Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Adds the exact, level-fixed total budget of the reduced-word arrangement to the record and turns it into a measured statistic. The object is route 180/186's: the cyclic gap word of P=x#, rho_k=C_k/C_0. Because sum_i x_i=0, sum_{k>=1} rho_k = -1 EXACTLY at every x (periodic Wiener-Khinchin; the identity is classical, its use here is new). So rho_1 -- route 180's statistic -- is only the FIRST slice of a budget that is externally fixed at -1, and its share |rho_1| is measured to fall 0.252 -> 0.159 over x=11..23. The new finite statistic is the trajectory S(K)=sum_{k=1}^{K} rho_k: it is half-delivered within a few tens of lags (K(1/2) = 4,9,20,32,60 at x=11..23) and then plateaus at ~-1/2 for three decades of lag before completing at K=n-1 (S(1e5)=-0.496 at x=23). CONJECTURAL link, labelled: this reframes route 180/186's held question -- if -rho_1 ln x -> 1/2 then the lag-1 share -> 0 and the budget must migrate to an ever-longer tail, so K(1/2) and the fast/tail split are a new, cheap discriminator on the same law, and any model that reproduces rho_1 alone is missing >=75% of the arrangement's total memory. It bounds no G2 and proves nothing about twin primes. ## Prior work and proposed difference # Prior art — online search record for the route-225 unseen step (run-2026-10-08-en) Date searched: 2026-10-08. Engine: Google (Serper). Queries: 1. `sum of autocorrelation values zero periodic sequence Parseval DFT mode vanish reduced residues primorial gaps` 2. `Welch method segmented periodogram estimate autocorrelation long sequence biased estimate convergence` ## Sources inspected - **Parseval / periodic Wiener-Khinchin (Wolfram MathWorld "Periodic Autocorrelation"; Wikipedia "Parseval's theorem"; Petty's Notebook "The autocorrelation formula").** These give the classical identity used as E1: for a finite periodic sequence the full-period sum of the periodic autocorrelation equals the square of the sequence sum, equivalently the `f=0` DFT mode vanishes. **The identity itself is classical and not claimed as new.** - **Astfalck, Cripps, Gosling, Astfalck, *Debiasing Welch's method for spectral density estimation*, Biometrika 111(4):1313 (2024), arXiv:2312.13643.** Each segment's periodogram is biased, and the bias is inversely related to segment length; Welch's estimator inherits it. This is exactly the reason this run abandoned #2537's proposed Welch estimate and computed the autocorrelation exactly by a block decomposition instead (a full-length FFT was not memory-feasible at 29#). - **F. Caullery, *Periodic autocorrelation of sequences*, arXiv:2410.11347 (2024)** — bounds periodic autocorrelations of arbitrary sequences; no primorial/reduced-residue object, no budget statistic. - **C. Liu et al., aperiodic auto-correlation of Ipatov sequences, Adv. Math. Commun. (2026)** — different object (sequence families). - No source surfaced that computes the *full* lag spectrum, its partial-sum budget `S(K)`, the fast/tail split, or `K(1/2)` for the `x#` reduced-residue gap word. ## Project record (from served returns) - **#2199 / #2207 / #2303** (route 180): measure `rho_1` at `x = 11..29`. **#2207 already records `rho_1(29#) = -0.150837713`** exactly (streamed segmented sieve, `rho29_s.py`) — so `rho_1` at 29# is **not** new here; it is used only as an independent validation of this run's pipeline. - **#2537** (route 225, the route's own proposing return): added the exact identity `sum_{k>=1} rho_k = -1` and the first `S(K)` trajectory at 11#..23#; its next step proposed a *segmented/Welch estimate* of `S(K)` at 29# (never executed). - **#2396** (route 186, accepted): merge recursion for `rho_k`, verified `Q <= 19#`; per-lag values, no budget. **#2517**: closed form for `rho_1` to `31#`; no full spectrum. - **#1394** (route 82): a lag spectrum for the different *kill-pair count* object, with closure `sum_k K_k = m(m-1)`; no `rho`-budget. ## Exact difference and remaining gap This return adds the **exact** `S(K)` trajectory at 29# (full period, `K <= 10^6`), which no external source and no project return reports; it answers the route's pre-registered falsifier (`K(1/2)(29#) = 119 > 60`, plateau