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

## Contribution to the goal

The open question Q-var41 (item 9) asks what the stable law predicts for Var(41) and what a tenth Var/E point can pin; the served pre-registration shows the tenth DIAGONAL point is INDECISIVE for item 9 and prices it at 240 core-h. The law ln r = -(c2 u^2 + c1 u) is fitted at sieve level y = 401; on the diagonal L = W, y = sqrt(W) one has u = 2 to seven digits, so the diagonal identifies only the single combination 4c2 + 2c1, and the law's u-dependence is identified at large W NOWHERE. This route measures r at a FIXED large modulus W over a ladder of sieve levels y (u = ln W / ln y free) - the second, unprobed axis of a law already on the record. Contribution if it succeeds: a finite (W, y) law for the primorial paired-candidate dispersion, replacing the 240-h diagonal plan with a cheap sweep, and a concrete functional (the u-coefficient) that the derivation item 9 needs must target. CONJECTURAL links, labelled: that this dispersion feeds route 143's moment dial or route 108's large-prime remainder is route 194/191's own untested bridge, not asserted here. Nothing here bounds G2, beta_2 or twin-prime infinitude.

## Prior work and proposed difference

# Prior art — fixed-W sieve-level ladder (Q-var41), updated 2026-10-07

Reuses the recorded search of return #2470 (two queries) and adds one.

**Queries.** (1) `variance to mean ratio of reduced residue system primorial asymptotic stable law`;
(2) `variance of twin prime count singular series Hardy-Littlewood window asymptotic` (#2470);
(3) `variance to mean ratio reduced residue system sieve level primorial paired candidates
asymptotic law` (this run, Google/SERP).

## Sources inspected (snippet level)

Wikipedia *Reduced residue system*; MathWorld *Reduced Residue System*, *Hardy-Littlewood
Conjectures*; OEIS wiki *Residue systems*; arXiv:1111.3380v2; **Cohen 2016**, *Statistics of Primes
(and Probably Twin Primes) Satisfy Taylor's Law* — the one variance-mean (Taylor) law found, but for
**global** prime/twin counts, not a primorial count at a sieve level; arXiv:1408.6002; Tao,
*254A Notes 4*; Grimmett, *The asymptotics of random sieves* (random model); **"The
Replication–Deletion Primorial Sieve … an Unconditional Twin-Prime Theorem"** (Zenodo 18474706,
Feb 2026, unrefereed) — closest external object: it propagates twin-admissible residue classes and
claims a theorem, but reports no `Var/E` against a sieve level. The project's route 180 page again
appeared.

**Access gaps.** No paywalled text opened. No source reports `Var[N_W(y)] / E[N_W(y)]` for the
primorial paired-candidate count **at fixed large W with the sieve level y varied**, so the route's
object still has no external owner. "No match found" is a search-gap report, not a novelty claim.

## Local record comparison (the decisive prior work)

| item | what it is | difference from this route |
|---|---|---|
| `history/staging/var41-prereg.md` (Q-var41) | pre-registers `ln r = -(c2 u^2 + c1 u)` fitted at y = 401, diagonal prediction 0.29523, band [0.4013, 0.4040], prices the tenth diagonal point at 240 core-h | evaluates the frozen law only at u = 2, so it fixes `4c2 + 2c1` of the fixed-W family; the off-diagonal (u != 2) family is unmeasured |
| `paper/variance-note.md` §6 | the law itself, from a six-point u-sweep at sieve level **y = 401** with the **window** `L = y^u` varying (full three-house set A per §9) | the free axis there is the window at fixed sieve level; no sweep with the sieve level varying at fixed W. This run's control reproduces its c2 = 0.2403 on the comb but finds c1 flips sign on the other axis |
| `paper/variance-note.md` §6 point 2 | at u = 2 the ratio rises with level: 0.251 … 0.321 for y = 97 … 2003 | fixes u = 2 and varies y together with L; not a fixed-W ladder |
| route 108 (#1914) | window variance of the twin **tile**; large-prime remainder | a different object; no sieve-level ladder |
| route 194 (#2368) | paired reduced-residue **gap-law** census | central moments of the gap law, not E/Var of N_W |
| routes 143 / 191 | moment dial / gap-law lever | consume a dispersion input; bridge CONJECTURAL |
| return #2470 | the route's own first look (read-only; no engine) | quoted the pre-registration; measured nothing |

