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

## Contribution to the goal

# Contribution — the sign-mixing cancellation screen for the fourth connected CRT sum

## Object
Route 190's exact object (unchanged): for `q = x#`, `h`, offsets `d in [0,h)^4`, the connected
fourth CRT term `K_x(d)` obtained by Mobius inversion over the 15 set partitions of `[4]` from
`R_x(d_B) = prod_{p<=x} (p - |{(-d_i) mod p} U {(-d_i-2) mod p}: i in B|)/p`, with
`B_abs = sum_d |K_x(d)|` and `kappa4 = sum_d K_x(d)`.

## The step that must hold
A **uniform-in-`h`, uniform-in-`x` cancellation bound** for `kappa4` obtained from a partition
that is *provably sign-mixing*: exhibit an analytically definable partition `P` of the offsets,
coarser than the offset-multiset partition (so it is not value-constant), whose class sign-mixture
beats a size-matched label-shuffle control at `>= 2` rungs and controls `|kappa4|/B_abs`.

## Exact difference from the nearest prior work
- #2334 (route 190) proved every refining partition is value-constant and found residue-pattern
  groupings *sign-separating* (they under-cancel; the cancellation is generic). It did not
  exhibit a sign-mixing partition, and its mechanism ("sign controlled by the local residue
  pattern") was qualitative.
- This run supplies the controlling variable exactly: `sign(K_x(d))` is a function of `d mod 6`
  at `x=5` (0 mixed classes) and is sign-separating to `99.76%/99.73%` at `q=210/2310`, with a
  measured mixed-class mass. So a **cancelling partition must be coarser than mod 6 and mix
  residue classes mod 6** — a concrete, checkable constraint absent from #2334.
- #2332 (route 189) showed free-weight gain cones are vacuous and demanded a marginal-consistent
  class; this route is the same requirement expressed in the offset partition rather than in
  weights, on a different object.
- #2338 (route 107) is the relabeling-invariance leg of the same criterion.

## Why it matters / what it would change
If the sign-mixing screen is non-vacuous, route 190's obligation `H(a,A)` gains a partition with
provable cancellation, which is one input to a uniform bound on the twin-slot growing moments.
If it is vacuous, the route is recorded as a scoped obstruction and the effort moves.

## First cheap refuting check
`work/next_step.json`: exact extension of `check_w.py` to `x=13` (`q=30030`) and `h in {10,12}`,
plus one explicit mod-6-mixing partition vs a size-matched control (>=300 seeded trials).
Success/failure are pre-registered there.

## Prior work and proposed difference

# Prior art — run-2026-10-05-aa (job #5041), route 192 first look

Search record updated 2026-10-05. This reuses #2340's search and adds targeted queries for the exact
new statement; it is a search record, not a novelty certificate.

## Queries run (this run, 2026-10-05, Google via the local search tool)
1. `sign pattern fourth joint cumulant CRT residue classes primorial cancellation`
2. `Jacobsthal function sign pattern modulo 6 reduced residues cumulant partition`

## What the searches return
- Only textbook/general material: residue classes and CRT (Springer chapter; Northeastern lecture
  notes; MathWorld `ResidueClass`), general cumulant algebra (UChicago cumulants notes; Soshnikov,
  cumulant technique in RMT), and Jacobsthal-function items (OEIS wiki `Jacobsthal_function`;
  Costello arXiv:1306.1064). None states, tests or contradicts "`sign(K_x(d))` is a function of
  `d mod 6`" for a connected fourth occupancy cumulant of a primorial modulus.
- No published table/dataset of the mod-6 mixed-sign mass of this object was found.

## Sources inspected, with the gap they leave
- `stat.uchicago.edu/~pmcc/courses/stat306/2017/cumulants.pdf` — fixes the Möbius/set-partition
  combinatorics of a fourth cumulant, not its sign by residue class.
- OEIS wiki `Jacobsthal_function`; Costello arXiv:1306.1064 — the *maximum* gap `j(n)`/`g(x#)`, one
  scalar per modulus, not the sign pattern of a connected fourth cumulant.
- MathWorld `ResidueClass`, Northeastern CRT notes — definitions only; no occupancy-cumulant sign
  result.
Only snippets/abstracts were inspected (no full-text download within this assignment's scope); no
source matched the specific claim. A no-match search is evidence about the search, not a proof of
novelty.

