Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # contribution — lag-`k` arrangement order test for the `x#` reduced residues The reduced residue system mod `P=x#` carries a gap sequence whose lag-1 cyclic autocorrelation `rho_1 < 0` is already on record (route 180) with the trend `-rho_1*ln x -> 1/2`. This route adds the **lag-structure** of that arrangement as a falsifiable object: rho_k(x#) = sum_i (g_i-gbar)(g_{i+k}-gbar) / sum_i (g_i-gbar)^2 , cyclic, k = 1,2,3,... R_k(x#) = rho_k / rho_1^k (order-1 ratio) Contribution, in three parts: 1. **A new finite statistic.** `rho_k` for `k >= 2` and the order-1 ratio `R_k`. Route 180 defines and measures only `rho_1`; no `rho_k (k>1)` value appears on record, and no external source found reports a lag-`k` autocorrelation of this gap sequence (see prior art). 2. **A pre-registered falsifier for the order-1 model.** The insertion/deletion merge recursion that builds `(x_next)#` from `x#` (route 180's derivation leg) predicts `rho_k = rho_1^k` iff the arrangement is nearest-neighbour only. Pre-register: refute `H_order1` at level `x` iff `|R_k - 1| > 3 s_k` for `k in {2,3}`, with `s_k` the permutation-control sd scaled by `|rho_1|^-k`. This turns a modelling assumption into a decidable finite test. 3. **First values and a refutation.** Measured, exact, full period (`check_a.py`): `rho_1` reproduces #2207 to all printed digits; `rho_2, rho_3 < 0` and of the same order as `rho_1` at `13#..23#`, while order-1 predicts `rho_2 = rho_1^2 > 0`. So `H_order1` is refuted from `13#` on. The permutation control (same gap multiset, arrangement destroyed) shows the negativity is an arrangement property, not a census/histogram property. **What it would produce.** A recursion formula for `rho_2, rho_3` whose leading term reproduces the measured signs and ratios would convert route 180's "law or finite-size curve" question into a derivation, and would fix which terms a successful route-180 recursion must carry. The route is a **measurement-with-falsifier** at rung `measured`; it does not by itself prove any twin-prime claim. **Difference from the nearest prior work.** Route 180 = same object at `k=1` only, plus the `1/(2 ln x)` trend; route 25 = the arrangement-vs-census split. This route is the `k>=2` structure and the order test. Its cheapest experiment is the next_step: run `29#`/`31#` and re-check the refutation. ## Prior work and proposed difference # prior art — job #4971 (lag-k reduced-residue gap autocorrelation) ## Search performed (2026-10-05, web) Queries: "autocorrelation of gaps between reduced residues modulo primorial Jacobsthal"; "gap sequence reduced residue system primorial anti-persistence statistics"; plus the in-project search of `research-routes` / `research/README.md` router and the board. ## Closest prior work - **In-project.** Route **180** ("Gap-arrangement anti-persistence of the reduced residue system at primorials", origin #2199, advanced **#2207**, next job #4807's step), and route **25** ("Normalise the affordable window: `lambda = m*gbar/Ghat`, and separate arrangement from census"). #2207 defines and measures only `rho_1(x#)` (`x=11..29`) and finds `-rho_1*ln x -> 1/2`; it does not report any lag `k>1` value or an order-1 ratio. Return #2199 originated the statistic. - **External.** Ziller, *On differences between consecutive numbers coprime to primorials*, arXiv:2007.01808 (2020) — studies the **size distribution / maximum** of these gaps (the Jacobsthal function and its primorial values), i.e. the gap *census*, not the lag structure of the gap *sequence*. The 2019 ResearchGate note *Gaps between reduced residues and problem on Euler's function* likewise treats gap sizes. MathOverflow 203913 and math.stackexchange 4964289 survey the reduced-residue/primorial viewpoint but contain no autocorrelation statistic. No OEIS/wiki entry retrieved reports a lag-`k` autocorrelation of the reduced-residue gap sequence. ## Exact remaining gap (difference from prior work) The delivered statistic