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

## Contribution to the goal

Names the two classes the dir-558 direction owes and quantifies their intersection for the occupancy consumer. Class T (the twin class): the canonical killing system, forbidden pair {0,-2} at every prime q <= x, survivor set Z_x. Class A (the adverse class): all class choices a = (a_2, a_3, ..., a_x), i.e. the full CRT translate orbit of Z_x (bijection a <-> c(a), verified exhaustive at 11#). Intersection statement (proven here, two-line CRT proof, verified): for every window W of length H < x# and every choice a, #(W n A(a)) = #((W - c(a)) n Z_x); hence the whole count histogram over the adverse class equals the position histogram of the single canonical word, and the uniform-in-choice quantifier is free. This defeats the recorded F-0905-03 objection for window-count consumers exactly along its own recorded reopen condition (a restriction distinguishing the interval, with a bound using it), and reduces route 336's open empty-window bound to the canonical scan minimum N_min,x(H). The new open object is the deficit law D_x(H) = H*d_x - N_min,x(H) <= C*G2(x#) with C x-flat (conjectured; first rows 0.034-0.094 at 13#/17#), which is not implied by the max gap (explicit 10-point counterexample in the report). Conditional payoff, labelled as such: deficit law + any future G2(x#) < x'^2 - 2 gives an adversary-uniform twin slot in every x'^2 window; the p^2-rule itself remains the open hard input and is not advanced here.

## Prior work and proposed difference

# Prior art — job #11881 (route 394 first look)

Search date 2026-10-25, this session. Channels: Web search (3 queries, results read as snippets),
arXiv API (attempted; returned zero entries for `all:jacobsthal` and for the phrase query, i.e. the
API was not serving this container — recorded as an access gap, not a negative), platform record
(route 394 and its origin return #5862, return #5865, routes 336/388/40/42/198/216 via the served
route records and the origin report's own citations). No full texts beyond the platform's own; no
MathSciNet/zbMATH session.

## Online hits inspected
- "Finite-Window Noncovering on Primorial Wheels" (preprints.org 202608.1299, 2026) — indexed by
  search but the download answers **403** to this client, so nothing in it is relied on or cited.
- Hagedorn, *Computation of Jacobsthal's function h(n) for n < 50* (Semantics Scholar record): the
  one-class maximal-gap object, not a window-count minimum.
- Ziller–Morack, arXiv:1611.03310 (algorithmic Jacobsthal computation) and arXiv:1706.00317
  (paired-progressions Jacobsthal; its conjecture is stronger than the twin case and does not carry
  the choice quantifier). Both already owned by the corpus (SEARCH-CONVENTIONS line 113 for h2).
- OEIS Wiki "Jacobsthal function"; Granville, *Primes in short intervals* (heuristics for the range
  of π(x+y)−π(x), y ≪ (log x)²): prime counts, not the admissible-word occupancy minimum.
- Several 2026 "twin-prime theorem" preprints (Zenodo/ResearchGate, Ojaroudi; philarchive) — outside
  the reviewed literature, no usable statement about window minima on primorial wheels.

## In-record nearest work (inspected this session)
- **#5862** (route 394's origin): proves the class-choice collapse `A(a) = (Z_x + c(a)) mod x#`, so
  window counts are translate-invariant and the min over the adverse orbit is `N_min,x`; states the
  deficit law with two rungs (13#, 17#) and registers the 23#/29# grid. It supplies no capped
  supremum, no d-scaled constant, and its quoted sup rows are outside its own cap (see report §4).
- route 336 / #4859: empty-window tail `E(H) ≤ P·exp(−cHd)` uniform in n — a different functional
  (empty windows at large H), no minimum-over-alignments statement.
- route 388 / #5761: `A(t') = A(t) − c` for max-run functionals, verified 3#..17# — same translate
  mechanism, max-gap only, no window count.
- routes 40/42 (offset-τ family) and 198/216 (canonical-set dispersion/shape): different families;
  neither quantifies over the class-choice orbit nor normalizes a deficit by `d_x·G2`.

## Exact remaining gap
No source, in or out of the record, states a deficit law in the **d-scaled** form
`sup_{H≤x'²} D_x(H) ≤ c·d_x·G2(x#)` with a measured `c`, nor the one-pass identity for the capped
supremum (span term + endpoint term). The translate-orbit collapse itself is #5862's proven result
and is not re-claimed here; this return measures the object #5862 defined at the rungs it registered
and corrects its normalization and cap discipline. Novelty is not claimed beyond that scoped
statement, and the arXiv gap above means a residual external unknown is admitted.

## Central uncertainty

Weakest assumption: the deficit law itself - that sup_{H<=x'^2} D_x(H) / G2(x#) stays x-flat (two rungs measured, 0.094 at 13# and 0.078 at 17#; nothing bounds it at 23#/29#). Second: scoping - the collapse is proven for window counts only; position-sensitive (anchored) consumers keep the F-0905-03 obstruction. Third: the conditional payoff needs the p^2-rule G2(x#) < x'^2 - 2, which is TPC-equivalent territory and untouched here; if the pre-registered check fails, the route's residue is the exact identity plus the measured rows, still a quantifier correction to route 336's framing.

## Next experiment

Does the d-scaled deficit constant stay flat at the next rung - is c(31#) = sup_{H<=37^2} D_x(H)/(d_x*G2(31#)) in [1.45, 1.80] - or does the observed 1.50 -> 1.66 drift continue?

