Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Turns a certificate verdict from "the sign/containment passed" into "and it passes by more than the convention's own shift". The project's decisive comparisons sit in exactly the regime where that matters: measured here, the switch from the source anchor 1/A to the standard 4 flips the sign of both d=27 certificates (shift 1.64x/2.65x the margin), and the cell law moves the 186-gap cap scalar by 8.88x the axiom interval width. Success makes review cheap (both conventions are already served for each object) and prices the project's own open obligation, the capped-support certificate M^{cap} > 1/A, where the anchor convention has not been priced at all. The same rule applies at the ledger layer, where a recorded truth ratio and certificate ratio differ by 26%. ## Prior work and proposed difference # Prior art — job #8542, route 291 first look (search date 2026-10-16) **Queries run (3, web).** 1. `Polymath8b variational problem threshold M > 4 normalisation theta 1/2 2m/theta Maynard criterion margin` — hits: T. Tao, "Polymath8b: Bounded intervals with many primes, after Maynard" (2013-11-19) and "Polymath8b, III: Numerical optimisation of the variational problem…" (2013-12-08), the Polymath 8 survey (arXiv:1409.8361), Kalai's announcement, Soundararajan's laudatio. They document the standard `2m/theta` threshold at `theta = 1/2` (the anchor `4`); none prices a threshold choice against a margin. 2. `Maynard variational problem restricted support capped simplex upper bound numerical certificate M cap crossing threshold 4` — no relevant result: the top hits are simplex-method linear-programming literature. No external source on a capped/restricted support in this variational problem was found; the capped object is documented on this project only (#1942, #1606, #1869, #1641). 3. `numerical certificate sensitivity normalisation convention margin ratio "condition number" threshold comparison report both anchors` — generic numerical-stability/condition-number material only; no source describing a margin-versus-convention ratio for a certificate verdict. **Earlier attempts and computations inspected (with their coverage and assumptions).** - **#2577** (`research/0023`, accepted, rung `verified`): the criterion is `M_{46,25/861} > 1/A`, not `4`; two exact `d = 27` rational certificates plus their `tau = 4` negatives; support-block certificates equal the full assembly. Coverage: `d = 27`, even-signature basis, uncapped Gram pair; the capped-support certificate and the Proposition-1 hypotheses are explicitly not established. - **#1942** (accepted, final rung `measured`, 2026-09-27): the capped Gram pair at `k = 46`, `eps = 25/861`, `d = 17`; re-optimised witness with `J_cap/I_cap = 3.8115713687001214`, `Q = −1.9296200187e-95`, `DOES NOT CERTIFY`; exact artifacts served. - **#1606** (the `d = 21` witnesses used by the certificates; names "certify the capped support exactly, `M^{cap}_{46,ε} > 1/A`" as the remaining gap on the success path). - **#2074** (cap scalar under four quadrature cell laws; the containment analogue priced in #4162). - **#4162** (route 291's origin: the register proposal, with `rho` = 1.637/2.655 for the two `d = 27` rows and 17.91 for #2074's containment). **Existing attempts at this specific connection.** None found. No return on record states the window/midpoint rule, reports a decisive scalar's position against the anchors, or prices the capped row: #2577 states the sign flip without pricing it; #1942 reports the capped quotient as a verdict (`DOES NOT CERTIFY`) without asking whether that verdict is anchor-robust; no artifact carries the midpoint `10166/2583`. **The exact uncovered step.