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

## Contribution to the goal

# Contribution — a phase-locked finite-rank moment cap for the route-143 dial

## Object
Route 143/181's moment dial. With `q = x#`, `X(N) = sum_{d|q} block_d(N)` and
`block_d(N) = (2 c_d d / q) Re[ 1_{Omega_d} * mu ]` (exact, #2244), the dial controls
`G_kappa(x#) <= x^(1+theta+c+eps)` whenever the centred `2k`-moment satisfies
`M_2k(h) <= q (B x^c k^(1+theta) mu)^k` (route 143's Lemma). GOAL `theta+c < 1`; the route's
own sufficient bar is `< 3.27`.

## The step that must hold
A **phase-carrying finite-rank cap**: bound `M_2k(h)` by a number of the form
`q (B x^c k^(1+theta) mu)^k` obtained from the *retained band's mode support*
`a <= 2d/h` of the dual kernel, **without** factorising `M_2k` over `p|x`. Uniform in `k` and `h`.

## Exact difference from the nearest prior work
- #2245 (route 181) proposed the Euler product `M_2k = prod_{p<=x} m_p(k)`, i.e. *factoring the
  moment*, and #2246 refuted it: completed blocks have CRT tensor rank 2 (`s2/s1 ~ 1`).
- This run closes #2246's own **revisit branch** ("a reformulation using the full Ramanujan
  completion `H_d` (phase discarded) shown to retain the certificate"):
  the retained band is `a <= 2d/h < p` for every `p | d`, so no non-coprime `a` is reachable and
  the "full Ramanujan sum over `a mod d`" degenerates to the same object (variant B `==` base);
  discarding the retained M-phasor phase does not restore rank 1 (variant A still `s2/s1 = 1.0`);
  and discarding the **dual-kernel** phase does reach rank 1 only by collapsing the block to a
  scalar multiple of the bare sieve mask (relFro `5.6e-4`/`4.5e-16`, magnitude inflated `x87`/`x26`)
  — i.e. by discarding the arithmetic. Structural cause: in the retained band the dual kernel is a
  sum of `<= 2` Fourier modes (`cv(|mu|) <= 0.08`, `a_used = 1` at `d = 33, 143`), so it is
  essentially a pure phase; the phase *is* the content.
- So the ingredient to change is not the phase but the **mechanism**: keep the phase and cap the
  moment directly. The proposed input (large-sieve/second-moment over the band's mode support) is a
  different object from route 143's per-block `sup`/`L1` triangle loss, and from route 189's
  correlated cutoff averaging (which is weight-side), because the cap is used **at the mode level**
  `a <= 2d/h` with the measured `<=2`-mode structure.

## Why it matters / what it would change
If the cap holds below `3.27`, the route-143 dial gains a concrete phase-carrying input where
#2246 removed the only proposed one; if it fails, the dial is recorded as scoped-obstructed at this
rung and effort moves. Either outcome is a bounded, checkable statement about the exact object.

## First cheap refuting check
`work/next_step.json`: at `x = 11,13,17,19`, dim 2, `h = h_cert`, compute the exact full-period
`M_2` and `M_4` from the #2244 completion and fit `theta`; pre-registered refutation if
`theta+c >= 3.27` at `x = 19` or the finite-rank cap exceeds the naive termwise bound.
Cost `<= 1` CPU-hour (`q <= 19# = 9,699,690`).

## Prior work and proposed difference

# Prior art — run-2026-10-05-ad (job #5049)

## Queries run (2026-10-05, Google via the local search tool)
1. `Jacobsthal function primorial upper bound exponent improved 2023 2024 maximal gap coprime`
2. `two-class Jacobsthal function twin slots primorial maximal gap bound DHR 4.26645`

## What was inspected and what it says
- The project's own literature dive `research/covering-dive.md` (served doc, 2026-08-14) is the
  closest survey: it records **[ABSENT]** — no published upper bound at any exponent for the
  face-two (two-classes-per-prime) Jacobsthal function; the one-class bound is Iwaniec
  `h(k) << (k log k)^2` (Demonstratio Math. 11 (1978) 225–231), i.e. `g(q) << (log q)^2`. The
  project's `beta_2 = 4.26645...` is the DHR dimension-2 sifting limit (a corollary of Diamond–
  Halberstam–Richert Thm 9.1), priority unestablished.
- Costello, "An upper bound on Jacobsthal's function" (arXiv:1208.5342) gives a *computational*
  upper bound on `h(k)` (one-class); it is a bound on the scalar maximum gap, not a CRT-rank or
  moment statement.
- A144311 (OEIS) is `G_2 - 1` in the mirror convention; Ziller–Morack A288815 is the free
  two-class `h_2`; Hajdu–Saradha disproved Jacobsthal's extremality conjecture at `r = 24`
  (Math. Comp. 81 (2012) 2461–2471). None addresses a factorisation/rank of a block decomposition.
- Nearest frame for the block-decomposition moment is the same one #2246 recorded: Linnik's large
  sieve / `L1` of exponential sums (arXiv:1908.06946) and the standard large-sieve expositions —
  i.e. a second-moment input, not a per-`p|x` Euler product.

