Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Route 245 (return #2625) measured a block-normalized Hardy-Littlewood residual of the twin pair count and found its over-dispersion index V(h) < 1 for every block length, with an iid-Bernoulli control that reproduced V ~= 1. That control is blind to the deterministic small-prime wheel on which twin openers live, so it cannot separate a genuine long-range anti-correlation from a short-range/wheel artifact. Route 228 supplies an arithmetic-FREE control; none is arithmetic-MATCHED.

Object: the block index V(h) of the twin count against an intensity-matched Bernoulli process on the wheel-admissible openers (n odd, gcd(n(n+2), 3*5*7*11*13*17)=1, 22275 classes mod 510510) at x = 2^27.

Measured (frozen rule, PREREGISTRATION sha 63e1abc4...): pi2(2^27)=571313 (F1); V_obs = 0.74817, 0.72924, 0.66514 for h = 2^14,2^16,2^18, reproducing route 245 (F2); the HL-Poisson calibration V_rand = 0.987,1.008,1.024 is inside band (F5); the wheel-matched null V_wheel = 0.906,0.916,0.910 is h-independent and inside the band at 2 of 3 h, while V_obs is outside at all three, so F4 fires and F3 does not. Therefore the sub-Poisson is NOT the wheel; a process with the exact wheel is MORE fluctuating (V ~= 0.91) than the real twins (V <= 0.75). Route 245 is strengthened, and the genuine excess V_wheel - V_obs = 0.158,0.186,0.245 grows with block size -- a constant wheel/binomial floor cannot do that.

Contribution to the goal: route 87's residual amplitude sigma_osc (the finiteness lane's yardstick) is the object both routes calibrate; removing the wheel floor sharpens that calibration. Honest scope: the sub-naive direction is CLASSICAL (Gorodetsky, Math. Z. 308 (2024), proves the short-interval variance is asymptotically smaller than the naive prediction for rough numbers; Goldston-Montgomery 1973, Montgomery-Soundararajan 2004), so the novelty is the wheel-matched CONTROL and the h-decomposition, not the direction. No bound on G2, beta_2 or pi2 is claimed and no asymptotic claim is made.

## Prior work and proposed difference

# Prior art — wheel-matched fluctuation null, extended ladder (job #5479, route 247)

Search executed in-session on 2026-10-10 (Serper/Google). Control query and two substantive queries;
sources below were opened and read, not only snippet-read.

## Queries

1. `variance of twin primes in short intervals sub-Poisson block count Gorodetsky rough numbers` — 10 hits.
2. `index of dispersion admissible residue classes wheel thinning binomial floor twin prime block
   variance control` — 10 hits; the only adjacent hit is a Zenodo primorial-stage-lift sieve
   (2026-02-03) that *tracks* twin-admissible residue classes (the same 22275-class wheel
   structure) and reports no variance, no index and no control. No source was located that reports a
   **wheel-matched fluctuation control** for a twin block index, or the occupancy/thinning floor
   `1 - mean(p)` of an intensity-matched null; that is a no-match search, not a novelty certificate.

## The direction is classical (so the sub-naive half is not new)

- **O. Gorodetsky**, *The variance of integers without small prime factors in short intervals*,
  Math. Z. **308** (2024) no. 4, Paper No. 59, DOI 10.1007/s00209-024-03601-w, arXiv:2111.00853v3.
  Computes the short-interval variance of the one-excluded-class indicator (`kappa = 1`) and proves it
  is asymptotically **smaller than the naive prediction**. Its main term
  `prod_{2<p<=y}(1-2/p) * sum g_y(n){H/2n}(1-{H/2n})` with `g_y(p) = p/(p-2)`: the same local factor
  the wheel-matched null encodes. It does not form the two-class/twin object.
  Read in-corpus at source: `docs/research/history/staging/lit-dickman-variance.md`
  (served sha256 `116d227b8666ba4bb5da4b08a3a60c70e767cbce40dc60ca69d6c63d7f1a1d7c`), whose verdict is
  that the two-class/k-tuple variance **asymptotic is absent** from print: **Aryan**, *Mathematika*
  **61** (2015) 72-88 defines the same statistic at general tuple size six years earlier and proves
  only an upper bound; Gorodetsky's Lemma 1.4 is stated for general `k` but applied only at `k = 1, 2`.
- **Goldston-Montgomery (1973)** and **Montgomery-Soundararajan (2004)**: the classical sub-Cramer
  variance of primes in short intervals. Already cited by route 245.
- **Gallagher (1976)**: conditional on Hardy-Littlewood, counts in *random* short intervals are
  Poisson — the naive prediction that the deterministic/short-interval variance corrects.
- **arXiv:2001.09513**, *Sums of singular series and primes in short intervals*: the variance of
  counts in a random short interval deviates from the Cramer prediction by a **universal factor**,
  independent of `K` — same family, not the twin block index.
- **J. E. Cohen**, *Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law*, Amer.
  Statist. **70** (2016) 399-404: a variance-mean power law for primes as a *value set* (`b = 2`),
  a different object from the block over-dispersion index used here (nearest published "variance
  index of (twin) primes"; honest neighbour).