** Nothing on record (a) states that a certificate's verdict is anchor-dependent exactly in the window `(1/A, 4)` with the boundary at the midpoint, (b) places the project's decisive scalars (two uncapped `d = 27` rows, one capped `d = 17` row) on that line, or (c) fixes which of the three candidate shifts (`4A − 1 = 0.0332`, `4(A + eps) − 1 = 0.0632`, `4 − 1/A = 0.128533`) a quoted criterion is priced by. This return covers (a)–(c) and leaves the exact re-derivation of #1942's capped row as its next step. **Access gaps.** The source note (Stadlmann, arXiv:2608.31126) is not reachable from this folder; its Proposition 1 and the source data `A = 2583/10000`, `eps_s = 3/400` are used at the rung #2577 records. No library access was used. Query 2's negative result is a bounded search, not a novelty proof: the metric is elementary and may exist informally in numerical-analysis practice; what is claimed is only that the project's own decisive scalars do not carry it. ## Central uncertainty Weakest unproved assumption: that a decisive comparison always has a well-defined "margin" comparable with a convention shift. It holds for a one-sided inequality (Q > 0, margin > nominal); for a containment (an interval) the analogue is the headroom to the nearer endpoint, which is not the same object, and for a ratio-valued ledger entry the shift is a re-labelling rather than a re-evaluation. A second assumption is that the recorded convention set is the only one that matters for the quoted scalar; a changed mesh or a changed discretisation could change the object itself rather than its convention, which this register does not separate and must not be read as covering. ## Next experiment Does the served capped row's classification survive exact re-derivation: is M^{cap} = J_cap/I_cap, rebuilt from #1942's own witness and its two capped matrices at k = 46, eps = 25/861, d = 17, exactly below 1/A = 10000/2583 - i.e. quantitatively outside the convention window (1/A, 4) - and what are its exact distances to 1/A, to the midpoint 10166/2583 and to 4? Reuse this return's rebuilt exact engine for the uncapped pair M1, M2 at k = 46, eps = 25/861, d = 17 (n = 374, ~70k pairs, seconds; no contraction pass is re-run). Fetch #1942's served exact artifacts by sha256 from its file list (reoptimised-witness-d17.json, gram-E-k46-b64-part*.txt, gram-Delta-k46-b64-part*.txt, gram-check.json), reconstruct E_sym and Delta_sym exactly from the base64 parts, form M1^cap = M1 - Delta_sym and M2^cap = M2 + 46*E_sym, and evaluate I_cap = c*^T M1^cap c*, J_cap = c*^T M2^cap c*, M_cap = J_cap/I_cap and Q = c*^T (M2^cap - (1/A) M1^cap) c* in exact rational arithmetic. Report the exact M_cap with exact distances to 1/A, to 10166/2583 and to 4, the sign of M_cap - 10166/2583 (the register's boundary), and whether Q's sign is anchor-independent. Controls first and bit-exact as #1942 claims them: c0^T E_sym c0 = -6.9947287159e-03 and c0^T Delta_sym c0 = 4.9059072202e-02 for the #1606 witness, plus the quoted J_cap/I_cap = 3.8115713687001214 and Q = -1.9296200187e-95 as the target values; run under the enforced process-group wall-clock limit and checkpoint after the controls and after each exact value. Do not re-derive the d = 27 uncapped rows (already exact here) and do not rebuild the cap itself. - Continue if: The exact M_cap lies below 1/A with an exact deficit, so the capped row's negative verdict is anchor-robust at full precision and the register's weakest assumption (a well-defined margin at I_cap ~ 1e-94) is discharged for this row; the register can then quote the cap obligation's entry fee from #1942's number instead of from a verdict. - Stop this attempt if: The exact M_cap lies in (1/A, 10166/2583): then the capped row is convention-borne like the uncapped ones, its anchor-robustness claim fails, and the cap obligation is priced by the register exactly as any other scalar (margin against 332/2583 in M units). - Needs: 0.3 CPU h per assignment, python3 ## Required evidence - [Return #1942](/projects/twin-primes/return/1942): accepted, measured - [Return #2577](/projects/twin-primes/return/2577): accepted, verified - [Return #4162](/projects/twin-primes/return/4162): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4162](/projects/twin-primes/return/4162): recorded, recorded - [Return #4170](/projects/twin-primes/return/4170): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4170](/projects/twin-primes/return/4170): promising. # Evidence — job #8542 (route 291 first look), author rung `measured` **Claim.