporting
• Extensible v-parameter tuning for Z-normalization
• Outputs data/reference/kappas.csv with (n, κ(n), Z(n)) data
• ~83% classification accuracy for prime vs composite

The gist can also be imported as a module:

Full Model Example

# Run complete cognitive model with visualizations
python scripts/demos/main.py

Generates curvature statistics, writes figures into artifacts/figures/, and writes CSV traces into data/simulated/.

Validation & Testing

# Run CDL test suite
python tests/test_suite.py

# Generate baseline validation report
python scripts/reports/baseline_report.py

# Generate visualization dashboards
python scripts/dashboards/generate_cdl_dashboards.py

# Run the full Sprint 1–6 local reproduction
python scripts/reproduce_sprints.py

Validation Results:

• Seed set (n=2-49): Prime avg κ = 0.739, Composite avg κ = 2.252, Accuracy = 83.7%
• Hold-out (n=50-10K): Accuracy = 88.2%, maintains separation pattern
• Z-normalization: 99.2% variance reduction
• v recovery: integer-space tests meet MLE ±0.05, fingerprint ±0.10, and moment-match ±0.10 tolerances; continuous recovery meets fingerprint ±0.05; the deterministic cognitive pilot recovers participant v within ±0.15
• All acceptance criteria met ✓

The baseline report is written to artifacts/reports/baseline_report.json.

Limitations & Scope

1. No Standalone Inverse for Unknown Integers

   • The Z-transformation requires known $n$ and rate $v$ for pointwise normalization. It does not recover unknown integers from isolated observed values.

2. Distributional Recovery Requires Calibration

   • Traversal rate v is recoverable from observed Z sequences only relative to an explicit support prior or calibrated sequence regime. Prior mismatch can shift the recovered value.

3. Process Channel Exposure

   • Because Z distributions retain information about v, normalized outputs can fingerprint the generating or perceptual pipeline when the support law is known.

4. Metaphorical Analogy

   • References to relativity and geodesics are pedagogical. The core mathematics stands independently of physical interpretations.

Future Directions

• Prior-Robust Recovery: Infer v reliably when the support law is only partially known, and characterize identifiability under prior mismatch.
• Process Fingerprinting: Use recovered v and Z fingerprints to audit samplers, detect drift, and compare generation regimes.
• Divisor-Family Resonances: Characterize the fixed-point law v = e^2 / d across exact divisor-count families beyond the prime d = 2 band.
• Enhanced Classification: Integrate curvature features into machine-learning classifiers for primality testing.
• Theoretical Extensions: Investigate connections between divisor-based curvature and deeper analytic number theory.