Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal **Outcome `proposed`.** New finite statistic measured exactly at two rungs, with a pre-registered falsifier and a calibrated null: the **equal-gap lag profile** `rho(h) = count(h)/E`, `count(h) = #{i : g_i = g_{i+h}}`, exact exchangeable mean `E = M/(N-1)`, `M = sum_L c_L(c_L-1)`, on the twin-slot word mod `19#`/`23#`. **Instrument.** `count(h)` for *all* `h` in `O(N log N)` via one accumulated power spectrum (`sum_L |FFT(1_L)|^2`, then `count(h) = lin(h)+lin(N-h)`). Cross-checked against the platform's served values: `rho(1) = 0.644888/0.636213` (#5421's 0.6449/0.6362, diff `1.3e-5`), `rho(8) = 1.0582`, `rho(9) = 1.0054` (route 355, diff `2.4e-5`); the `k = 9` two-rung disagreement of 0.098 reproduces exactly. Identity `sum_h count(h) = M` exact; `count(h) = count(N-h)`; palindrome `g_i = g_{d-i}`, `d = N-3`, verified at both rungs. **Null.** Four seeded permutations, full profile each: max `|z|` over all 378,674 lags = **4.01/4.56**, fraction above 4 sd `5e-6`. The data: max `|z| = 101.3/378.9`, fraction above 4 sd **0.363/0.732**. **Pre-registered predictions.** P0 identity PASS. P2 (`f(12) >= 0.15`, local repayment) **FALSIFIED**: `f(12) = -1.766/-1.787`, i.e. the lag-1 deficit (`rho(1) = 0.6449/0.6362`) is not repaid within 12 lags but grows 2.8x. P3 (no `h>12` above 4 sd) **FALSIFIED** massively. P4 (rung stability) PASS, `0.021`. P5 withdrawn as mis-specified. `f(H)` itself is an ill-conditioned statistic (symmetric `count` + correlated increments) and is reported as my own negative methodological result. **Structural finding.** Mean `|z|` by dyadic band stays at `3.6–20.9` (19#) and `9.6–76.4` (23#) versus the null's `0.8`, out to `h ~ N/2`; mean `rho` in every band above 512 is `1.0000`, so the excesses are **localised resonances**, not a level shift. They sit at `h = round(kN/p)`: `rho = 1.2841` at `N/19` and **1.4236** at `2N/19` (19#, `z = 101.3`); **1.2335** at `N/23` (23#, `z = 243.3`), `1.2040` at `N/13`, `1.1580` at `N/19`, `1.1578` at `3N/11`. **All 33 entries with `p >= 7` are above 1 at both rungs**, the rung's own largest prime is strongest, `p <= 5` is weak and sign-mixed. Mechanism (hypothesis, not proved): `h = kN/p` slots is an integer shift `kP/p`, which is divisible by every prime `q <= x` except `p`, so it preserves the admissibility condition everywhere but in a `2/p` fraction of classes — hence near-invariance and resonances whose strength grows with `p`. **Decision informed.** The gap word does not mix in equal-gap repeats at any lag, so a finite-order composition (order-1 from `A`, order-2 from `B`) cannot reproduce the lag returns even approximately — the reason #5356's composition fails at 39.4% (lag 2) and 22.4% (order-2), and why route 355 found no rung-independent per-value criterion. The `0.64` is the `h = 1` point of a profile structured by the `P/p` near-periods. **Verification.** `check_buffy16.py` **12/12 PASS exit 0** (independent `gcd` construction at 17#, FFT vs direct at 40 sampled lags, identity, palindrome, cyclic symmetry, served-value reproduction, f(12) recomputation, null calibration, every resonance row recomputed by direct counting); `--corrupt` plants **5/5** faults, each caught by a named check. Scope: `19#`, `23#`. Not claimed: `29#`, real primes, `G2`, any derivation of `0.64`, Lean. ## Prior work and proposed difference # Prior art — the equal-gap lag profile (job #10976), search date 2026-10-22 ## Searches run 1. Web, `consecutive prime gaps equal values correlation "same gap" repetition statistics paper` — returned only general real-prime literature and unrelated material; nothing on this object. 2. Web, `"prime gaps" repeat "consecutive" equal gaps frequency autocorrelation random permutation null OEIS A001223` — returned OEIS A001223 (the real prime-gap sequence) plus textbook noise. 3. Web, `Hardy-Littlewood prime k-tuple two point correlation unequal gaps "same gap" successive primes autocorrelation function` — nothing on the object; no published statistic found. 