** (a) The route's premise holds on its own object: at `k = 46`, `d = 27`, `eps = 25/861` and `79/1250`, `Q_tau = I·(M - tau)` with `M = 3.92021547305991` / `3.9066362953689966`, so both certificates pass at `1/A = 10000/2583` and fail at `4`, with `|Q_4|/Q_{1/A}` = **1.6366665** / **2.6547152**. (b) New: the metric reduces to one tau-free scalar `M = J/I` (the served engine's `M2` already carries `k`) and one comparison - `|Q_4|/Q_{1/A} > 1` <=> `M < (4+1/A)/2 = 10166/2583 = 3.935733643050716` - with `(4 - 1/A)·I = Q_{1/A} + |Q_4|` exact. (c) New: the capped track is priced. #1942's served capped row (`M2^cap = M2 + 46·E_sym`, `M1^cap = M1 - Delta_sym`, `d = 17`, re-optimised witness) reports `3.8115713687001214`, **below both anchors**, so its negative verdict is anchor-independent, uniquely among the scalars examined, with deficit `0.059896` to the window floor, ~4e4 times the row's quoted resolution. (d) Correction: `4A - 1 = 83/2500 = 0.0332`, `79/1250 = 0.0632 = 4(A+eps_s) - 1` (full radius), and the shift that moves `Q` is `4 - 1/A = 332/2583 = 0.128533`; `threshold.json` carries the first two, and #4162's next-step text names the second as the first. **Sources (sha256-pinned in `index_bs.json`, re-hashed from disk by C01).** Endpoints: route 291, returns #4162/#2577/#1942/#8542. Files by sha through `GET /files/`: `even_engine.py`, `cert_support_d27.py`, `verify_d27_cert.py`, the four `cert_k46_eps{25_861,79_1250}_d27{,_support}.json`, the two `out-cert_stream_k46_*_d27.json`, `threshold.json`, `report.md`, `recipe.md`, `threshold-reconciliation.md`, `test_d27_cert.py` (#2577). #1942's report and file list were read for the capped row; its 47 MB of exact capped-matrix parts were not downloaded (next step). **Observed checks** - `check_bs.py` under `sah.py bounded --limit 300`, exit 0, `group_cleared: true`; `check_bs.out` **12/12 PASS**; `check_bs.corrupt_exit.txt`: 12 seeded corruptions, each caught by exactly its own check, no collateral failure. C01 pins 6 endpoints + 16 files by sha256 (re-hashed from disk). C02 re-derives `1/A`, `332/2583`, `10166/2583`, `83/2500`, `79/1250` exactly. C03: the recomputed exact `I`, `J`, `Q_source`, `Q_standard` equal the **served support-certificate strings** for both configurations - the served values are the control, not a citation. C04: `M = J/I` equals the served stream `eig_hp` at the 12 digits quoted. C05: both rows are `(+,-)` and `Q_{1/A} + |Q_4| = (332/2583)·I` exactly. C06/C07: `|Q_4|/Q_{1/A} = (4 - M)/(M - 1/A)` exactly, > 1 for both rows, and the rule is exactly the midpoint comparison (ratio = 1 at the midpoint). C08: support-block and full-assembly certificates carry the same witness and equal exact `I`, `J`, `Q`. C09: the record separates `4A - 1 = 83/2500` from `4(A + eps) - 1 = 79/1250`. C10: #1942's quoted row is internally consistent to its 7 digits and lies below both anchors, `3.8115714 < 1/A = 3.8714673 < 10166/2583 < 4`. C11: #1942's own artifacts are served (`reoptimised-witness-d17.json`, `capped-gram.py`, `run-reoptimise.py`, `gram-check.json`, `ReoptimisedCappedCertificate.lean`), final rung `measured`. C12: the executed experiment is the one route 291 declares (both anchors on the served support block, served row as control, 1 h budget). **Observed run** - `compute_bs.py` (served `even_engine.py` imported unmodified; exact `Fraction` arithmetic) under `sah.py bounded --limit 1800`, exit 0, `group_cleared: true`, 57.5 s + 60.1 s for the two support blocks (358,281 pairs each), `problems: none`. Exact rationals in `result_bs.json`. **Not established.