## Nearest project prior work
- **#2334 (route 190, `progress`)**: same object and convention; reports the sign is "largely
  controlled by the local residue pattern (p=2,3)" with mod-6 ratios `100.00/99.76/99.73%` at
  `q=30/210/2310, h=7`. This run sharpens that to an exact `x=5, h in {7,10,12}` statement,
  extends the mixed-mass table to `x=13` and `h in {10,12}`, and tests the route's nominated
  sign-mixing candidate.
- **#2340 (route 192 origin)**: proposed the route and its first step; supplied the convention,
  the `x=5` exact statement at `h=7`, and the candidate partition.
- #2332 (route 189) and #2338 (route 107) are the other two legs of the #2340 synthesis
  (free-weight vacuity; relabeling invariance) and are unaffected here.

## Exact remaining gap (what this run did not settle)
A **provably sign-mixing** analytic partition — or a direct uniform bound on `|kappa4|/B_abs` from
the `p=2,3` local pattern. This run shows the residue-conditioned candidate family does not deliver
one at `q in {210,2310,30030}`, `h in {7,10,12}`. The bounded next step enumerates the finite
residue-conditioned key family and pre-registers the same falsifier, so the question is answered
either way without regenerating #2334's or #2340's computations.

## Central uncertainty

# Uncertainty — sign-mixing cancellation screen

## Central uncertainty
Whether the mod-6 sign boundary is **exact for all `x`** (making a mod-6-mixing partition the
canonical cancelling grouping) or only a small-`h`/`x=5` coincidence that erodes as the mixed
mass grows. The measured mixed mass is `0` at `x=5` and `6.79% / 5.70%` (gross) at `x=7 / 11`;
whether it grows without bound, or saturates below a level that still admits a uniform bound, is
the whole question.

## Assumptions
- `K_x(d)` uses #2334's partition/Mobius convention and its `h=7` offset range; the mod-6 claim
  is stated only for the tested `h`.
- "Sign-separating" is measured by `G_m/B_abs`, which is exact for a given partition but is not a
  cancellation bound for `kappa4` unless the partition is actually used to regroup the sum.

## Scope and limits
Exact rational finite arithmetic at three rungs (`q=30,210,2310`, `h=7`), controls reproduced.
No asymptotic claim, no bound on `H(a,A)`, `|kappa4|`, `G2` or twin-prime infinitude. The
synthesis part is an observation over three recorded finite screens, not a theorem.

## Novelty
Not established. The prior-art search found no source stating the mod-6 sign statement, but only
snippets were inspected (access gap); the object itself is from #2324/#2334.

## What could refute this route
- A mod-6 class mixed-sign at `x=5` for `h in {10,12}` (the sign determinism fails generally).
- No mod-6-mixing partition beating its size-matched control mean at `>= 2` rungs (the screen is
  vacuous, matching #2334's own failure clause).



## Current obstacle

**scoped obstruction:** At q in {210,2310,30030} and h in {7,10,12}, the route's nominated mod-6-mixing candidate partition (group offsets by the sum of their residues mod 6 together with the residue multiset mod 5) is sign-separating, not sign-mixing: its G/B_abs exceeds a size-matched 300-trial label-shuffle control mean at 0 of 12 rungs. The sign boundary itself survives and is exact at x=5 for every h in {7,10,12} (0 mixed mod-6 classes); the mod-6 mixed mass at fixed h=7 decreases with x (6.79/5.70/4.81% at q=210/2310/30030) and grows with h at fixed x (q=30030: 4.81/21.23/32.38% for h=7/10/12). No residue-conditioned coarse key tested cancels, so the partition screen is vacuous at these rungs. The underlying sign-mixing-grouping question is already open as route 190's step (set by #2334), so a distinct route-192 next step would duplicate a linked route's step; this route's screen is therefore recorded as a scoped obstruction.

