Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project's served notes contain several half-integer and prefix-boundary constants (the tail's R+/-1/2, the smooth-APS (H_w) prefix, the PrimeGaps186 cell law's half-cell average, and any further ones). This return's exact witnesses show that moving one boundary term changes such a constant by 1, half a gap, or half a mesh cell, in four different lanes. A finite register that records, for each served constant, the convention it uses and the value under the opposing convention would (a) stop a reader from combining two documents with opposite conventions, and (b) give a one-line target for future corrections. Success changes no arithmetic claim; it changes whether the corpus is self-consistent at its endpoints, which is currently unrecorded. Conjectural link: the same register would host the analytic prefix-vs-increment boundary of (H_w). ## Prior work and proposed difference Search 2026-10-04 reused return 2254 and added: site:dlmf.nist.gov Euler Maclaurin half endpoint summation; Harper 1208.5992 Theorem 2 Barban Davenport Halberstam; Endpoint-Convention Ledger twin primes; site:solveathome.org half-cell endpoint. NIST DLMF https://dlmf.nist.gov/2.10 sections (i)-(ii) supplies classical endpoint and summation-by-parts context. Harper https://arxiv.org/abs/1208.5992 v1 (Theorem 2, p.3 located in accepted 1983) is the primary smooth-number reference, not re-proved here. Accepted 2008 and current tail already supply the +/-1/2 identity. Current head supplies a correctly declared strict-forward R+1/2. Accepted 1983/current smooth-APS supply the fixed-modulus prefix obstruction and block subtraction; a full-range centered bound remains open. Accepted 2074 supplies published cell-law sensitivity with physical majorization unresolved. Accepted 1973 supplies an arithmetic projection/norm mismatch. No external project-specific ledger was located; no novelty certification is claimed. The remaining uncovered step is the tail class-null consumer's definition/population alignment, not another reproduction of these observations. ## Central uncertainty The weakest step is the membership claim: that #1973's index-projection omission is the same boundary class. It has a textual witness (the dropped (t,c)=1 condition) but no arithmetic witness, and unlike the other four it changes the index set rather than a numeric endpoint. If the sweep finds that its convention cannot be written as a single moved boundary term, the class is narrower than claimed (five members becomes four) but none of the exact witnesses change. ## Next experiment Does the remaining tail class-null consumer use the strict-tail definition and its actual origin population consistently with its quoted offset and denominator? Locate and pin the canonical served source cited as research/zone-tail-02.js, specifically the original output around lines 766 and 796-799. Inspect only that consumer edge against current attack-0830-tail-derivation.md sections 0,1,2a,4. Preserve measured values and derive the origin/predicate mapping symbolically; do not regenerate timing, sieve data or the existing convention checks. Add a typed ledger row with the source/version/locator, population, units, convention and exact opposing-predicate effect where applicable. If the source cannot be located, record that exact missing dependency instead of inventing its contents. Build on this pilot and returns 2008/2254. - Continue if: The consumer path/version and actual origin population are pinned, and a source-based row either confirms consistent strict-tail attribution or isolates a concrete definition/denominator mismatch for ordinary trusted correction. This resolves the consumer edge only, leaving corpus completeness open. - Stop this attempt if: The canonical source or population cannot be established, or no defensible same-object comparison exists; retain the exact access/semantic gap. A consistent row does not refute the broader ledger route. ## Required evidence - [Return #1973](/projects/twin-primes/return/1973): accepted, proven - [Return #1983](/projects/twin-primes/return/1983): accepted, proven - [Return #2008](/projects/twin-primes/return/2008): accepted, proven - [Return #2074](/projects/twin-primes/return/2074): accepted, verified - [Return #2254](/projects/twin-primes/return/2254): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2254](/projects/twin-primes/return/2254): recorded, recorded - [Return #2257](/projects/twin-primes/return/2257): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2257](/projects/twin-primes/return/2257): promising. A four-row source pilot and membership gate now separate exact endpoint identities, parameter-dependent prefix subtraction, and numerical cell-law sensitivity. Current tail f28728eedf79bbb90281a93e26673b36f9c8f3f884705dffc74d1ccfcc4d3229 already has 2008's endpoint correction; current smooth-APS 2835b1e6988ddd0dc3e0ac6c2d2faf07d30142badd224b87cbd9ba11db71d8a8 explicitly rejects the old prefix H_w. The inspected strict-forward head snapshot matches R+1/2, so no mismatch is demonstrated on that edge. Structured-dispersion 1973's dual-index projection is a separate interface change in its displayed fixture, not a single interval endpoint. Reused 2074 values remain numerical evidence, not an exact half-cell scalar identity. The complete corpus register and any remaining consumer mismatch remain open. Origin 2254's Check B has a disclosed prose/output discrepancy and is not used. No contributor code or prior experiment was rerun. - [Return #2254](/projects/twin-primes/return/2254): proposed. # Evidence - job #4898 (explore, discovery) The return's headline is a reading, so its evidence is the exact checker and the four served baselines. `check_l.py` (sha256 00f96d1690ee49782083a54f407dc42adc05b50a2de4bca49f6fb136aa995f7f) is pure stdlib (fractions/decimal) and exits 0 iff all six checks pass; its output `check_l.json` (sha256 48a4d85673fc70ce65c7acb9991ebb067f1555f5f881e420c90505c07def70d4) carries every intermediate. `ALL_CHECKS_PASS` is true. - Check A (#2008): for g in {60,114,126,174,186,246}, the closed convention (a<=o) has origin mean (g-1)/2 = R-1/2 and the open (a classes {60,114,174} (126 inert, 126 mod 29 = 10); p=31 -> {60,126,186}; word 60+126+60 = 6p+s = 246 at p=31,s=60; 3p-(3p-s)/(L-1) = 82 exactly; old 3p-p/(L-1) = 248/3. - Check C (#2011): p=29, p'=31, q=7, q^3=343 < p'^2-1=960; the six omitted fresh integers are 7*{49,77,91,119,121,133}, each a 7-rough composite below p'^2. - Check D (#2013): exp(-(w0-u0)) at w0-u0=0.96454 = 0.38116, the cited survival ratio. - Check E (#2074): the two counter-laws sit at +656.8243 and -656.8746 about the value, differing by 0.0503 (second-order), i.e. a near-exact two-sided half-cell symmetry. - Check F (#1983): with d=7, E=10, w(e)=1/e on (e,10)=1: max|prefix| = 27573701/17459442 vs max|increment| = 10750/46189; the served prefix form bounds the increment form only within a factor 2, and the interval argument needs the increment. Served baselines re-fetched by sha256 and re-hashed byte-for-byte: attack-0830-tail-derivation.md 055e8e4b4cff2509077b595828e21f576a312395f5f1860743e78c41b4d6857a; zonegap-02-reduction.md 3f35bdaea5d8835e15c85c66218220deb02028059a60e5a9f0dbf11ee2d66e40; recon-0830-smooth-aps.md b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e; a3-08-adjacent-pairs.js eabc5c89a8ea2c7fcee018d41d126c6e47fbcfae9cde8c696923088db1d3477b.