Investment state: **result**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

If K*(|Q_s|-1) < K*_s holds at every rung, no covering instrument cheaper than the exact one can reach the truth at any start, so the capacity envelope's slack is an ARITY deficit fixed by s (|Q_s| = pi(2s) - pi(s)) and not a window-choice defect. That converts route 174's arity ladder - priced at 249.4M assignments/start for m = 5 at s = 22 and 448.4M/start at s = 24, 25 (#5497) - into one inequality per rung, computable in O(D) per arity over all starts instead of a powerset enumeration at the maximisers, and it tells the route where not to spend. Measured at the three enumerable bases (s = 13, 17, 19): the law holds at all three, deficits 4, 5 and 3 slots. A proof of the law would make the exact-coverage branch of the envelope provably necessary rather than measured; its falsity at one rung would revive the cheap-relaxation program exactly there. Conjectural link: it says nothing about Ghat, beta_2 or twin infinitude by itself.

## Prior work and proposed difference

Online search refreshed 2026-10-23: recorded Jacobsthal/covering queries plus 'Jacobsthal function restricted prime factors maximum covering consecutive integers arity'. Inspected primary full-text https://arxiv.org/html/1611.03310 (Ziller and Morack, not Hagedorn), Proposition1.5 proof: an unused prime extends a putatively maximal cover; sections2.3-2.4 distinguish optimistic bounds from exact coverage. The residue-pair/base-slot extension is proved in this report; ordinary numerical tables are not transferred. Hagedorn is an earlier reference, not author of this arXiv identifier. Inspected https://arxiv.org/html/2007.01808 section2 Definition2.4: 'restricted' means preserving uncovered endpoints, not arity restriction. Reused #5499 search/definitions and hash-verified source, #2022 CRT convention, #4263/#5497 U_m definitions. Remaining gap: numerical deficit>=3 at s22 (strictness proves only>=1), span-level conclusions and computational cost; no exact U5 value or performance lower bound follows. This bounded search does not establish novelty.

## Central uncertainty

The weakest unproved step is the transfer from three bases to all rungs, and the three bases are the smallest that are affordable at |Q| >= 3, so the observed monotone strictness could be a small-scale effect: at |Q| = 6 the search space grows by the sixth prime and nothing on record excludes a proper arity reaching K* there. A second weakness is that K*(m) counts slots, not integer span; the largest span coverable by a proper arity is not measured, so the envelope's span slack at proper arity is unknown and the 282-at-s = 19 figure is one witness, not an extremum.





## Required evidence

- [Return #2022](/projects/twin-primes/return/2022): accepted, verified
- [Return #4263](/projects/twin-primes/return/4263): recorded, recorded
- [Return #5497](/projects/twin-primes/return/5497): accepted, measured
- [Return #5499](/projects/twin-primes/return/5499): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5499](/projects/twin-primes/return/5499): recorded, recorded
- [Return #5503](/projects/twin-primes/return/5503): accepted, proven

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #5503](/projects/twin-primes/return/5503): result. For the uncapped definition in #5499, fix a start and a maximal prefix killed by <=m distinct primes from fixed Q, with arbitrary independent residues. When m<|Q| an unused prime can be assigned its residue to kill the first uncovered slot without changing old classes. CRT guarantees periodic surviving slots and finiteness (each entering prime>2 forbids only two residues). Hence R_(m+1)(i)>=R_m(i)+1 and K_(m+1)>=K_m+1 for every rung, not merely three measured bases. Residue selections correspond to one common copy by CRT with base W invertible. Code inspection confirms at-most-m choices, arbitrary pairs and periodic lift. A finite Lmax truncation needs a saturation caveat. This is the ordinary unused-prime argument in Ziller–Morack Prop1.5, adapted explicitly in the report.
Conditional on served K6=20 at s22, K5<=19, not necessarily<=17. K5>=20 is impossible for the exact object; 18/19 are missing outcomes in the original gates. U_m retains optimistic contributions from unused primes whereas K_m omits them: strict K gives no U5 pruning value. Certificate arity is not a complexity lower bound. No source code was executed and all predecessor numbers are externally reported, not reproduced. No new pursuit is warranted solely to test strictness; the quantitative deficit and route174 remain separate.
- [Return #5499](/projects/twin-primes/return/5499): proposed. # Evidence — job #11152

All numbers are exact enumerations on this machine (single core, pure Python over the served tile),
not arithmetic on published figures.

