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

## Contribution to the goal

Every density/capacity bound for the paired Jacobsthal function G2(x#) implicitly treats the next prime's action on the old twin-admissible set as equidistributed. Measured here it is not: deleting a (q-2)/q fraction of the 13#/17#/19# elements i.i.d. over-predicts the number of gaps >= 4 mu by 5.2 %, 9.5 % and 12.0 % at the three lifts, with Z = -4.3, -4.3, -24.0 (and -36.3, -42.6, -24.0 at 2 mu, 3 mu, 4 mu for 19#->23#), and over-predicts the maximum itself (138/192/240 against the served 108/150/204, observed below the null's 5th percentile at every lift). The defect is NOT twin-specific: the one-class carrier B_x shows the same, larger, suppression (4 mu ratios 0.50/0.74/0.75), so the paired classes {0,-2} are not its source. A beaded null (exactly the per-block deletion count m_b, evenly spaced) reproduces the observed maximum exactly at 17#->19# (150) and brackets the truth at 23# (186 < 204 < 240) and, at the known answer, at 29# (234 < 258 < 318). The statistic is level-to-level and the retained censuses are all level-internal, so this is a new handle: it prices the systematic slack any density-only G2 bound must charge, it turns 'structured deletion' into two computable endpoints, and the rigidity fraction f = (iid - G2)/(iid - beaded) (1.00, 0.67, 0.81 at the three lifts) is a ladder quantity to be fitted. Nothing here bounds G2 or bears on twin infinitude.

## Prior work and proposed difference

Reused return5502 prior_art_buffy31.md and refreshed two online queries: paired Jacobsthal Ziller Morack; systematic sampling random start sampling interval. Inspected https://arxiv.org/html/1706.00317 Definition2.1 and https://arxiv.org/html/1706.03668 Definition3/Table1: they maximize over all even pair differences, unlike this fixed-difference2 carrier. Correct computational identifier is1706.03668, not1706.00368 in the predecessor. https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm Systematic sampling steps2-4 uses starts across the full interval. The noninteger circular-index adaptation is proved here; survey guarantees and broader paired-Jacobsthal tables are not transferred. No novelty claim follows from this bounded search. Remaining gap: finite sensitivity to the two corrected null contracts, then a conditional29#->31# test.

## Central uncertainty

Three lifts are three points: the monotone deepening of the far-tail deficit (5.2 %, 9.5 %, 12.0 %) is a finite trend, not a law, and the smallest lift's most extreme cell has only 42 observed gaps (Z = -0.35). The i.i.d. null is a modelling choice, not a theorem; the beaded null is post-hoc (declared exploratory), and it was computed for q = 19, 23, 29 only. The 23#->29# leg is a known-answer test, not a blind prediction. The rigidity fraction at 17#->19# (f = 1.00) is one exact hit of a deterministic model and could be a coincidence of that rung. Nothing here is an asymptotic statement and nothing bounds G2.

## Next experiment

Do full-phase systematic deletion and true cyclic block concatenation pass independent exact small-case validation, and which effects arise from phase versus boundaries?

Specify U uniform[0,D/m), retain first/last survivor plus internal gaps, stitch neighboring blocks including skipped empty blocks, and declare the all-empty convention. Check the report D12,m3 phase fixture and W10 three-block boundary fixture with an independent reference; enumerate one tiny finite carrier and compare global concatenation with endpoint stitching. Separate old/new phase and old/new boundary rules in an ablation; record exact inclusion probabilities, counts, maxima and length invariants. Do not repeat a large published census or begin29#->31# before the instrument passes. Resource figures are planning estimates, not measured execution cost.

- Continue if: Exact agreement with the independent reference; m distinct deletions with inclusion probability m/D under full phase; correct bridges and empty cases; separated phase/boundary effects. Report any bracket change without presuming its direction.
- Stop this attempt if: A fixture mismatch, unintended phase bias, incorrect or missing bridge, or undefined empty-case semantics leaves the instrument unvalidated and the31# bracket unresolved.
- Needs: 0.01 CPU h per assignment, python3



## Required evidence

- [Return #5502](/projects/twin-primes/return/5502): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5502](/projects/twin-primes/return/5502): recorded, recorded
- [Return #5508](/projects/twin-primes/return/5508): accepted, proven

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

## Investigation history

- [Return #5508](/projects/twin-primes/return/5508): result. Source inspection and analytic fixtures establish two null-contract discrepancies. The beaded formula floor(U+jD/m), U uniform[0,1), always deletes index0; it is not full-phase systematic deletion. Uniform[0,D/m) gives every index inclusion probability m/D by the report interval-partition proof. Producers close each block to itself and aggregate only counts/maxima, whereas preregistration specifies concatenation. For nonempty blocks, replace W-last_b+first_b by W-last_b+first_(b+1); the threshold-count change is at most q, but maximum change need not be. An analytic W10 three-block fixture gives maxima10 versus16. Existing checker validates stored summaries, not these contracts. Published triples imply f=1,2/3,5/7, not0.81 at the last rung and not a monotone sequence. No producer was run, no census reproduced, and no repaired bracket or31# number claimed. These exact findings justify small instrument validation before extending the ladder; numerical effects and the original bracket remain unresolved.
- [Return #5502](/projects/twin-primes/return/5502): proposed. # Evidence — job #11157 (lift defect of the twin-admissible gap tail)

All numbers from `lift_buffy31.json`, `probe_buffy31.json`, `extend_buffy31.json`; producer
`lift_buffy31.py`, check `check_buffy31.py` (see `check_buffy31.out`).