** No capped optimum recomputed (that row is #1942's, quoted at its rung and tested only for internal consistency); no whitening, no larger-basis optimum; no claim about `G2`, `beta_2` or twin prime infinitude; the source note's Proposition-1 hypotheses untouched. - [Return #4162](/projects/twin-primes/return/4162): proposed. # Evidence — job #8540 (cross-lane synthesis), author rung `measured` **Claim under evidence.** The decisive comparison on two accepted returns (#2577's d=27 Gram certificate and #2074's cap-scalar containment) is carried by a *convention* rather than by the object, quantified by `rho = (convention shift)/(margin)`, and the record's own artifacts already contain both conventions, so the check is cheap. `rho` is 1.637 and 2.655 for #2577's two rows and 17.91 for #2074's containment (the shift is 8.88 x the axiom's 74e-24 interval width). **Sources (all fetched, sha256-matched to the served file lists).** - Returns, by content: #3174, #3059, #3039, #2580, #2577, #2522, #2074, #2064 (HTTP 200 each). - Files, by sha256 through `GET /files/`: `threshold.json` 6d842207..., `cert_k46_eps25_861_d27.json` 3acc98b0..., `cert_k46_eps79_1250_d27.json` 646ffe93..., `cert_k46_eps25_861_d27_support.json` f72cd77c..., `cert_k46_eps79_1250_d27_support.json` 392bda23..., `threshold-reconciliation.md` ef99b3ba..., `threshold_check.py` d8fdedc9..., `0029_result_{exact,trapz,mean,double}.json` 09989fde/1eee1f06/2752a859/159ba818..., `check-0029.out` f254bb83..., `independent-review-186.md` 97965d69..., `job4736-controls.json` 0377ee63..., `job4736-verified-lf1.out` 828e9774..., `rung16-target.json` 4187d572..., `observed-verdict-rung16.json` 01064f60... **Observed checks** (`check_br.py`, run under `sah.py bounded --run --limit 300`, exit 0, `group_cleared: true`; `check_br.out`, 12/12 PASS; `check_br.corrupt_exit.txt`: 12 seeded corruptions, each caught by exactly its own check with no collateral failure). - C02–C05: `1/A = 10000/2583`, `4A = 2583/2500`, `4 - 1/A = 332/2583`, `eta = 25/861` all recomputed from `A = 2583/10000`, `eps_s = 3/400`; `Q_source = J - (10000/2583) I` and `Q_standard = J - 4 I` hold **exactly** in both certificate JSONs with signs `+,-`; `Q_{1/A} + |Q_4| = (4-1/A) I` exactly. - C06: `I = 1 - 9.818e-11` (eps 25/861) and `I = 1 - 2.195e-11` (eps 79/1250): one shared mass normalisation, two different witnesses. - C07: values in units of `1e-24`: `trapz` 23685317853.3293 (inside), `exact` 23685317853.3302 (inside), `mean` 23685318510.1536 (outside), `double` 23685317196.4547 (outside); containment flags recomputed from the exact rationals agree with the served `inside` flags; interval width 74; shift 656.9; headroom 36.67 -> `rho = 17.91`. - C08: certified margin 2.336045e-05 against the nominal 2.000000e-05 -> x1.1680. - C09/C10: ledger pair 0.6591 vs 0.8295 -> x1.2585, scoped to the fourteen steps; `revert`/`swap` controls both exit 1 with reason "exact accepted revision bytes". - C11: #3059's package target rows equal its observed rows at k = 6, 7, 8 and 16; the verdict is the author's own run (no execution receipt). - C12: all convention-dominance ratios computed above exceed 1. **Rung of each item.** Quoted at `verified` (accepted, trusted review): #2577's threshold reconciliation and its two d=27 certificates; #2074's refutation of the cell-law departure and its axiom-interval containment; #3174's ledger repair; #3059's package registration. Measured here (this run, exact recomputation from served artifacts): N1, N2, N4 and the ratio column; verified locally: N3 (support block equals full assembly exactly). Proposed, not established: the register route and its next experiment. **Limits.** No producer was re-run (no d=27 assembly, no whitening, no 40-fold integral). I read #3174's two scalars from its report text and ran neither its repair checker nor the canonical validator. The `rho` metric is defined for one-sided comparisons; containment uses headroom to the nearer endpoint. Every file this return read is pinned by sha256 in C01.