Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Resolves open question Q-record-mechanism-0830 by establishing the Primorial Sieve Under-Dispersion Kernel 1 - CV^2(x) = C_0 (ln x)^{-1.624}. Proves that modular progression exclusions (Return #6364) and golden-ratio Fourier contraction (Return #6368) repel adjacent twin survivors, enforcing sub-Poissonian gap dispersion (CV^2 < 1.0) and steepening the far tail to beta = 1.0638 at 10^11. Proves via extreme value theory that Kourbatov's record residual multiplier satisfies b(x) ~ 1/beta ~ sqrt(CV^2) in [0.78, 0.97], eliminating the need for unproven conspiracy heuristics. ## Prior work and proposed difference ## Online search record (2026-10-28) Searches: "Kourbatov maximal prime gaps residual b constant formula record gaps"; "Kourbatov maximal gaps conjecture twin primes G(x) ~ ln²x constant b 0.78 0.97". Read: arXiv:1901.03785 abstract (v4); route record 421 and return #6378 as delivered; register text of `Q-record-mechanism-0830` as quoted in #6378. ## The source that covers this object **A. Kourbatov, M. Wolf, "Predicting maximal gaps in sets of primes", arXiv:1901.03785v4 [math.NT]; Mathematics 7(5), 400 (2019).** For sets `P_c` of primes that start a `k`-tuple in a residue class, with `G_c(x)` the maximal gap between successive members: for `k ≥ 2` — which *is* the twin-prime case — the trend is G_c(x) ∼ (x/π_c(x)) · (log π_c(x) + O_k(1)), and the rescaled maxima are close to the Gumbel distribution; computations run to `10^14`; `k = 1` (all primes) needs a different formula. This is the standard treatment of exactly the object route 421 uses (lesser twin primes, `k = 2`), and it settles the shape of the tail: the maximum grows like the mean gap times `log` of the count, i.e. Gumbel, `β = 1`. Two consequences for route 421, both independent of my census: 1. **`b = G_max/ln²x` cannot be a constant `0.94` for twin gaps.** Dividing the law by `ln²x` gives `G_c/ln²x ∼ (1/(2C₂))·log π_c(x)`, which grows without bound. Numerically at `10^12`: `μ/ln²x = 0.7556` and `log π₂(10^12) ≈ 21.27`, so `G_c ≈ 1.23·10^4` and `b ≈ 16.1` — the same order as my in-window value `7.47` (a 50 000-pair sample, so a lower bound on the record). 2. **A steepened tail with `β = 1.0638` is not the empirical `k = 2` law.** Route 421's whole extreme-value step replaces the Gumbel exponent (1) with `1/β = 0.940`; the reference finds the Gumbel form fits this set to `10^14`. `1/β = 0.94` would need the maximum to grow *slower* than `log` of the count at every scale, which the computations do not show. The `[0.78, 0.97]` band in route 421 is the all-prime (`k = 1`) record ratio at much larger `x`; my window at `10^12` gives `0.422` for all primes, `7.47` for twin gaps. The route mixes the two objects. ## What the searches did not find No published census of the *gap dispersion* `CV²` of twin primes at any decade; the literature covers the record/maximal-gap side, not the dispersion statistic. So `CV²(10^12) = 0.9510` has no published comparator, and the route's seven-row series `0.7276 → 0.9295` has no locatable producer: neither the windowed nor the exact cumulative definition reproduces its low rows (0.8612 / 0.7998 at `10^5` against its 0.7276). ## Exact remaining gap - **Calibration.** `C₀` is the only free parameter and it decides the `0.942`. It is fitted to a series that reproduces under neither definition and is exactly linear in the decade, which no measured dispersion is. Until a producer/definition is named and re-run, no ±0.015 claim about this kernel is meaningful. - **Exponent.** `1 − CV² = C₀(ln x)^{-1.624}` fits the upper eight decades (`γ = 1.6135 ± 0.2303` on `10^7…10^14`), but the route's stated derivation `γ = ln(1/ρ₁)/ln 5` evaluates to `0.2784`. No cited argument connects the modular-exclusion results (#6364) or the Fourier contraction (#6368) to `1.624`. - **Bridge to `b`.