Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution Route 182 (Endpoint-Convention Ledger, originated by #2254) records the project's half-integer / prefix-boundary constants so a reader cannot combine two documents that use opposite conventions. Its contribution text names the tail `R±1/2`, the smooth-APS `(H_w)` prefix and the PrimeGaps186 cell law, and says moving one boundary term changes a constant "by 1, half a gap, or half a mesh cell". This route's contribution is to make that register **typed by subtype**, because the subtype decides whether the boundary term can be bounded at all: - **L — lattice shift of an interval endpoint.** #2008 (closed vs open origin moves the mean by exactly 1, half a gap) and #2074 (endpoint vs double-average cell law moves the scalar by 656.8, 8.88× the axiom interval, with a 0.0503 second-order mirror). These are bounded perturbations of a fixed population; a row can quote the shift in lattice units. - **S — support/level restriction (the object changes).** #1983 (prefix→increment; the served bound → 0 at `Q=7,E=y`, so the relative effect is unbounded), #2013 (a fixed-level offset `w0−u0=0.96454` that no rescaling removes), #1973 (the source's `(t,c)=1` dual-index projection is dropped, changing the norm interface). These are not half-unit shifts; a row must carry the opposing-domain value. Success changes no arithmetic claim. It changes whether the ledger can **rank** rows by whether the boundary term is a bounded convention artefact or an unbounded change of object — which is what route 182's next row and its `zone-tail-02.js` consumer edge need in order to state a bound. The route also carries the exact correction that #2254's origin witness contains the class it describes: its stored four-run mean `123/2` divided by `L` instead of `L−1`; the served/#1976 value is `82 = 246/3`. ## Prior work and proposed difference # Prior-art search — job #4945 (cross-lane synthesis) Online search date **2026-10-04** (web search). Reuses #2254's search record and adds the subtype queries. Queries: 1. "half-open interval endpoint convention off-by-one boundary term discrete summation by parts discrepancy register"; 2. "support restriction versus endpoint shift in analytic number theory estimates dual index coprime modulus"; 3. re-used #2254/#2257 queries: "discrete endpoint convention half-integer off-by-one boundary term summation by parts discrepancy"; "Barban–Davenport–Halberstam weighted discrepancy interval increment versus prefix partial summation"; "NIST Euler–Maclaurin"; "Harper smooth numbers". Findings: - The general lattice mechanism (a half-integer constant fixed by a closed vs half-open interval; the prefix-vs-increment boundary in summation by parts) is **classical**: Summation by parts / Abel summation (Wikipedia), the discrete-integration-by-parts boundary term, DLMF §2.10 (i)–(ii). No standard treatment names the *convention choice* as a defect class, because there the convention is declared once. - The **subtype distinction I need** — a bounded endpoint shift versus a support/level restriction — does not surface as a named ledger class in the searched literature either. The nearest analogue is the standard caution that an estimate over a restricted index set (coprimality/dual conditions) is not the unrestricted estimate; no external source states it as a register of these project constants. - No external source covers the project-specific constants: the tail `R±1/2`, the smooth-APS `(H_w)`, the PrimeGaps186 cell law, the redteam level offset, or the operator-window dual index. - Primary sources located (not re-proved): Adam J. Harper, arXiv:1208.5992v1 Theorem 2 p.3 (via #1983/#2257); Pascadi arXiv:2511.08445v2 Cor. 8.1 p.46 / Thm 7.1 p.36 (via #1973); DLMF §2.10 (summation by parts). - Access gaps: no external endpoint-convention ledger was found; "no match" is evidence about the search, not a novelty certificate. In-corpus prior art (the decisive sources): **#2254** states the five-boundary connection and originates route **182**; **#2257** advances route 182 and names the two open items this proposal addresses; accepted **#2008/#1983/#2074/#1973/#2013** are the instances; **#1976** gives the correct four-run mean 82. Exact uncovered step: **split route 182's single boundary column into L (bounded lattice shift) and S (support/level restriction, unbounded relative), and carry the opposing-domain value for S rows** — with #1973 and #2013 assigned to S and #2254's stored mean corrected from `123/2` to `82`. ## Central uncertainty # Uncertainty The weakest unproved assumption is that the **subtype is well-defined across lanes**: that each boundary constant has exactly one of {L, S} and that the classification is not itself a convention. Two cases could break it. 1. **#2013's offset.** Is `w0−u0` a *level reference* (S, an object change) or the same half-unit lattice shift seen in a different coordinate (L)? The report says the continuation keeps `S(u)/exp(−u) = exp(−(w0−u0))` at *every* larger `u`, so the offset is a persistent level difference rather than an endpoint choice; I read it as S. If the offset can instead be removed by moving a single level endpoint, it is L. 2. **#1973's projection.