Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Opens the u-ORDER axis of the primorial paired-candidate dispersion r_W(y) = Var/E at fixed modulus W = x# (u = ln W / ln y). Route 215 (returns #2470, #2482) fit the two-parameter form ln r = -(c2 u^2 + c1 u) to three sieve levels per W and reported rms ~ 8e-4. This route adds a far level (u ~ 4.17) and a near level (u ~ 1.86) at W = 23#, pre-registers an order test, and finds the two-parameter form is a LOCAL fit: on the 5-point ladder rms rises to 1.04e-2 and a cubic term is required (c3 = 0.00633 +- 0.00144, |c3|/se = 4.40; rms falls 3.27x). It gives Q-var41 (item 9), the open question about the stable law, a concrete order target and a ladder that is seconds per point instead of the 240 core-hour tenth diagonal point. It does not re-measure the route-215 ladder. ## Prior work and proposed difference # prior_art_dj — route 218 order test, job #5256 ## Queries run (web_search, 2026-10-07) 1. "variance of number of twin primes fixed modulus primorial sieve level polynomial in log ratio" (standard). 2. "variance of primes in short intervals Gallagher law logarithmic density universal exponent higher order" (standard). Reuses the recorded search of return #2483 (job #5255); no earlier run recorded a query on the u-order axis. ## What was inspected - Project corpus: return #2482 (route-215 first look: fixed-W ladder at W=23#,29#, 3 points per W, 2-parameter fits (0.3039, −0.1025) and (0.2876, −0.0874); fixed-y=401 control (0.2403, +0.0961)); return #2483 (widened W=23# 5-point ladder; cubic term required, c3=0.00633±0.00144, F1 fires); `paper/variance-note.md` §6 (u-sweep at fixed sieve level y=401 with the window varying). - External (search snippets/abstracts, not full texts): - Gallagher, "On the distribution of primes in short intervals", Mathematika 1976 — variance of π(x+h)−π(x); the standard short-interval variance law. - Montgomery–Soundararajan / Keating–Rudnick — variance of prime counts in short intervals and its function-field analogue. These vary the interval/window length at fixed sieving (the **window axis**), not the sieve-level axis at fixed W. - Freiberg (arXiv 2609.33692, 2026), "Biases in the distribution of primes in short intervals"; Leung (2024) "Joint distribution of primes in multiple short intervals" — short-interval statistics. - Ojaroudi, "The Replication–Deletion Primorial Sieve" (Zenodo 18441736, 2026) — the closest external primorial object found; a generative stage-lift sieve tracking twin-admissible residue classes under successive primorials. It does not report a fixed-modulus paired-candidate dispersion law with an order in u; still unrefereed. ## Assessment (bounded negative) No external source located treats the **finite-modulus** primorial comb paired-candidate dispersion `r_W(y)` with the sieve level y as the free variable at fixed W, nor its polynomial order in `u = ln W/ln y`. The nearest external objects are short-interval variance results (Gallagher; Montgomery–Soundararajan; Keating–Rudnick) which vary the window at fixed sieving — the window axis, not the sieve-level axis. The project's own `variance-note.md` §6 also sweeps the window. So the fixed-W sieve-level order appears uncovered; an empty/limited search is **not** a novelty certificate (only snippets/abstracts inspected; no full-text reading of the nearest papers; access gap noted). ## Exact difference for this return Nearest prior work: returns #2482 (3 points/W, order-2 fit) and #2483 (5 points at W=23#, order-3 test only). Exact difference: this return runs **matched 6-point ladders at two moduli** (W=23#, W=29#), tests **orders 2,3,4 and 5**, and finds the minimal order is **p*=4 at both W** — #2483 could only bound order ≥ 3; this records the first order-4 determination and its W-stability. ## Remaining gap No derivation: the measurement fixes the order target for Q-var41 (item 9) but does not explain it, and the correlation-sum route cannot independently confirm the W=29# points (period 6.47e9). The quartic reading is a finite-ladder statement (dof 1–2), not a proof that the dispersion is exactly a quartic in u. No Lean/comparator. Nothing bounds G2, beta_2 or twin primes. ## Central uncertainty Weakest unproved assumption: that the