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v0.2.0 - Multi-boundary mechanism + amplification + regime map

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@Hashevolution Hashevolution released this 13 Jun 13:27
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v0.2.0
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v0.2.0 — Multi-boundary mechanism observation

Date: 2026-06-13

This release upgrades the §3.6 stochastic-resonance observation from a single-cell "Goldilocks" framing to a universal trial-level mechanism observation based on 13-seed × 12-σ high-statistics measurement at (N, d) = (437, 4).

The main theoretical contribution — Theorems 1–5 on noise-invariant determinism, logarithmic coverage time, exact noise scaling, conditional Regev compatibility, and a hybrid (C) + Regev b-trick factoring algorithm — is unchanged. The mechanism observation has been substantially deepened.

What's new

§3.6 — Universal trial-level boundary-flip mechanism

  • 13 independent base sets × 200 trials × 12 σ values = 31,200 trial-measurements at (N, d) = (437, 4)
  • 13/13 seeds exhibit boundary-flip mechanism — universal at the base-set level
  • K-bin boundary distribution: 76.9% K=1/K=2, 15.4% K=2/K=3, 7.7% K=3↔K=1 long-jump
  • σ-curve direction asymmetry identified: positive-direction seeds saturate + decline, negative-direction seeds monotonically worsen (consequence of K-distribution skew at this cell)
  • Direction independence from K_baseline: seeds with identical K_baseline = 1.720 can show opposite SR directions, confirming that direction is determined by base-set-specific K-distribution structure, not by aggregate K_mean
  • Mechanism follows the classical Benzi–Buchleitner stochastic resonance shape: sub-threshold + saturation plateau + overload decline
  • Cross-cell verification: ceiling cells (4) and noise-floor cell (1, multi-seed) and active-boundary cell (1, multi-seed) are all consistent with regime predictions

★ Mechanism diversification at higher K_baseline (new — (1147, 2))

A 5-seed σ-scan at (N, d) = (1147, 2) with K_baseline ≈ 2.92 reveals a richer mechanism than the K = 1 / K = 2 boundary flip at (437, 4):

  • High-K rescue ★ : trials at very high K bins (K = 8, 11, 15, even K = 20) move to moderate K bins (K = 4, 5) under noise (seed 3: K = 15 → K = 5, SR = +9.44%; seed 4: 3 × (K = 8 → K = 4), SR = +8.56%)
  • Per-seed |SR| amplifies with K_baseline: max +9.44% at (1147, 2) vs +1.93% at (437, 4)
  • Boundary distribution shifts: only 40% K = 1 / K = 2 at (1147, 2) vs 77% at (437, 4); the remainder includes K = 8 → K = 4, K = 15 → K = 5, etc.
  • Cross-seed mean: +3.35% (sd 5.37, SE 2.40, t = 1.39, p ≈ 0.12 with t-distribution / 0.082 normal approx) — borderline at 5-seed sample, dominated by high-K-rescue seeds

★ Engineered amplification — (C) augmentation as a noise buffer

Two thinned variants of the hybrid test the structural prediction that SR magnitude is bounded by the borderline-trial population:

  • Over-thinned (smallest convergent only + no (C)): K_baseline = 19.87 (9.55× sub-functional), SR = 0.00% — sub-functional alone does not amplify (no borderlines left)
  • Mild thinned (all convergents + no (C) augmentation): K_baseline = 2.92 (1.4× sub-functional), per-seed |SR| amplifies to 4.03–4.44% (~5× larger than full hybrid)

The (C) augmentation acts as a noise buffer: its removal exposes the underlying boundary-flip mechanism in amplified form, while leaving the direction stochastic.

★ Algorithm-structure regime map for noise-as-resource (testable conjecture)

Algorithm structure SR magnitude Source
Single-base Shor (1994) small (≤ 1%) predicted
Multi-base Regev (LLL) negative SR predicted
Hybrid full (this work) +0.14% at (437, 4) measured
Hybrid mild-thinned ~5× amplified measured
Hybrid over-thinned zero measured

The regime map predicts that noise-as-resource in quantum factoring is naturally maximized in a multi-base + per-coordinate independent recovery + no buffer variant. Standalone Shor and Regev predictions remain testable with the same protocol; we leave their verification to follow-up work.

