The full manuscript is 129 pages. Propositions 20.22–20.26 establish the weighted prime bound, diagonal tail inverse comparison, remote moment enclosure, signed certificate and complete fixed-window coercivity at lambda=3. G2 for unbounded windows and RH remain open.
The certified arithmetic used Python 3.12, python-flint 0.9.0 and FLINT 3.6.0. Install python-flint in a suitable Python environment if it is not already present. The certificate itself uses only Python’s standard library and python-flint. Earlier exploratory pilots additionally use NumPy/SciPy, but they are not dependencies of the final verifier.
Run from the bundle’s g2_schur_directional directory:
python3 certify_weighted_prime_tail.py
python3 certify_infinite_schur.py --N 256 --M 512 --J 4096 --r 80 --parity even --bits 768 --diagonal --witness schur_witness_N256_M512_even_b768.json.gz
python3 certify_infinite_schur.py --N 512 --M 1024 --J 4096 --r 100 --parity odd --bits 768 --diagonal --witness schur_witness_N512_M1024_odd_b768.json.gz
For the independent precision replays, replace --bits 768 by --bits 896 while retaining each original 768-bit witness filename. Both replays have already passed and their reports are included. The witness files contain exact dyadic mantissa/exponent pairs, not rounded decimal approximations. The verifier checks that their entries remain exact at the chosen precision.
To generate a fresh witness from the verified intermediate solve, omit --witness. Positivity is checked after the result is frozen, so every intermediate residual remains in the calculation. Regenerating an identical witness is not required to establish positivity; replaying the supplied one makes the precise certified object reproducible.
The weil_sequences_n4096_b768_k128.json and corresponding 896-bit cache store outward enclosures of exact coefficients. assembly.py automatically recomputes them when absent. They may be renamed or removed in a separate reproduction directory to force fresh function evaluations. No cache contains an assumed positive sign or an omitted-tail assertion.
T>=diag(g_n)>0, so its inverse is bounded above by the diagonal inverse. The base cut remains fixed in every loss term.K=K_Z-R*T^(-1)R. All finite residual rows through J4096 are computed with the true diagonal.The two final parameter sets and hashes are recorded in fixed_window_certificate_report.md, and every positive pivot enclosure appears in the associated infinite_schur_..._diagonal.json report. A PASS is a fixed-window result for the stated sector. Both sectors are necessary for the asserted whole-space result. Pivot magnitudes are not spectral lower bounds.
adversarial_schur_report.md and directional_tail_derivation.md contain the analytic review and derivation. The proof inserts are integrated into the complete root LaTeX source. schur_pilot_report.md, the pilot JSON reports and the failed full-majorant reports preserve the exploration history. A negative pivot in those lower majorants does not establish that the true Weil form has a negative direction. The large pilot block caches were not retained because the supplied pilot code regenerates them; they are unnecessary for the final proof.
Only the original two exact witness archives are retained: the 896-bit replay witnesses have identical uncompressed hashes and need not be duplicated. All prior releases’ reproducibility material remains in the cumulative bundle.
At the bundle root:
latexmk -pdf -interaction=nonstopmode -halt-on-error fixed_space_prime_action_v1.tex
The verified output has 129 pages, 407 unique labels and no LaTeX warnings or unresolved references. All pages were rendered for review; the new proofs were also inspected at higher resolution. The next mathematical target and all remaining qualifications are recorded in RH_G1_G2_research_log.md.