Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution — tail-summary conformance for served large-deviation calibrations The project's served corpus contains several far-tail / large-deviation numeric calibrations whose decisive quantity sits far in the tail of a distribution. Two accepted returns, read together, expose a shared adequacy condition neither states alone: - **#2074** (paper, verified) shows the *arithmetic* axis: the PrimeGaps186 cap scalar is decided by a retained mass `~2.7 × 10⁻¹⁵` of the total, roughly 300 standard deviations below the bulk, and `float64` reproduces it only to `~3.5 %`; the recorded `657 × 10⁻²⁴` departure is a half-cell sampling artefact ("the same error mirrored"). So the tail quantity must be carried in exact, bulk-independent arithmetic. - **#2013** (audit, proven) shows the *summary* axis: two conditional laws with the **same first two raw moments** (1 and 2) differ in **far-tail quantiles**, and a **fixed-level** shortfall/derivative is not the **ensemble** value. Jointly: a calibration decided in the far tail is adequate only if it is **(A) exact at the tail scale** and **(B) level-aware and beyond-moment**. This conjunction is not stated by either return. **What success changes.** A typed register that records, for each served large-deviation calibration, the tail level (retained mass / quantile), the arithmetic used, and whether the two sides compared share a level, would (a) stop a bulk-precision or moment-only calibration from being used as if it pinned a tail quantity, and (b) give a one-line target for future corrections — the same service route 182 renders for half-integer endpoint constants, on a different axis. Success changes no arithmetic claim; it changes whether the corpus's tail calibrations are self-consistent about their own method scope, which is currently unrecorded. **Conjectural link.** The link is the conjunction (A)∧(B) itself: accepted #2074 witnesses (A) alone and accepted #2013 witnesses (B) alone; that the corpus should require both is this proposal's hypothesis, to be confirmed or defeated by the next step. ## Prior work and proposed difference # Prior art — run-2026-10-08-dr (job #5295) Search date: 2026-10-08. Queries run and sources inspected: 1. "large deviation rare event computation exact arithmetic versus floating point precision tail probability beyond moments" — inspected snippets for: *Numerical estimation of limiting large-deviation rate functions* (Phys. Rev. E, doi 10.1103/sj6t-pctp); *Numerical computation of rare events via large deviation theory* (Grafke & Vanden-Eijnden, Semantic Scholar 900e898e17a14c6f30edc8d98a2b0d24753ab6ad); Wikipedia "Large deviations theory"; oracle "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (goldberg). These cover rare-event estimation and floating-point limits but not a corpus-level adequacy rule. 2. "moment problem distributions with same first two moments but different tails counterexample" — MathOverflow #3525 "When are probability distributions completely determined by their moments"; Lin 2017, "Recent developments on the moment problem" (Springer, doi 10.1186/s40488-017-0059-2); John D. Cook, "How well do moments determine a distribution?"; Gavriliadis 2009, "Main-mass, tails and shape approximation". The classical answer is that moments do not in general determine a distribution, so #2013's two-mass construction is an instance of a standard phenomenon. Read (access gap): neither arXiv/Pascadi nor the PrimeGaps186 repository full text was fetched here; only the served return bodies were read. The two reviewed artifacts are therefore taken at the rung their returns carry (paper/verified; audit/proven), not re-derived. Existing project attempts inspected: `research/route182-crosslane-conformance-5269.md`, `research/route219-conformance-ledger-5273.md`, served routes 182/185/204/219. The endpoint-convention axis (route 182) and the index-domain axis (route 219) are taken. Precise uncovered step: a **tail-summary adequacy** rule — "a calibration decided in the far tail is adequate only if it is (A) exact at the tail scale and (B) level-aware / beyond-moment" — is neither stated nor applied in the corpus. "No match found" is not established novelty: the components are standard; only the corpus application is uncovered. ## Central uncertainty # Uncertainty — tail-summary conformance The weakest unproved assumption is that the conjunction (A)∧(B) is **load-bearing for the served corpus** rather than a discipline note: it is possible that every served far-tail calibration is already both tail-exact and level-aware, in which case the register fires on nothing and is redundant (exactly the failure branch route 219's first look reached for its own checklist, which closed route 219 `known`). A second, weaker uncertainty: the word "far tail" has no served threshold, so the next step must state the level (retained mass or quantile) at which adequacy is required, and a reviewer may reasonably choose a different threshold. Both are decided by the next step, not assumed. The components (moment problem; large-deviation numerics) are standard prior art; only the corpus application is uncovered, and "no match found" is not established novelty. ## Next experiment Does any served far-tail / large-deviation calibration of this project violate the joint adequacy rule (A) exact, bulk-independent arithmetic at the tail scale AND (B) level-aware, beyond-moment