Statistical caveat

  • Net SR direction across 13 seeds at (437, 4): mean = +0.144%, t = 0.51, p = 0.31 — not statistically significant
  • Sign test: 8/13 positive (p = 0.29)
  • The mechanism is universal; the net direction is base-set-stochastic at our sample size
  • (1147, 2) 5-seed mean +3.35% (p ≈ 0.12 t-dist / 0.082 normal approx) is marginal; high-K-rescue seeds dominate

Retractions

Six earlier claims have been retracted with explicit footnotes:

  1. 17.86% peak at (1147, 2, σ=0.01) — single-seed direction fluctuation
  2. Polynomial scaling SR ∝ N^α — rejected when N=2491 cells gave near-zero or negative SR
  3. σ_opt ∝ N^α ("small lock, small wiggle" intuition) — rejected by σ-scan finding σ_opt ≈ 0.010 independent of N
  4. Anti-Optimization Principle (d=1 universal positive SR) — undermined by multi-seed re-measurement at (1147, 1) giving −0.53% ± 4.28%
  5. V3 sign-test p = 0.03 as significance — invalid because σ values within the saturation plateau are perfectly correlated (same boundary trials flip in all)
  6. Goldilocks single-cell robust — refined: K_baseline ≈ 2 marks the regime where mechanism is detectable, but direction is stochastic, not systematically positive

Reproducibility

New scripts to reproduce §3.6 results:

python -m experiments.sigma_scan_437                # (437, 4) baseline σ-scan (3 seeds × 12 σ × 200 trials)
python -m experiments.sigma_scan_437_extend         # (437, 4) extended seeds 4-13 + K-histogram backfill
python -m experiments.analyze_histograms            # per-seed K-bin flip identification
python -m experiments.sigma_scan_general 1147 2 5 100 compact  # (1147, 2) cross-cell σ-scan
python -m experiments.sr_amplification              # over-thinned (null amplification result)
python -m experiments.sr_mild_amplification         # mild thinned (5× amplification)

Raw measurement data is committed:

  • experiments/sigma_scan_437_d4_results.txt — K_mean at (437, 4)
  • experiments/sigma_scan_437_d4_extended.txt — extended seeds K_mean
  • experiments/sigma_scan_437_d4_histograms.txt — per-seed K-histograms at (437, 4)
  • experiments/sigma_scan_N1147_d2_compact_results.txt — (1147, 2) K_mean
  • experiments/sigma_scan_N1147_d2_compact_histograms.txt — (1147, 2) K-histograms
  • experiments/sr_mild_amplification_results.txt — mild thinned amplification data

All scripts are resumable (immediate per-cell save, skip-existing on re-run).

Unchanged from v0.1.0

  • Theorems 1–5 (paper §3.1–3.5) with proofs and empirical verification
  • Lemma 5.1 (per-b nontrivial-sqrt probability ≥ 1/2 for any semiprime)
  • 17,700 measurements across 6 composite sizes verifying Theorem 1
  • Cross-cell verification of Theorem 5 hybrid algorithm at N ∈ {437, 1147, 2491, 4087}
  • Hardware-calibrated noise simulation (Appendix E)

Connection to noise-as-resource literature

The mechanism we identify is the discrete analog of stochastic resonance in continuous systems. It provides the first explicit, mechanism-level bridge between integer factoring quantum algorithms and the noise-as-resource paradigm (Benzi et al. 1981 on classical SR; Wellens–Buchleitner 2004 on quantum SR; Plenio–Huelga 2008 on ENAQT).

The effect magnitude (1–7 trial flips per 200 trials, ±0.3–2% K change per seed, no statistically significant cross-seed net direction) is too small to enable cryptographic advantage; the contribution is conceptual.

Open questions / future work (v0.3 candidates)

  1. Direction bias at larger sample sizes (30+ seeds) — would a small net positive bias emerge?
  2. Active-boundary determinants at the base-set level — what predicts which K-boundary flips?
  3. Extended (1147, 2) verification — script sigma_scan_N1147_d2_extended.py (3 seeds × 7 σ × 100 trials, ~2 hours) tests high-K rescue universality at seeds 2, 3, 6
  4. Algorithm-structure regime map verification — measure SR at standalone Shor and Regev with LLL post-processing; confirms or refines the predictions in §3.6
  5. Universality across N — verification at (1147, 3), (4087, 4) at V3-style scale
  6. Long-jump events — true noise-induced trajectory divergence or cascade through adjacent boundaries?
  7. Hardware verification — survives structured (non-iid) phase noise?

How to cite

See CITATION.cff (GitHub will auto-format BibTeX from this file).

Hashevolution. (2026). A Noise-Invariant Determinism Theorem for Multi-Base
Post-Processing in Shor's Order Finding (Version 0.2.0). Zenodo.
https://doi.org/[TO BE ADDED]

License

MIT (see LICENSE)

Companion code

~1,000 lines of numpy. No quantum libraries required. Verified on Python 3.13.

Acknowledgements

This work benefited from extensive iteration: initial single-seed observations gave way to multi-seed validations that overturned several over-claims (see Retractions above). The published narrative reflects the corrections, not the original mistakes.

🤖 Generated with Claude Code

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