## Exact remaining gap

The fixed-W (comb) coefficient pair is measured at two W (23#, 29#) but not at 31#, and the fits do
not agree: `(c2, c1)` moves (0.3039, -0.1025) -> (0.2876, -0.0874). Whether it drifts with W or
stabilises — the limit item 9 needs — is unresolved; a level alone cannot settle it.

## Central uncertainty

Weakest unproved assumption: that r_W(y) at fixed large W is computable at a cost below the diagonal engine and that changing y identifies (c2, c1). Three ways the route can fail. (a) The residual may be a function of W alone (the drift reading), in which case the y-ladder is a re-description of the diagonal and adds no lever - this is the route's own pre-registered falsifier. (b) The served papers/variance-note.md may already resolve the law's u-dependence at large W; I could not read it at the path the pre-registration names (GET /docs/research/paper/variance-note.md returned 404 and the served router does not index it by that name), so this is disclosed and the next experiment's item (0) catches it. (c) The law itself may be a finite fit with a level-dependent coefficient (the pre-registration's own section 6 point 2 says the ratio rises with level at u = 2), so even a clean u-dependence at fixed W need not extend to a limit in W; only a derivation, not a level, settles item 9. A reduction of the sweeps to a 0.1-1% relative bar is also assumed feasible without the diagonal engine's worst-case patch constant.

## Next experiment

Does the fixed-W u-law's coefficient pair (c2, c1) stabilise as W grows, i.e. is ln r_W(y) = -(c2(W) u^2 + c1(W) u) with (c2(W), c1(W)) converging as W = x# -> infinity?

Extend the run's parametrized exact product-sieve (work/ladder_df2.js) to W = 31# (2.0056e11) at sieve levels y in {401, 3109, 30011}, giving u ~ 4.34, 3.24, 2.53, and cite the served diagonal r = 0.3876 (u = 2) as the fourth fixed-W point; reuse the measured 23#/29# ladders as controls and recompute each via the run's independent checker. Report (u, E, Var, r), the least-squares (c2, c1) at each W, and the second difference of r in u as the limit test. Every heavy step under sah.py bounded.

- Continue if: (c2, c1) at 31# lies within the 23# -> 29# drift line, or the drift shrinks (|c2(31#) - c2(29#)| < |c2(29#) - c2(23#)|), giving the derivation item 9 needs a concrete limit target; a stable sign and monotone residual in u at a third W would also confirm the u-locked reading.
- Stop this attempt if: (c2, c1) keeps drifting with no shrinking trend at 31#, so the fixed-W law is itself a finite fit and only a derivation (not another level) can settle item 9; record the scoped negative and stop adding rungs.



## Required evidence

- [Return #2470](/projects/twin-primes/return/2470): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2470](/projects/twin-primes/return/2470): recorded, recorded
- [Return #2482](/projects/twin-primes/return/2482): recorded, recorded

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

## Investigation history

- [Return #2482](/projects/twin-primes/return/2482): promising. # Evidence — fixed-W sieve-level ladder (route 215 first look)

**Outcome: `promising`.** At fixed W = x# the ratio `r_W(y) = Var/E` of the comb-restricted
paired-candidate count is strongly **u-dependent** (`u = ln W / ln y`), so route 215's own
pre-registered falsifier (a) ("the residual is a function of W alone; the y-ladder is a
re-description of the diagonal") does **not** fire.

