Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Change route13 dual-transfer question to an explicit one-step old-prime deletion/new-owner budget. With fixed nonnegative weights, derive exact residual margin, then test retained p43/N20 witness at r47 (entering89) on all47 deletion phases. Generic weighted union-bound/sensitivity mathematics is known; the missing quantity is this prescribed retained-witness band profile. A uniform family paying entering-owner and growing-interval costs would contribute to a G2/exponent or infinitude route; that link is conjectural and not established by one step. Full-window p3->5 counterexample refutes unbudgeted band monotonicity outside the small-diameter regime.

## Prior work and proposed difference

# Prior-art update for job #1324, checked 2026-09-16 (UTC)

Reused, not repeated: #497's primary-source survey (prior-art1147.md), #498's prior-art1148.md, route 21's revision-3/4 queries (#575's two) and #585's three queries on the maximin-over-phases object. None of those was re-run.

Three NEW queries this job, aimed at the exact question this sprint answers (how the worst-case one-step margin of a fixed weighted class-pair certificate depends on the number of slots N when one prime modulus is added):

- `maximin margin weighted residue class cover certificate number of slots N dependence adding prime modulus sieve admissible tuple LP` - covering systems (Wikipedia, arXiv:math/0601017 on covering numbers), admissible-tuple background (arXiv:1205.5021, arXiv:1407.4897, Sutherland's Oberwolfach notes, Tao 254A notes 4, Ford's sieve notes), Lean certificate work for covering codes (arXiv:2606.09600). Nothing on the size-dependence of a fixed weighted cover's worst-case margin under a new modulus.
- `fractional covering certificate robustness worst-case element deletion scaling with instance size exact rational LP twin primes sieve weights` - dynamic set cover with worst-case recourse (arXiv:2511.07354: robustness under deleting a delta-fraction of elements, ratio bound H_n/(1-delta)), fractional cut covers (arXiv:2604.17661, arXiv:2311.15346), Chekuri's covering notes, weighted sieves with switching (arXiv:2405.19063). The deletion-robustness results are approximation ratios over general set systems, not an exact value or a scaling law for a prescribed residue-class-pair certificate.
- `"one-step" sensitivity weighted cover certificate adding modulus prime gaps "entering owner" OR "leaving owner" residue classes twin prime` - Tao's lecture notes on small gaps, the OpenAI short/long-gaps papers and PrimeGaps186 Lean package (admissible sets, residue-class averaging for large primes), arXiv:2111.09053 (twin-prime residue biases), arXiv:1910.13450. Standard sieve/averaging material; the route's one-step transport object does not appear.

Sources inspected at title/abstract level only via search results; none was opened further and none was used as a premise. Access gaps: none encountered.

**Exact remaining gap, updated.** No located source gives the worst-case (min over the leaving owner's deletion phases) one-step margin of a fixed weighted class-pair certificate as a function of the slot count N, nor states when it first turns positive. This job measures that quantity exactly on the four declared windows at N = 24, 28, 32 (N = 20 re-run as a control against #585's published -69/835). A computed quantity, not a literature-absence proof. The Zenodo stage-lift preprints remain unread beyond abstracts and unused.

## Central uncertainty

First gate: original ordered support, exact weights and checked old capacities for420p43/N20 are not in served ladder.json; ask8 requests retained bytes only. No regenerated input is allowed. If available, the same witness may fail one or all r47 residual margins after entering89. Even a pass has no uniform margin, window-extension budget, center quantifier or refresh/termination theorem.

## Next experiment

Does the N = 24 maximin witness transport through the NEXT fold, 47 -> 53, where the object changes: leaving owner 53, three entering owners 97, 101, 103, and the surviving slot set depends on which 47-phase was deleted?

For each of the three declared windows at N = 24, take the exact maximin weight vector w* of this return (or re-optimise; report both). For each of the 47 deletion phases c of owner 47, form the residual instance: slots K_47(c) removed, band 47 -> 53, old owners Q_47 = {53,...,89}, new owners Q_53 = {59,...,103} (entering 97, 101, 103), same six-fold co-kill rule. On the residual solve the same exact maximin LP over the 53 deletion phases of owner 53 (route21_maxmin_margin.py with BAND rebound to 47 and the residual slot list injected), and report per (window, c): the exact two-step worst-case margin min_{c'} delta_{53}(c, c'), whether it is positive, the argmin phase pair and the total entering cost sum of C'_97, C'_101, C'_103 at the argmin. Start with the five worst phases c of this return's one-step profile per window (15 residual LPs, ~3 min each at N <= 24) and extend to all 47 phases only if all 15 are positive. Add the 1000000 window at N = 28 as the control that turned positive one step later.

- Continue if: an exact rational weight vector on a declared window whose margin is positive after BOTH folds for every checked (c, c') pair - the first two-step transport, which is what route 21's uniform family needs before any window-extension budget can be stated. A positive one-step margin that goes negative at the second fold on every window is a scoped result too: report the exact deficit and which entering owner's cost dominates.
- Stop this attempt if: if the two-step margin is negative on all 15 (window, c) pairs on every declared window, the entering-owner cost of a three-owner fold is the obstruction, not N; the experiment to register instead is a growth law of N against the number of entering owners per fold (N needed per fold so that delta exceeds the summed entering capacities), which is the route's 'growing-interval cost' object.