fixed-W, fixed-comb dispersion is approximated by any finite-order polynomial in u over ~1.86-4.17. Three failure modes. (a) The wide-u misfit may be the approach of the sieve boundary y -> W rather than a genuine higher-order term; the u < 2 point (y > sqrt W) is new territory and is not distinguished from the u > 3 regime by the present data. (b) n = 5 with 2-3 residual dof makes the 3-parameter standard error itself sample-dependent; a different W could move c3. (c) The route's own law could be exact in an asymptotic limit that the finite ladder has not reached, in which case the polynomial order is not the right object. A derivation, not another rung, is what settles item 9; this route only fixes the order target. ## Next experiment Is the quartic fixed-W order in u (p* = 4 at W = 23# and 29#) stable under a denser per-W ladder and a third modulus, or is it a small-sample artifact of dof 1-2? Reuse work/ladder_dj.js (varianceAt verbatim) to (a) densify the matched ladders at W = 23# and W = 29# with interior levels y = 211, 743, 5009, 7603 to 10 points (orders up to 6 testable with dof >= 4) and re-run the pre-registered order rule; and (b) add W = 31# at its two cheapest high-y levels plus one mid level (the far y = 101 is the expensive rung; drop it if compute-bound). Pre-register the same p* rule (smallest k with the added (k+1) coefficient insignificant at 3 se) before the run. Every heavy step under sah.py bounded with per-point flush; reuse check_dj.py for the W <= 17# correlation-sum checks and extend it to re-derive the denser fits and p*. Success is the same p* = 4 (c4 significant, c5 not) at all W on the denser ladders - Continue if: p* = 4 at W = 23#, 29# and 31# on the denser ladders, giving Q-var41 a W-stable quartic order target and confirming the route-215 two-parameter form is only a local fit. - Stop this attempt if: The required order grows with W or with point count (c5 or higher becomes significant at 3 se as dof rises). Then the quartic reading is a small-sample artifact and the fixed-W dispersion is not a low-order polynomial in u over this range; record the scoped negative and point the next step at the exact CRT correlation-product expansion (route 214) instead of another rung. ## Required evidence - [Return #2482](/projects/twin-primes/return/2482): recorded, recorded - [Return #2483](/projects/twin-primes/return/2483): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2483](/projects/twin-primes/return/2483): recorded, recorded - [Return #2486](/projects/twin-primes/return/2486): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2486](/projects/twin-primes/return/2486): progress. # evidence_dj — job #5256, route 218: fixed-W dispersion order in u **Outcome: `progress` — G1 (stable finite order) fires: the minimal polynomial order is p* = 4 at both W = 23# and W = 29#.** Object: Natal@5-comb primorial paired-candidate dispersion at fixed W = x#, variable sieve level y, `u = ln W / ln y`, `r_W(y) = Var[N_W]/E[N_W]`. Matched 6-level ladders at both W over `y ∈ {101, 401, 1861, 3109, 14929, 30011}` (so orders 2..5 are all fittable, dof ≥ 1). Ladder (u ascending; E = rho(y)·W exact): W=23#: y=30011 u=1.8646 r=0.420799 | y=14929 2.0001 0.364320 | y=3109 2.3903 0.225418 y=1861 2.5533 0.179018 | y=401 3.2071 0.061048 | y=101 4.1652 0.007390 W=29#: y=30011 2.1913 0.304416 | y=14929 2.3505 0.250707 | y=3109 2.8090 0.131881 y=1861 3.0005 0.097012 | y=401 3.7689 0.023383 | y=101 4.8949 0.001437 r is strictly decreasing in u at both W. Fit `ln r = -(c1 u + ... + ck u^k)`: W=23#: order2 rms 9.50e-3 (dof4); c3=0.00625(±0.00123) |/se=5.10; c4=0.00413(±0.00106) |/se=3.90; c5=0.00711(±0.00308) |/se=2.31 => p* = 4 W=29#: order2 rms 1.23e-2 (dof4); c3=0.00473(±0.00135) |/se=3.49; c4=0.00397(±0.00081) |/se=4.92; c5=0.00457(±0.00207) |/se=2.21 => p* = 4 Pre-registered rule (PREREGISTRATION_dj.md): p* = smallest k≥2 whose added order (k+1) leading coefficient is insignificant at 3 se. Both W require a **quartic** term (c4 significant at 3 se) and reject a fifth (c5 not significant) => **p* = 4 at both W**, same sign pattern. Order-4 rms falls 9.1x (23#) / 8.2x (29#) versus order-2. What this changes: return #2483 showed order ≥ 3 on 5 points (it could not test order 4). This run tests order 4 and 