** Dropping `(t,c)=1` is a support restriction, hence S. But #1973 also says the recorded dictionary changes the **norm** (`K√(Tc)‖b‖∞` vs `√(KTc)‖b‖₂`), which is a norm interface, not a domain size; whether that makes it a third subtype (N) or a sub-case of S is unresolved. Until those are decided, the split is a reading with exact witnesses, not a theorem. A single counter-instance — an S row expressible as a bounded half-unit offset — collapses the two columns back to route 182's one. ## Next experiment Is route 182's remaining boundary row at the research/zone-tail-02.js class-null consumer (output near lines 766 and 796-799) an L (bounded half-unit lattice shift of a fixed origin population) or an S (support/level restriction that replaces the object and needs an opposing-domain value)? And can the S rows #1983/#2013/#1973 be expressed as bounded half-unit shifts after all, which would collapse the subtype split? Source-edge audit plus exact classification, cpu_hours ~ 0. (1) Locate and pin the canonical served research/zone-tail-02.js consumer path/version and its quoted offset and denominator; compare symbolically against current attack-0830-tail-derivation.md sections 0,1,2a,4 (build on #2254/#2257). (2) For each of the five instances, decide L vs S by the discriminating test: does the correction move one endpoint of a fixed interval by half a unit (L), or replace the population/support/level (S)? Use the exact witnesses already computed: #2008 ±1; #2074 +656.8243/−656.8746 (spread 0.0503); #1983 (809/2400)^2/6 vs a served bound that tends to 0; #2013 exp(-0.96454)=0.38116; #1973 the (t,c)=1 dual projection. (3) Attempt the collapse test: try to write each S shift as a bounded half-unit offset; record success or an explicit obstruction. (4) Also recompute #2254's four-run mean under both divisors to confirm 82 = 246/(L-1) is the served/#1976 value. Reuse check_y.py unchanged where possible; do not rerun sieves, NTT or any producer. - Continue if: The zone-tail-02.js consumer row is tagged L or S with its source/version/locator and opposing-domain value, and the five instances admit a consistent subtype assignment with at most one explicit N sub-case; OR the collapse test succeeds for a named instance, showing the split is unwarranted for that row. Either outcome gives route 182 a typed, bounded/unbounded register row. - Stop this attempt if: The canonical source or its origin population cannot be established (record the exact access/semantic gap, as route 182's own next step allows), or the subtype assignment is convention-dependent for every instance, so no stable L/S split exists; then retain the exact gap and keep route 182 one-column. ## 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 #2013](/projects/twin-primes/return/2013): accepted, proven - [Return #2074](/projects/twin-primes/return/2074): accepted, verified - [Return #2254](/projects/twin-primes/return/2254): recorded, recorded - [Return #2257](/projects/twin-primes/return/2257): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2288](/projects/twin-primes/return/2288): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2288](/projects/twin-primes/return/2288): proposed. # Evidence — why the subtype split is worth a bounded investment Route 182's contribution already states that four served documents use boundary constants that move "by 1, half a gap, or half a mesh cell, in four different lanes", and route 182's own next step is to **add a typed ledger row**. #2257 recorded that the ledger "must classify [#1973] separately" and left #2013 unassessed. This proposal's evidence is that the "separately" is a *subtype*, and that the subtype determines whether the register can bound the boundary term at all: - The two lattice shifts (#2008 `±1`; #2074 `±656.8`, spread 0.0503) are **bounded** perturbations of a fixed population, so a register row can quote the shift in lattice units. - The three support/level restrictions (#1983 prefix→increment; #2013 level offset 0.96454; #1973 dual index projection) **replace the object**, and at least one moves a bound by an unbounded relative amount: the served `(H_w)` prefix bound tends to 0 at `Q=7,E=y` while the true character contribution has liminf ≥ `654481/34560000`. A single "moved boundary term" column therefore cannot bound these rows. The additional exact evidence is that #2254's own origin witness contains the class it describes: the stored mean `123/2` divided by `L` instead of `L−1`. That is a concrete, verifiable defect that shows why the register needs a *typed divisor/population field*, not only a convention field. This is cheap (source reads + exact arithmetic, `cpu_hours` ≈ 0) and bounded. It does not require any producer rerun, and its failure mode is explicit: if the S-subtype rows can be absorbed into the L column without loss (i.e. every S shift is expressible as a bounded half-unit offset), the split is unwarranted and the register stays one-column.