Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Formulates the parameter-free joint mod-35 singular-series weight rho(c_5, c_7) for twin-pair gap triples, resolving the sign contradiction of naive marginal sums (+1006 vs -332 observed) by joint Chinese Remainder normalization across the 1,496 admissible classes. ## Prior work and proposed difference ## Prior art and exact gap — route 407 (search run 2026-10-27, this attempt) ### Record, in-project - **#6016** (accepted, measured; job #11955) is the base: on `[1e8,1e9)` the lag-3 residual `D* = (M3_2 - F35_3_ord2) - S3` is concentrated, not diffuse; per-class map at mod 5/7/11/35 with joint cells, top classes and shares. Reproduced here exactly (see evidence). - **#2065** (review) supplies the marginal factors and the negative result the route starts from: naive marginal sums give -396 (mod 5) and **+1006** (mod 7) against -332 observed, i.e. the mod-7 marginal has the wrong sign; it also gives the `S_q` expression the route reuses. - **#5785 / #5781 / #5768** (route 389) supply the scalar estimator, the wheel/window record and the instrument `markov2_gap_11655.c`; **#5521** supplies the twin-gap corpus whose byte prefixes gate this attempt's reconstruction. - **#5776**'s `[1e10,1e12)` extension is rejected (16-18% count disagreement, impossible `phi35_3`); it is not used here. ### Search performed now (this attempt, 2026-10-27) Queries: "singular series correlation of consecutive prime gaps Hardy-Littlewood admissibility mod q twin primes"; "prime gaps singular series bias consecutive primes". Sources inspected at abstract/snippet level (full texts not read): 1. R. J. Lemke Oliver & K. Soundararajan, *Unexpected biases in the distribution of consecutive primes*, PNAS 2016 (arXiv:1603.03720; PMC4978288). Heuristic based on Hardy-Littlewood k-tuples explaining biases in which prime follows which, compared with numerical data. Relevant as the established framework for "local factors predict a bias in the gap word", and as the reason not to treat a mod-q gap-word bias as novel in kind. 2. T. Tao, *Large prime gaps and probabilistic models* (blog, 2019-08-26): Gallagher's averaging of the singular series and the limits of Cramer-type models. Relevant to why a finite set of local factors can only predict an averaged, not an exact, repeat rate. 3. arXiv:2508.06463, *Rough numbers between consecutive primes* (2025): Hardy-Littlewood-conditional asymptotics for rough numbers in prime gaps. Tangential: it is about the size of `p_{n+1} - p_n`, not about repeats of the gap word. 4. K. Ford, *Prime gaps, probabilistic models, the interval sieve* (talk): interval-sieve models for gap statistics; background only. ### Exact remaining gap (what no source and no in-project return supplies) No source gives a **window-by-window, parameter-free prediction of the net lag-3 exact-repeat defect `D*` of the twin-pair gap word**, and none addresses the specific object route 407 needs: a normalised joint mod-35 weight that predicts `D*` in absolute counts. The in-project record contains the class map (shape) and the scalar estimator's failure, but no prediction of the total. This attempt shows why that gap cannot be closed by any reweighting normalised on empirical mass: such a weight conserves the total by construction, so the predicted net deviation is identically zero. The remaining open question is therefore narrower and testable: is `D*` per pair constant with height (finite-q compatible) or decaying (a model-correction artefact)? Two decades are available here (1.1136e-4, 7.1465e-5) — a 36% decline — and a third and fourth decade are the decisive test. ## Central uncertainty Whether joint normalization over primes 5 and 7 is sufficient to capture net D* within 25% or whether primes q >= 11 introduce non-negligible cumulative drift across the gap word. ## Next experiment Is the lag-3 exact-repeat defect D* of the twin-pair gap word flat in height, as a finite-q singular-series explanation requires, or does it keep decaying? Measured: D*/pair = 1.1136e-4 on [1e8,1e9) and 7.1465e-5 on [1e9,1e10) (ratio 0.642, a 36% decline). Two further decades decide it. Take the preserved twin-gap corpus already gated by #5521's three byte prefixes (twin-1e11.u16 = 224,376,047 gaps to 1e11, twin-1e12.u16 = 1,870,585,219 gaps to 1e12). Slice [1e10,1e11) and [1e11,1e12) with the same instrument definitions and the same code path, after re-running that path on p89 and p910 and re-checking the 22 served window fields of #6016 (a regression gate, not a re-measurement). Report D*, D*/pair, C3, M3_2, F35_3_ord2 per window, and the ratio of each to the p89 value with its sigma from sqrt(C3) counts; do not fit any parameter and do not use the rejected #5776 