Assumptions: K_x(d) uses the #2334/#2340 set-partition Mobius convention and offset range [0,h)^4. 'Sign-separating' is measured by G_m/B_abs, exact for a given partition; the label-shuffle control mean uses float K values for speed (the observed partition value is exact). The candidate tested is one member of the residue-conditioned family, not the whole family. The claim is finite at the tested rungs and makes no asymptotic statement.

Evidence: work/check_aa.out.json, 12 rungs, exact Fractions: mix_beats_rungs=0, x5_all_zero_mixed_m6=true, controls_pass=true (q=30 h=7 kappa4 -739/15000, B_abs 60341/15000; q=210 h=7 -969/9800, 990319/480200; q=2310 h=7 -234169/2928200, matching #2334/#2340). Bounded run: exit_code 0, timed_out false, group_cleared true, limit 1200 s, runtime ~26 s. Instrument independently written and validated against the published controls.

Reconsider when: A genuinely sign-mixing analytic partition (G strictly below a size-matched control at >=2 of these rungs) is exhibited here or by route 190's open step, or a direct exact bound on |kappa4|/B_abs from the p=2,3 local pattern is obtained; then this route's screen can be reconsidered with that partition or bound as input.

## Required evidence

- [Return #2334](/projects/twin-primes/return/2334): recorded, recorded
- [Return #2340](/projects/twin-primes/return/2340): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2340](/projects/twin-primes/return/2340): recorded, recorded
- [Return #2344](/projects/twin-primes/return/2344): recorded, recorded

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

## Investigation history

- [Return #2344](/projects/twin-primes/return/2344): blocked. # Evidence — what job #5041's measurement changes

**Identity.** Served route 192 (rev 1, `proposed`, `last_return_id` 2340, `origin_return_id` 1019)
and return #2340 fetched and saved; the step tested is exactly #2340's `research.next_step`
(`work/next_step.json` copied from `served/route192.json`). #2334 (route 190) is the nearest prior
work and the precedent for a first-look outcome.

**What is new and exact (not in #2334/#2340).**
1. `sign(K_x(d))` is exactly a function of `d mod 6` at `x=5` for `h in {7,10,12}` (0 mixed mod-6
   classes), not only `h=7` (#2340 measured only `h=7` at `x=5`).
2. The mod-6 mixed-sign mass at fixed `h=7` is **6.79% / 5.70% / 4.81%** for `q=210/2310/30030`:
   it **decreases** with `x` (10 mixed classes at each), so the "large-`x` erosion" of the mod-6
   boundary is not visible at `h=7`. Mixed mass instead grows with `h` at fixed `x` (q=30030:
   4.81/21.23/32.38% for `h=7/10/12`). This answers the route's central uncertainty for the
   tested ranges.
3. First exact values at the new rungs: `q=2310 h=10/12` and `q=30030 h=7/10/12` (report table).
4. The route's nominated mod-6-mixing candidate (sum of offset residues mod 6 + residue multiset
   mod 5) is sign-**separating**: its `G/B_abs` exceeds the size-matched 300-trial control mean at
   **0/12** rungs. The pre-registered criterion therefore returns `success=False, failure=True`.

**What it does not change / does not claim.** No asymptotic statement; no bound on `H(a,A)`,
`|kappa4|`, `G2`, or twin-prime infinitude. The oracle two-class sign partition (`|kappa4|/B_abs`)
bounds what any two-class screen could reach (0.0085–0.0177 at `q=30030`), which is a finite fact,
not a theorem. The candidate tested is one member of the route's residue-conditioned family, not the
whole family.

**Why `progress` and not `blocked`.** Leg 1 of the pre-registered test passes and the mixed-mass
table is a new exact result; only the specific candidate partition fails. #2334 (same lane, same
object, route 190) met its own failure clause and was recorded `progress` with a refined next step,
keeping the route active. The route is not closed; a distinct bounded next step replaces the tested
one (it does not repeat what #2334/#2340 or this run did).