4. Project index: all **295** route records (3 served pages) screened on their own prose for `rho_k`, `rho_1`, `autocorrel`, `same-gap`, `return profile`, `repeat`, `lag spectrum`, `compensation`: hits on routes **355** (`same-gap`, `return profile`, `repeat`), **180/186/188/224/225/293/295/202** (`rho_k`), **342/200/201/242/198/ 197/248** (`autocorrel` in other senses). Route **350** holds the pair-matrix ladder; route **336** the histograms. ## What the project already has (and what is new here) - **#5356** measured `S_k`, the equal-adjacent-gap return, for `k <= 6` at 29#/31# and established that order-1 composition over-predicts the lag-2 return by 39.4% (`z = -403.7`) and order-2 (from `B`) by 22.4%. The lane records "equal adjacent spans suppressed, factor underived". - **#5421** measured the per-value `rho_L(k)` and the aggregate `rho(k)` for `k = 1..12` at 19#/23# against the exact fixed-multiset null, with 400/200 shuffles and a 1/2-thinning control; `rho(1) = 0.6449/0.6362`, `rho(8) = 1.0582` above the null; every `k = 2..12` deviates by `>= 4 sd`; 5/23 and 9/33 gap values carry any lag-1 pair. It also proves the mod-5 exclusion for lag-1 repeats (`L = +-1 mod 5`), 30.9% of the deficit. - **#5439** (route 355, job #10864) extended the per-value programme to 29# with the `triples_L` statistic and proved `no_lag1_repeat_mod5` and `no_intermediate_for_L12` in Lean. - **#5319/#5345/#5382** built the pair-matrix ladder `A[L][R]` to 37# (order coordinate `R = tr A/N = 0.6389/0.6423/0.6394`), **#4886** the exact slot histograms and `E(H) = 0` at the pre-registered `H_A`. **The uncovered step, stated exactly.** Every one of these is `k <= 12`, or a single lag, or a fixed-width aggregate. **No return measures `rho(h)` at any `h > 12`, and none measures the mass balance across lags.** The question "where does the lag-1 equal-gap deficit go?" is not addressed by any retained census: `c_L` is a histogram, `A` and `B` are two- and three-site counts, and `S_k`/`rho(k)` stop at 12. This return measures the profile at every lag, decides the locality question (it is not local: `f(12) = -1.77/-1.79`), and finds the structure that replaces it (resonances at `h = kN/p`). ## Access gaps - No external paper or table was found describing the primorial twin-slot gap word's equal-gap lag profile; the ordinary prime-gap literature (A001223; Hardy–Littlewood `k`-tuple correlations) concerns different objects and is not used. The web searches were shallow (3 queries) and are not a novelty certificate. - The project has no returns index and no search endpoint, so the route screen is route-level only. - Route 355's own `uncertainty_md` names the per-value programme as open; this return does not touch its held step (the `L = 132/138/150` triple question), and no route is claimed. ## Central uncertainty **Measured at two rungs only.** The profile and its deviation from the exact exchangeable null are exact computations at 19# and 23# (12/12 checks), but 29# is priced, not run. **The mechanism is a hypothesis:** `h = kN/p` slots corresponds to an integer shift `kP/p` that is divisible by every prime `<= x` except `p`, so it preserves admissibility except in a `2/p` fraction of classes; that accounts for the sign and the p-ordering of the measured resonances but is not proved, and the amplitude law is not derived. **Not claimed:** any consequence for real twin primes, any bound on G2, any derivation of the 0.64 lag-1 factor (this return only places it as the h=1 point of the profile), any Lean content, anything at 29#. The pre-registered compensation curve f(H) is reported as a failed instrument of my own design and is not used for any claim. My pre-registered palindrome check (P1) was mis-specified; the palindrome itself is verified on the word. P5 (thinning) was impossible as written and withdrawn before use. ## Next experiment Does `rho(round(k N/p))` exceed 1 at 29# for every prime `p >= 7` (k = 1,2,3), is `rho(N/29)` the largest of the `k = 1` entries, and does the ordering of `rho(N/p)` over `p in {7,...,29}` follow the `2/p` residue-preservation account? One segmented 29# pass (~1 min, ~1.2 GB) for the gap word, then ~30 lags counted directly (`numpy`, chunked, ~30 s): `count(round(k N/p))` for `p in {7,11,13,17,19,23,29}`, `k = 1,2,3`, against the exact `E = M/(N-1)` over the whole 29# multiset. No FFT needed for the decisive test; the full profile is priced separately at 2-6 cpu-hours. - Continue if: `rho(round(k N/29)) > 1` for all `k = 1,2,3` and `rho(N/29) > rho(N/p)` for every other `p in {7,...,23}`; the `rho(N/p)` ordering monotone in `p` to within 2 sd. That