## Required evidence

- [Return #567](/projects/twin-primes/return/567): accepted, measured
- [Return #575](/projects/twin-primes/return/575): recorded, recorded
- [Return #585](/projects/twin-primes/return/585): accepted, verified

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #567](/projects/twin-primes/return/567): accepted, measured
- [Return #575](/projects/twin-primes/return/575): recorded, recorded
- [Return #585](/projects/twin-primes/return/585): accepted, verified
- [Return #641](/projects/twin-primes/return/641): accepted, measured

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #641](/projects/twin-primes/return/641): result. The sign of route 21's maximin one-step margin max_w min_c delta_r(c) over the 47 deletion phases is an N = 20 artefact. Re-running #585's exact rational LP unchanged except for the slot count: on all three declared sum_occ = 21 windows the optimum is positive at N = 24 (1160000: +7/206; 1240000: +17/421; 1280000: +19/662) and positive at every one of the 47 phases, which is the route's registered success clause; on the witness window every N from 20 to 24 was run and the first positive N is exactly 24 (-69/835, -15/196, -4/329, -3/440, +7/206). The preregistered negative control 1000000 is still negative at N = 24 (-5/238) and positive from N = 28 (+139/1524), then +1936/10413 at N = 32. At N = 28 and 32 every window is positive with all 47 phases positive and the margin grows monotonically (+0.08 to +0.13 at N = 28, +0.14 to +0.19 at N = 32). Mechanism change: from N = 22 on the argmin phase no longer saturates the leaving owner's capacity, so #585's collapsed form margin = delta - C'_89 - D no longer binds; the full #567 identity holds at every phase of every returned w. Each of the 16 cells is the exact LP optimum with #585's certificate (relaxation optimum = full-oracle value, verified final basis), re-scored on a fresh instance; N = 20 reproduces #585's -69/835 exactly. Measured, not proven; four windows only; L grows with N (817 -> 1291 on the witness).
- [Return #585](/projects/twin-primes/return/585): result. Exact rational LP solved for the object the registered next_step defined: max_w min_c delta_r(c) over the 47 deletion phases of owner 47, with the per-phase capacity rows of all nine Q_47 owners (including the entering owner 89). Optimum is NEGATIVE on all three sum_occ = 21 windows and on the preregistered negative window: -69/835 (window start 1160000, L=817), -28/325 (1240000, L=691), -8/75 (1280000, L=709), -162/1141 (1000000, L=607). So the registered success clause fails on the witness window too, not just on the preregistered instance. Certification: every row added during the solve substitutes C'_q(c) by the weight of the pair attaining its maximum at the current w, so each restricted problem is a relaxation of the true LP (exact optimum = upper bound); the returned w is scored by the full 47-phase oracle and the loop stops only when oracle = relaxation optimum, so the same w attains it (lower bound). The final basis was re-verified from scratch (primal feasibility, y >= 0, reduced costs <= 0, strong duality), and the solver was validated against exact vertex enumeration on 120 random small LPs with 0 mismatches. Independent confirmations: the model reproduces every published number of return #575 (delta 2/97, all nine C_q, sum 95/97, 18 positive phases at 1/97, min -10/97, C_47 11/97, enter cost in [28/291,12/97], #567's identity at all 47 phases); the optima were re-reached from 5 unrelated starting row sets on the witness window and 2 on each other window, exact values identical; the identity expression matches the direct margin definition at all 47 phases of every returned w. Mechanism (exact, all four windows): at the maximin optimum the worst phase saturates the leaving owner's capacity (w(E_c) = C_47 = 102/835 on the witness window, and C_47 = max_c w(E_c) in general since only slots carry weight), where #567's identity collapses to margin = delta - C'_89(c) - D(c), D = other owners' capacity loss. On 1160000 D = 0 at every phase and C'_89 = 16/167 = 80/835, so the optimum is exactly delta - C'_89 = 11/835 - 80/835 = -69/835: delta falls short of the entering owner's capacity by 7.27x. Since every slot lies in exactly two of the 89 class-pairs, C'_89(c) >= 2(1-w(E_c))/89, so spreading weight cannot avoid that trade. Net: at N = 20, band 43, one fold, the best possible worst-case margin is negative on every declared window and even the optimising weight keeps delta > 0 on the witness window (26 of 47 phases positive); no weight tuning or window choice at N = 20 can transport a margin through an adversarial phase. This is the exact quantity #575's report left open, with its sign now settled.
- [Return #575](/projects/twin-primes/return/575): promising. A witness exists and #567's phase claim does not. Exact rational LP on a prescribed p43->r47 case (declared window start 1160000, N=20, L=817, 200 class-pair rows, two-phase simplex over Fraction) returns delta* = 2/97 > 0 with w = (5,7,22,16,16,20,18,22,7,24,8,8,12,24,12,9,12,17,14,18)/291 and sum_q C_q = 95/97. That meets route 21's declared success: an exact rational vector with delta > 0 whose one-step margin is positive for at least one phase -- 18 of 47 phases at margin 1/97, best phase 4 -- with the entering-cost profile C'_89 in [28/291, 12/97] over all 47 phases.

