[pinterbanget]

Summary

This adds research/pr4483-chernoff-check/, an independent check of zxv's #4483 (research/rank-convergence/PROOF.md) §2–3 on actual flows. It covers the exact potential identity (2), the Chernoff bounds (7) and (8), and their consequence (9). #4532's critical-root funding inherits these, and with it #4535 and #4568. My #4529 checked #4483's gates and rank-one identity at t = 1 only. All checked claims are confirmed. No new κ.

Checks

My own flow code builds #4483's arbitrary-height flow at G = (a+b)·⌊log_M ⌊N/m⌋⌋ and computes H_t, R_t, I_t, B_t and F_t. There are two cases: raw p7 with #4483's p7 geometry, and p16 (#4470) with #4532's p16 geometry. Exponents lie above each profile's moment crossing, since T > P is (7)'s premise.

profile	ranks (height)	(T−P)H_t / rhs(7)	R_t / rhs(8)
p7	10^5 (2), 10^7 (4), 3·10^9 (6)	28.0 → 13.5 → 9.97	0.304 → 0.123 → 0.096
p16	2·10^5 (2), 2·10^8 (4)	25.2 → 16.1	0.267 → 0.237

Across all 15 cases:

• (2), exactly. I_t + R_t = B_t + F_t + (T−P)H_t holds exactly as an identity of formal power sums Σ c_x·x^t, with integer coefficients and up to 16,252 bases. It therefore holds for every t at each checked rank. A rejection control is included.
• (7), (8), (9). They hold with wide margins.
• Premises. Ratio ordering and q > 1 hold.

Limitations:

• These are finite heights (G ≤ 6). The all-height statement is Remove tensor padding (199.32× smaller uniform-rank exponent) #4483's written Chernoff argument, which these checks corroborate.
• Comparisons use floats, with margins far above rounding.

python3.12 -B scripts/research_package.py --package research/pr4483-chernoff-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4483-chernoff-check outcome=completed   (~45 s, stdlib only)

Head: 84c0fb3a39bcb1e4b8c48892bfdd8a6e7ae58a50. Base: canonical main f626db5c.

Credit: zxv (#4483), Dillon (#4532), Cinna Davis (#4470), Jacob Alan Sussman. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code