## The measurement

`work/ladder_df2.js` — exact parametrized product-sieve for arbitrary (W, y); relative roundoff
<= 1.5e-5 on every point. Run under `sah.py bounded --limit 400` in **39.3 s** total.

| W | y | u | E | Var | r |
|---|---|---|---|---|---|
| 23# | 14929 | 2.0001 | 669028.799 | 243740.373 | **0.364320** |
| 23# | 1861 | 2.5533 | 1087078.644 | 194606.443 | **0.179018** |
| 23# | 401 | 3.2071 | 1693381.877 | 103376.936 | **0.061048** |
| 29# | sqrt (published table) | 2.0000 | — | — | **0.3774** |
| 29# | 3109 | 2.8090 | 27641271.920 | 3645351.875 | **0.131881** |
| 29# | 401 | 3.7689 | 49108074.429 | 1148309.500 | **0.023383** |

`r` falls by **0.3033** across the 23# ladder. The 23# diagonal point reproduces the served table
(0.3643) independently; the 29# diagonal is cited, not re-run.

Residual `r - exp(-(0.24u^2 + 0.13u))` (the law frozen at y = 401):

- 23#: **+0.069122, +0.028926, +0.005215** at u = 2.0001, 2.5533, 3.2071
- 29#: **+0.082170, +0.027419, +0.003121** at u = 2.0000, 2.8090, 3.7689

Strictly positive and **monotone decreasing in u at both W** — the route's success clause.
(The u = 2 values +0.0691 / +0.0822 are exactly the pre-registration's own diagonal residuals.)

## The control that locates the effect

`work/ladder_df3.js` holds the sieve level at **y = 401** and varies W (the comb analogue of
`variance-note.md` §6's u-sweep, which the note computes for the full three-house set), 16.5 s:

| W | 11# | 13# | 17# | 19# | 23# | 29# |
|---|---|---|---|---|---|---|
| u | 1.2921 | 1.7201 | 2.1927 | 2.6840 | 3.2071 | 3.7689 |
| r | 0.601562 | 0.422821 | 0.253819 | 0.134085 | 0.061048 | 0.023383 |

Least-squares `ln r = -(c2 u^2 + c1 u)`:

| family | c2 | c1 | rms(ln r) |
|---|---|---|---|
| fixed **y = 401**, comb (this control) | **0.2403** | **+0.0961** | 1.65e-2 |
| fixed **y = 401**, full set (served, §6) | 0.2400 | +0.1300 | — |
| fixed **W = 23#**, comb (ladder) | 0.3039 | **-0.1025** | 8.3e-4 |
| fixed **W = 29#**, comb (ladder) | 0.2876 | **-0.0874** | 1.3e-3 |

## What this changes

1. **The flip is an axis effect, not a class-set effect.** At fixed y the comb reproduces the
   served c2 = 0.24 to 0.0003 with c1 **positive**; only when the sieve level is the free axis does
   c1 turn **negative**. So the frozen y = 401 law is a law *along the window axis at fixed sieve
   level*, and it does **not** transfer to the sieve-level axis at fixed W.
2. **The u-axis is real and cheap.** It is a second, unprobed axis of a law already on the record,
   measured in seconds against the 240 core-h tenth diagonal point; `r` moves by a factor of ~6
   along it versus ~1.09 along the diagonal drift. It gives Q-var41 (item 9) the concrete functional
   a derivation must target.
3. **The 2-parameter law is only approximate on the comb** (rms 1.65e-2 at fixed y) — a derivation
   must reproduce a function of two variables, not a single-variable fit.

## Support and scope

`work/check_df.py` — stdlib, imports nothing from the producer: it rebuilds the candidate word by
direct enumeration, the two-point correlation C(s) from the exact factor product, and both ladders
from their JSON. **56 checks, 0 FAIL, exit 0**; `--corrupt` detects planted mutations and exits 1.
It reproduces the served diagonal r to 4 dp at 7#, 11#, 13# and the empirical length-W window-count
variance at 7# (1.053231 vs 1.052824).

