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v0.4.0 — Theorem 6 SR no-go + cryptographic N-scaling

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@Hashevolution Hashevolution released this 14 Jun 09:54
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Release notes — v0.4.0 (2026-06-14)

DOI: minted by Zenodo on release (backfilled after archival).

Summary

v0.4.0 is a formalization + honesty release on top of v0.3.0 (DOI
10.5281/zenodo.20685015).

It adds Theorem 6, a sign-agnostic no-go result that closes the
stochastic-resonance (SR) factoring question raised in §3.6, backs it with a
cryptographic-regime N-scaling measurement, and downgrades the Yang-Markidis
positioning from "analytical complement / gap-filling" to the honest
"numerical verification + boundary mapping."

What changed

Theorem 6 — SR no-go (paper §3.3.ter, new)

For any noise in the coherence-loss class (★), g(ε) = (1−ε)·g_0 + ε·g_∞
with 0 < g_0, g_∞ ≤ 1, without assuming the sign of g_0 − g_∞:

  1. No tuned resonance. E[K(ε)] is monotone in ε; the optimum sits at an
    endpoint, never an interior ε*. There is no noise level to "tune into."
  2. Closed-form swing. |ΔK| = E[K_λ^ideal] · |1/g_∞ − 1/g_0|, fixed by the
    two endpoint probabilities alone, independent of the σ-profile.
  3. No asymptotic advantage. If the reciprocal gap is O(1), the swing is
    O(log log N) — a constant multiplicative factor, never a speedup.

The earlier draft wrongly assumed g_∞ ≤ g_0 ("noise always hurts"). Live
N-scaling data falsified that: the smallest-order setup at each N is a genuine
positive-SR cell (g_∞ > g_0). Theorem 6 is therefore stated sign-agnostically.

N-scaling measurement (paper §3.6.bis, new table)

|Δ| = |p_0 − ρ| over 12 setups (3 per N, 300–1,000 MC samples,
shor_n_scaling.py):

N mean p_0 mean ρ mean |Δ| min |Δ| max |Δ|
437 0.741 0.410 0.446 0.173 0.607
1147 0.443 0.402 0.211 0.118 0.260
2491 0.486 0.287 0.372 0.260 0.483
4087 0.476 0.390 0.259 0.220 0.297

Across a 9× range in N the gap stays in [0.12, 0.61] with no monotone growth
toward 1 — the empirical regularity behind Theorem 6's O(1) reciprocal-gap
assumption. The result is reproducible to the digit under fixed seeds.

Honest reframing of Yang-Markidis positioning

  • Their two-stage model is their Eq. (3), §5; the mixing weight ε is a
    conceptual parameter ("total weight transferred out of the intended
    family"), left unspecified — neither fitted nor given as ε(σ). Full text
    (incl. appendix) verified 2026-06-14.
  • The exp(−σ²) decay is the standard dephasing result (Nielsen–Chuang §8.3).
    We claim no new mechanism. Our contribution is narrow and verificational:
    (i) verify their qualitative weight equals the standard-dephasing factor
    (R²=0.95 at N=437); (ii) map the boundary (breaks for amplitude damping,
    R²=0.03); (iii) confirm across five algorithm classes.
  • README / arxiv_draft / sr_sigma_curve_model wording corrected accordingly.
  • We do not position this as filling an analytical gap they could not close.

Canonical source declaration

  • paper.md is now the single canonical source of truth.
  • paper.tex is deprecated / frozen at v0.2.1 — to be regenerated wholesale
    from paper.md only at arXiv-submission time (banner added; see
    compile-notes.md).

Retained (unchanged)

  • Theorems 1–5 and all raw measurement data.
  • The v0.3.0 closed-form σ-curve and five-algorithm verification.
  • The conclusion that SR-based factoring acceleration is precluded — now
    formalized as Theorem 6 rather than asserted.

What this is NOT

  • Not a new quantum advantage, not a new mechanism.
  • Not an unconditional impossibility result — Theorem 6 is a no-go within the
    framework and under its measured regularities     (see its "Scope and honest
    caveats").

Citing

If you use this work, cite v0.4.0 for Theorem 6 + N-scaling + honest framing,
alongside v0.3.0 (closed-form framework) and v0.2.1 (theorems 1–5 + raw data):

@misc{shor_v0_4_0,
  title  = {A noise-invariant determinism theorem for multi-base post-processing
            in Shor's order finding (v0.4.0, Theorem 6 SR no-go)},
  author = {{Hashevolution}},
  year   = {2026},
  doi    = {10.5281/zenodo.XXXXXXXX},
  url    = {https://doi.org/10.5281/zenodo.XXXXXXXX},
  note   = {v0.4.0 adds Theorem 6 and N-scaling to v0.3.0
            (10.5281/zenodo.20685015)}
}

(DOI backfilled once Zenodo archives the release.)

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