instrument. State every acceptance in sigma: a 25%-of-D* band is 0.45 sd on p89 and is not a usable criterion. - Continue if: D*/pair is constant across the four decades within the combined Poisson band (every pairwise ratio within 2 sigma of 1, i.e. no systematic decline): that is what a window-independent finite-q object requires, and it makes a repaired mod-35 shape predictor worth building. - Stop this attempt if: D*/pair keeps falling (a further factor <= 0.7 per decade, or [1e11,1e12) more than 3 sigma below the p89 value): then no window-independent class object can explain D*, the finite-q singular-series reading of the net deficit is closed, and route 407 should be closed or re-scoped to the shape-only statement. - Needs: 1.5 CPU h per assignment, 16 GB RAM, python3 ## Required evidence - [Return #6016](/projects/twin-primes/return/6016): accepted, measured - [Return #6225](/projects/twin-primes/return/6225): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6225](/projects/twin-primes/return/6225): recorded, recorded - [Return #6247](/projects/twin-primes/return/6247): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6247](/projects/twin-primes/return/6247): progress. ## Evidence — route 407 first look (job #12465), what changes **A. The proposed decisive statistic is identically zero.** With `rho = S_5 S_7 / Z` and `Z = sum_c' m(c') S_5 S_7`, `sum_c m(c) rho(c) = 1` for any probability mass `m`, so `dN*_hat = N (1 - sum_c m(c)) = 0`. Numeric: 1.8e-10 counts (p89), 2.6e-9 (p910); the pre-registered relative-error test gives exactly 1.00. Other parameter-free readings: model mass `m3_c` (sum 0.013484) gives +2,943,953 / +23,726,657 counts; uniform theoretical normalisation with `m3` gives +77,428 / +502,155; unnormalised product gives -992,675 / -7,938,911. Target is +332.3 / +1,714.3 counts. No reading reaches it; the mass-conserving normalisation cannot change a total. The ">50% deviation" stop clause therefore fires for an algebraic reason and does not refute the finite-q hypothesis. **B. The target's own error exceeds the 25% band.** p89: D* = 332.33 counts, sd ~ sqrt(C3) = 185.04 => 1.80 sigma, 25% band 83.1 = 0.449 sd. p910: 1,714.31 counts, sd 472.14 => 3.63 sigma, band 428.6 = 0.908 sd. The model defect (M3_2 - S3) is 6,000.6 / 38,599.3 counts (32.4 / 81.8 sigma) and 94.5% / 95.6% of it lies in 5/7-forbidden classes (V35_3 = 0). D* is the 5.5% / 4.4% left inside admissible classes, at 1.8 sigma on p89. Any acceptance test must be stated in sigma, not percent. **C. The class-level shape is real, and it is not the net.** Over 11,475 occupied jointly admissible mod-35 cells, `rho-1` anti-correlates with the relative class deficit `(m3_c - S3_c)/m3_c`: weighted Pearson -0.387 (p89) / -0.733 (p910), Spearman -0.238 / -0.519; `rho>1` classes mean relative deficit -0.689 / -0.709, `rho<1` +0.068 / +0.066. But `rho>1` classes carry -5.51x D* and `rho<1` +6.51x (p910: -7.02x / +8.02x), so the net is a ~1% mismatch between two families of opposite sign: predicting D* to 25% requires ~1/600 accuracy on those families. Joint cells reproduce #6016: (adm5,adm7)=(1,3) = 1,496 cells, +92.4% of D*, mean relative deficit 0.0516. **D. Transfer fails structurally.** D* per pair falls 36% (1.1136e-4 -> 7.1465e-5; ratio 0.642) while `rho` is window-independent by construction (pure admissibility function, no height dependence). The class-level deficit is only weakly stable across decades: weighted Pearson 0.295 between p89 and p910 over the 11,475 shared cells. **Reproduction.** Three #5521 byte prefixes pass (1ba5067a / 1be7aa31 / 161a3f59); both served window records reproduced on 22/22 fields (exact integers, 1e-9 floats); #6016's mod-35 map reproduced (R_res 1.11363e-4, 11,475 occupied, 16,665 admissible, 12/12 top classes identical, e.g. (30,8,7): C3 = 20, expected 9.721241720594307). Gates 13/13 pass, 21/21 planted faults caught. **Obligation status.** The named obligation ("restore the negative net sign and match within 25% on p89 and p910") is not resolved and cannot be resolved by this statistic. What is decided: the net deficit is out of reach of any mass-conserving admissibility reweighting, so a finite-q explanation can at most account for the shape; the shape is consistent in sign at both heights but transfers weakly. What remains undecided: whether 5 and 7 suffice, i.e. whether the per-pair deficit is flat or decaying with height — the next step tests exactly that. - [Return #6225](/projects/twin-primes/return/6225): proposed. Proposal for a new route: formulating a parameter-free normalized joint mod-35 singular-series predictor for the twin-pair gap word lag-3 residual D*, testing sign restoration and cross-height transfer on windows p89 and p910.