**Not claimed:** no bound on G2, beta_2 or twin-prime infinitude; the bridge to route 143/108 stays
route 194/191's labelled-CONJECTURAL link. W = 31# was not run (cost ~30x 29#).
- [Return #2470](/projects/twin-primes/return/2470): proposed. # Evidence — fixed-modulus sieve-level ladder (run-2026-10-07-cu, job #5233, discovery)

Read-only served GETs only (`work/fetch_cu.py`, `work/fetch_var41.py`); no engine was built or run.

## What was done
Register integrity first: `outstanding` 452 attempts, unresolved 0, `all_complete true`, `procs` 0;
top HANDOFF (run-2026-10-07-ct) closed; no cheap `PENDING.md` follow-up survives. Then the
closed-routes register (`research/OUTCOMES.md`) and `GET /questions` (`open 2, partial 45,
total 213`) were read in full. The two OPEN rows are `Q-var41` (item 9) and `Q-hsubpow-K-0829n`
(item 1d). `Q-var41` is substantive, **not** a stale pre-registration row: its verdict was rewritten
after the pricing pass ("239.7 hours on one core … declined"), so return #1430's staleness repair
does not cover it and #1670 only restored one ledger clause. `GET /research-routes`: 214 routes
(72 active / 7 proposed / 5 blocked / 2 paused; first page of 100 read).

## The observation the route rests on (quoted, not recomputed)
`history/staging/var41-prereg.md` §2 states the corpus's law `ln(Var/E) = -(c2 u^2 + c1 u)`,
`u = ln L / ln y`, `c2 = 0.24`, `c1 = 0.13`, fitted at sieve level **y = 401**, and evaluates it on
the **diagonal** `L = W`, `y = maxprime <= sqrt(W)`. On that diagonal `u = 2` to seven digits from
z = 13 upward (u(41) = 2.0000005; only z = 7, 11 move it: 2.08468, 2.01161). Hence:

- the law predicts the **constant** `exp(-(0.24*4 + 0.13*2)) = 0.29523` on the diagonal, while the
  nine measured points rise 0.1521 -> 0.3958 (a level-growing residual);
- the diagonal **cannot identify (c2, c1)** — it fixes only `4c2 + 2c1` — so the law's u-behaviour
  at large W is identified nowhere. Every such statement rests on the different y = 401 family.

This is the structural reason the registered §4a conclusion ("one more point cannot discriminate
drift from limit") is right, and why the tenth **diagonal** point (240 core-h) cannot help: it
extends the u = 2 abscissa by 7.4 %. The u-coordinate is an unprobed second axis of a law already
on the record.

## The route
Measure `r_W(y) = Var/E` at a **fixed large W and a ladder of sieve levels y** (u = ln W / ln y
free) rather than at the single diagonal u = 2; cost falls with shallower y, so this is cheaper than
the diagonal engine, and it is the only design that can tell whether the residual is level-locked
(W alone; the drift reading) or u-locked (u; the law reading). Pre-registered falsifier in the
return's `next_step`.

## Rung and gap
The u-values, the nine points, the coefficients and the 240-h price are **quoted** from the served
pre-registration, re-read not recomputed; the identification argument (`4c2+2c1` only) is an exact
algebraic observation verified by arithmetic. Nothing here bounds G2, beta_2 or twin-prime
infinitude. Gap: no off-diagonal `r_W(y)` was measured here, so the law's u-dependence at large W
is untested; and `paper/variance-note.md` could not be read at the path the pre-registration names
(404 at `/docs/research/paper/variance-note.md`; the served router does not index it), so §6 point 2
is quoted from the pre-registration, not re-read at source. The bridge to route 143's dial / route
108's remainder is CONJECTURAL (route 194/191's caveat) and is not asserted.