Re-verify PCF note against arXiv:2508.01581v1 (2026-07-20): verified clean

[OpenCnid]

Re-verification: verified clean against arXiv:2508.01581v1 on 2026-07-20

Independent, source-first re-verification of density-chain.md against the pinned source (arXiv:2508.01581v1, submitted 2025-08-03; v1 is still the latest). Every number was studied at the arXiv v1 PDF this session — none from memory. No discrepancies found; verification date refreshed only.

Ledger — {claim} · {note said} · {source says} · {locator}

Claim	Note said	Source says	Locator
OLS satisfaction coefficient	2.3370 (SE 0.004, t 589.442, CI 2.329–2.345)	2.3370 (SE 0.004, t 589.442, CI 2.329–2.345)	Table 1
OLS intercept	2.9239 (SE 0.022, t 134.440, CI 2.881–2.967)	2.9239 (SE 0.022, t 134.440, CI 2.881–2.967)	Table 1
OLS R² / adj R²	0.217 / 0.217	0.217 / 0.217	Table 1
OLS F / n / Durbin-Watson	3.474e+05 / 1,250,000 / 1.581	3.474e+05 / 1,250,000 / 1.581	Table 1
Spline intercept / Star_Level	3.1489 / 0.3122	3.1489 / 0.3122	Table 2
Spline basis coefficients	0.4011; 1.3377; 1.4408; 1.8278; 2.6972	0.4011; 1.3377; 1.4408; 1.8278; 2.6972	Table 2
Spline R² / F / n	0.297 / 1.408e+04 / 200,000	0.297 / 1.408e+04 / 200,000	Table 2
Tier CI level	99% (Tables 1–2 use 95%)	"Descriptive + 99% Confidence Intervals"	Table 3
5-star time / sat	22.0249 / 6.5675	22.0249 / 6.5675	Table 3
4-star time / sat	16.4997 / 5.3738	16.4997 / 5.3738	Table 3
3-star time / sat	16.4948 / 5.3759	16.4948 / 5.3759	Table 3
2-star time / sat	10.9785 / 4.6852	10.9785 / 4.6852	Table 3
1-star time / sat	10.9938 / 4.6868	10.9938 / 4.6868	Table 3
Tier separation	three levels, not five: 4★≈3★, 2★≈1★, 1★ slightly > 2★	4★ (16.4997/5.3738) ≈ 3★ (16.4948/5.3759); 1★ (10.9938/4.6868) > 2★ (10.9785/4.6852) on both	Table 3
Adaptable vs non-adaptable	condition declared; no statistic reported anywhere	§2.2 declares the condition; no contrast is reported anywhere in the paper	§2.2
Single-agent worked example	4×3×3×3×2 = 216	4×3×3×3×2 = 216	§2.3.1
Frameworks undergirding parameter system	exactly two (topos, rough fuzzy); triad only in Abstract	"Two primary mathematical frameworks undergird…" (§2.5); triad in Abstract	§2.5; Abstract
Released artifacts	github.com/cadavid1/PCF	embedded "See GitHub here" link → github.com/cadavid1/PCF	Abstract
Iterations per tier / scale	250,000 / 1.25M	250,000 per tier / 1.25M	§4.2

Discrepancies: none — verified clean.

Changed

• verified_against_source: 2026-07-19 → 2026-07-20 (frontmatter + index.json); index.json updated → 2026-07-20.
• Provenance: appended the 2026-07-20 clean re-verification; the 2026-07-19 correction history is preserved untouched.
• No note-body facts changed. Owner prose, the Murphy disclosure, and "Our take" are untouched.

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