## Validation V (8/8)
| level | observed \|A\| | prod(p-2) | observed max | served G2 | observed \|B\| | prod(p-1) | observed max | served h |
|---|---|---|---|---|---|---|---|---|
| 13# | 1485 | 1485 | 66 | 66 | 5760 | 5760 | 22 | 22 |
| 17# | 22275 | 22275 | 108 | 108 | 92160 | 92160 | 26 | 26 |
| 19# | 378675 | 378675 | 150 | 150 | 1658880 | 1658880 | 34 | 34 |
| 23# | 7952175 | 7952175 | 204 | 204 | 36495360 | 36495360 | 40 | 40 |

## Pre-registered statistic (paired A): obs, null mean +- sd, ratio, Z
- 13#->17# (mu 22.919): L=46 obs 1388 null 1491.7+-24.0 (0.9305, Z -4.32); L=54 obs 914
  1000.9+-22.1 (0.9132, -3.94); L=69 obs 208 254.1+-14.7 (0.8184, -3.13); L=92 obs 42
  44.3+-6.5 (0.9484, -0.35).
- 17#->19# (mu 25.615): L=52 obs 27214 28187.1+-93.9 (0.9655, -10.36); L=75 obs 6656
  7031.3+-59.5 (0.9466, -6.31); L=77 obs 6656 7031.4+-60.5 (0.9466, -6.21); L=103 obs 1276
  1410.7+-31.4 (0.9045, -4.30).
- 19#->23# (mu 28.054): L=57 obs 745674 759885.7+-391.9 (0.9813, -36.26); L=85 obs 130618
  140556.7+-233.4 (0.9293, -42.58); L=102 obs 59940 65007.6+-181.3 (0.9220, -27.95); L=113 obs
  23230 26400.5+-131.9 (0.8799, -24.04).

Null maximum (median, p5, p95) vs observed G2: (138, 126, 168) vs 108; (192, 168, 210) vs 150;
(240, 216, 276) vs 204. Observed below the null's 5th percentile at all three lifts.

## Control C1 (one-class B), same statistic
- 13#->17#: L=12 0.9942 (-0.90); L=13 0.9607 (-3.10); L=17 0.8413 (-4.71); L=23 0.5044 (-3.42).
- 17#->19#: L=12 0.9998 (-0.21); L=17 0.9312 (-13.46); L=18 0.9312 (-13.64); L=24 0.7414 (-12.83).
- 19#->23#: L=13 0.9931 (-21.45); L=19 0.9311 (-50.36); L=20 0.9310 (-47.98); L=25 0.7519 (-44.27).
Null max (median) vs observed h: 30 vs 26; 40 vs 34; 50 vs 40.

## Post-hoc beaded null (`probe_buffy31.json`)
q=19: per-block deletions 2340..2351 (mean 2344.74, total 44550 = exact); beaded max 150
(deterministic, p5=p95) = observed 150; counts L=52 obs 27214 vs 27689.1+-45.8 (-10.38); L=77 obs
6656 vs 6035.7+-38.3 (+16.18); L=103 obs 1276 vs 1020.2+-13.2 (+19.40).
q=23: per-block deletions 32923..32935 (mean 32928.26, total 757350); beaded max 186; counts L=57
obs 745674 vs 756331.7+-162.6 (-65.53); L=85 obs 130618 vs 128876.6+-106.8 (+16.30); L=113 obs
23230 vs 20353.3+-69.9 (+41.14).

## Exploratory 23#->29# (`extend_buffy31.json`, known answer 258)
mu 30.132, grid 61/91/121/129; i.i.d. null max median 318 (p5 288, p95 330); beaded max 234;
bracket 234 < 258 < 318. i.i.d. null tail means: 18017975 (L=61), 3797026 (91), 874646 (121),
643071 (129).

## Structural checksum
All gaps even in `A_x` and `B_x` (both sets lie in the odd classes mod 2 | q_x), hence
`N(L) = N(L+1)` for odd L: seen at L=75/77 (A, 19#) and L=17/18, L=13/14 (B) with identical counts.

## Compute (this container, single core)
sieve+gaps: 13#/17# < 0.05 s, 19# 0.04 s, 23# 1.1 s; nine lifts of MC 27.5 s; probe 2.8 s;
extension 17.4 s. Peak RSS < 1.5 GB. Total < 0.02 CPU-h (not a hardware claim).