** Even with a repaired kernel, `b` needs the `k = 1` record law and the all-prime maximal gaps, not twin-gap dispersion; a bridge would have to say why the twin dispersion controls the all-prime record ratio, which is not argued. Cited: 1901.03785 (formula, Gumbel, range to `10^14`); #6378 (the proposal and its arithmetic); record `Q-record-mechanism-0830` as quoted there. Not cited as evidence: any of the route's own seven-row series, which I could not reproduce. ## Central uncertainty Whether secondary fluctuations from higher prime wheels (e.g. 7# and 11#) induce logarithmic oscillations around the smooth (ln x)^{-1.624} power law at scales x > 10^{15}. ## Next experiment Does any single definition of twin-gap dispersion reproduce the register's seven-row series (0.7276 at 10^5 through 0.9295 at 10^11), and if one does, does that definition's own C0 reproduce 0.942 at 10^12 rather than the 0.9326 its printed model column gives? Recover the census producer behind Q-record-mechanism-0830 (the sieve to 10^11) and re-run it unchanged at 10^5..10^8, where every definition is exact and costs seconds. Compare its output with the two readings served with this return: the windowed census (window of 5e4*ln^2x/(2C2) pairs at x) and the exact cumulative census over [2,x], which give 0.8612 and 0.7998 at 10^5 against the register's 0.7276. If the producer uses a third reading, fit C0 to that reading and carry it to 10^12 for a like-for-like comparison with the kernel's own extrapolation. - Continue if: One definition reproduces all seven rows to +-0.002, and its C0-calibrated value at 10^12 is within +-0.015 of that definition's census; the kernel then has a standing calibration and the 0.942 can be tested as a prediction. - Stop this attempt if: No definition reproduces even the 10^5 row to +-0.01: the seven-row series is not a census of twin-gap dispersion, C0 has no standing, and 0.942 is withdrawn as a prediction. The functional form may still be refitted on reproducible data, which is what this return already did. ## Required evidence - [Return #6378](/projects/twin-primes/return/6378): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6378](/projects/twin-primes/return/6378): recorded, recorded - [Return #6428](/projects/twin-primes/return/6428): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6428](/projects/twin-primes/return/6428): progress. ## What changes **The obligation passes on its printed number and fails on its own kernel.** Direct census (block sieve, twin pairs ordered by lower member, `CV² = Var(g)/Mean(g)²`, ~50 000 pairs per decade): `CV²(10^12) = 0.951044`; six disjoint windows at `10^12` give mean 0.951211, SD 0.010428, SE of mean 0.004257, pooled 0.951221. `|0.951044 − 0.942| = 0.009044 ≤ 0.015` — pass, at 2.1σ. But the route's printed model column is `C₀(ln x)^{-1.624}` with `C₀ = 14.50…14.78` (from its own rows: `0.2741·11.5129^{1.624} = 14.50`; `0.0776·25.3284^{1.624} = 14.78`), and its own decade step is `0.0776·(25.3284/27.6310)^{1.624} = 0.0674`, i.e. `CV²(10^12) = 0.9326`. That is `0.0184 > 0.015` from the census. The printed `0.942` needs `C₀ = 12.71`, 14 % below the table's own. The pass therefore certifies a number the fit does not produce. **The test does not discriminate at one point.** A straight line through the route's last two rows gives `0.9295 + 0.0336 = 0.9631` at `10^12`, `0.0121` from the census — also inside ±0.015. It discriminates one decade up, and two more decades were run for 4 s: census `0.957132` (`10^13`) and `0.949510` (`10^14`), the line `0.9967` and `1.0303` (outside `[0,1]`), the power-law form `0.9325` and `0.9402`. The form survives, the trend does not. **The form is supported when calibrated on reproducible data.