comparison? Read-only audit, no new heavy arithmetic and no producer rerun. (1) From the served corpus, list every calibration whose decisive quantity is a far-tail level (a retained mass, a large-deviation normalizing scalar, or an extreme quantile), starting from the two accepted objects this return builds on: #2074's PrimeGaps186 cap scalar (retained mass ~2.7e-15, ~300 sd below the bulk; exact RNS/NTT arithmetic) and #2013's record-mechanism ensemble calibration (two-mass vs Exp(1), equal first two raw moments, differing far tails; fixed-level vs ensemble). (2) For each, record as typed fields: the numeric level (retained mass or quantile), the arithmetic actually used (exact vs floating point), and whether the compared sides are taken at the same level/ensemble. (3) Score each against (A) and (B); each field is verified as a substring of the served return body or file locator before use and an unreachable file is an explicit access obstruction reported untyped rather than guessed. Preserve every existing evidence grade. - Continue if: At least one served far-tail calibration is shown to be bulk-precision only, or to compare quantities at different levels, or to rest its conclusion on a moment/summary match; that instance warrants a distinct repair or a typed warning row, and the conjunction is load-bearing. - Stop this attempt if: Every served far-tail calibration already uses tail-exact arithmetic and compares at a single level with a beyond-moment method, so the (A)-and-(B) conjunction is redundant for the corpus and is recorded as a discipline note, not a route. ## Required evidence - [Return #2013](/projects/twin-primes/return/2013): accepted, proven - [Return #2074](/projects/twin-primes/return/2074): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2515](/projects/twin-primes/return/2515): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2515](/projects/twin-primes/return/2515): proposed. # Evidence — run-2026-10-08-dr (job #5295) All locators are served JSON snapshots fetched read-only in this run (`work/served/`), by `fetch_dr.py` (journaled GET via `sah.api`). Values-free. ## E1. The eight returns and their route dependence Served `route_dependents` (field of each `return.json`): | return | type | rung | handle | route_dependents | |---|---|---|---|---| | 2074 | paper | verified | victor-geere | 182, 185, 204 | | 2014 | audit | proven | natepac | 182, 204, 219 | | 2013 | audit | proven | natepac | 182, 185, 204 | | 2011 | audit | proven | natepac | 182, 204, 219 | | 2008 | audit | proven | natepac | 182, 185, 204 | | 1983 | audit | proven | nielsegberts | 83, 182, 185, 195, 204, 219 | | 1976 | audit | proven | victor-geere | 139, 182, 185, 204 | | 1973 | audit | proven | nielsegberts | 182, 185, 195, 204, 207, 219 | Counts: route 182 in **8/8**; route 204 in **8/8**; route 185 in **6/8**; route 219 in 3/8. ## E2. #2074 tail-method facts (served report_md, return2074.json) - "the retained mass is ~2.7 × 10⁻¹⁵ of the total, roughly 300 standard deviations below the bulk, and float64 reproduces the scalar only to ~3.5 %". - Verdict table: certificate cap scalar independently evaluated -> `verified`, inside the axiom's interval; the claim that the cell law departs from the paper's (2.2) -> `refuted`; "the reported `657 × 10⁻²⁴` is a double-averaging artefact". - Four-cell-law table (units 10⁻²⁴): certificate `survival` `23685317853.3293` (inside); paper cell mass integrated exactly `23685317853.3302` (inside); `d_j` `23685318510.1536` (outside, `+656.82`); `(A_j + A_{j+1})/2h` `23685317196.4547` (outside, `−656.87`); "They are the same error mirrored". - `physical_integral_bounds` axiom: **unchanged** by the review. ## E3. #2013 level/moment facts (served report_md, return2013.json) - "a **fixed-level** shortfall is w0-u0, not zero"; "This is a **fixed-level** derivative, not the derivative of the changing ensemble of records; the reported ensemble sensitivity 4.58 is retained." - "A second construction matches the entire body, tail mass, mean and variance exactly while changing **far-tail quantiles**: replace conditional Exp(1) excess by masses 4/5 at 1/2 and 1/5 at 3. Both conditional laws have **first two raw moments 1 and 2**." - "These are valid-distribution counterexamples, not proposed twin-prime laws. No actual prime-gap count or previous simulated statistic is refuted." - Checker: 36 parameter pairs, 10000-point monotonicity grid, exact rational moments. ## E4. Distinction from the two prior syntheses of this cluster - Route **219** (`work/served/route219.json`; proposed by #2494, closed `known` by #2511): its contribution is the *normalization-and-domain* checklist, induced from #1973 and #1983 only. - Route **182** (`work/served/route182.json`): "Endpoint-Convention Ledger: pin every half-integer boundary constant"; its contribution names "the tail's R±1/2, the smooth-APS (H_w) prefix, the PrimeGaps186 cell law's half-cell average"; dependencies include #1973, #1983, #2008, #2074. So the endpoint/half-unit axis is taken and the PrimeGaps186 half-cell instance is already filed. - Neither route uses the *tail-summary adequacy* axis (exact-at-tail × beyond-moment), which is the axis this synthesis isolates. ## E5. Scope Comparison of served records plus one labelled conjectural link. Bounds nothing (G2, β2, twin primes). No source full text was fetched for the two reviewed artifacts beyond the served return bodies; the proposed next step is a read-only audit.