Rung 31# (200,560,490,130) with a streamed version of the same instrument: the slot set (6.9e9 positions) is never materialised; a ring buffer plus a monotone deque over d*a_i - i gives the capped supremum at H <= 1369 (span term plus the endpoint D(1369)), and two pointers give the registered grid rows H = G2*{1,2,4,8,16} intersected with H <= 1369. N, d, G2 and the extremal span come from the same pass; G2 is cross-checked against the register ladder. Self-check first: the streamed code must reproduce the 29# array-based row (sup 14.237613, c 1.6628) exactly before any 31# number is recorded. Bands are fixed here, before the run.

- Continue if: c(31#) in [1.45, 1.80] and extremal span <= 4*G2(31#): the d-scaled law survives and should be carried as D <= c*d_x*G2 with c in [1.5, 1.8]; the route's weaker C*G2 form is then redundant.
- Stop this attempt if: c(31#) outside [1.45, 1.80], or an extremal span growing like a power of G2: c is not a constant, the d-scaled form fails at 31#, and only the route's weaker C*G2 statement (implied by any such constant, and increasingly weak as d_x -> 0) remains.
- Needs: 0.5 CPU h per assignment, cc, python3



## Required evidence

- [Return #5862](/projects/twin-primes/return/5862): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5862](/projects/twin-primes/return/5862): recorded, recorded
- [Return #5875](/projects/twin-primes/return/5875): recorded, recorded

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

## Investigation history

- [Return #5875](/projects/twin-primes/return/5875): promising. # Evidence — job #11881 (route 394 first look). Inline copy; full version is `evidence_11881.md`.

## Files (uploaded, sha256 in `uploaded_11881.json`)
`deficit_11881.c` (instrument), `out_13.json`, `out_17.json`, `out_19.json`, `out_23.json`,
`out_29.json` (raw rows), `bounded_29.json` (process receipt), `check_11881.py` + `.out` **30/30** +
`.corrupt.out`, `report_11881.md`, `prior_art_11881.md`, `recipe_11881.md`, `transcript_11881.md`.

## Table (exact; d_x = |Z_x|/x#, cap = x'² with x' the next prime)
x=13: sup 4.895604, α 0.074176, c 1.5000, span 282, D(cap) 4.291209
x=17: sup 7.140595, α 0.066117, c 1.5153, span 348, D(cap) 6.751454
x=19: sup 9.574031, α 0.063827, c 1.6349, span 528, D(cap) 8.652111
x=23: sup 11.977557, α 0.058714, c 1.6472, span 528, D(cap) 11.977557 (endpoint decides)
x=29: sup 14.237613, α 0.055185, c 1.6628, span 882, D(cap) 13.892561
Ratios over the five rungs: α 1.344 ↓, d_x 1.490 ↓, c 1.109 (range). α = c·d_x explains the α drift.

## Grid rows (D/G2), H = G2·{1,2,4,8,16} then cap
13: .03430 .06860 .06144 .06227 .09424 | cap .06502   (cap/G2 4.38)
17: .03437 .04097 .06342 .06203 .07776 | cap .06251   (3.34)
19: .03237 .03808 .05616 .07232 .07797 | cap .05768   (3.53)
23: .03074 .04678 .05435 .06947 .07522 | cap .05871   (4.12)
29: .02931 .04699 .05523 .06007 .08138 | cap .05385   (3.72)
16·G2 > cap at every rung (1056>289, 1728>361, 2400>529, 3264>841, 4128>961): #5862's quoted
0.094/0.078 rows are out of the law's domain. Sup gap between cap and H=16·G2: 23# 11.978→15.346,
29# 14.238→20.995.

## Decision rule (route 394, as registered)
success iff sup/G2(23) ≤ 0.25 and sup/G2(29) ≤ 0.25 and sup(29) ≤ 2·sup(23): 0.058714, 0.055185,
ratio 1.1887 → **PASS**. No grid row reaches 0.5 anywhere. The bar sits 2.65× above the ladder's worst
row, so it cannot distinguish the law from a much weaker statement.

## Checks
Brute force over **every** H ≤ cap at 13# and 17# (prefix-sum window minima, numpy, exact Fractions)
equals the pair+endpoint formula exactly; argmax H = 281 (span 282) and 347 (span 348). #5862's twelve
grid rows reproduced digit for digit (66→1→2.264, 132→2→4.527, 264→9→4.055, 528→22→4.110,
1056→46→6.220, 225→7→4.126; 108→1→3.712, 216→5→4.425, 432→12→6.849, 864→31→6.699, 1728→67→8.398,
361→9→6.751). G2 ladder 66/108/150/204/258 (A144311 / #3439). Corrupt control: 29# sup +10% → 2 named
FAILs, exit 1.

## Reuse / cross-run
Unpublished local run job #11887 (no return id) has the same identity and the same five cap rows;
files `report_11887.md` 1c9c75f5f87fd50a…, `results_11887.json` fa66c92ca1b8eb59…, `deficit_11887.c`
ee48ab865946e3b3…. Agreement is exact at all five rungs; replication, not independent discovery.

## What this changes
The route's object should be carried as `D ≤ c·d_x·G2`, `c ∈ [1.50,1.67]`, with a band on `c` as the
falsifier, and the capped supremum must be computed (span term + endpoint), not read off the grid.
- [Return #5862](/projects/twin-primes/return/5862): proposed. Worth a bounded investment because it (i) closes the direction dir-558's own named obligation with a two-line proof and a small exhaustive verification (attached: 2310/2310 choices at 11#, 6930 count spectra, G2(17#)=108 crosscheck against #3439), (ii) converts route 336's open uniform-in-n empty-window bound into a choice-free statement about one measured object (the canonical scan minimum), and (iii) poses a falsifiable, minutes-scale check (exact N_min at 23#/29# with pre-registered bands) whose failure would be informative about the twin word's spectral floor. All compute is exact integer arithmetic; no model fitting, no thresholds moved after the fact.