[unpaidmoon]

Change and mathematical scope

Scoped proof. No new κ. The selected frontier stays 781451/250000000.

For the ordinary BIT stage retained by certificates/frontier6030/CLAIM.json (admitted saving 17/5000, rank N = 2^H1 with H1 = 25986965159937), the inherited nested-width bound used by research/lockstep-live-role-audit/constant_c.py gives an integer

X = 471715391583208384

such that the stock formula's productive role count satisfies log2(B) < N*X. Combined with the recorded prime rule q > 2^26*N^5,

c_B = ceil(log_q B) <= 2^(H1+13) <= N^2.

PR #4954's sufficient guard at log2(e) >= 52182660963735 therefore no longer needs a role count beyond that inherited bound, once the cited reading S_old = 1 from PR #4970 is used. The six-bit gap below the guard is unchanged. This does not materialize B, re-prove the compiler census, admit a supplier, or discharge the five framework assumptions.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. Task selected-stock-envelope, scope refined: one selected-stage coefficient, the role digit of this BIT stage, rather than a new stock envelope or a cutoff rerun.

The band note (PR #4954) asserts that c_B <= N^2 follows from the nested-width bound and does not compute the digit. PR #4970 derives S_old = 1 and leaves c_B open. The m=73 envelope, the log_q parent correction, and the two-round cube stock are different objects. Local overlap search covered the working-tree notes at this base and did not find another derivation of this digit; open-PR and Discussion coverage was not refreshed in that cache.

Validation

CPython 3.12, standard library. From the repository root:

python3 -B scripts/research_selected_baseline.py
python3 -B research/bit-role-digit-bound/verify.py
python3 -O -B research/bit-role-digit-bound/verify.py
python3 -B scripts/research_package.py --package research/bit-role-digit-bound
python3 -B scripts/check_research_commit.py --package research/bit-role-digit-bound --check-source-history --check-modes

The verifier recomputes X from the pinned BIT invoice, checks X < 2^13*(5*H1+26) and H1+13 <= 2*H1, and rejects rank 12 and an upper bound shifted by the full slack. Marker:

PASS_BIT_ROLE_DIGIT c_B<=2^(H1+13)<=N^2 X=471715391583208384 no_new_kappa

research_selected_baseline.py reports bindings only (PASS_RESEARCH_BASELINE_BINDINGS_ONLY). make frontier6030-verify was not run. No Lean proof and no supplier replay.

Upstream main was fetched again before the commit and was still 17469fff78c22941280f04d39dc5a1faee09c840.

Credit

JoyboyBrian: PR #4954, the band theorem and the open c_B <= N^2 hypothesis. James Reed: PR #4970, S_old = 1. Mark (Matarener): the nested-width formula as used in research/lockstep-live-role-audit. zxv: the delayed-cover stock formula. The selected claim is PR #6030.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or proof-kernel acceptance is claimed.

Open next questions

Can a D = 2 depth lemma, rather than a smaller c_B, move the sufficient guard below t = 52182660963735? The role digit is now at most 2^(H1+13), negligible beside N^2*w, so it does not close the six-bit gap. The smallest follow-up is an exact comparison of D*(N^2*w + c_B) with e/4 at D = 2 and w = ceil(e^(1/250)). It is blocked by the slack the published band proof spends on the exponent arithmetic.

Workflow feedback (optional)

The package generator and pinned-source replay made the digit comparison easy to check. Overlap search against open PRs needed a live refresh that this session did not complete; a cached open-PR snapshot at the current base would have made that check local. Feedback discussion: #1200