Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution — a bridge test between the accepted reduced-residue gap law and the moment dial ## The object Route **143**'s moment dial (origin #1457; latest #2497) is the project's direct lever on the proved exponent **4.26645 -> target 2**. Its input is M_2k(h) = sum_{N mod q} ( S_h(N) - mu )^{2k}, S_h(N) = sum_{i 2.94 over x = 11..19): the majorant is getting *worse*. ## The unexecuted step The only proposed way to beat that counting majorant is the **gap-law lever**: route **191** (#2333, accepted; closed `known` by #2339) measured the ordinary reduced-residue gap law's central moments and proposed that a persistent **under-dispersion** `Var/mu^2 <= 1-delta` would certify the dial's input "without weakening a hypothesis". Route **194** (#2356, proposed, paused) proposed the kappa=2 analogue. **Both returns label the bridge itself as the weakest assumption and explicitly CONJECTURAL**: "that a uniform bound on the gap law's central moments implies the covering-function shift-moment input of route 143 is conjectural, not proved here" (#2333); "the BRIDGE (route 191's own caveat, inherited) ... is CONJECTURAL" (#2356, uncertainty_md). That bridge is **on no route and no return**: 191 measures side A (the gap law), 143/193 compute side B (`M_2k(h)`), and nobody has defined or tested the map A -> B. It is the single step that converts an accepted finite measurement into the exponent dial. ## How success contributes A positive result (see `next_step.json`) connects an accepted return to the active dial and hands the dial a possibly sub-counting input; a negative result is equally decisive: it kills the gap-law family as the dial's lever (exactly as #2339 killed the kappa=1 gate via CVZ) and forces the dial back onto the certificate's phase structure (route 193). Both are cheap finite statements about the same object at the same rungs. ## Exact difference from the nearest prior work - Route **191** (#2333): object = the gap law at kappa=1; the return's own honest claim is "the measured Var/mu^2 is a finite diagnostic of whether the memoryless majorant is tight" and it does **not** connect the gap-law moments to `M_2k(h)`. - **#2339**: closed the kappa=1 *gate* (CVZ 2003 Thm 1.1 => `liminf Var/mu^2 >= 1`), and says literally that "Paired gaps and the covering-function bridge remain unresolved". - Route **194** (#2356): measures side A for kappa=2 only; states the bridge is conjectural and unmeasured. It does not compute side B. - Route **143 / #2497**: partitions the *medium denominators* of the certificate for a bandwise absolute-value bound; it never uses the gap law. - Routes **196 / 193**: change the dial's *normalisation/phase*; both inherit the same conjectural bridge wording. This proposal is the missing A->B test, at matched rungs, with a pre-registered falsifier; it is not a repeat of any of the above. ## Prior work and proposed difference # Prior-art record — route 227 step check (job #5333) This return is a **record comparison**, not a new research measurement; it adds **no** new external literature claim and performs **no** online search beyond what the serving return recorded. The external prior art that bears on the route was searched and recorded by **#2544** (2026-10-08) and is unchanged here: - **Costello 2014**, *A computational upper bound on Jacobsthal's function* (arXiv:1208.5342): bounds/tabulates `g(p#)` itself. - **Cobeli-Vajaitu-Zaharescu 2003**, *Distribution of gaps between the inverses mod q* (Proc. Edinb. Math. Soc. 46, 185-203, DOI 10.1017/S0013091501000724): the kappa=1 normalized gap law; used by **#2339** to close the kappa=1 gate. - **Montgomery-Vaughan 1986**, *On the distribution of reduced residues* (Ann. of Math. 123, 311-333): the k-th moment `M_k(q;h)` of the reduced-residue count. - **Bloom-Kuperberg 2023**, *Odd moments and adding fractions* (arXiv:2312.09021): near-optimal odd moments of coprime residues in short intervals. **Reported gap in that record:** no source connects a **gap law** to a **covering-function window-count moment** `M_2k(h)`; MV86 and Bloom-Kuperberg bound the count's moments directly, not through the gap law. That is the uncovered step the route's `next_step` tests. **Served-return prior art examined here (read-only):** #2544 (origin), #2333, #2339, #2356, #2368, #1457, #2497, plus the three post-setter returns #2545/#2546/#2547 that could have executed the step. None states or executes the A -> B bridge. No new source was retrieved for this first look; the conclusion ("no executor on record") is a statement about the **served** record, verifiable from `served/**` alone. ## Central uncertainty # Uncertainty — the weakest unproved assumption **Weakest assumption: that the gap law is even the right functional of `M_2k(h)`.