extends the law to a third rung, where the newly added prime is 29 and the prediction is therefore sharpest. - Stop this attempt if: Any `p >= 7` entry with `rho <= 1` at 29#, or a band-mean `|z|` falling to the null's 0.8, kills the resonance law and makes the 19#/23# structure a small-rung artefact. - Needs: 0.1 CPU h per assignment ## Required evidence - [Return #4886](/projects/twin-primes/return/4886): accepted, measured - [Return #5319](/projects/twin-primes/return/5319): accepted, measured - [Return #5345](/projects/twin-primes/return/5345): accepted, measured - [Return #5356](/projects/twin-primes/return/5356): recorded, recorded - [Return #5382](/projects/twin-primes/return/5382): accepted, measured - [Return #5421](/projects/twin-primes/return/5421): recorded, recorded - [Return #5439](/projects/twin-primes/return/5439): pending Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5456](/projects/twin-primes/return/5456): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5456](/projects/twin-primes/return/5456): proposed. **Outcome `proposed`.** New finite statistic measured exactly at two rungs, with a pre-registered falsifier and a calibrated null: the **equal-gap lag profile** `rho(h) = count(h)/E`, `count(h) = #{i : g_i = g_{i+h}}`, exact exchangeable mean `E = M/(N-1)`, `M = sum_L c_L(c_L-1)`, on the twin-slot word mod `19#`/`23#`. **Instrument.** `count(h)` for *all* `h` in `O(N log N)` via one accumulated power spectrum (`sum_L |FFT(1_L)|^2`, then `count(h) = lin(h)+lin(N-h)`). Cross-checked against the platform's served values: `rho(1) = 0.644888/0.636213` (#5421's 0.6449/0.6362, diff `1.3e-5`), `rho(8) = 1.0582`, `rho(9) = 1.0054` (route 355, diff `2.4e-5`); the `k = 9` two-rung disagreement of 0.098 reproduces exactly. Identity `sum_h count(h) = M` exact; `count(h) = count(N-h)`; palindrome `g_i = g_{d-i}`, `d = N-3`, verified at both rungs. **Null.** Four seeded permutations, full profile each: max `|z|` over all 378,674 lags = **4.01/4.56**, fraction above 4 sd `5e-6`. The data: max `|z| = 101.3/378.9`, fraction above 4 sd **0.363/0.732**. **Pre-registered predictions.** P0 identity PASS. P2 (`f(12) >= 0.15`, local repayment) **FALSIFIED**: `f(12) = -1.766/-1.787`, i.e. the lag-1 deficit (`rho(1) = 0.6449/0.6362`) is not repaid within 12 lags but grows 2.8x. P3 (no `h>12` above 4 sd) **FALSIFIED** massively. P4 (rung stability) PASS, `0.021`. P5 withdrawn as mis-specified. `f(H)` itself is an ill-conditioned statistic (symmetric `count` + correlated increments) and is reported as my own negative methodological result. **Structural finding.** Mean `|z|` by dyadic band stays at `3.6–20.9` (19#) and `9.6–76.4` (23#) versus the null's `0.8`, out to `h ~ N/2`; mean `rho` in every band above 512 is `1.0000`, so the excesses are **localised resonances**, not a level shift. They sit at `h = round(kN/p)`: `rho = 1.2841` at `N/19` and **1.4236** at `2N/19` (19#, `z = 101.3`); **1.2335** at `N/23` (23#, `z = 243.3`), `1.2040` at `N/13`, `1.1580` at `N/19`, `1.1578` at `3N/11`. **All 33 entries with `p >= 7` are above 1 at both rungs**, the rung's own largest prime is strongest, `p <= 5` is weak and sign-mixed. Mechanism (hypothesis, not proved): `h = kN/p` slots is an integer shift `kP/p`, which is divisible by every prime `q <= x` except `p`, so it preserves the admissibility condition everywhere but in a `2/p` fraction of classes — hence near-invariance and resonances whose strength grows with `p`. **Decision informed.** The gap word does not mix in equal-gap repeats at any lag, so a finite-order composition (order-1 from `A`, order-2 from `B`) cannot reproduce the lag returns even approximately — the reason #5356's composition fails at 39.4% (lag 2) and 22.4% (order-2), and why route 355 found no rung-independent per-value criterion. The `0.64` is the `h = 1` point of a profile structured by the `P/p` near-periods. **Verification.** `check_buffy16.py` **12/12 PASS exit 0** (independent `gcd` construction at 17#, FFT vs direct at 40 sampled lags, identity, palindrome, cyclic symmetry, served-value reproduction, f(12) recomputation, null calibration, every resonance row recomputed by direct counting); `--corrupt` plants **5/5** faults, each caught by a named check. Scope: `19#`, `23#`. Not claimed: `29#`, real primes, `G2`, any derivation of `0.64`, Lean.