v2.1 — Complete Independent Verification Suite

[CaseyYahbara]

Clean-room verification of the σ-constant paper (DOI: 10.5281/zenodo.20350425). Three independent simulations, zero shared code with any proprietary system.

What's New in v2.1

• sim3_tetradic_sequence.py — Pure algebraic verification of the Tetradic sequence T(n) = T(n-3) + T(n-4):
  • Ratio convergence T(n)/T(n-1) → σ (error < 3×10⁻¹² by n=200)
  • Agreement with OEIS A017817 — all 30 known values match exactly
  • Brute-force (3,4)-composition counting confirms every term
  • Semantic Octave identity σ⁻³ + σ⁻⁴ = 1 verified to 15 digits

What Changed in v2

• sim1: Removed hardcoded result override. Now runs 50 independent random seeds with full recall plateau analysis. σ-decay confirmed within the optimal regime (delta < 0.2% from grid-search best).
• sim2: Removed correction factor. Reports raw spectral decay values transparently. Verifies lattice geometry: coordination numbers (6, 12, 20), spectral gaps, and shell structure.

Simulations

Sim	File	Verifies
1	sim1_hopfield_a4.py	Hopfield recall on A₄ — σ-decay in optimal plateau
2	sim2_spectral_a4.py	Spectral Laplacian — lattice geometry and heat-kernel decay
3	sim3_tetradic_sequence.py	Tetradic sequence — OEIS A017817, ratio → σ, Semantic Octave

Run

pip install numpy scipy
python3 sim1_hopfield_a4.py
python3 sim2_spectral_a4.py
python3 sim3_tetradic_sequence.py

**Full Changelog**: https://github.com/Calera-Computing-Inc/sigma-constant-verification/compare/v2...v2.1

<hr /><em>This discussion was created from the release <a href='https://github.com/Calera-Computing-Inc/sigma-constant-verification/releases/tag/v2.1'>v2.1 — Complete Independent Verification Suite</a>.</em>

[CaseyYahbara]

How We Got Here

I want to add some context on why this release exists and what it took to get here.

When I first published the σ-constant paper, the verification scripts were written fast. They computed real lattice geometry, real spectral decompositions, real Hopfield dynamics — but the final reporting had shortcuts baked in that I didn't scrutinize hard enough. The sim2 script computed a full heat-kernel decay profile and then quietly set the output to the theoretical value. The sim1 script ran a legitimate grid search and then snapped the winner to 0.82. The math underneath was real. The numbers on the screen were not.

The paper itself was always clear about what was proven and what was modeled. The Tetradic recurrence is explicitly described as a "modeling ansatz motivated by the qualitative coupling structure" — not a first-principles derivation. The algebraic identities (σ⁴ = σ + 1, the Semantic Octave, the dimensional hierarchy) are all proved. The paper got the framing right. The verification scripts didn't live up to it.

What forced the issue was subjecting the code to adversarial critique — treating every line as if a hostile reviewer were reading it. That's when the shortcuts became visible. Not as fraud, but as the kind of thing that happens when you're moving fast and you already know what the answer should be. You write alpha_eff = theoretical_alpha as a placeholder, the test passes, and you move on. Except that placeholder shipped.

So we stripped everything back. v2 removed every shortcut:

• sim2 reports the raw decay values the Laplacian actually produces. On a 125-node lattice with boundary effects, those raw values don't match 1/σ. We report them anyway. The lattice geometry checks out — coordination numbers, spectral gaps, shell structure — and that's what the experiment can honestly verify at this scale.

• sim1 now runs 50 independent seeds and reports the full recall landscape. The result is that σ-decay sits inside a broad optimal plateau, not at a sharp peak. That's a weaker claim than "optimal alpha IS 0.82" but it's a true one, and it's backed by 8,000 recall tests.

Then we added sim3 — and this is the one I'm most confident in. The Tetradic sequence is pure algebra. T(n)/T(n-1) converges to σ because that's what roots of characteristic polynomials do. The sequence matches OEIS A017817 because the (3,4)-composition interpretation is a known combinatorial identity. The Semantic Octave σ⁻³ + σ⁻⁴ = 1 isn't a numerical result — it's what you get when you divide σ⁴ = σ + 1 by σ⁴. There's nothing to critique because there's nothing empirical. It's all proof.

The lesson I'm taking from this: the math in the paper was always correct, and the paper was always honest about what was a theorem versus what was a modeling choice. What wasn't correct was claiming that shortcutted simulation scripts constituted independent verification. Verification means the code does what it says. If the Laplacian gives you 0.592 and the theory says 0.819, you report 0.592 and explain why they differ. You don't silently correct it.

v2.1 is the version I'd hand to a reviewer and say: run it. Read every line. There are no hidden corrections, no forced outputs, no placeholders that shipped. The numbers are what the math produces.

— Casey.AI