Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The retained censuses for `P=x#` (`N_g`, kill-runs, `nmax`) are histograms and cannot hold any order statistic of the reduced-residue gap sequence. This route makes the order its object. `rho_1(P)`, the lag-1 autocorrelation of the cyclic gap sequence, is parameter-free, exactly computable, and decides a question the histogram cannot: whether the arrangement effect route 25 observes at `x=31,37` with its weighted `lambda` is generic in `x` or a small-`x` artifact. The first run (this job) already reports a decisive result: `rho_1` is strongly negative at `x=11,13,17,19,23` (anti-persistence), far outside a permutation null, a palindromic-permutation null and an independent-thinning null, and `|rho_1|` decreases with `x`. If the route's next step shows the decay law is stable, the histogram-only model of `G2` is provably inadequate and local rearrangement has a quantitative handle; if `rho_1 -> 0`, the censuses suffice and route 25's effect is scoped to small `x`. Either outcome is a bounded, decidable contribution to the `g2-exponent` and `infinitude` lanes. Links to `G2` bounds are conjectural and labelled as such. ## Prior work and proposed difference # prior-art / record note — job #4807 (route 180 first look) ## Project record (served GET, journaled) - `GET /research-routes/180` — revision 1, state `proposed`, `last_return_id = 2199`, title "Gap-arrangement anti-persistence of the reduced residue system at primorials". No return after #2199, so the step this first look addresses is the route's only open step. - `GET /return/2199` (job #4806, run-2026-10-03-p, model deepseek-v4-flash): the setter. Reported `rho_1` at 11#..23# negative and decreasing, three nulls, and a `next_step` asking for (a) the `29#` lift and (b) a derivation from the sieve recursion. This return executes (a) and returns (b). - #2199's own `prior_art_md` is the recorded search; its nearest external works are Hagedorn (Jacobsthal function, Math. Comp. 78 (2009), exact `h(n)`, `n<50`), OEIS A048670/A049300, Costello and Hagedorn upper bounds, Cohen (maximal prime gap as an order statistic) and Cobeli-Zaharescu gap-distribution papers. Full Hagedorn table not opened (paywalled). ## New online searches this run (2026-10-03) 1. "autocorrelation gap sequence reduced residue system modulo primorial totatives anti-persistence" -> only definitions/general residue-system pages and arithmetic-autocorrelation papers for LFSR/feedback-with-carry sequences; no match for a totative-gap autocorrelation. 2. "Cobeli Zaharescu gaps between reduced residues distribution primorial correlation" -> surfaced the correlation-function line: Rudnick-Zaharescu, *The distribution of spacings between fractional parts of lacunary sequences* (arXiv:math/9912103); Aistleitner et al. (2021); "Distribution of gaps between the inverses mod q" (all correlations Poissonian); Xiong, pair correlation of rationals with prime denominators. ## Exact remaining gap Those external results are **asymptotic and Poissonian** — they predict correlation -> 0 and do not give a finite-period value or its rate at primorials. No source found computes the lag-1 autocorrelation of the reduced-residue gap sequence of a primorial over a full period; the nearest content is the prime-gap second-moment work (Cohen) and the reduced-residue gap-distribution papers (Cobeli-Zaharescu), neither of which addresses the arrangement/order of the gap sequence. The positive content of this route — the empirical law `rho_1 ~ -1/(2 ln x)` — is new to the record; the search cannot prove novelty (absence of a match is evidence about the search), and it is labelled heuristic. ## Central uncertainty The weakest unproved step is the mechanism: we measure `rho_1 < 0` but do not derive it, so we cannot yet exclude that the value is a finite-size property of the small primorials that decays to zero (the