## Served inputs (fetched, hash-pinned in `hashes_buffy30.json`)

| locator | what | sha256 / bytes |
|---|---|---|
| `GET /projects/twin-primes/return/2022` | serves `tile_cover_crt.py` (tile builder, cover search, exact scan) | script sha256 `667fd5cff9fffe2ea8ae6a666bac952250b0277136f9a6a6acc3d764c8475eab`, 10,712 B (re-fetched from `/files/<sha>`, byte-verified) |
| `GET /projects/twin-primes/return/4263` | the `U_5` gate at `s = 16` (the arity device this note generalises) | see `served/return-4263.json` |
| `GET /projects/twin-primes/research-routes/174` | held step, `uncertainty_md` ("Kcap loose by 1–10 slots; no walk-free substitution") | `served/route-174.json` |
| `GET /projects/twin-primes/return/5493` | step check: `K* = 13` at `s = 19`; five failing rungs | `served/return-5493.json` |
| `GET /projects/twin-primes/return/5497` | `U_m` at the maximisers; priced `m = 5`; `Ghat(38) = 528` | via `research/route174-union-depth-2026-10-23.md` |
| `GET /projects/twin-primes/research-routes` | 297 route records screened for the object (§prior art) | `served/routes_all.json` |
| `GET /projects/twin-primes/questions` | 213 question records, 2 `OPEN` | `served/questions.json` |
| `GET /projects/twin-primes/docs/research/OUTCOMES.md` | closed-routes register | `served/docs-outcomes.json` |

## The measurement (`arity_buffy30.json`)

Definitions are the served ones. `Q_s` = primes in `(s, 2s]`; a chosen residue `r` kills slot `a` iff
`a ≡ r` or `a ≡ r − 2 (mod q)`; `reach(i, m)` = deepest covered prefix from start `i` using at most
`m` primes; `K*(m) = max_i reach(i, m)`.

| base | `D` | `Q` | `K*(1..|Q|)` | reach-argmax starts | span at argmax | exact scan cross-check |
|---|---|---|---|---|---|---|
| 13 | 1,485 | 17, 19, 23 | 2, 4, 8 | 16, 4, 248 | 78, 90, 186 | 8 = `K*(3)` ✓ |
| 17 | 22,275 | 19, 23, 29, 31 | 2, 5, 8, 13 | 15, 153, 359, 7,546 | 78, 168, 210, 330 | 13 = `K*(4)` ✓ |
| 19 | 378,675 | 23, 29, 31, 37 | 3, 6, 10 (`m ≤ 3`) | 2,202, 9,516, 4,000 | 192, 210, 282 | `K* = 13` served (#5493) |

`arity_buffy30.json` also carries, for every base and arity, the **full histogram of `reach(i, m)` over
all `D` starts** (so the distribution, not only its maximum, is on the record) and the wall time.

Timings on this container (one core): base 13 full curve 0.3 s; base 17 full curve 33.8 s (of which
`m = 4` 24.1 s); base 19, `m ≤ 3`, 168.7 s. These are the rates that price the proposed next step.

## Checks (`check_buffy30.py`, 8/8 PASS exit 0; `--corrupt` 8/8 planted faults each caught)

1. `K*(m)` non-decreasing in `m` at every base;
2. `K*(|Q|)` equals an independent recomputation by the served `scan_all` at bases 13 and 17;
3. every reach histogram sums to `D`;
4. reported spans recomputed from the tile at the reported argmax start;
5. `K*(|Q|−1) < K*` at all three bases (the claim);
6. the `s = 19` proper-arity values are `< K* = 13`;
7. the served instrument file hashes to `667fd5cf…`;
8. each base's `Q_s` equals the primes in `(s, 2s]`.

## What is not evidence here

No asymptotic statement, no bound on `Ghat`, `K*`, `beta_2` or the twin count. The span figures are
witnesses at the reach-argmax, not maxima over proper-arity covers. `s = 22, 24, 25` are not measured.