Classification: fixed-claim validation and scoped gain audits. Same open gap — NB-GAIN / NS53/61/87. Six separate agents completed two groups of three. This index is a forward publication note; the original 31 manifested files and their manifest are preserved byte-for-byte. Their local paths and earlier unpublished-status fields record the original execution environment, not the current publication status.
The direct 192/320-bit Arb method certifies
-0.640027118160897146421661394744768547319369747
< beta_(log4)(1) <
-0.640027118160897146421661394744768547319369746 < -3/5.
The independent 128/256-bit digamma-series method certifies the broader interval (-0.640150, -0.639904), also inside the historically reported enclosure and below -3/5. Only the direct method establishes the 45-place interval. The methods use different analytic evaluations but share Arb as the interval backend. The third reviewer passed all 37 recorded replay/integrity checks.
This is REPLACEMENT evidence for the numerical premise of NS38 / prop:v132-global-index, not recovery of certify_beta4_negative.py or its output. The beta certificate does NOT transfer to the physical lambda=4 form: its tests are not compactly supported in that physical window. It supports the existing full-line negative-index implication, including the even sector and finitely many homogeneous linear constraints. Both original-evidence recovery groups remain OPEN.
Gain01 checks the previously derived weighted coefficient inequality, retaining full old refit. L <= B S gives S^-1 <= B L^-1, hence an upper gain bound in corrected-load coordinates. The valid lower bound in optimizer coordinates still needs an arithmetic estimate of those coordinates. A hypothetical Mobius-sized new block contributes only an N^-3 certificate, whose normalized dyadic sum converges. This is a limitation of that certificate, not an upper bound on actual gain or a route closure.
Gain02 assesses the literal structured-phase import for the complete projected kernel. It leaves the representation/complexity, a factor-N allowance and the companion actual signed numerator unestimated. A fixed theorem exponent cannot become A(N) with unchanged constants. Gain03 accepts both scoped audits after correction G03-01: little-o model error is sufficient, not necessary; fixed strict signal/error separation also suffices. None supplies a nonsummable actual signed gain.
The dependency note makes the previously local coefficient-corollary dependency explicit from public NS78. Historical references to local six-review working files are provenance references, not additional claimed public evidence. The arXiv source addendum records the structured-phase source and its exact scope. No new research row, theorem node, thaw or manuscript version is claimed. RH, G2 and all cofinal arithmetic inputs remain open.
The preserved original programs point to their historical local checkout. Use the new wrapper to replay the mathematics from this public checkout without modifying those records:
python3 -m venv .venv
.venv/bin/python -m pip install -r audits/bounds-six-agent-2026-09-30-v1/publication/requirements-replay.txt
.venv/bin/python audits/bounds-six-agent-2026-09-30-v1/publication/replay_publication.py --output .local-runs/beta-six-publication.json
Choose a fresh output path. The wrapper verifies every original manifested file and the unchanged manuscript source hash, recomputes the same four fixed-precision runs, compares every numerical field with the sealed records, checks cross-enclosures, and confirms the originals remain unchanged. It does not call the old output-writing entry points, fetch data, or require any private/local historical path. The output and environment from the publication replay are recorded under publication/.
The original manifest binds the historical round. The additive publication manifest and validation record bind the portable wrapper, forward notes, and completed build. The wrapper’s first failed display-ball check is preserved and its correction recorded; the final replay passes all 85 checks. Section 9 disposition: this is replacement validation of the same previously reported scalar at the same parameters, not a newly selected certificate or a new mathematical result. Publish as an audit without a version bump; preserve the manuscript and its historical conditional wording, with the forward ledger link identifying the now-available replacement evidence.