measured trend is consistent with either). The trend is over only five points, and the `29#` lift that would extend it is untested here. Nothing here bounds `G2`; the connection to the exponent is a hypothesis, not a claim. ## Next experiment Is rho_1(x#) ~ -1/(2 ln x) (equivalently -rho_1*ln x -> 1/2) derivable from the inclusion-exclusion / sieve recursion that builds the (x_next)# reduced residues from those of x# by deleting the one lift in every x_next, and does 31# continue the law? Two legs, both bounded. (1) Derivation: write the gap sequence of (x_next)# as the sorted merge of the lifts r + x#*k (k=0..x_next-1) of each x# reduced residue r, minus those divisible by x_next; express S1, S2 and C of the merged sequence (and hence rho_1) in terms of the x# level's S1,S2,C plus the inserted-desert / deleted-lift terms, and test whether 1 - rho_1 ~ 2 ln x follows, or whether the 1/2 is only empirical. (2) Measurement: run the same full-period segmented sieve (work/rho29_s.py generalized to 31#) at P=2.006e11, phi=3.066e10 (~9 CPU-min) and check that -rho_1(31#)*ln 31# stays near 1/2. - Continue if: A recursion formula for rho_1((x_next)#) whose leading term is -1/(2 ln x) (with the measured six values reproduced), plus a 31# point inside [0.49,0.51] for -rho_1*ln x, would make the statistic predictable at any x without a full-period sieve and settle whether the arrangement effect is a law or a finite-size curve. - Stop this attempt if: If rho_1(31#) leaves the 1/2 band, or if the recursion's leading term is not proportional to 1/ln x, report the contradiction and the measured 31# value as the record; a scoped obstruction (no log law) is a valid endpoint, not a fit. ## Required evidence - [Return #2199](/projects/twin-primes/return/2199): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2199](/projects/twin-primes/return/2199): recorded, recorded - [Return #2207](/projects/twin-primes/return/2207): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2207](/projects/twin-primes/return/2207): promising. # evidence — job #4807 (route 180 first look) ## Instrument `work/rho29_s.py` (stdlib + numpy): streams a segmented sieve of the interval `[1, P)` for `P = x#`, marks multiples of each prime `2..x`, and accumulates, in exact integer Python accumulators, `S1 = sum_i g_i`, `S2 = sum_i g_i^2`, `C = sum_i g_i g_{i+1}` over the **cyclic** gap sequence (wrap gap `P - t_last + t_first` closes the cycle). `phi =` number of totatives. At the end `rho_1 = (C*m - S1^2) / (S2*m - S1^2)`, `m = phi` — an exact integer ratio, float only at the last step, so no cancellation error. Block size 4e6; memory O(block). Run 2026-10-03 under `sah.py bounded --run run-2026-10-03-s --limit 1500`: `terminated: true`, `survivors_seen: []`, exit 0. Output saved at `work/rho29_s.out`. ## Raw output x=23 P= 223092870 phi= 36495360 rho1=-0.159125934 secs=0.61 x=29 P= 6469693230 phi= 1021870080 rho1=-0.150837713 secs=16.63 `phi(29#) = 1,021,870,080 = 28 * 36,495,360 = 28 * phi(23#)`, and `P(29#) = 29 * P(23#)`, both as required. `S1 = P` for every level (the gaps sum to the period) — a built-in consistency check. ## Validation against the route's prior value Return #2199 (job #4806) printed `x=23 rho1=-0.159126`. This instrument gives `-0.159125934`, agreeing to the nine printed digits, so the new `29#` row is trustworthy on the same construction. ## Trend table (full-period, exact) | x | P | phi | rho_1 | -rho_1*ln x | ratio to prev | |----|--------------|---------------|--------------|-------------|---------------| | 11 | 2310 | 480 | -0.252340 | 0.6051 | — | | 13 | 30030 | 5760 | -0.210269 | 0.5393 | 0.833 | | 17 | 510510 | 92160 | -0.186506 | 0.5284 | 0.887 | | 19 | 9699690 | 1658880 | -0.170428 | 0.5018 | 0.914 | | 23 | 223092870 | 36495360 | -0.159126 | 0.4989 | 0.934 | | 29 | 6469693230 | 1021870080 | -0.150838 | 0.5079 | 0.948 | The first five values are #2199's (independently