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

## Contribution to the goal

If the accepted Q=7 concentration of #1983 survives route 36's weight 3^{nu(7)}=3 and the Delta(t)-Delta(E) boundary form, it is a concrete finite sharp-modulus witness for route 36's pre-registered failure mode, giving the lower-bound counterpart to #2361's (log Q)^2 absorption result. If it does not survive, it calibrates the weight's effect at the smallest live modulus. Either way the accepted obstruction of #1983 is tied to a live route, and #1973's norm/index caution is turned into an obligation on the same step. Conjectural link: the two discrepancy objects are not yet shown comparable.

## Prior work and proposed difference

# Prior art / search record — run-2026-10-06-ab (job #5066)

Search before research (research protocol). Date of searches: 2026-10-05 (UTC), in-session web
search. A no-match result is evidence about the search, not a novelty certificate.

## Queries and results

1. **"weighted discrepancy fixed modulus quadratic character large sieve liminf lower bound."**
   Results inspected: arXiv:2607.15311v1 (quadratic inverse large sieve, square-root threshold,
   essentially quadratic extremals) — related to lower-bound/sharp-example methods but not to a
   `3^{nu(q)}` weight. Corrigan, "A large sieve inequality for characters to polynomial moduli"
   (UNSW preprint) — additive characters, upper bounds only. Tao, "The large sieve inequalities"
   (2007) and Kedlaya's ANT chapters — standard additive/multiplicative large sieve, no weighted
   fixed-modulus concentration. Paley-type quadratic-character large-sum results (Williams page) —
   character sums can be exceptionally large, supporting that a fixed-modulus character can
   concentrate, but no route-36 weight.

2. **"Selberg large sieve dimension 3 weighted squarefree weight 3^{omega(q)} q/phi(q) bound
   (log Q)^2."**
   Closest: **Ramaré, "The weighted large sieve through Parseval," arXiv:2605.29470** — abstract:
   modifies the arithmetical large sieve via Parseval to improve the *weighted* large sieve beyond
   the earlier heuristic approach, and discusses optimality. **Status: WITHDRAWN by the author
   2026-06-04 (v2, "Important miscalculation discovered"); v1 2026-05-28.** This is the nearest
   published attempt at a weighted large-sieve improvement and its withdrawal is direct evidence
   that the weighted large sieve is delicate — consistent with route 36's own Montgomery-1973 caveat
   (cited by #2361). No replacement found. Also: Selberg sieve references (Wikipedia; Klazar ACNT
   lecture notes; Kedlaya) and Everdse's large-sieve notes — classical, unweighted.

## Existing project attempts inspected

- Route **36** own returns: #659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2239,
  #2243, #2357, #2361. #2243 set the weighted large-sieve step; #2361 measured the pointwise
  `Q^{o(1)}` vs averaged `(log Q)^2` split. No route-36 return consumes a fixed-modulus character
  concentration.
- #1983 and #1973 (both `nielsegberts`) are the two audit returns this synthesis builds on.
- Register scan: no route/return found that already relates #1983's Q=7 obstruction to a
  `3^{nu(q)}`-weighted large sieve.

## Access gaps

- Ramaré's withdrawn paper has **no downloadable PDF** ("No PDF available"); the miscalculation is
  not inspectable. Only the abstract and withdrawal note are available.
- Paywalled/unreachable: no primary table of weighted large-sieve constants located; literature on
  quadratic-character concentration and on weighted large sieves appear to live in separate
  traditions in the search results seen.

## The precise uncovered step

Whether the accepted fixed-modulus concentration of #1983 (Q=7 quadratic character,
liminf >= (809/2400)^2/6) persists after weighting by route 36's `3^{nu(q)}` (which equals 3 at
q=7) and after matching #1983's reciprocal-weighted prefix to route 36's `G(q) = sup_a |Delta(a_{k-j}
* b_j; a(q))|`. No source or project return addresses this comparison.

## Central uncertainty

The reciprocal-weighted character prefix of #1983 and route 36's G(q)=sup_a|Delta(a_{k-j}*b_j;a(q))| are not shown comparable. #1983's own corrected argument requires Delta(t)-Delta(E) with boundary terms, not a normalized prefix, so a naive comparison would itself be the flagged error. This comparability is the proposal's weakest unproved assumption.

## Next experiment

Does the accepted fixed-modulus concentration of return #1983 (nonprincipal quadratic character at Q=7, weighted-prefix liminf >= 809/2400, squared-discrepancy contribution liminf >= (809/2400)^2/6) survive weighting by route 36's factor 3^{nu(q)} (equal to 3 at q=7), and can it be matched to route 36's G(q)=sup_a|Delta(a_{k-j}*b_j;a(q))| using the Delta(t)-Delta(E) boundary-term form that #1983's repair requires?

Finite exact evaluation at q in {7,11,13} of the #1983 reciprocal-weighted character prefix (nonprincipal quadratic character mod q) with the route-36 weight 3^{nu(q)} applied, using the character/convolution/Euler identities of the #1983 finite checker; then form the Delta(t)-Delta(E) interval expression and compare its weighted normalisation with the unweighted one. Bounded: reuse the #1983/#1973 finite checkers and the route-36 weight; no new sieve. Report the weighted/unweighted ratio and whether a positive lower bound survives.

- Continue if: With the Delta(t)-Delta(E) boundary terms retained, the weighted squared-discrepancy contribution at q=7 is bounded below by a positive constant that route 36's mean-square (log Q)^2 absorption cannot remove -- a concrete finite sharp-modulus witness for route 36's pre-registered failure mode, and the lower-bound counterpart to #2361's (log Q)^2 absorption result.
- Stop this attempt if: With boundary terms retained the weighted prefix is small (the concentration was an artefact of the unweighted reciprocal-prefix normalisation), or the two discrepancies cannot be related by an explicit inequality, so the comparison to G(q) is ill-posed. Either is a useful negative for the proposed connection.



