Why the inherited repair dictionary freezes the weight at 20

Date: 2026-09-06 (local investigation date).

Status and inputs

This is a new analysis by the originating ChatGPT session, separate from the completed review on PR #200. It is not included in that review, has not been independently refereed, and makes no novelty, LP-optimality, asymptotic-rate, prime-counting-record, RH, or total-CHHL-E claim. No repository file or PR was changed during this calculation.

The reviewed input is PR #200, head 85a42e7c0c7dad8163dcab8d239bac5bb6fe1b38, in teal-sea/zeta-lab. Its joint_correction_candidate/JOINT_REVIEW.md reports that the candidate survives; its JOINT_CORRECTION.md, script, and JSON are the unchanged three members of original_archives/joint_correction_candidate.zip. That ZIP has SHA-256 828a4d85d471b51331b5f0a32c30818ccff4de95e9e4c91dbb836536376e8260.

The construction and the proposed update use M=15, R=100000, and the mask 210. original_archives/certificate_route_test.zip supplies the prior package and its fixed starting and repair seeds. Nothing in these original archives was rewritten. Extracted source inputs are under inputs/.

1. A structural statement about the full permitted LP family

Define

b_n(t) = floor(t/n) - floor(t/(n+1)) - floor(t/[n(n+1)]), h_q(t) = sum_{d|210} mu(d) b_q(t/d), lift(f)(t) = sum_{k>=0} f(t/15^k).

Write H_q=lift(h_q), B_n=lift(b_n), and let W0 be the lift of the period-30030 initial seed. The original joint search uses

W(t) = W0(t) - sum_{q=13}^{36} y_q H_q(t)

• sum_{n in S} lambda_n B_n(t)

on 1<=t<R, with all y_q and lambda_n nonnegative. The repair set S is the 411 distinct sites in the preceding baseline. Its least element is 189, and 220 is absent. The tail shield vanishes below R.

The following are exact rows of this constraint system, not values only at its chosen optimum:

These values were evaluated directly with Fraction arithmetic and checked against every one of the 24 permitted correction columns and all 411 repair columns. An additional elementary check of the last repair column assertion: for 189<=n<220, both floor(220/n) and floor(220/(n+1)) are 1 and the third floor is 0; for n>220 all are 0. Lower dilation arguments are at most 14. The only missing case that could contribute is n=220, which is not in S.

Since W(13)>=1 and y_13>=0, the first row forces y_13=0. The fourth row now requires

1 - y_16 - y_20 >= 1,

which, by nonnegativity, forces y_16=y_20=0. The third row therefore gives

W(20)=3

for every feasible choice in the original joint LP, not only the observed numerical vertex. Its excess on [20,21) is necessarily 2.

Consequently this fixed allowed family has

C-1 >= integral_20^21 2 dt/t^2 = 1/210,

whenever its global coverage and integrability are established as in the reviewed construction. This is a restricted-family statement, not a bound on all possible factorial constructions. It is not a global optimality certificate for the particular value of C in PR #200.

The obstruction is specific: the permitted correction can act at 20, but its induced deficit at 220 cannot be repaired inside the inherited repair dictionary. Merely choosing a different objective within that same family cannot remove the excess at 20.

2. A bounded diagnostic allowing new repair sites

To test this explanation, retain the reviewed joint candidate and subtract alpha*h_20, with alpha in {1/108, 1/2, 1}. Do not rerun an LP. Instead apply the existing lifted-prefix greedy repair, now allowing a repair at any newly deficient integer n<R, not only at the old 411 sites.

For each trial, form its finite balanced floor sum

D_alpha = D_joint - alpha*h_20 + sum lambda_n b_n.

For a deficient lifted weight at n, take lambda_n=1-W(n)>0, update all later lifted cells exactly, and continue in increasing n. The repair is nonnegative, zero before n, and one at n, so it cannot undo prior coverage.

The full seed is

g_alpha(t) = D_alpha(t) + (701/36 + 3*alpha) W_*(t/R).

Above R, this is the old full seed minus alpha*h_20 plus nonnegative patches and 3*alpha W_*(t/R). The old full seed is nonnegative there, h_20<=3 over its complete period, and W_*>=1. Thus the new seed remains nonnegative above R. Prefix coverage plus that tail property gives global lifted coverage by the already-reviewed descent argument.

The finite sum is balanced, the full seed is O(1+log t), and its weighted absolute integral exists. The same factorial identity and error bound therefore apply. The check explicitly verifies kappa(D_alpha)>0 before dropping geometric tails. That is the assertion omitted from the original joint proposer and supplied by its reviewer.

3. Results, keeping costs separate

Every trial passed exact balance and the full 99,999-cell lifted-prefix check, rebuilt from its combined coefficients. Rational log enclosures showed a strict decrease of C for every trial.

The independent checker, importing neither proposer nor refine.py, fully reconstructs the alpha=1 trial and establishes:

• 99,999 lifted prefix inequalities via divisor-increment tables;
• complete period checks of the initial and repair seeds;
• all 88,200 cells of the h_20 period, with supremum 3 and infimum -3;
• exact coefficient equality, balance, mass, and the tail coefficient;
• kappa(D)>0.97694 with outward interval logs;
• an interval separation showing a decrease in C greater than 0.0006439199809990653434.

This independent code path is a self-check by the same originating session, not an independent-agent proof review.

For alpha=1, with S1 and S2 defined below, the all-cutoff sufficient bound is

psi(N) <= B_new(N) <= C_new*N + (1405/2) S1(N) + (4045/12) S2(N).

S1(N) = sum_{k:15^k<=N} [1+log(N/15^k)], S2(N) = sum_{m:100000*15^m<=N} (m+1)[1+log(N/(100000*15^m))].

The coefficient costs INCREASE relative to the reviewed joint candidate. This is not another simultaneous decrease in mass and C.

High-precision evaluations, not directed interval comparisons, give:

Thus the complete conservative ceiling is worse at the two smaller cutoffs, better at the two larger cutoffs. The actual factorial value is lower in all four diagnostics. The strictly smaller C with fixed logarithmic error allowances also gives eventual improvement of the sufficient ceiling, without claiming a computed threshold or monotonicity at every finite N.

4. Interpretation and boundaries

This test identifies an inherited-support restriction and removes it in one diagnostic. It does not establish a useful uniform refinement law. The new W(20)=2 still leaves excess at 20, and other early excess remains. There is no inference that repeated removal has a controlled aggregate cost, no claim that C tends to 1, and no bound of RH strength.

The useful structural requirement for future candidate families is that a newly introduced correction must be allowed to introduce the repair sites its own constraint violations require. Reusing an old repair list can create an artificial mathematical barrier even when the new correction is explicitly present in the search.

A prospective family still needs a proof that weighted excess removed exceeds all repair and tail costs, with useful control uniform in the family parameter. This note neither assumes nor proves that statement.

5. Reproduce

From this package directory, after installing numpy and mpmath:

OPENBLAS_NUM_THREADS=1 python test_q20_direction.py OPENBLAS_NUM_THREADS=1 python check_support_direction.py

inspect_support.py prints the exact short-row constraints. The original archives and extracted inputs remain read-only inputs; these commands write only this package's generated diagnostic outputs.

The new analysis and diagnostics are saved here, not committed to Zeta Lab. PR #200 has not been merged or changed by this calculation.