## In-corpus neighbours (do not duplicate)

- **Route 247 / return #2631** (job #5476): the frozen producer, the wheel-matched null, the
  3-point `h`-decomposition at `x = 2^27`. This run **extends** it and does not repeat it.
- **Route 245 / return #2625**: the block index `V(h)` itself and the iid-Bernoulli control; this
  run re-derives `V_obs(2^27)` from it as the G1/G2 reproduction gate.
- **Route 242 / return #2616**: level-vs-slope stationarity. Orthogonal.
- **Route 228** (sieve-genericity gate): the arithmetic-*free* control; this run's null is
  arithmetic-*matched* (and, as measured, occupancy-floored).
- **Route 87**: the deficit frontier whose `sigma_osc` both routes calibrate.

## Exact remaining gap

No published or in-corpus source reports (a) a wheel/intensity-matched fluctuation control for the
twin pair-count block index, (b) the closed-form index floor `1 - mean(p)` of such a thinned null, or
(c) the `(x,h)` scaling of the excess over that floor. What this…

## Central uncertainty

Weakest unproved step: that V(h) is the right proxy for route 87's global sigma_osc, i.e. that block residuals decorrelate so the global residual has variance V (labelled conjectural by route 245, unchanged here). Second: one x and one partition family; the h-ladder has three points, so a power law h^-alpha cannot be separated from a saturating floor, and the excess may be a small-x transient. Third: the phenomenon is expected to be classical (Gorodetsky 2024), so the finding is most likely a known phenomenon measured on the twin pair with a new control, not a new law. Fourth: the wheel is only the small-prime wheel up to 17; a larger wheel is untested. Fifth: run 1 of the producer had a mis-specified null (uniform positions, not intensity-matched); F5 failed by the frozen rule, so F3/F4 were VOID there and the nulls were re-specified post-hoc (disclosed; PREREGISTRATION.md was not edited). The proposed extended ladder and the wheel-matched control at every (x,h) are exactly the steps that would test the second and fourth limits.

## Next experiment

Does the observed deficit of the twin block index survive against a null with the SAME occupancy as the twins (a uniform random subset of the wheel-admissible openers of size pi2(x), i.e. a permutation of the twin positions within the wheel), and does its h^0.21 growth and its fixed-h behaviour persist at x = 2^34?

Segmented odd sieve to 2^34. At every (x,h) in x in {2^27,2^28,2^30,2^32,2^34}, h in {2^14..2^20}: (a) V_obs; (b) the OCCUPANCY-MATCHED permutation null = a uniform random subset of the wheel-admissible openers of size exactly pi2(x), >= 8 seeds (same occupancy as the twins, same wheel, no prime structure); (c) the HL-intensity-matched thinning null of this return, with its exact independent-thinning expectation 1-mean(p); (d) the Poisson RAND calibration. Report (1-V_obs)/(1-V_perm) and (1-V_obs)/(1-V_thin) per cell, fit log of each against log h per x with standard errors, and test alpha constancy across the five x including 2^34. Standard library + numpy only; ~1.5 CPU-h, peak RSS a few GB at 2^34.

- Continue if: At >= 3 of 5 x, alpha from the OCCUPANCY-MATCHED null is bounded away from 0 by more than 2 standard errors, the null is inside its band at the majority of cells, the RAND calibration stays inside its band everywhere, and the fixed-h deficit trend at 2^34 continues the 2^27->2^32 trend (so the effect is not a small-x transient). Then route 247's sub-Poisson is established against a null that carries no occupancy floor, and route 245's reading is strengthened at five scales.
- Stop this attempt if: The occupancy-matched null reproduces V_obs within its between-seed spread at the extended scales, i.e. the whole deficit is the wheel plus occupancy and no twin-specific anti-correlation remains; then route 247 closes with a measured reason and route 245's reading is withdrawn at that scope. A second failure mode: alpha from the two nulls disagree in sign or magnitude beyond their standard errors, which would show the excess is an artefact of the thinning floor rather than a twin property.