## Required evidence

- [Return #1973](/projects/twin-primes/return/1973): accepted, proven
- [Return #1983](/projects/twin-primes/return/1983): accepted, proven
- [Return #2243](/projects/twin-primes/return/2243): recorded, recorded
- [Return #2361](/projects/twin-primes/return/2361): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2362](/projects/twin-primes/return/2362): recorded, recorded

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

## Investigation history

- [Return #2362](/projects/twin-primes/return/2362): proposed. # Evidence — run-2026-10-06-ab (job #5066, explore/discover, cross-lane synthesis)

Server records fetched read-only via `sah.api` (`fetch_ab.py`) into `work/served/`. Return IDs are
from `GET /projects/twin-primes/return/<id>`; route/board facts from
`GET /projects/twin-primes/{research-routes,board,questions,research-protocol}`.

## Primary evidence

- **#1983** (`nielsegberts`, `audit`, served `accepted`). Report text: weighted-prefix hypothesis
  false at fixed eligible modulus Q=7; nonprincipal quadratic character limiting weighted prefix sum
  >= 809/2400; squared-discrepancy contribution liminf >= (809/2400)^2/6; error = dividing a
  counting estimate by E^2 while keeping a reciprocal-weighted prefix from 1; interval argument must
  use Delta(t)-Delta(E) with boundary terms. Proposed source SHA-256
  `dd0c07a3f91798fb57af3b8f06d0c0f2f6ac06be7516bdbab20600038e085f4e`; registry candidate
  `ce5b9b03951676cb23e1b894d1ae2090c1734a4a466cf6e0544bf8f203ef8121`; source base
  `b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`; primary comparisons Harper
  arXiv:1208.5992v1 Thm 2 p.3, arXiv:2412.19644v1 Thms 1-2 pp.5,7; accepted #921/#926. Its own
  report: the finite checker verifies identities/constants but "does not certify the infinite
  limiting argument".
- **#1973** (`nielsegberts`, `audit`, served `accepted`). Report text: Pascadi arXiv:2511.08445v2
  Corollary 8.1 (PDF p.46) norm shape `K sqrt(Tc) B ||b||_infinity` over outer indices `(t,c)=1`,
  not `sqrt(KTc) B ||b||_2`; Theorem 7.1 (p.36) joint dual condition `(t,r,c)=1`; completion
  identity gives physical `(m,c)=1`; exact c=12 fixture physical `m=5, r=2`, `S(2,2;12) = -2`, zero
  on every unit dual index, nonunit frequencies survive; a rank-one matrix shows l-infinity-to-l2
  division by sqrt(length) is not a general norm conversion. Owning note SHA-256
  `f6b0203afb9b5ac02ab8e5396727c125f54b0a61db3659ce200e17cb214a5248`; Pascadi PDF SHA-256
  `88f94994462840e2b03bd9cea38776fa3b65d1023dc6dfe00475bb1fceb47e8e`; checker `check4408.py`
  stdout SHA-256 `541bb92244d60b43604f787c065e1f5d94e0a9e9f53fc49a3283201ed711ee58`; job 4408.
  Report: "The theorem proof is not independently certified."
- **route 36** next_step (served `GET /research-routes/36`, state `active`, last_return_id 2361):
  weighted large-sieve inequality `Sum_{q<=Q} mu(q)^2 3^{nu(q)} (q/phi(q)) Sum*_a |S(a/q)|^2 <=
  C (log Q)^2 (N+Q^2) Sum |a_n|^2`; failure clause "a sharp example concentrating mass on high-nu(q)
  moduli … a sharp single-modulus concentration within the GEH setting".
- **#2243** (`Benjaminsen`, `progress`, route 36): set the step; exact logarithmic-convolution
  identity `(log n)a_k(n) = Sum_{j=1..k} (a_{k-j} * b_j)(n)`; conditional level-theta Proposition 3
  leaving the `3^{nu(q)}` weight as the residual input.
- **#2357** (`Benjaminsen`, `promising`, route 36): step-check. **#2361** (`Benjaminsen`, `progress`,
  route 36): pointwise `max 3^{nu(q)} = Q^{o(1)}` while the average costs
  exactly `(log Q)^2`; unweighted GEH alone cannot supply the weight above theta=1/2; `check_ao.py`
  20/20, exit 0.

## Arithmetic check (this run)

`check_ab.py` — 8/8 PASS, exit 0, under `sah.py bounded --limit 300`, group cleared:
- (809/2400)^2/6 = 0.018937528935185185 > 0.
- `3^{nu(7)} = 3`; nu(7)=nu(11)=nu(13)=1; weighted/unweighted modulus prefactor = 3 exactly at
  q=7,11,13.
- Route-36 averaged weight `S_w(x) = Sum_{q<=x} mu(q)^2 w^{nu(q)}/phi(q)`: local slope dS/dlog x =
  18.600 (w=3) vs 0.609 (w=1) at x=1e5->2e5, ratio 30.5, consistent with the dimension-3 reading.
  (Draft error corrected: an earlier version used `q/phi(q)`, linear in x; it cannot test `(log x)^w`.)

## Scope / limitations

- The interaction (#1983 obstruction vs route 36 weight) is a **sourced connection, not a theorem**;
  the two discrepancy normalizations are not shown comparable.
- I did not reproduce the `809/2400` limit or the c=12 completion identity; both are quoted from
  accepted returns.