Wall check: Same open gap. Closest result: NS38 / prop:v132-global-index.
What changes: replayable evidence for the single existing numerical premise
beta_(log4)(1)<-3/5. No new arithmetic input or cofinal estimate is attempted.
This is a conclusions/dependency review plus a fresh numerical certificate,
not a replay of all historical proofs and not recovery of an original file.
Read-only source: /private/tmp/riemann-prime-folding-review-20260930, commit
769359298fada57711b6895c4bde3fabe2cc5168, refreshed by the coordinating agent.
Reviewed current AGENTS, all research-map node status/title/notes and the
continuation-input scopes, current README status and dependent summaries,
NEXT_STEPS relevant rows, the full missing-evidence ledger and NS38 claim
changes. Read manuscript eq:v130-r-symbol, eq:v130-beta,
prop:v132-global-index, their surrounding proofs and the appendix disclosure
of absent certify_beta4_negative.py. Exact source hashes are in outputs.json.
The source formula was checked before computation. No independent agent’s implementation or result was read before sealing this result. Only the coordinator’s fixed-claim brief was used to select the task; its reported numerical interval was a claim to verify, not an input to evaluating beta.
Put L=2 log4=log16. The current manuscript defines
beta = Re psi(5/4+i/2) - log(pi)
- 2 sum_(1<m<16) Lambda(m) m^(-1/2) cos(log m)
+ 2 integral_0^L exp(t/2) cos(t) dt.
The strict integer endpoint excludes 16 even though 16 is a prime power.
Exact integer factorization gives the included prime powers
2,3,4,5,7,8,9,11,13; Lambda(p^k)=log(p), not log(p^k).
An antiderivative of exp(t/2)cos(t) is
F(t) = (4/5) exp(t/2) ((1/2)cos(t)+sin(t)).
Differentiation gives F’(t)=exp(t/2)cos(t): the sine cross terms cancel, and (1/4+1)/(5/4)=1. Since exp(L/2)=4 and F(0)=2/5,
2(F(L)-F(0)) = (16 cos(L)+32 sin(L)-4)/5.
This retains the continuum term’s positive sign in beta and its lower endpoint contribution -4/5. A second complex-exponential evaluation of the same integral overlaps at each precision; the elementary antiderivative is the analytic justification, not overlap alone.
verify_beta.py evaluates the formula directly with python-flint 0.9.0:
acb.digamma() at the exact rational argument 5/4+i/2, and Arb pi/log/sqrt/
cos/sin for the other terms. No floating midpoint is used in a proof gate.
It recomputes at 192 and 320 bits, exports each term’s directed dyadic
endpoints exactly as rational numerator/denominator pairs, then rounds the
beta interval outward to denominator 10^45. Strict ball comparisons confirm
both decimal-rational endpoint inequalities at both precisions:
-0.640027118160897146421661394744768547319369747
< beta_(log4)(1) <
-0.640027118160897146421661394744768547319369746.
In particular the manuscript’s historical broad interval
-0.64784909 < beta_(log4)(1) < -0.6321938 < -3/5
is verified. This is fresh replacement evidence, not an assertion that the unavailable historical computation used these methods or returned these precise values. Arb/acb’s certified-function implementations are the numerical library trust dependency of this direct-method certificate.
Approximate decomposition for orientation only (exact balls are in JSON): archimedean -1.2251461931; signed prime sum -2.0613235404; continuum -1.4762044655; beta -0.6400271182. The signed prime sum is SUBTRACTED. Its negative value therefore contributes positively to beta.
Continuity and evenness of the finite-sum symbol provide symmetric open frequency intervals on which it is negative. Arbitrarily many disjoint smooth symmetric frequency supports there yield arbitrarily many mutually orthogonal negative directions for the full-line multiplier; any fixed finite number of linear constraints reduces this dimension by at most that finite number. These inverse Fourier transforms are Schwartz, but they are not supported inside (-log4,log4). Thus this supplies the numerical premise of the existing full-line negative-index implication, not a negative physical-window Weil direction. Certified physical W4 positivity is intact.
Both historical original-evidence groups remain recovery OPEN. No source manuscript, registry, primary configuration, research state, gallery or publication file was changed. RH, G2 and the full cofinal signed-arithmetic floor remain open. No candidate scan, new theorem row or thaw occurred.
From the repo root:
.venv/bin/python audits/bounds-six-agent-2026-09-30-v1/certificate/01-direct/verify_beta.py
pyright --project audits/bounds-six-agent-2026-09-30-v1/certificate/01-direct/pyrightconfig.json
Replay passes both precision gates and exact outward comparisons. Scoped CLI Pyright diagnostics: zero errors, zero warnings (no LSP tool available in this session). Every Python function has type annotations. The owned configuration selects the existing project virtual environment; no working import, dependency or global configuration was changed.