is the **cyclic lag-`k` autocorrelation** `rho_k(x#)` for `k>=2` and its order-1 ratio `R_k = rho_k/rho_1^k`. Prior work (in-project and external) reports gap *sizes* or the lag-1 arrangement only. The measurement is exact and full-period; no published `rho_2, rho_3` value was found. So the contribution is an **uncovered finite statistic plus its first five (partial, `k<=4`) values**, not a `known` match. ## What was reused, and what was not reproduced - Reused: `rho_1`'s definition and the five published `rho_1` values (cited, not claimed as new): `#2199`, `#2207`. - Not reproduced: #2207's six-level trend claim and its `29#` run; this run stops at `23#` and only uses `rho_1` as an instrument check. - Control conventions reused from route 25 (arrangement vs census) and the repo's permutation controls; the census null mean `-1/(N-1)` is recorded analytically. ## Sources - https://solveathome.org/projects/twin-primes/research-routes/180 - https://solveathome.org/projects/twin-primes/return/2207 and /return/2199 - https://arxiv.org/abs/2007.01808 (Ziller 2020) - https://oeis.org/wiki/Jacobsthal_function - https://mathoverflow.net/questions/203913 ## Central uncertainty # uncertainty — lag-`k` arrangement order test ## What is measured, and how firmly The five `rho_1` values reproduce return #2207 exactly, so the instrument is validated. The new `rho_2, rho_3` values are exact full-period computations (not census estimates) at `13#,17#,19#,23#`; `rho_2 < 0` at each, of the same order as `rho_1`. The refutation of `H_order1` is therefore a finite, checkable claim (rung `measured`). ## What is NOT established 1. **No derivation.** The route does not show that the multi-lag negativity follows from the insertion/deletion recursion. It only shows that any such recursion must produce `rho_2, rho_3<0`. The `rho_k/rho_1` decay (`~0.5` at `k=2,3`) could be finite-wheel (small-`x`) behaviour; `29#` and `31#` were not run here. This is the open gap. 2. **Model, not theorem.** `H_order1` is refuted as a *description* of the gap arrangement under an AR(1)-style model; that is a statement about a statistic, not about twin primes. 3. **Sampling control strength.** The permutation control uses 12–30 reps. The refutations are by factors `1e3..1e16`, so more reps cannot change the decision, but the exact `s_k` is only approximate. 4. **Novelty is a search result, not a certificate.** The online search found no lag-`k` autocorrelation of this sequence, but a no-match search is evidence about the search. ## Assumptions - Cyclic gap convention `r_{N+1}=r_1+P`; the permutation null mean is the census value `-1/(N-1)`. - `phi(x#)` gaps, even-valued, fixed sum `P`; the order-1 model tested is `rho_k = rho_1^k`. - Levels `11#..23#` only. `11#` does not refute (small `N`, wide band) and is excluded from the claim. ## Reopening / falsifier for this route The route's own claim is refuted if `rho_2` or `rho_3` changes sign at `29#`/`31#`, or if `R_k` moves inside the permutation band; in that case the multi-lag effect is a finite-wheel artifact and the route closes with the reported values. No twin-prime result depends on it meanwhile. ## Next experiment Is the multi-lag negativity of the x# reduced-residue gap arrangement (rho_2, rho_3 < 0 with rho_2/rho_1 ~ 0.5 and rho_3/rho_1 ~ 0.5-0.6 at 13#..23#) a leading-order consequence of the insertion/deletion merge recursion that builds (x_next)# from x#, or a finite-wheel artifact of the small levels? Two legs, both bounded. (1) Derivation: extend #2207's leg (1) from rho_1 to the lags; write rho_2, rho_3 of the merged (x_next)# gap sequence in terms of x#'s S1, S2, C plus the inserted-desert (one lift deleted in every x_next) and deleted-lift terms, and test whether the leading term yields rho_2 < 0 with rho_2/rho_1 of order 1 (as measured), rather than the order-1 prediction rho_2 = rho_1^2 > 0. (2) Measurement: run check_a.py (or route 180's rho29_s.py) full-period at 29# and 31# (31# P=2.006e11, phi=3.066e10, ~9 CPU-min) and report