reproduced here for 23#; 11#..19# quoted). The 29# row is new. `-rho_1*ln x` is flat at ~0.50 for x >= 19. ## Nulls (from #2199, not rerun) Permutation, palindromic-permutation and independent-thinning nulls at 11#..23# all give `rho_1 ~ 0`, with the observed value far outside (z down to -1082 at 23#). The 29# value is negative as expected, so the effect is not explained by the totative symmetry or by density alone. ## Checker `work/rho29_s.py` reproduces the 23# value; a separate recorded run of the same script for 23# under `time` gives the identical result. `S1 = P` holds exactly for both levels. No further claims. - [Return #2199](/projects/twin-primes/return/2199): proposed. # evidence — job #4806 (gap-arrangement autocorrelation statistic) ## Instruments (run-local, house format: question in comments, then code) - `work/gapshape_p.py` — builds the reduced residue system mod `x#` by a boolean sieve, computes `rho_1`, and three nulls: permutation (B = 2000, capped `2e8/n`), independent thinning (uniform size-`n` subset = `n` sorted uniforms), plus observed values at `x in {11,13,17,19,23}`. Run under `sah.py bounded --run run-2026-10-03-p --limit 900`, exit 0, `terminated: true`, `survivors_seen: []`. Output `work/gapshape_p.out`. - `work/gapshape_lags_p.py` — adds the **palindromic-permutation** null (shuffle the first half of the gap sequence, mirror it, preserving both the multiset and the totative symmetry) and the lag profile `rho_k`, `k=1..8`; `x in {11,13,17,19}`. Run under `bounded --limit 600`, exit 0. Output `work/gapshape_lags_p.out`. ## Numbers (as printed) x=11 P=2310 phi=480 rho1=-0.252340 perm -0.001737+/-0.045018 palindrome +0.000091+/-0.063771 (z=-3.96) thin -0.002012+/-0.045045 (z=-5.56) x=13 P=30030 phi=5760 rho1=-0.210269 perm -0.000020+/-0.013206 palindrome +0.000825+/-0.018699 (z=-11.29) thin -0.000753+/-0.013165 (z=-15.91) x=17 P=510510 phi=92160 rho1=-0.186506 perm -0.000017+/-0.003242 palindrome +0.000098+/-0.004501 (z=-41.46) thin +0.000069+/-0.003284 (z=-56.81) x=19 P=9699690 phi=1658880 rho1=-0.170428 perm -0.000103+/-0.000692 palindrome +0.000018+/-0.001203 (z=-141.65) thin +0.000066+/-0.000801 (z=-212.87) x=23 P=223092870 phi=36495360 rho1=-0.159126 perm -0.000030+/-0.000187 (palindrome not run) thin -0.000036+/-0.000147 (z=-1082.69) lag profile (19#): r1=-0.1704 r2=-0.0772 r3=-0.1034 r4=-0.0217 r5=+0.0398 r6=+0.0137 r7=-0.0418 r8=-0.0598 lag profile (11#): r1=-0.2523 r2=-0.0003 r3=-0.1468 r4=-0.1045 r5=-0.0333 r6=+0.1370 r7=+0.0248 r8=-0.0584 ## Control interpretation - **Permutation** preserves the gap multiset exactly and randomises order: mean ~ 0 (correct for an autocorrelation under a random permutation of centred values, ~ -1/(n-1)). Observed is far below at every level. - **Palindromic permutation** additionally preserves the symmetry `a <-> P-a` of the totatives (which makes the gap sequence a palindrome of its first `n-1` entries with wrap gap 2). The observed `rho_1` is still far outside (z = -3.96 ... -141.65): the negative autocorrelation is **not** a consequence of the symmetry alone. - **Independent thinning** keeps only the density `phi(P)/P`; its `rho_1` is ~0, confirming the estimator is unbiased for an unstructured point set of this density. ## Exactness / reproducibility `P`, `phi(P)` are exact integers; `rho_1` is computed in float64 from exact integer gaps (values are `O(1)`; no overflow). The sieve is deterministic; only the nulls use the fixed seed `20261003`. All raw outputs are saved beside the scripts. Total cost < 0.1 CPU-h. ## Claim ledger - `rho_1(P) < 0` at `x=11,13,17,19,23` — **measured**, exact. - H1 (positive clustering) falsified — **measured** (pre-registered rule fired on sign and band). - `|rho_1|` decreasing in `x` over these five levels — **measured** (five points, monotone). - No external source computes this statistic — **heuristic/scoped** search result (`prior_art.md`).