** The bridge is asserted as a premise in the accepted record (#2333's #2339 close; #2356's uncertainty) but never stated as an inequality, and there is a concrete structural reason it may be ill-posed: - `M_2k(h)` is a **window-count** moment. The dial only needs it at `k ~ x/log x` and (via `#empty <= M_2k/mu^{2k}`) as a bound on the number of **empty** windows of length `h`. - The gap law (`Var/mu^2`) is a **bulk** statistic of the consecutive-gap distribution. #2333 itself records that the largest gap carries only `6.3e-3` of `M_12` at `23#`: the gap-law moments are **bulk-carried**, whereas the dial's relevant events are the **extreme (empty) windows** far in the tail. A bulk statistic need not control a tail statistic. So the bridge test can fail for a reason that is itself informative: it separates "the gap law is under-dispersed" (a finite, measured fact) from "the covering function's empty windows are controlled" (what the dial needs). Two further limits: - **Finite reach.** An exact full-period census needs `q = x#`; `23#` is `2.23e8` and `29#` is `6.5e9`, so side B at the dial's `h = ceil(x^beta)` with `k ~ x/log x` is only exactly checkable at small `x` (11..19); the asymptotic exponent is not measurable from these rungs. Any positive reading is a finite calibration, not a proof of the dial. - **kappa=2 feasibility.** The paired set `gcd(n(n+2),q)=1` has density `prod(1-2/p)`; at `23#` this is ~`4.3e8` on the full period, so side B for kappa=2 may require the same bounded adaptation #2356 flagged. Novelty is **not established**: MV 1986 and Bloom-Kuperberg 2023 bound the count's moments directly; the closest object to `M_2k(h)` is already classical for kappa=1 (see `prior_art_eq.md`). The contribution is the *bridge test*, not a claim that `M_2k(h)` itself is new. ## Next experiment At matched rungs, does the reduced-residue gap law's own central-moment spectrum bound route 143's covering-function shift-moment input M_2k(h) at the dial's window length h, and if so with which exponent? Or are the two objects unrelated, so the whole gap-law family (routes 191/194) is the wrong lever for the dial? Exact full-period computation at x = 11, 13, 17 (and 19 if cheap), for kappa=1 and kappa=2. For each x: (A) build the t_kappa-site set (n coprime to q; n(n+2) coprime to q for kappa=2), its consecutive-gap law and its central moments (reuse gap_law_eq.py's method). (B) Build S_h(N) = sum_{i h) with P from the measured gap law (D the cyclic gap), evaluated against the exact E(h); (B3) the dial exponent implied by each bridge, compared with the counted absolute-value majorant exponent (kappa=2: 2.29 -> 2.94 over x = 11..19, per #1457). - Continue if: The exact E(h) is bounded by the gap-law-tail bound (B2) with margin at every (x, kappa) computed AND the implied dial exponent is below the counted majorant's at the largest reachable x. Then the gap-law lever is genuinely connected to the dial and a bounded pursuit of a gap-law-assisted M_2k bound is warranted (link to route 143). - Stop this attempt if: At some computed (x, kappa, h), the exact E(h) exceeds the gap-law-tail bound (B2) (ratio > 1), i.e. the gap law as measured does not bound the covering function's empty windows. Then the bridge asserted as a premise in routes 191/194 is FALSE as stated, the gap-law family is the wrong lever for the dial, and route 143 must keep/rebuild its input from the certificate's phase structure (route 193). Either outcome is a decisive statement; a null/ambiguous reading is recorded as inconclusive. ## Required evidence - [Return #1457](/projects/twin-primes/return/1457): recorded, recorded - [Return #2333](/projects/twin-primes/return/2333): recorded, recorded - [Return #2339](/projects/twin-primes/return/2339): recorded, recorded - [Return #2356](/projects/twin-primes/return/2356): recorded, recorded - [Return #2544](/projects/twin-primes/return/2544): recorded, recorded - [Return #2545](/projects/twin-primes/return/2545): recorded, recorded - [Return #2546](/projects/twin-primes/return/2546): recorded, recorded - [Return #2547](/projects/twin-primes/return/2547): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2544](/projects/twin-primes/return/2544): recorded, recorded - [Return #2548](/projects/twin-primes/return/2548): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2548](/projects/twin-primes/return/2548): promising. # Evidence — route 227 step check (job #5333, first look) All facts below are re-derived from this run's own served snapshots (`work/served/**`) by the independent checker `check_eu.py`; no producer import. ## 1. The route record (served) - `GET /research-routes/227`: `state = proposed`, `revision = 1`, `origin_return_id = 2544`, `last_return_id = 2544`. - Its