Skip to content

v0.2.1 - Regime map 5/5 validated + Universal direction stochasticity

Choose a tag to compare

View all tags
@Hashevolution Hashevolution released this 13 Jun 17:37
· 80 commits to main since this release
v0.2.1
a45f507

v0.2.1 — Regime map empirically validated (5/5 measured) + Universal direction stochasticity

Date: 2026-06-14

This patch release validates the Algorithm-structure regime map from v0.2.0 through direct measurement of all five entries, including new measurements of Pure Shor SR, Pure Regev SR with a faithful LLL implementation, (2491, 2) cross-cell verification, and (1147, 2) extended scan. A new main finding emerges: universal direction stochasticity across algorithm structures.

What's new

★ Algorithm-structure regime map (5/5 measured)

All five entries are now empirically measured:

Algorithm structure Per-seed |SR| Mean SR K_base Source
Single-base Shor (1994) 0–1.10% −0.04% (3+/2−) 10.38 measured (5 seeds × 100 trials × 3 σ)
Multi-base Regev (LLL) 1.27–3.95% −0.31% (2+/3−) 2.44 measured (5 seeds × 50 trials × 3 σ, faithful LLL)
Hybrid full at (437, 4) 0–1.93% +0.14% (8+/5−) 1.82 measured (13 seeds × 200 trials × 12 σ)
Hybrid mild-thinned 4.03–4.44% −1.43% (1+/2−) 2.92 measured (3 seeds × 100 trials × 3 σ)
Hybrid over-thinned 0% 0% 19.87 measured (3 seeds × 100 trials × 3 σ)

★ Universal direction stochasticity across algorithm structures

The 5-algorithm regime map measurement reveals an unexpected unifying pattern: cross-seed direction is base-set-stochastic (not algorithm-determined) in every tested algorithm structure. Mean SR magnitudes are all small (|mean SR| ≤ 1.5%) and none reach statistical significance at our sample sizes. The regime map's original directional predictions (Shor "small", Regev "negative") are qualitatively confirmed but with smaller magnitudes than the naive "LLL is fragile" picture suggested.

Pure Shor σ-scan (regime map verification)

Single-base Shor (d = 1, no (C) accumulation) at N = 437:

  • K_baseline = 10.38 (~50% trials reach max_runs = 20, dominated by failed b-trick conditions)
  • Mean SR at σ = 0.050: −0.04% (sd 0.85%, SE 0.38%, t = −0.11, p ≈ 0.46)
  • Per-seed |SR| range: 0–1.10%
  • Sign test: 3 positive / 2 negative

→ Regime map prediction "Shor: small SR" confirmed.

Pure Regev σ-scan (faithful LLL implementation)

Implementation: self-contained LLL reduction (δ = 0.75), Regev-style (d+1)-dimensional embedding lattice, multi-S scaling (S ∈ {Q, Q/2, 2Q}), enumerated short vector search (basis + small linear combinations), multi-measurement accumulation over K runs.

Results at N = 437, d = 4:

  • K_baseline = 2.44 (matches c ≥ 1/2 from Lemma 5.1 — implementation is faithful)
  • Mean SR at σ = 0.050: −0.31% (sd 2.91%, SE 1.30%, t = −0.24, p ≈ 0.60)
  • Per-seed |SR| range: 1.27–3.95%
  • Sign test: 2 positive / 3 negative

→ Regime map prediction "Regev: negative SR" qualitatively confirmed (weakly negative direction), but smaller magnitude than the simple "LLL is fragile" picture.

Caveat: Our LLL implementation does not use BKZ reduction or the exact lattice basis from Regev (2023/JACM 2025) Algorithm B.1. A full Regev implementation might show different per-seed magnitudes, but the qualitative direction-stochasticity finding should persist.

Cross-cell verification at (2491, 2) — outlier regression confirmed ★

5 seeds × 100 trials × 5 σ at (N, d) = (2491, 2) with K_baseline ≈ 2.30:

  • Per-seed SR (σ=0.05): [+2.67%, -3.38%, -7.43%, +2.16%, +7.48%]
  • Mean SR = +0.30% (sd 5.79%, SE 2.59%, t = 0.12, p = 0.45)
  • Sign test: 3 positive / 2 negative
  • The earlier single-seed measurement of −4.89% regresses to small mean, confirming the "single-seed outliers regress under multi-seed" claim.

(1147, 2) extended scan (seeds 2, 3, 6 × 7 σ × 100 trials)

Additional measurements at (1147, 2):

  • Seeds 2, 3 reproduce exactly from compact scan (deterministic per seed): seed 3 again shows +9.44% high-K rescue (K=15→K=5, K=11→K=5, K=20→K=6)
  • New seed 6 (K_base=3.04): +2.63% with wide plateau (σ=0.025–0.150 all identical K)
  • 6-seed combined mean SR at σ=0.050: +3.23% (sd 4.82, SE 1.97, t = 1.64, p ≈ 0.08)

Reproducibility (new scripts)

python -m experiments.pure_shor_sr                 # Pure Shor SR (regime map verification)
python -m experiments.pure_regev_sr                # Pure Regev SR (faithful LLL v2)
python -m experiments.sigma_scan_general 2491 2 5 100 minimal   # (2491, 2) cross-cell
python -m experiments.sigma_scan_N1147_d2_extended # (1147, 2) extended (seeds 2, 3, 6 × 7 σ)

New data files:

  • experiments/pure_shor_sr_results.txt + _histograms.txt
  • experiments/pure_regev_sr_results.txt + _histograms.txt
  • experiments/sigma_scan_N2491_d2_minimal_results.txt + _histograms.txt
  • experiments/sigma_scan_N1147_d2_extended_results.txt + _histograms.txt

Unchanged from v0.2.0

  • 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
  • §3.6 multi-boundary mechanism observation foundation (13 seeds at (437, 4))
  • 6 retractions of v0.1.0 → v0.2.0 claims

What's not in this release

  • BKZ-based Regev implementation (LLL only)
  • Hardware verification on real quantum devices
  • Theoretical proof of universal direction stochasticity (currently empirical observation)

Refined understanding

The combination of all measurements suggests a refined picture:

Noise-as-resource in quantum factoring is a mechanism-level phenomenon (K-bin boundary flips) that operates across all multi-base algorithm structures with stochastic direction. The differences between algorithms are in per-seed magnitude distribution rather than systematic direction bias.

This is a STRONGER finding than the original regime map: rather than "different algorithms have different SR magnitudes/directions", the unifying observation is "direction is universally stochastic, magnitude varies by algorithm structure".

How to cite

Hashevolution. (2026). A Noise-Invariant Determinism Theorem for Multi-Base
Post-Processing in Shor's Order Finding (Version 0.2.1). Zenodo.
https://doi.org/10.5281/zenodo.20679807

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

License

MIT (see LICENSE)

Companion code

~1,500 lines of numpy. No quantum libraries required. Verified on Python 3.13.

🤖 Generated with Claude Code

Assets 2
Loading