Certified computation + proved implications. G2 and RH remain open.
The first positive zero of the complete lambda=4 ground transform is simple
and lies in (gamma_1 - 8.752082e-33, gamma_1 + 8.752082e-33).
The two precisions, 1024 and 1280 bits, give a displacement budget
8.75208157392882826...e-33, hence the simpler outward radius 9e-33.
This is the best enclosure extracted here from the archived comparison;
no optimality claim over every possible use of the archive is made.
The limiting quantity is C_ell * rho: the evaluator dual constant
C_ell = 79920.06156776... times the upper ground-energy budget
rho = 2.45361754930714...e-75. It is not the old ordinary projection
error sqrt(rho/b) = 1.5664...e-4. The new bound is sqrt(C_ell*rho)/d,
with d=0.0016. All complete tail cross terms are retained.
A hypothetical lower bound 0 <= l <= mu0 would instead control the
trial-to-ground error energy by b*(rho-l)/(b-l), where b=1e-67.
For its value-to-zero uncertainty to reach 1e-71, the scalar budget
requires rho-l <= 3.20320065696758...e-153 (relative gap about 1.3055e-78).
The outward condition rho-l <= 3.2031e-153 suffices for that transfer term.
The archive does not contain such a lower bound. It would not alone
produce a gamma-centered enclosure at this scale: the frozen trial has
fhat(gamma_1) in (1.0641,1.0642)e-40. The former ordinary-norm transfer
would require a much smaller Rayleigh gap, about 9.233248e-216.
The complete discrepancy has no certified sign or nonzero magnitude
lower bound. The separately certified N=120 finite benchmark is about
+2.92504e-71; it is not a complete-ground value. Its scale lies below
the resolution of the displayed archived comparisons. This is a failure
of those bounds, not a failure of the object or a universal impossibility
theorem for the data. The revised task closes NS-19’s accounting and ends
the pursuit of the former sign/factor-ten target.
sharpening_section.tex: complete proof and explicit hypotheses.archive.py: loads the unchanged v1.42 coarse verifier by reference.certify_discrepancy_budget.py, discrepancy_budget_b1024.json,
discrepancy_budget_b1280.json: evaluator dual bound and all 304 earlier-zero exclusions.certify_resolution.py, resolution_b1024.json, resolution_b1280.json:
precise energy-gap thresholds, full two-sided endpoints and trial-center caveat.check_discrepancy_budget.py, discrepancy_cross_checks.json: second
implementation of the dual solve, rational dyadic coverage, resolution
algebra, saved signs and cross-precision agreement; no external audit.certify_finite_lambda4.py, finite_lambda4_N120_b1024.json,
finite_lambda4_N120_b1280.json: the finite benchmark prepared during
the original Task B, retained separately and replayed after relocation.scope_review.md: claim and proof review.build_report.json, integration_checks.json: actual build and preservation checks.research-report-2026-09-21-v1.html: user-facing report.The planning midpoint diagnostic is labelled under ../diag_ns18_sharpening/.
It supplies no proof bound. Tagged v1.42 artifacts are unchanged.
Python 3.14.6, python-flint 0.9.0, mpmath 1.4.1. Run from the repository root. Every command requires a fresh output path; saved outputs are not overwritten.
.venv/bin/python -B evidence/v143/certify_discrepancy_budget.py --bits 1024 --output /tmp/ns19-budget-1024.json
.venv/bin/python -B evidence/v143/certify_discrepancy_budget.py --bits 1280 --output /tmp/ns19-budget-1280.json
.venv/bin/python -B evidence/v143/certify_resolution.py --bits 1024 --output /tmp/ns19-resolution-1024.json
.venv/bin/python -B evidence/v143/certify_resolution.py --bits 1280 --output /tmp/ns19-resolution-1280.json
.venv/bin/python -B evidence/v143/certify_finite_lambda4.py --bits 1024 --output /tmp/ns19-finite-1024.json
.venv/bin/python -B evidence/v143/certify_finite_lambda4.py --bits 1280 --output /tmp/ns19-finite-1280.json
.venv/bin/python -B evidence/v143/check_discrepancy_budget.py --output /tmp/ns19-second-check.json
python3 tools/verify_manifest.py v143
./manuscript/build.sh
The resolution script and second checker read the saved reports and bind them by hashes; the first two commands independently recompute the complete small gates. Full historical residual assemblies at 768/896 bits are inherited, not freshly recomputed. Their analytic bounds cover the infinite tail. No window, tail metric, cutoff or trial support is changed. NS-1’s two missing historical evidence groups remain OPEN and are not dependencies.