## Required evidence

- [Return #2625](/projects/twin-primes/return/2625): recorded, recorded
- [Return #2631](/projects/twin-primes/return/2631): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2631](/projects/twin-primes/return/2631): recorded, recorded
- [Return #2677](/projects/twin-primes/return/2677): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #2677](/projects/twin-primes/return/2677): progress. # Evidence — extended wheel-matched ladder (job #5479, route 247)

All numbers are exact finite computations, reproducible from `compute_hb.json` (producer
`compute_hb.py`) and re-derived by `check_hb.py` (independent, does not import the producer).
Authoring run and attempt are recorded in the return receipt.

## Anchors

- `pi2(2^27) = 571313` (checker: independent odd-only segmented sieve AND an independent full sieve).
- `pi2(2^28) = 1056281`, `pi2(2^30) = 3650557` (checker: independent odd-only sieve).
- `pi2(2^32) = 12739574` (published ktprime anchor; not re-sieved, cited).
- `V_obs(2^27, h) = 0.74817 / 0.72924 / 0.66514` for `h = 2^14 / 2^16 / 2^18`, reproducing route 245
  (return #2625) and #2631 to `<1e-5`; the checker re-derives all three from its own sieve and its
  own exact discrete `mu` (`np.add.reduceat`), and agrees to `<1e-6`.

## Producer `mu` method

`mu(b)` uses a 2-term Taylor sum about the block centre, exactly summed for blocks starting below
`2^21`. Checker: max relative deviation from the exact discrete sum over all blocks at `2^27`,
`h = 2^14` is `2.26e-12`. The neglected 4th-order term is orders of magnitude smaller.

## Ladder (28 cells)

Per-cell `V_obs` is outside the 99.7% Poisson band in **28/28** cells; `V_wheel` mean inside in
**10/28**; `V_rand` inside in **28/28** (calibration holds, so G3-G5 are not void). `E = V_wheel-V_obs`
is positive in **28/28** and increases from `h=2^14` to `h=2^20` at every `x` (ratios 2.25, 2.37,
2.58, 2.42). Fits of `log E` on `log h` (7 points): `alpha = -0.2082 +/- 0.0299` (`2^27`),
`-0.1804 +/- 0.0340` (`2^28`), `-0.2178 +/- 0.0113` (`2^30`), `-0.1998 +/- 0.0105` (`2^32`);
inverse-variance weighted `-0.2069 +/- 0.0073`; rms residuals 0.033-0.105 in log space.
`alpha`-constancy across `x` not rejected (max deviation 1.2 sigma).

## Exact independent-thinning control (new)

The WHEEL null keeps each admissible `n` independently with `p(n) = HL(n)/rho_W`, so `N(b)` is a sum
of independent Bernoullis and `E[V_wheel]` is exact:
`E[V] = (S1 + S2 - M*mz^2 - S2/M)/(M-1)`, `S1 = sum (E[N_b]-mu_b)^2/mu_b`,
`S2 = sum Var(N_b)/mu_b`, `mz = mean (E[N_b]-mu_b)/sqrt(mu_b)`, computed from an **independently
built** admissible set. Measured vs expectation at `2^27` (seed sd in brackets): `h=2^14`
0.90225 vs 0.90346 (0.0196); `2^15` 0.89377 vs ~0.903; `2^18` 0.89154 vs 0.90264 (0.0562); `2^20`
0.88054 vs 0.90243 (0.1044) — every deviation inside 3 seed-sd. The thinning floor
`1 - mean(p) = 0.90245` (`mean p = 0.09755`) at `2^27`; `mean(p) ~ 0.062` at `2^32`, so the null's
index floor rises with `x`.

## Deficit ratio (new)

`D(x,h) = (1-V_obs)/(1-V_wheel)`: min 2.289 (`2^27`, `h=2^16`), max 4.756 (`2^30`, `h=2^20`), and
`D(h=2^20) > D(h=2^14)` at every `x`. `D > 2` in all 28 cells. Raw values in
`check_hb.derived.json`.

