Full-cost versus leading-constant optimization of the factorial-certificate seed

Experiment, 2026-09-06, a follow-up to the preserved pilot in ../factorial_certificate_pilot/ (read PILOT.md, pilot.py and REVIEW.md there first). The pilot directory is untouched; everything here is new. Script full_cost.py, record full_cost_results.json, pinned by tests/test_factorial_full_cost.py.

Label. A finite comparison of two 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, with every accepted seed rechecked exactly.

The question

The pilot chooses a seed a = (a_j)_{j | L} by minimizing the leading constant C = kappa/(1 - 1/M), kappa = -sum_j a_j log(j)/j, and then proves (PILOT.md section 3)

psi(N) <= B_N <= U_N := kappa N (1 - M^{-K-1})/(1 - 1/M) + A [(K+1)(1 + log N) - log(M) K(K+1)/2],

with A = sum_j |a_j| and K = floor(log_M N). Does minimizing all of U_N (constant and error term together) give a better certificate than minimizing C alone, at the same L, M, N?

Method

• Same constraints as the pilot, unchanged: j | L; sum_j a_j/j = 0; g(r) >= 0 for 0 <= r < L; g(r) >= 1 for 1 <= r < M. Same sizes: L in {30, 210, 2310}, M = 2..30. Cutoffs N in {10^4, 10^6, 10^8, 10^12}. 87 x 4 = 348 rows.
• U_N = c1 kappa + c2 A with c1 = N (1 - M^{-K-1})/(1 - 1/M) and c2 = (K+1)(1 + log N) - log(M) K(K+1)/2, both positive. A enters the LP through auxiliaries u_j >= |a_j|; the objective handed to HiGHS is kappa + (c2/c1) A, tolerances 1e-10. The LP only proposes: each proposal is reconstructed with Fraction(x).limit_denominator(10^6) (the pilot's rule) and every constraint is rechecked in integers. A failed recheck is a rejection, recorded, never rounded toward feasibility.
• Baseline: the pilot's recorded leading-constant seed for the same (L, M), re-verified. The leading-constant LP was also re-solved here; it returned the recorded seed in all 87 cases (leading_constant_resolve_mismatches: 0).
• Both seeds are evaluated on the same U_N at 50 digits from their exact rationals. The record also carries the certificate value itself, B_N = sum_{k<=K} sum_j a_j log(floor(N/(j M^k))!), for every row where the two seeds differ. Log-factorials only. No prime data anywhere.

Results

Counts. 348 rows; 348 accepted, 0 rejected; full cost strictly better than its own baseline in 30 rows, tied in 318, worse in 0. Every one of the 318 ties is the identical seed. Runtime 5.2 s.

Where full cost wins. 29 of the 30 wins are at N = 10^4: one at (L, M) = (210, 7) and 28 at L = 2310 (every M except 6). The 30th is (2310, 7) at N = 10^6. At N = 10^8 and N = 10^12 the full-cost LP returns the leading-constant seed in all 87 pairs.

Size of the wins at N = 10^4. Relative improvement of U_N from 0.07% ((210, 7)) to 5.4% ((2310, 2)); the L = 2310 wins other than M = 2, 3 are 0.8% to 2.9%. The one win at 10^6 is 7.8e-6 relative.

The mechanism is a trade of constant for coefficient mass. In every winning row the full-cost seed has a larger C and a smaller A. At N = 10^4 and M = 15 the LP weighs one unit of A against c2/c1 ~ 2.3e-3 units of kappa; at 10^6, 10^8, 10^12 that price falls to about 4.5e-5, 7.4e-7, 1.5e-9, below the cheapest available trade, and the optimum reverts to the leading-constant seed.

The headline row, and the per-L winners. For L = 30 and L = 210 the two objectives pick the same seed at every N (Chebyshev's M = 6 seed with A = 5, and the pilot's M = 6 seed with A = 10). For L = 2310 they differ only at N = 10^4:

Full cost lowers the proved bound by 91.28 (0.82%). Winning seed:

a_1 = 1; a_2 = a_3 = a_5 = -1; a_6 = 1; a_7 = -1; a_10 = 1; a_11 = -1; a_154 = 1/2; a_210 = -1/2; a_2310 = -1.

Baseline seed (the pilot's): a_1 = 1; a_2 = a_3 = a_5 = -1; a_6 = 1; a_7 = -1; a_10 = 1; a_11 = -1; a_30 = -1; a_33 = a_105 = 1; a_210 = a_330 = -1; a_385 = 1; a_1155 = -1. At N >= 10^6 the L = 2310 winner under both objectives is the baseline.