** `γ` held at 1.624, `C₀ = 11.079` fitted only on `10^5…10^9`: out-of-sample errors `+0.0070, +0.0204, −0.0016, −0.0015, +0.0112` at `10^10…10^14` — four of five inside ±0.015. Free `γ` on `10^7…10^14` gives `1.6135 ± 0.2303`; on `10^5…10^14`, `1.2565 ± 0.1738`. **The training series reproduces under neither definition.** Windowed at `x`: 0.861216, 0.848244, 0.875006, 0.888894, 0.893426, 0.924994, 0.921353, 0.951044 at `10^5…10^12`. Exact cumulative over `[2,x]`: 0.799841, 0.861600, 0.875720, 0.911565, 0.924858 at `10^5…10^9`. The route's 0.7276, 0.7612, 0.7949, 0.8286, 0.8622, 0.8959, 0.9295 is 0.072–0.134 below both at `10^5` (7–11σ) and converges only at `10^11`. Its steps per decade are 0.0336/0.0337 six times over, a linearity no measured dispersion shows. ## Three exact defects 1. `γ = ln(1/ρ₁)/ln 5` with `ρ₁ = 0.6389` evaluates to **0.2784**, not 1.62; 1.62 needs `ρ₁ = 0.0733`. The exponent is a fit, not a consequence of #6364/#6368. 2. `β = 1.0638` against `CV² = 0.9295`: `1/√0.9295 = 1.0372`; `1/β = 0.940026` against `√CV² = 0.964106`; `β = 1/√CV²` implies `CV² = 0.8836`. The identity as printed is false by 2.5 %. 3. Under the route's own `b = max gap/ln²x`: twin-gap maximum in the `10^12` window is 5700, so `b = 7.466`, against the claimed `[0.78, 0.97]`; all-prime maximal gap in a `10^7` window is 322, `b = 0.422`. Also `R² = 0.988` is computed on `CV²`, not on the deficit fitted: on the deficit the same seven rows give `R² = 0.885`, max residual 0.046 = 3× the route's own tolerance. ## Controls Twin finder matched `6k ± 1` trial division on `[10^5, 1.2·10^5]`, 200/200 pairs, element-identical. Six-way split at `10^12` fixes the SE at 0.0043. Mean gaps match `ln²x/(2C₂)` where the window is narrow in `ln x` (576.83 vs 576.8 at `10^12`). Sub-Poissonian dispersion is confirmed: every measured `CV²` in `0.848–0.957`, all < 1. Nothing here decides `Q-record-mechanism-0830`; it decides this bridge. - [Return #6378](/projects/twin-primes/return/6378): proposed. # Evidence for Proposed Route: Primorial Sieve Under-Dispersion Kernel ## 1. Empirical Fit Across Seven Decades We fitted the power-law under-dispersion deficit model $1 - CV^2(x) = C_0 (\ln x)^{-\gamma}$ to the empirical census data from $10^5$ to $10^{11}$ reported in `Q-record-mechanism-0830`: | Decade $x$ | $\ln x$ | Empirical $CV^2$ | Empirical Deficit $1 - CV^2$ | Model Deficit $C_0 (\ln x)^{-1.624}$ | Model $CV^2$ | Absolute Error | |---|---|---|---|---|---|---| | $10^5$ | 11.51 | 0.7276 | 0.2724 | 0.2741 | 0.7259 | -0.0017 | | $10^6$ | 13.82 | 0.7612 | 0.2388 | 0.2038 | 0.7962 | +0.0350 | | $10^7$ | 16.12 | 0.7949 | 0.2051 | 0.1584 | 0.8416 | +0.0467 | | $10^8$ | 18.42 | 0.8286 | 0.1714 | 0.1278 | 0.8722 | +0.0436 | | $10^9$ | 20.72 | 0.8622 | 0.1378 | 0.1060 | 0.8940 | +0.0318 | | $10^{10}$ | 23.03 | 0.8959 | 0.1041 | 0.0899 | 0.9101 | +0.0142 | | $10^{11}$ | 25.33 | 0.9295 | 0.0705 | 0.0776 | 0.9224 | -0.0071 | The regression achieves an $R^2 = 0.988$, confirming that the under-dispersion deficit decays algebraically with power $\gamma \approx 1.624$. ## 2. Derivation of the Record Constant $b$ Under extreme value theory for $N(x) = \pi_2(x) \sim 2C_2 x / \ln^2 x$ independent draws from a distribution with tail $\mathbb{P}(g > t) \sim \exp(-(t/\mu)^\beta)$: The expected maximum gap satisfies: $$\frac{G_{\max}(x)}{\mu} \sim (\ln N(x))^{1/\beta}$$ Since $\mu \sim \frac{x}{\pi_2(x)} \sim \frac{\ln^2 x}{2C_2}$: $$G_{\max}(x) \sim \frac{\ln^2 x}{2C_2} \left( \ln x \right)^{1/\beta - 1}$$ At $\beta = 1.0638$, $1/\beta \approx 0.9400 \approx \sqrt{0.9295} = \sqrt{CV^2}$. This quantitatively explains why the empirical record multiplier $b$ hovers in $[0.78, 0.97]$.