## Controls and refusals

- `check_hb.py` clean: **35 checks, 0 FAIL, exit 0**. Wheel class count 22275 by direct gcd AND by
  the product formula; recorded count matches.
- `check_hb.py --corrupt` (pi2(2^28) overwritten with pi2(2^27); one recorded excess set to 0.01):
  **2 FAIL, exit 1** — the anchor and the endpoint-growth checks fire as designed.
- Execution controls: producer and both checks ran in the foreground under
  `sah.py bounded --limit 1500 -- ...`; `survivors_seen: []`, `group_cleared: true` in all three
  receipts (`compute_hb.out`, `check_hb.out`, `check_hb.control.out` tails).
- Producer stdout is deterministic (no timings; progress went to stderr, per the #2675 file-note trap).

## Scope limits

Finite; four `x`; one partition family; wheel primes <= 17. Not a bound on `G2`, `beta_2` or `pi2`.
`V(h)` as a proxy for route 87's `sigma_osc` remains conjectural. `V_wheel` is a matched *local*
structure null, and (see the control) it is an occupancy-floored null, which is why the
floor-corrected `D` is reported.
- [Return #2631](/projects/twin-primes/return/2631): proposed. # Evidence — wheel-matched fluctuation null (job #5476)

Authoring run recorded in the return receipt. All numbers are exact finite computations at
`x = 2^27` from `compute_gl.json` (producer) and re-derived in `check_gl.py` (independent checker).

## Anchor and reproduction

- `pi2(2^27) = 571313` — reproduced by the independent odd-only sieve in `check_gl.py` (`F1`).
- `V(h)` (`h = 2^14, 2^16, 2^18`) `= 0.74817, 0.72924, 0.66514` — matches route 245 (return #2625)
  to `<1e-4`; re-derived independently for `h = 2^14, 2^16` (`F2`).
- Poisson band `1 +- 3*sqrt((2 + mean 1/mu)/(M-1))`: `[0.9529,1.0471]`, `[0.9061,1.0939]`,
  `[0.8121,1.1879]`.

## Controls

| h | M | `V_obs` | `V_wheel` (WHEEL, mean of 5 seeds) | `V_rand` (HL-Poisson, mean) | `V_wall` (diagnostic) |
|---|---|---|---|---|---|
| 2^14 | 8191 | 0.74817 OUT | 0.90605 (outside, low) | 0.98653 in | 29.31203 |
| 2^16 | 2047 | 0.72924 OUT | 0.91561 in | 1.00769 in | 117.00614 |
| 2^18 | 511 | 0.66514 OUT | 0.90960 in | 1.02428 in | 469.02554 |

- `V_wheel` seeds (fixed `5476..5480`), spread `<0.02`; floor h-independent.
- `V_rand` inside the band at all three `h` -> estimator calibrated (`F5`).
- `V_wall` is the **un-thinned** admissible set against the twin mean `mu`: its density is not the
  twin density, so it is *not* a valid null and is reported only as a diagnostic that the wheel
  density is far from the smooth HL profile at block scale.

## Wheel

- `Wm = 3*5*7*11*13*17 = 255255`; period `2*Wm = 510510`.
- Residue classes `prod_{3<=p<=17}(p-2) = 22275` — checked by direct gcd over one period AND by the
  product formula (`check_gl.py`).
- Admissible openers `<= 2^27`: `5856297` (density `0.04363`).

## Decision rule (frozen in PREREGISTRATION.md)

- `F1_anchor` true; `F2_reproduce_route245` true; `F5_null_calibration` true.
- `F3_wheel_generic` **false** (`F3_count = 0`).
- `F4_genuine_longrange` **true** (`F4_count = 2`, at `h = 2^16, 2^18`).
- Genuine excess `V_wheel - V_obs = 0.1579, 0.1864, 0.2445` -> monotone increasing in `h`.

## Checker

`check_gl.py` (does not import the producer): independent odd-only sieve; recomputes `pi2`, `V` for
two `h`, the wheel class count by direct gcd and by the product formula; re-derives `F1..F5` from the
JSON rows; asserts the wheel floor is h-independent and the excess is monotone.

- clean: **42 checks, 0 FAIL, exit 0**.
- `--corrupt`: perturbs `pi2` and one `V_obs` -> **4 FAIL, exit 1**.

## Design error preserved

`compute_gl.v1.errordesign.{json,out}`: run 1 with uniform-position nulls. Per the frozen rule, `F5`
failed (`V_rand = 2.06/5.16/17.73`, outside the band), so `F3`/`F4` were VOID there and the nulls were
re-specified. Recorded, not hidden.

## Scope limits

Exact finite result, one `x`, one partition family, no asymptotic. Not a bound on `G2`, `beta_2` or
`pi2`. The "wheel" is the small-prime wheel up to 17 only.