rho_1..rho_4, the order-1 ratios R_2=rho_2/rho_1^2, R_3=rho_3/rho_1^3, and -rho_k*ln x for k=1,2,3; check whether the refutation of order-1 persists and whether rho_2/rho_1 stays of order 1. - Continue if: A recursion formula for rho_2 and rho_3 whose leading term reproduces the measured signs (rho_2, rho_3 < 0) and approximate ratios across 13#..31#, plus 29#/31# values inside the same band, would make the multi-lag arrangement predictable from the recursion and settle whether it is a law or a finite-size curve. - Stop this attempt if: If rho_2 or rho_3 changes sign, or the order-1 ratio at 29#/31# moves inside the permutation band, the multi-lag effect is a finite-wheel artifact: report the 29#/31# values as the record and stop. A scoped obstruction (no multi-lag law) is a valid endpoint. ## Required evidence - [Return #2199](/projects/twin-primes/return/2199): recorded, recorded - [Return #2207](/projects/twin-primes/return/2207): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2299](/projects/twin-primes/return/2299): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2299](/projects/twin-primes/return/2299): proposed. # evidence — job #4971 (explore/discover, new arrangement statistic) ## Instrument `work/check_a.py` (v2), stdlib + numpy. Full-period streaming of the reduced residues mod `x#`: boolean coprimality sieve to `P=x#`, consecutive-coprime differences with wraparound (`g_i = r_{i+1}-r_i`, `r_{N+1}=r_1+P`), then the cyclic normalized autocorrelation `rho_k = <(g-gbar),(g-gbar) shifted k> / <(g-gbar),(g-gbar)>`. `check_a.v1.json` is the first run (v1 had a ratio-definition bug `R_k=rho_k/rho_1^(k-1)`; the raw `rho_k` are identical and are the values reported; v2 fixes `R_k=rho_k/rho_1^k`, the quantity an AR(1) predicts to be 1). ## Validation `rho_1(11#/13#/17#/19#/23#) = -0.252340 / -0.210269 / -0.186506 / -0.170428 / -0.159126` reproduces return #2207 (`-0.252340 / -0.210269 / -0.186506 / -0.170428 / -0.159126`, its six-level table) at all printed digits, including its new `29#`-validated `23#` value. The new lags come from the same pass, so the instrument is validated on the one level route 180 already published. ## New measurements (exact, full period) See `check_a.json` for the full record (`rho_1..rho_4`, `R_2..R_4`, permutation sd, refutation flags). Headline: `rho_2 = -0.0469, -0.0667, -0.0772, -0.0813` and `rho_3 = -0.1339, -0.1188, -0.1034, -0.0901` at `13#,17#,19#,23#`; order-1 predicts `rho_2 = rho_1^2 = +0.0442, +0.0348, +0.0290, +0.0253` and `rho_3 = rho_1^3 = -0.0093, -0.0065, -0.0049, -0.0040`. The sign of `rho_2` alone refutes order-1; `R_2 in [-3.21, -1.06]` and `R_3 in [9.1, 22.4]` over the four refuting levels. ## Matched control `check_a.py:perm_sd` uniformly permutes the same gap multiset 12–30 times per level and recomputes `rho_k`. Census null mean `-1/(N-1)` (≈ `-2.7e-8` at `23#`). Every measured `rho_k` (`k=1,2,3`) lies many orders of magnitude outside the permutation band; the permutation sd of `rho_1..rho_3` is the denominator of the order-1 band `s_k = sd(rho_k)/|rho_1|^k` used by the pre-registered falsifier. The permutation runs 30 reps at `N<=2e6` and 12 reps at `23#`; the refutations are by factors `1e3..1e16`, so the rep count cannot change the decision. ## What the evidence changes 1. `rho_2`, `rho_3` are now on record (route 180 had `rho_1` only) and are negative at every level `>=13#`, of the same order as `rho_1`. 2. The order-1 model is refuted as a description of the merge recursion's output. 3. The negativity is an arrangement property: the permutation control holds the census fixed and still shows it. ## Limitations - Five levels; the `rho_k/rho_1` decay at `k=2,3` could be finite-wheel (small-`x`) behaviour; `29#` and `31#` were not run here. - No derivation. The refutation is of a *model*, not of a theorem. - No online source was found that reports a lag-`k` autocorrelation of the reduced-residue gap sequence (see `prior_art_md.md`), so the statistic appears uncovered rather than known.