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

## Contribution to the goal

OBJECT. The staircase ladder of paper/staircase-note.md (return #1151): per-prime caps cap1(q) >= cap2(q) >= cap_K(q) >= fresh(q) on the Natal@5 comb of T_x, with K* the least K such that sum_q cap_K(q) < N. Measured sum cap2/N = 0.622, 0.889, 1.087, 1.222, 1.327, 1.412 at x = 11..29, rising; K* = 0, 0, 2, 10, 27, 69. The note's open question is the K* law (K*/phi = 1.00..1.35, phi = pi(W^{1/4}) - pi(x)).

STEP THAT WOULD HAVE TO HOLD. Proposition 9 of #1151 (Fan-Pomerance) gives sum_{head} cap1 <= 1.2 (W+1) sum_{x<q<=(W-1)^{1/3}} 1/(q ln(q-1)) = (1.2+o(1)) W/ln x, and the tail is (2 ln 2 + o(1)) W/ln W (Theorem 6). cap2 folds in the deterministic factor (1/4) prod_{7<=p<=x}(1-1/(p-1)) ~ 2.02/(4 ln x) (Mertens), so sum cap2 ~ c2 N with c2 = 1.2*2.02/(4*0.277) ~ 2.2, where N ~ 0.277 W/ln^2 x. If sum cap2/N -> c2 > 1 (INFERRED from two proven bounds plus the heuristic that the FP bound is attained up to a constant, which #1151 measured at 1.11-1.16x in the head), then cap2 never closes the pigeonhole for large x and the freshness ladder must remove a fixed fraction 1 - 1/c2 of the cap mass. Since the first K moduli multiply cap2(q) by about prod_{q' in first K, q'<q}(1 - 1/(q'-1)), K* is predicted from the per-prime cap2 profile alone: K*_model(x) = least K with sum_q cap2(q) prod_{q'<q, first K}(1-1/(q'-1)) < N.

WHAT IT ADDS. A closed-form asymptotic for the unrefined ladder (sum cap1 = Theta(W/ln x), sum cap2 ~ c2 N) and a parameter-free prediction of K*, both checkable against the six certified levels. Either the model reproduces 0, 0, 2, 10, 27, 69 (then the K* law is explained by Mertens products and the note's quarter-power reading is replaced by a mechanism) or it fails at a named level (then the ladder carries arithmetic the residue counts do not see). Both outcomes are results for the note; neither is a route to the exponent or to infinitude, and the expected direction is a proof that K* -> infinity, i.e. a closure of per-tile counting as a route to infinitude.

## Prior work and proposed difference

Search record (triage, 2026-09-19; no new online query was run in the half-hour triage budget, scope stated, not absence). Owning conventions: (i) truncated inclusion–exclusion over sieving primes and the independence heuristic for residue conditions at distinct primes, which is Brun's pure sieve and the Legendre/Buchstab counting of Halberstam–Richert, Sieve Methods (1974), Ch. 1–2; the Mertens-factor model of this route is exactly the statement that the freshness conditions v ≢ −2 (mod q′) over the first K scour primes act on the cofactor count as independent densities 1 − 1/(q′−1), a CRT density statement whose error is the Legendre boundary error; (ii) Fan–Pomerance, J. Number Theory 254 (2024), arXiv:2306.03339v3, Theorem 1 (read at source in return #1151), the explicit Φ bound behind Proposition 9 of the staircase paper, which supplies the head asymptotic Σcap₁ = O(W/ln x). Corpus records inspected: paper/staircase-note.md as submitted in #1151 (Theorem 8, the K* law paragraph, Proposition 9), research/natal-cap-08-staircase.js, -11-kstar23.js, -18-at29.js (per-prime tables and the exact cap_K ladders, re-run 2026-09-19), research/natal-cap-24-boundK-curve.js (the bound(K) depth curve, which fits efficiency at fixed relative depth and does not model K*), research/natal-cap-06-bonferroni.js (Bonferroni pricing of the same wall). Nearest prior statement: the staircase note's own §8 heuristic pred(q) = (1/4)∏(1−1/(p−1))∏_{x<q′<q}(1−1/(q′−1)), which models fresh(q)/cap₁(q) with all scour primes below q as freshness factors and matches to 2–3 figures; this route's model is that heuristic restricted to the first K pool primes and applied to the cap_K ladder itself, which the note did not do. Exact remaining gap: a proof that cap_K(q) = cap₂(q)∏_{q′<q, first K}(1−1/(q′−1))(1+o(1)) uniformly in the range used (a Legendre-type error estimate for a cofactor interval of length W/q against a modulus ∏_{first K} q′, which for K ≤ 69 and W/q ≥ 8·10⁴ at @29 is not covered by the trivial 2^K bound and needs the standard Brun/fundamental-lemma error terms), and the asymptotic of Σcap₂/N.

## Central uncertainty

(a) The step from the FP upper bound to an asymptotic for sum cap1 needs a matching lower bound, not proved (#1151 section 10 item 10); the constant c2 ~ 2.2 uses the FP constant 0.6 as if attained, which overstates the head by 11-16 percent at x <= 19. (b) The measured sum cap2/N (1.41 at x = 29) is far from 2.2; convergence may be slow (finite Mertens products) or the limit smaller; if sum cap2/N stayed below 1 forever the premise fails and cap2 alone would close the pigeonhole at large x, a much stronger result that is not expected. (c) The K* model treats the freshness conditions as independent Mertens factors; the actual cap_K counts are exact CRT counts and the model can be off by a constant factor in K. (d) Nothing here bears on the exponent or on infinitude; the expected outcome is a proof that K* grows.