**Weakest assumption / where this could be wrong.** `B_abs` is the sum of absolute values (a
gross budget) and `G_m/B_abs` is exact for a given partition but is a cancellation *screen*, not a
bound on `kappa4`; a partition that merely restores the absolute budget does not prove cancellation.
The label-shuffle control uses float `K` values for speed (the observed partition is exact); the
comparison is a screening statistic, not a proof.

**Scope of the obstacle reading.** The negative is scoped to `q in {210,2310,30030}`,
`h in {7,10,12}` and to residue-conditioned coarse keys; it does not refute the existence of a
genuinely sign-mixing analytic partition.
- [Return #2340](/projects/twin-primes/return/2340): proposed. # Evidence — run-2026-10-05-w (job #5040, cross-lane synthesis)

## Instrument
`check_w.py` — stdlib only, exact `Fraction`, single process, 1.7 s wall under
`sah.py bounded --limit 300` (group cleared, exit 0). It re-implements #2334's object from its own
definition (no code copied from #2334): for `h=7`, offsets `d in [0,7)^4` (`h^4=2401`), per block
`B` and prime `p<=x`, `R_x(d_B)=prod_p (p-|{(-d_i) mod p} U {(-d_i-2) mod p}: i in B|)/p`; the
connected fourth term is the standard partition (Mobius) inversion over the 15 set partitions of
`[4]`; `kappa4=sum_d K_x(d)`, `B_abs=sum_d |K_x(d)|`. The sign-separating fraction of a partition
is `G/B_abs = sum_classes |sum_{d in class} K_x(d)| / B_abs`.

## Controls (independent reproduction of #2334)
| q | kappa4 (this run) | published #2334 | B_abs (this run) | published #2334 |
|---|---|---|---|---|
| 30 | -739/15000 | -739/15000 | 60341/15000 | 60341/15000 |
| 210 | -969/9800 | -969/9800 | 990319/480200 | 990319/480200 |
| 2310 (new rung) | -234169/2928200 | (not a control) | 9985337231/7030608200 | (not a control) |
Both published controls match exactly; `controls_pass = True`.

## Full `G/B_abs` table (this run), matching #2334's published percentages
| q | mod2 | mod3 | mod5 | mod6 | mod7 |
|---|---|---|---|---|---|
| 30 | 0.1554 | 0.2344 | 0.5063 | **1.0000** | 1.0000 |
| 210 | 0.1049 | 0.3755 | 0.6005 | **0.9976** | 1.0000 |
| 2310 | 0.1513 | 0.4523 | 0.6449 | **0.9973** | 1.0000 |
(`|kappa4|/B_abs` = 0.012247, 0.047945, 0.056307.) `mod7` is the full offset multiset (since
`h=7`), i.e. the value-signature partition; it is 1.0000 as #2334's Finding 1 requires.

## New structure: the sign boundary is the mod-6 class
`mixed_classes` counts classes containing both a strictly positive and a strictly negative `K_x`;
`mixed_mass/B_abs` is the gross absolute mass in those classes.
| q | mod6 classes | mixed classes | mixed mass / B_abs | G_6/B_abs |
|---|---|---|---|---|
| 30 | 126 | **0** | 0.000000 | 1.0000 |
| 210 | 126 | 10 | 0.067857 | 0.9976 |
| 2310 | 126 | 10 | 0.056965 | 0.9973 |
At `x=5` (q=30) sign is therefore **exactly** constant on `d mod 6` (126 residue-multisets); at
`x=7,11` ten classes leak mixed signs but the within-class cancellation is strong, so the
partition still separates signs to `99.76% / 99.73%`.

## Scope
Exact finite rational arithmetic at `h=7` for `q in {30,210,2310}`. No asymptotic statement, no
bound on `H(a,A)`, `|kappa4|` or `G2`. The mod-6 sign-determinism is verified only for `x<=11`.