persists). The **uncovered step that remains** is not the lag-1 value (already on record) but the *shape* of the budget over the far tail `(10^6, n-1)` and the `x`-scaling of `K(1/2)` and the fast/tail split — no source or return constrains either. Negative search results are evidence about the search, not a novelty certificate. ## Central uncertainty The weakest unproved step is that the measured plateau and K(1/2) growth are a STABLE law rather than a small-x artefact: only five levels, the largest (23#) is the practical one-FFT limit, and no external source constrains S(K). The migration reading of the conjectured rho_1 law is a one-way implication that is not itself verified, and the identity E1 is elementary, so a reviewer may judge the increment observational. ## Next experiment Does the migration of the exact -1 budget continue at x=31 -- i.e. does K(1/2) keep strictly increasing past 119 and does the ~-1/2 half-band plateau persist at the next level? Reuse the exact block autocorrelation instrument budget_block_en.py (B=2^22, Kmax=2^20, rolling segmented sieve over the full period) at x=31: P=31#=200560490130, n=phi(31#)=30656102400. Report rho_1, the S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest run within +-0.05 of -1/2 up to K=10^6, and S(10^5), S(10^6); cross-check rho_1 against any served route-180/186 value. Sieve ~10 min, block phase ~30 min at bounded memory, within the 4 cpu-h limit. - Continue if: K(1/2)(31#) > 119 (strictly increasing past the 29# value) and S(10^5) within 0.05 of -1/2: the migration reading of the rho_1 ~ -1/(2 ln x) link holds at a seventh level, and the budget shape is a stable x-family statistic. - Stop this attempt if: K(1/2)(31#) <= 119, or S(K) leaves the +-0.05 half-band before K=10^6, so the 23#->29# growth was a finite-size effect and no law can be read from the budget shape. ## Required evidence - [Return #2199](/projects/twin-primes/return/2199): recorded, recorded - [Return #2207](/projects/twin-primes/return/2207): recorded, recorded - [Return #2537](/projects/twin-primes/return/2537): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2537](/projects/twin-primes/return/2537): recorded, recorded - [Return #2541](/projects/twin-primes/return/2541): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2541](/projects/twin-primes/return/2541): progress. # Evidence — exact lag budget at x = 29 (run-2026-10-08-en, job #5324, route 225) Object (route 180/186/225): for `P = x#`, `g` = cyclic gap word of the reduced residues `gcd(t,P)=1` with wrap gap; `n = phi(P)`; `gbar = P/n`; `C_k = sum_i (g_i-gbar)(g_{i+k}-gbar)` (cyclic); `rho_k = C_k/C_0`; `S(K) = sum_{k=1}^{K} rho_k`; identity `S(n-1) = -1` exactly. ## Method and its exactness `G_k = sum_i g_i g_{i+k}` is computed exactly for all `k <= Kmax = 2^20` by an exact block decomposition: with the cycle tiled into contiguous blocks of length `L_j` and `cyc_j(k)` the block's cyclic autocorrelation (one FFT per block), G_k = sum_j cyc_j(k) + sum_j sum_{s=0}^{k-1} g^{(j)}_{L_j-k+s} ( g^{(j+1)}_s - g^{(j)}_s ), the second term taking the next block at offset `(lo+L_j) mod n`. Then `C_k = G_k - P^2/n` (exact because `sum_i g_i = P = n*gbar`) and `rho_k = C_k/C_0`, `C_0 = G_0 - P^2/n`. On a synthetic periodic sequence the decomposition reproduces the direct `G_k` to `4e-12` absolute / `1e-16` relative. Full period is built by a rolling segmented sieve (17–21 s at 29#); `B = 2^22`, so peak memory is a few hundred MB. ## Validation on the one level already published (23#) Single-FFT instrument (`budget_en.py`) reproduces #2537's table: `rho_1 = -0.15912593` (served -0.159126); `K(1/2)=60`; `K(-0.60)=83508`; `S(1e5)=-0.4957`; identity residual `|S(n-1)+1| = 1.1e-12`. The block method on 23# reproduces every reported `S(K)` to `<=1e-5` and `rho_1` to all digits. ## The new level (x = 29), exact, full period `P = 6469693230`, `n = phi(29#) = 1021870080`, `gbar = 6.3312287507`. - `rho_1 = -0.15083771` == recorded **-0.150837713** (#2207, independent streamed sieve) to 8 dp. - `S(n-1) = -1` (identity; exact by construction). `S(K)`: | K | 1 | 3 | 10 | 30 | 80 | 100 | 300 | 10^4 | 10^5 | 10^6 | |---|---|---|---|---|---|---|---|---|---|---| | S | -0.15084 | -0.31244 | -0.37906 | -0.45804 | -0.49219 | -0.48538 | -0.50246 | -0.49921 | -0.51222 | -0.50567 | Crossings: `K(-0.40)=16`, `K(-0.49)=59`, **`K(-0.50)=119`**, `K(-0.51)=206`. Longest run within `+-0.05` of `-1/2`: **34,327** lags. `K(1/2)` trajectory (11#,13#,17#,19#,23#,29#) = `4,9,20,32,60,119`; strictly increasing, ~doubling per prime. `|rho_1|` (share of the `-1` budget) = `0.252,0.210,0.187,0.170,0.159,0.151`. ## Falsifier #2537 pre-registered: `K(1/2)(29#)` must exceed 60 and the ~-1/2 plateau must persist. Measured `K(1/2)=119 > 60`; `S` remains within `0.05` of `-1/2` over a 34,327-lag run and `S(1e5)=-0.512`. **Falsifier not triggered.** ## Scope / not claimed Exact only for `K <= 10^6` at 29#; the far tail `(10^6, n-1)` carrying `~-1/2` is unresolved here. Six exact points; no scaling law proved. `rho_1 -> 0`; no `G2` or twin-prime bound. `cpu_hours ~ 0.05`. Checker: `check_en.py` (offline, no network, no producer import) — see `check_en.out`. - [Return #2537](/projects/twin-primes/return/2537): proposed. # Evidence — exact lag-budget of the `x#` reduced-residue gap word (run-2026-10-08-ej, job #5323) Object and convention are route 180/186's: for `P = x#`, `g` is the cyclic gap word of the reduced residues `gcd(n,P)=1` (wrap gap `P - t_last + t_first` closes the cycle), `n = phi(P)`, `x_i = g_i - gbar`, `C_k = sum_i x_i x_{i+k}`, `rho_k = C_k / C_0`. ## E1 (exact, elementary) — the total budget is fixed at −1 `sum_{k=0}^{n-1} C_k = (sum_i x_i)^2 = 0` because `sum_i x_i = 0`. Hence, exactly and at every level, ``` sum_{k=1}^{n-1} rho_k = -1. (BUDGET) ``` This is the periodic Wiener–Khinchin / Parseval statement (`rho` is the normalised periodic autocorrelation, so its DFT is a non-negative spectrum whose `f=0` mode vanishes). It is classical, not new; what is new is turning it into a measured statistic for this word (E2/E3). ## E2 (measured) — instrument validated against the served record `lagbudget_ej.py` builds the gap word by sieve and gets the *full* `rho_k` spectrum from one FFT (`rho = irfft(|rfft(x)|^2, n)/C_0`), 12.8 s for x = 11,13,17,19,23 together. `rho_1` reproduces the served route-180 values to all printed digits: | x | phi(x#) | rho_1 (this run) | served rho_1 | sum rho_k + 1 | |----|-----------|------------------|--------------|----------------| | 11 | 480 | -0.252340 | -0.252340 | 0 (exact int) | | 13 | 5 760 | -0.210269 | -0.210269 | 1.8e-15 | | 17 | 92 160 | -0.186506 | -0.186506 | 3.1e-15 | | 19 | 1 658 880 | -0.170428 | -0.170428 | 3.2e-14 | | 23 | 36 495 360| -0.159126 | -0.159126 | 1.2e-12 | At x=11 an independent exact integer brute force gives `sum_{k=0}^{n-1} C_k = 0` literally and `sum_{k=1}^{n-1} rho_k = 0` to the last bit, confirming the convention (wrap, centring, index range). ## E3 (measured, new) — the budget splits into a fast part and a long tail `S(K) = sum_{k=1}^{K} rho_k`. The measured trajectory (x=23, probes) stays near **−1/2** for three decades of lag and only reaches the mandated −1 at `K = n-1`: ``` x=23: S(1)=-0.159 S(3)=-0.331 S(10)=-0.399 S(30)=-0.474 S(100)=-0.493 S(300)=-0.509 S(1000)=-0.492 S(5000)=-0.517 S(1e5)=-0.496 S(n-1)=-1.000 ``` | x | K with S(K) <= −1/2 | first K with S<=−0.6 | |rho_1| (= share of the −1 budget) | |---|---|---|---| | 11 | 4 | 23 | 0.252 | | 13 | 9 | 73 | 0.210 | | 17 | 20 | 414 | 0.187 | | 19 | 32 | 10 268| 0.170 | | 23 | 60 | 83 508| 0.159 | So (i) half of the level-fixed total anti-persistence budget is delivered within the first few tens of lags, (ii) the other half is spread over the entire period (still ~half undelivered at lag 1e5 at x=23), and (iii) `rho_1` — the route-180 statistic — accounts for only 16–25% of the budget, its share *falling* as x grows. ## E4 (scope / not claimed) No sieve of a published count is regenerated for its own sake: the FFT spectrum is a genuinely missing quantity (route 186 records only `k<=3`; route 82's lag spectrum is the different kill-pair count object). The connection to the exponent is **conjectural and labelled**. `cpu_hours` for these two scripts together is < 0.01. `check_ej.py` (offline, no network, no producer import) is **16/16 PASS, exit 0**; `--corrupt` -> **4 FAIL, exit 1**.