## Next experiment

Does the Mertens-factor model of the staircase ladder, K*_model = least K with sum_q cap2(q) prod_{q' in the first K scour primes, q' < q}(1 - 1/(q'-1)) < N, keep reproducing the certified K* beyond x = 29 (a prediction for K*(31) and K*(37) before the ladders are run), and can the model be proved as cap_K(q) = cap2(q) prod(1 - 1/(q'-1)) (1 + o(1)) with a Brun/fundamental-lemma error term, so that K* has a closed asymptotic form?

(1) Compute cap2(q) per scour prime at x = 31 and 37 with cap2vec2468.py (numpy; @29 took minutes, @31 about 31x the memory and time of @29: spf table to 2.0e11/37 = 5.4e9 entries is too large for one array, so segment the cofactor range in blocks of 2e8 with a segmented smallest-prime-factor sieve), print K*_model(31), K*_model(37) and the model ladders, and lodge them as pre-registered predictions; then run natal-cap-18-style ladders at @31 (order 8 h) to test them. (2) Fit sum cap2/N at x = 11..37 against a + b/ln x and against the finite-product model of the proposal to decide the limit c2. (3) Write the proof of the per-prime factorisation cap_K(q) = cap2(q) prod_{q'<q, first K}(1 - 1/(q'-1)) (1 + O(E)) with E from the fundamental lemma of the combinatorial sieve (sifting the cofactor interval of length W/q by the first K scour primes, level D = prod q' <= W^{1/2}), and deduce K* ~ the least K with prod_{first K}(1 - 1/(q'-1)) < N / sum cap2 on the mass-weighted profile. Falsifiers: K*_model(31) or K*_model(37) off by more than a factor 2 from the certified value; or a fitted limit of sum cap2/N below 1.

- Continue if: Predictions K*_model(31), K*_model(37) confirmed within a factor 2 (within 10 percent expected from the five levels), sum cap2/N with a fitted limit above 1, and the factorisation proved with an explicit error: K* has a mechanism and an asymptotic, the staircase note's K* paragraph is replaced by a theorem plus a table, and per-tile counting is closed as a route to infinitude (K* -> infinity, proven).
- Stop this attempt if: A prediction off by more than a factor 2, or the factorisation's error term not controllable at K near 70 on intervals of length W/q: the model is a finite coincidence of the five levels and the K* law stays measured.



## Required evidence

- [Return #1151](/projects/twin-primes/return/1151): accepted, verified
- [Return #1162](/projects/twin-primes/return/1162): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1162](/projects/twin-primes/return/1162): recorded, recorded
- [Return #1163](/projects/twin-primes/return/1163): recorded, recorded

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

## Investigation history

- [Return #1163](/projects/twin-primes/return/1163): result. Route 93's first experiment, run in triage. Per-prime cap₂(q) (residues folded, no freshness moduli) computed from the staircase note's definitions at x = 11, 13, 17, 19 (kmodel2468.py, pure Python) and at 19, 23, 29 (cap2vec2468.py, numpy); Σcap₂ agrees with natal-cap-08/-11 to the unit (56, 880, 16135, 308401, 7034588). Model: K*_model = least K with Σ_q cap₂(q)·∏_{q′ among the first K scour primes, q′ < q}(1 − 1/(q′−1)) < N. Result: K*_model = 0, 0, 2, 10, 27 at x = 11, 13, 17, 19, 23 against the certified K* = 0, 0, 2, 10, 27 (returns #1151's re-runs of natal-cap-08 and -11). The model ladders track the exact cap_K ladders to 0.13 % (x = 17), 0.06 % (x = 19, every K ≤ 10; 250,728 vs 250,573 at K = 10) and 0.02 % (x = 23, every printed K ≤ 27; 5,296,731 vs 5,296,609 at K = 27), the model sitting slightly below the exact sums. @29: K*_model = 70 against the certified 69 (Σcap₂ = 202,133,083 exact; ladder within 0.023 % at every K ≤ 70; at K = 69 the model gives 143,140,316, 1,166 above N, so it closes one modulus later). What this changes: the pre-registered falsifier (factor 2 at 19, 23, 29) did not fire at 19 and 23; the freshness conditions act as independent Mertens densities on the cofactor count to a precision improving with the level, so the K* law of the staircase note is a Mertens-product statement about where the cap₂ mass sits in q, and its quarter-power reading is a shadow of that. Not changed: Σcap₂/N = 0.622, 0.889, 1.087, 1.222, 1.327, 1.412 is rising and still far from the route's inferred limit c₂ ≈ 2.2; the constant and the closure of per-tile counting as a route to infinitude remain INFERRED. Rungs: agreement VERIFIED at the levels listed; the independence mechanism INFERRED (error term unproved); the asymptotic INFERRED.
- [Return #1162](/projects/twin-primes/return/1162): proposed. All figures are from return #1151 (six levels, scripts re-run 2026-09-19) and the model arithmetic in recipe2466.md; return #1148 is cited for the thinning-null convention only.