But the certificate itself gets worse, in every winning row. The last column above is the point of this experiment. In all 30 rows where full cost lowers U_N, the actual B_N of the full-cost seed is higher than the baseline's: by 29.88 in the headline row (0.28% of B_N), and by 12.4 to 311.0 across the 30. Zero rows lower B_N. The reason is in REVIEW.md section 3.6: the error term A (K+1)(1 + log N) is a valid bound on |B_N - main term| but is loose by a factor of ten or more at these N, and the true deviation is negative (the pilot seeds sit 10 to 40 below C N). So U_N charges A at a rate the true B_N never pays, and an optimizer that believes the charge trades away constant it did not need to. Full cost optimizes the proof, not the certificate.

Does the coefficient pattern persist across cutoffs? No. For each of the 29 (L, M) pairs that differ at 10^4, the full-cost seed at 10^6 is already the baseline, except (2310, 7), which passes through a third seed (A = 22 against the baseline's 170/7, C larger by 6e-5) and rejoins the baseline at 10^8. At 10^8 and 10^12 all 87 pairs are the baseline. The small-N winners are lower-complexity seeds, and three are literally sub-lattice seeds: at (2310, 2) the winner is the L = 30 seed a_1 = 1, a_2 = -2; at (2310, 4) and (2310, 5) it is the pilot's L = 210, M = 6 seed. What persists as N grows is the leading-constant seed, with the full-cost objective acting only as a tie-break toward smaller A among leading-constant optima, and in this record that tie-break changes nothing at 10^8 and beyond.

Answer

• Does full-cost optimization improve the bound? At N = 10^4, yes, in 29 of 87 pairs, by up to 5.4% of U_N (0.82% at the L = 2310 winner). At 10^6, in 1 pair by 7.8e-6. At 10^8 and 10^12, never: the two objectives return the same seed.
• Does it improve the certificate? No. In every row where the bound improves, B_N gets worse. The improvement is entirely in the slack of the error term.
• Winning coefficients and both costs: the table and seeds above; all 348 rows with both seeds, both U_N, and B_N where they differ, are in full_cost_results.json.
• Pattern persistence: none from 10^4 upward; the leading-constant seed is the full-cost optimum from 10^8 on in all 87 pairs, and from 10^6 on in 86.

What this suggests testing mathematically next

1. The stabilization is a parametric-LP fact and can be made exact. U_N/c1 = kappa + t A with t = c2/c1 -> 0 as N -> inf, over a fixed polytope. The optimal vertex is piecewise constant in t with finitely many breakpoints, so for each (L, M) there is a threshold t*(L, M) > 0 below which the full-cost optimum is the minimum-A vertex among the leading-constant optima, and an N_0(L, M) beyond which nothing changes. The record shows N_0 <= 10^8 for all 87 pairs and <= 10^6 for 86. The concrete test: compute t*(L, M) exactly (rational LP, or minimum A subject to kappa = kappa* over the optimal face), derive N_0, and check that the pilot's recorded seeds are the minimum-A optima, which the 10^8 and 10^12 rows suggest but a float solver cannot establish.
2. Replace the proxy before optimizing it. The experiment shows that optimizing a valid but loose error term moves the seed in the wrong direction for B_N. The next lemma to try is a sharper per-term inequality: with Stirling's second-order term, log(floor(y)!) - (y log y - y) = (1/2) log(2 pi y) - {y} log y + O(1/y), so the a_j-weighted sum has an explicit (1/2)(log(2 pi N) sum_j a_j - sum_j a_j log j) piece (note sum_j a_j, not sum_j a_j/j, so balance does not kill it) plus signed fractional-part terms -sum_j a_j {N/j} log(N/j). Whether an error term of that shape, still a proved bound, makes the full-cost ordering agree with the B_N ordering on this record is a bounded question with the same 348 rows as its test.
3. Consequence for the pilot's section 5. Any varying-seed program that aims at B_N - N = O(N^{1/2 + eps}) has to control the true deviation, not A (K+1)(1 + log N); this record is a small, concrete instance of the proxy and the target pulling in opposite directions, at the one scale (10^4) where the proxy is not yet negligible.

Checks run

full_cost.py (348 rows, 0 rejections, 0 re-solve mismatches); tests/test_factorial_full_cost.py (headline counts, every distinct full-cost seed rechecked exactly, every row's U_N recomputed at 40 digits from the recorded rationals, large-cutoff rows equal the baseline, the headline row's numbers); 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 exact LP-optimality proof, any search beyond the pilot's sizes, any prime computation at these N.