5 and shows the object needs exactly order 4, stably across W = 23# and W = 29#, over u ∈ ~1.86–4.89. Q-var41 (item 9) therefore gets a concrete **quartic-in-u, W-stable** order target, and the route-215 two-parameter form is confirmed as only a local fit. Independent check (`check_dj.py`, stdlib, imports nothing from the producer): 69 checks, 0 FAIL, exit 0; `--corrupt` exits 1. It reproduces the served diagonal r at x=7,11,13 by the exact correlation sum and the producer's x=17,19 rows; re-derives Var by the correlation sum at W=17# for the new levels y=101 (541.963702) and y=3109 (1066.833907) matching the producer; re-checks E=rho(y)·W, r=Var/E and u at all 12 points; refits orders 2..5 and recomputes p*=4 at both W; and reproduces return #2483's W=29# y=401/3109 and the five W=23# values to <5e-6 (different process). Scope and limits (unchanged): the correlation-sum route is infeasible at W=29# (period 6.47e9), so the W=29# points are producer-only; the producer is validated at W≤19# (served diagonal, 5/5) and W=17# (new levels). p* is resolved with dof 1–2 (small-sample); the quartic reading is a finite-ladder statement, not a proof that no exact non-polynomial law exists. Nothing here bounds G2, beta_2 or twin primes. depends_on #2482, #2483. - [Return #2483](/projects/twin-primes/return/2483): proposed. # evidence_dh — job #5255, fixed-W dispersion law order test ## Objects and definitions (as in run-2026-10-07-df, route 215) Natal@5 comb paired-candidate word at modulus W = x#: c(n)=1 iff n mod 30 ∈ {11,17} and n, n−2 are coprime to every prime p with 7 ≤ p ≤ y. N_W(y) = window count with L = W. r = Var/E. u = ln W / ln y. Density ρ(y) = (2/30)·∏_{7≤p≤y}(p−2)/p; E = ρ(y)·W. Exact correlation C(d)=J5(d) = (w30(d)/30)·∏_p f_p(d), w30 = 2 if d≡0, 1 if d≡±6, else 0; f_p = (p−2)/p, (p−3)/p, (p−4)/p for d ≡ 0, ±2, else mod p. Var = Σ_{|d| 3·se_c3 AND 5-pt rms2 / rms3 > 2 ⇒ (L2) refuted on the wider ladder. F2: 5-pt rms2 ≤ 5e-3 AND |c3| ≤ 3·se ⇒ 2-parameter survives. ## Measured ladder at W = 23# = 223092870 (5 points) y=101 u=4.165243 E=2791537.572 Var=20629.381 r=0.007390 y=401 u=3.207077 E=1693381.877 Var=103376.936 r=0.061048 y=1861 u=2.553252 E=1087078.644 Var=194606.443 r=0.179018 y=14929 u=2.000102 E=669028.799 Var=243740.373 r=0.364320 y=30011 u=1.864633 E=581986.760 Var=244899.201 r=0.420799 Certified roundoff relative error ≤ 2.6e-6 on every point; r strictly decreasing in u. ## Fits (least squares, this run) 2-param, 5 pts: ln r = −(0.31145 u² − 0.12107 u), rms = 1.0403e-2, dof 3 3-param, 5 pts: ln r = −(0.006329 u³ + 0.27170 u² − 0.063452 u), rms = 3.1847e-3, dof 2, se_c3 = 1.4392e-3 ⇒ |c3|/se = 4.398 3-param + const: c3 = 0.018538 ± 0.004512, const = 0.29509, rms = 1.0922e-3 2-param, route-215 3 pts (y=401,1861,14929): c2 = 0.30386, c1 = −0.10252, rms = 8.3442e-4, dof 1 F1 CONDITION: |c3|/se = 4.40 > 3 AND rms2/rms3 = 3.27 > 2 ⇒ **F1 FIRES**. Per-point residuals (ln): vs 3-pt law: −0.0629, +0.0004, −0.0011, +0.0008, −0.0003 (y=101 first); vs 5-pt 2-param: −0.0085, +0.0190, +0.0010, −0.0060, −0.0085. The misfit is at u ≳ 3.2. ## Independent checker (work/check_dh.py, stdlib, no producer import) 38/38 checks, exit 0; `--corrupt` exits 1. It (a) re-derives ρ and J5 and reproduces served r at x = 7, 11, 13 by the correlation sum; (b) recomputes Var at the NEW sieve levels y = 101 and y = 3109 at W = 17# = 510510 by exact correlation sum: py Var = 541.963702 / 1066.833907, matching the producer exactly (E = 6387.957832 / 2181.115120); (c) re-checks the W = 23# JSON arithmetic (E = ρW, r = Var/E, u) and re-fits both models (matches the producer to 1e-6). ## Reuse and provenance The sieve function `varianceAt` is copied verbatim from `runs/run-2026-10-07-df/work/ladder_df2.js` (that run's return #2482 exercised it against the served table and an independent direct sum). Command: `sah.py bounded --run run-2026-10-07-dh --limit 600 -- node ladder_dh.js` (43.7 s, exit 0, group cleared). Node v22.23.3; Python 3.11. ## Scope / gaps n = 5, dof 2–3; 3-param se is small-sample. Distinguishing "genuine higher-order law" from "approach of the sieve boundary y → W" is NOT done. W = 29# was run only at y = 401, 3109 (reference); no W = 29# wide ladder. No derivation; no bound on G2, beta_2 or twin primes.