Minimizing the factorial certificate B_N itself

Comparison, 2026-09-06, the third objective on the pilot's seed family. It is separate from ../factorial_full_cost/, whose record and conclusions (COST_COMPARISON.md) are left unchanged, and from the preserved pilot in ../factorial_certificate_pilot/. Script direct_bn.py, record direct_bn_results.json, pinned by tests/test_factorial_direct_bn.py.

Label. A finite comparison of LP objectives on one seed family. Not an asymptotic theorem, not an LP-optimality proof, not a prime-counting record, and nothing here bears on RH (docs/08). Grade: measured, every accepted seed rechecked exactly.

The question

The two earlier objectives were proxies for the certificate: the pilot minimized the leading constant C, and full_cost.py minimized the proved bound U_N, which turned out to move the certificate the wrong way. This comparison minimizes the certificate value itself,

B_N(a) = sum_{k=0}^K sum_{j|L} a_j log(floor(N/(j M^k))!), K = floor(log_M N),

which for fixed (L, M, N) is linear in the seed: B_N(a) = sum_j w_j a_j with w_j = sum_k log(floor(N/(j M^k))!). Same constraints as the pilot, unchanged; same sizes (L in {30, 210, 2310}, M = 2..30); same cutoffs N in {10^4, 10^6, 10^8, 10^12}; the same 348 rows. Compared on B_N against BOTH the pilot's seed and the full-cost seed.

Method

• The LP minimizes sum_j (w_j/N) a_j over the pilot's polytope (HiGHS, tolerances 1e-10); the w_j are computed at 50 digits and handed to the solver as floats. The LP only proposes: Fraction(x).limit_denominator(10^6) reconstructs, and every constraint is rechecked in integers by the same exact_verify that full_cost.py uses, imported from it so the constraint set is literally the same code.
• Fallback rule: the pilot's seed is the certificate of record. The direct proposal is adopted only if it re-verifies exactly AND its B_N is strictly below the pilot seed's. A worse or rejected proposal leaves the pilot seed in place and the row says so. Nothing worse was ever accepted, and no row needed the fallback for a worse or rejected proposal.
• All three seeds are evaluated on B_N at 50 digits from their exact rationals. Log-factorials only; no prime data used or produced.

Results

Counts. 348 rows; 0 rejected reconstructions; 334 rows where the direct LP returns the pilot's seed exactly; 14 rows where it strictly improves B_N; 0 rows worse than the pilot's seed; 0 ties with a different seed. Against the full-cost seed the direct seed is never worse and strictly better in 31 rows (the 30 rows where the full-cost seed differs from the pilot's, by 15 to 312, plus one row where it does not). Runtime 13 s.

Where and how much. All 14 improvements are at L = 2310: 13 at N = 10^4 (M = 2, 4, 5, 8, 13, 14, 16, 17, 18, 19, 20, 21, 22) and one at N = 10^6 (M = 3). None at 10^8 or 10^12. Absolute gains 0.09 to 2.76; relative 4.6e-7 to 2.5e-4. The largest, (2310, 17, 10^4): B_N 10871.5278 to 10868.7700, gain 2.758. The one at 10^6: (2310, 3), 1148784.2045 to 1148783.6798, gain 0.525.

The per-(L, N) winners do not move. Under B_N, the best M for each L and N is the same seed whether one takes the pilot's seeds or the direct ones:

Two things in that table are worth a sentence. At L = 2310, N = 10^4, the pilot's own seeds already put the smallest certificate at M = 10, not at the M = 15 seed the pilot selected by C (and which full_cost.py also selected by U_N); selecting by the constant or by the proved bound picks a seed whose B_N is 31.7 higher at this N. And the direct optimizer, given the freedom to exploit the actual fractional parts {N/(j M^k)}, changes none of these winners: its 14 gains land on non-winning M.

What the improved seeds look like. Each is the pilot's seed with its small denominators re-tuned (thirds to halves at M = 2, 3, 4, 5; new quarter-weights on 21, 22, 33, 35, 55, ... at M = 13, 14; a 2310 coefficient moved to -3 at M = 8), with the leading constant C equal or larger by at most 3e-5. The gain is entirely in the lower-order terms, which is why it is a few units at 10^4 and nothing measurable beyond. For scale, REVIEW.md records psi(10^4) = 10013.397, so the winning L = 2310 certificate at 10^4 sits 666.0 above psi; the best direct gain anywhere at 10^4 is 2.76, under half a percent of that gap. Within a fixed (L, M), the certificate is determined by the leading constant to within a few units, and the constant is what the pilot already optimized.

Does any pattern persist across N? No. For every one of the 14 improving (L, M) pairs the direct seed at the other three cutoffs is the pilot's seed, so each direct seed is specific to its N. At 10^8 and 10^12 the direct LP returns the pilot's seed in all 87 pairs. One caveat on those two cutoffs: differences between candidate seeds with equal leading constant are of order log^2 N, which relative to B_N ~ 10^12 is about 1e-9, close to the solver's 1e-10 tolerance; the exact evaluation is at 50 digits, but the LP may not resolve a gain that small, so "no improvement at 10^12" means none the float LP could find, not a proof that none exists in the polytope.

Answer

• Does direct minimization improve B_N? Marginally and only at small N: 14 of 348 rows, all at L = 2310, 13 of them at 10^4 and one at 10^6, by 0.09 to 2.76 (at most 2.5e-4 relative). Never at 10^8 or 10^12. It never changes the best seed for a given (L, N). It beats the full-cost seed wherever that seed differs from the pilot's, by 15 to 312, which restates COST_COMPARISON.md's finding from the other side.
• Does any pattern persist? No. Every improvement is N-specific and disappears at the next cutoff; from 10^8 on, minimizing B_N directly returns the leading-constant seed in all 87 pairs. Of the three objectives, the pilot's is the one that agrees with the certificate at every large cutoff, and at 10^4 the best seed per L was already among the pilot's, at a different M than the constant selects.

Checks run

direct_bn.py (348 rows, 0 rejections, fallback never needed); tests/test_factorial_direct_bn.py (headline counts, every distinct direct seed rechecked exactly, every row's three B_N values recomputed at 40 digits from the recorded rationals, the fallback rule re-derived row by row, the headline row); tests/test_factorial_full_cost.py, tests/test_factorial_pilot_archive.py, tests/test_frontier_archive.py, tests/test_hunt_probe_discipline.py, tests/test_docs_numbering.py, tests/test_doors.py; scripts/make_context.py --check. Not done: any LP-optimality proof, any search beyond the pilot's sizes, any prime computation at these N, any edit to the earlier records.