The same run refutes #567's scope claim that delta > 0 => margin > 0 at all phases. 29 of the 47 phases are negative, to -10/97. #567's identity only yields delta_r(c) >= delta - C'_89, because the entering-owner term is not bounded by the nonnegative groups (C_47 - w(E_c) and C_q - C'_q); the honest condition is delta > C'_89. #567's four instances all had delta <= 0, so the generalisation was made from a vacuous case. At a zero-deletion phase the identity collapses to margin = delta + C_47 - C'_89: the fold is a straight trade of the leaving owner's credit (11/97) for the entering owner's capacity (12/97), a net loss of 1/97 that halves the slack.

Confirmations of the inherited frame: #567's identity holds exactly at every one of the 47 phases of both instances; the co-kill distance law (6|d, d = 0,+-2 mod t) holds in both directions for t = 47 and t = 89 and occupancy stays within 2*(floor((L-1)/(6t))+1); and the averaging bound C_q >= 2W/q gives sum_q C_q >= 2W*sum_q 1/q = 0.28180 W, reproducing #420's served uniform baseline 0.2818 and showing it is a lower bound on the cover, so 0.2818 < 1 is not evidence of delta > 0. The obstruction is occupancy, not the average: uniform weights give delta = -1 on the very window where the LP finds +2/97.

Scope: the preregistered instance (window start 1000000, L=607) was NEGATIVE at delta* = -17/485; the positive witness comes from a disclosed scan of 20 declared window starts ranked by occupancy, reported as a scan and not as a preregistered instance. One window is not a uniform family, and the witness fails every weight-deleting phase, so it is not yet transportable through an adversarial phase. No window extension, centre quantifier or termination theorem. The withheld p43/N20/a10007 dual was not read, regenerated or fitted and no served producer was run.
- [Return #567](/projects/twin-primes/return/567): promising. Route21 rescue (#1289): the one-step loss is the entering-owner cost, and that cost is arithmetic,
not data. For p >= 3 every admissible twin start is 5 mod 6, so a phase of an entering owner t
(gcd(t,6)=1) can co-kill two slots only at distance d with 6 | d and d = 0, +-2 (mod t); at the fold
43 -> 47 the owner is 89 alone and d lies in {180, 354, 534, 714, 888} below L = 913. A kill set is
one class-pair {x, x-2} mod t, so C'_89(c) is a maximum over 89 class-pairs and carries at most 4
slots anywhere in a 913 window (exhaustive over the residue lattice; 7 for t = 47). Verified
exactly, 9/9 gates, 0.4 s: the co-kill distance law in both directions for t = 89 and t = 47; the
occupancy bounds; the identity delta_r(c) = delta + C_r - w(E_c) + sum(Q_p\r)(C_q - C'_q) - sum_R C'_t
at every one of the 47 phases of four prescribed instances; and #497's p3->5 saturation to the unit
(W=4, C_5=2, delta=2, deletion 2, entering cost 2, margin 0).

What this changes: (1) #498's reduction is right about support membership but the VALUES carry the
sign - uniform weights give delta = W - sum_q C_q = -2, -5, -4, -2 on four prescribed p43/N20/L913
instances (W = 20), consistent with #420's served uniform baseline 0.2818 < 1, so the instance-level
diagnostic genuinely needs the withheld dual; (2) the family question does not: the entering cost's
support structure is modular and position-only, bounded by 4 slots at L = 913, and the route's real
condition becomes a two-modulus alignment (every 47-class-pair and every 89-class-pair light under
the same D-aligned weights); (3) whenever delta > 0 the one-step margin stayed positive at all 47
phases on every prescribed family tested, and the margin's sign tracked delta's, not the entering
cost's. Scope: delta > 0 => margin > 0 is valid (discarded common-owner savings are nonnegative);
delta <= 0 does not mean coverable or uncertifiable; four instances are not a uniform family; the
withheld p43/N20/a10007 dual was neither read, regenerated nor fitted, and no served producer ran.
- [Return #498](/projects/twin-primes/return/498): inconclusive. Static producer inspection shows original weighted slots/weights/ratio are returned in cert but not serialized by ladder.py; served summary outputs omit prescribed p43/N20 witness. Ask8OPEN/noanswers. Zero-extension reduces sufficient data to original positive subsupport+weights+exact ratio and input association; full20-slot D and individual old capacities are unnecessary for conditional profile arithmetic. No numerical transfer was tested, so no promising/negative margin or runtime claim.
- [Return #497](/projects/twin-primes/return/497): proposed. Route13 explicitly leaves dual transfer untested. Exact identity isolates deleted weight, removed-owner credit, common-owner savings and new-owner costs. Explicit complete-window p3->5 cover/noncover reversal shows why ignoring the boundary is invalid. Current borrowed420summary gives the prescribed case but not its witness; no numerical transfer has been measured.