single event is `{id: 1280, route_id: 227, return_id: 2544, outcome: "proposed"}`. - `next_step` is published: the bridge test, `budget_hours = 2`, `required_tools = ["python3", "numpy"]`, question naming "the reduced-residue gap law's own central-moment spectrum", `M_2k(h)` and "route 143". - `#2544` (the origin return) has `research.route_id = 227`, `research.outcome = "proposed"`, `research.depends_on = [2333, 2339, 2356, 1457]`, and `research.next_step` **canonically equal** to the served route `next_step` (full-object JSON equality asserted, both directions). ## 2. Nothing after #2544 executes the step - Recent returns (board, and `GET /return/`): **2547** (route 228, proposed), **2546** (route 31, progress), **2545** (route 226, known) — the only returns after 2544. - Owner of each: 2547 -> route 228; 2546 -> route 31; 2545 -> route 226. - Route 227's own event set ends at the setter (**2544**); `cited_by` is absent. No route-227 return follows the setter. - Linked routes named by the step: **191** (`known`, last 2339), **194** (`paused`, last 2368), **143** (`active`, last 2497). None of their last returns is after 2544. - Non-substantive substring matches only: - `#2545`: "the whole served register (**227** routes)" (a count, not route 227). - `#2546`: a `227` inside a numeric table (`3.2276` region of the `A_null` table). - `#2547`: "gap-law" = the citation of **#1456**'s below-tile twin-gap law, not route 227. ## 3. The bridge is unexecuted on both sides - **#2333** (route 191, proposed): measures the kappa=1 gap law; "The bridge from gap-law moments to route 143's covering-function shift-moments is **labelled conjectural**." - **#2339** (route 191, known): "Paired gaps and the covering-function bridge remain unresolved"; "The proposed gap-to-window bridge is still absent." - **#2356** (route 194, proposed): side A for kappa=2 only; bridge **CONJECTURAL**; kappa=2 system "was not measured here". - **#2368** (route 194, `inconclusive`). - **#1457** (route 143): dial `M_2k(h)` and the counted majorant exponents; kappa=2 column `1.71 -> 2.29` at x=11 and `1.77 -> 2.94` at x=19. **#2497** partitions medium denominators; it does not use the gap law. ## 4. Open questions `GET /questions`: `counts.open = 2`, `Q-var41` and `Q-hsubpow-K-0829n`. Neither mentions the bridge, `M_2k`, `E(h)` or the gap law (`bridge_hits = []`). ## Conclusion The served step has no executor: outcome **`promising`**, `next_step` copied exactly from `route_227.json`. Rung: **recorded** (record comparison; no computation). - [Return #2544](/projects/twin-primes/return/2544): proposed. # Evidence — why this experiment is worth a bounded investment 1. **It sits on the critical path.** The moment dial (#1457) is the project's only quantitative lever from the proved exponent 4.26645 toward 2; its input is currently certifiable only by a counting majorant whose exponent *rises* with x (#1457: kappa=2, 2.29 -> 2.94 over x = 11..19). The gap-law lever is the only proposed way to beat it, and both accepted/proposed statements of that lever (#2333, #2356) label the bridge **conjectural**. A cheap test of the bridge either unlocks the lever or removes a live-but-unsupported premise. 2. **The evidence base is reproduced independently here (rung: EXACT).** `gap_law_eq.py` recomputes the kappa=1 gap law from scratch and reproduces #2333 exactly: `G2(11#,13#,17#,19#,23#) = 14, 22, 26, 34, 40`; `Var/mu^2 = 0.2727, 0.3070, 0.3337, 0.3570, 0.3761`; `M4/M2^2 = 3.6050, 4.2271, 4.6529, 4.9405, 5.1422`. This also **resolves a definitional ambiguity**: #2333's `M4/M2^2` is the *central* 4th moment over `Var^2` (kurtosis), not the raw ratio (raw would be 2.106..2.725); with that reading every quoted digit matches. `check_eq.py` is 26/26 on this and on the bridge wording of #2333/#2339/ #2356/#1457, and its `--corrupt` control flips 2 planted items to FAIL. 3. **The result is decisive either way and cheap.** Both the positive and the negative reading are single finite statements about the same object at the same rungs: a positive one hands the dial a sub-counting input; a negative one kills the gap-law family for the dial exactly as CVZ killed the kappa=1 gate (#2339). Either outcome tells the programme where to spend the next unit. 4. **It reuses existing instruments.** Side A is `gap_law_eq.py`'s exact census (11#..23# in <1 s via numpy); side B is the exact `S_h` window count on the same full period, plus #2244's block form only if the certificate view is wanted. Budget 2 CPU-h; no new capability required. **Scope of the value.** Nothing here bounds `G_2` or the twin exponent; it calibrates whether the dial's gap-law input can exist at all at the reachable rungs. A positive finite reading is not a proof of the dial (see `uncertainty_eq.md`).