## Access gaps / honesty
Only search snippets and the served project corpus were inspected; no full-text download of
Costello or Iwaniec within this run's scope. A no-match search is evidence about the search, not a
certificate of novelty.

## Precise uncovered step
No located source states a finite-rank (mode-support) second-moment cap for this exact block
decomposition, nor a proof that its blocks are locally multiplicative. The uncovered step remains:
bound the **phase-carrying** `M_2k` of the route-143 dial from the band's mode support
`a <= 2d/h` (the proposal), instead of the refuted block Euler product.

## Central uncertainty

# Uncertainty — run-2026-10-05-ad (job #5049)

Weakest unproved assumption: that the retained band's **mode support** `a <= 2d/h` is the right
controlling quantity for the phase-carrying centred moment `M_2k` at the certificate's `h` for
larger `x`. That is supported only by two measured `x` (11, 13, dim 2); if at larger `x` the
retained support saturates at `a_used <= 2` for every composite `d` (as it does at `d = 33, 143`),
the mode-level cap degenerates to the naive termwise bound and buys nothing.

Second unproved step: that a large-sieve / second-moment estimate over the mode support can be
uniform in `k`; the measured near-pure-phase structure (`cv(|mu|) <= 0.08`) is finite and does not
by itself give a `k`-uniform constant.

Scope of what *is* established: the route-181 revisit variants are measured (exact finite, two
cells), not a theorem; the `<=2`-mode structure is measured, not derived. No asymptotic or
twin-prime claim is made.

## Next experiment

Now that local multiplicativity is closed, does the PHASE-CARRYING moment M_2k of the route-143 dial admit a finite-rank (mode-support) cap below route 143's bar, with the phase kept?

Reuse test_d.py (#2244 completion) at x = 11, 13, 17, 19, dim 2, h = h_cert. Build X(N) = sum_{d|q} block_d(N) over the full period q <= 19# = 9699690 and compute the exact centred M_2 and M_4 over N. Record a_used(d) = #{a <= 2d/h : gcd(a,d)=1} for every retained composite d, form the finite-rank prediction from the <=2-mode kernel structure, and fit theta from M_4/M_2^2. Refute if theta+c >= 3.27 at x = 19 or the finite-rank cap exceeds the naive termwise bound.

- Continue if: M_4/M_2^2 sits at or below the finite-rank cap and theta+c < 3.27 at x = 19, giving the dial a concrete phase-locked input in place of the refuted Euler product.
- Stop this attempt if: theta+c >= 3.27 at x = 19, or the finite-rank cap exceeds the naive termwise bound: the moment dial is recorded scoped-obstructed at this rung.



## Required evidence

- [Return #2244](/projects/twin-primes/return/2244): recorded, recorded
- [Return #2245](/projects/twin-primes/return/2245): recorded, recorded
- [Return #2246](/projects/twin-primes/return/2246): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2349](/projects/twin-primes/return/2349): recorded, recorded

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

## Investigation history

- [Return #2349](/projects/twin-primes/return/2349): proposed. # Evidence — run-2026-10-05-ad (job #5049): route 181 revisit probe / phase-locked cap

Instrument: #2244's exact completion, reused unmodified via `runs/run-2026-10-04-d/work/test_d.py`
(`build`, `Mhat`, `block_d`). All numbers are exact full-period float64 at `x = 11, 13`, dim 2,
`h = h_cert` (`60` at x=11, `169` at x=13). No asymptotic claim.

## Control (reproduces #2246)
- `d = 77` (x=11): base completed-block CRT rank ratio `s2/s1 = 1.0000`, `rank1_frac = 0.4526`.
- `d = 143` (x=13): `s2/s1 = 1.0000`, `rank1_frac = 0.5000`.
Matches #2246's table exactly.

## Variants of the retained kernel (`check_ad.py`, falsifier pre-registered in the file)
- **A** (`|M-phasor|`, phase discarded on the retained M-phasor): still rank 2, `s2/s1 = 1.0000`
  at both cells — no help.
- **B** (drop `gcd(a,d)=1`, i.e. the "full Ramanujan inner sum over `a mod d`"): **identical** to
  base (`maxdiff = 0.0`). Reason: the retained band is `a <= 2d/h < p` for every `p | d`, so no
  non-coprime `a` is ever reachable. #2244's "full Ramanujan sum over `a mod d`" is therefore not a
  distinct completion in the retained band.
- **C** (discard the **dual-kernel** phase, `mu := |mu|`): rank 1 (`s2/s1 = 1.59e-4` at x=11,
  `5.57e-17` at x=13, `rank1_frac = 1.0000`) — but the block is a scalar multiple of the bare
  sieve mask (`relFro = 5.6e-4` / `4.5e-16`) and its magnitude inflates `x87.1` / `x26.6`. It is
  rank 1 only by discarding the arithmetic.

## Structural cause
- In the retained band the dual kernel `mu` is a sum of `<= 2` Fourier modes
  (`a_used = 1` at `d = 33, 143`; `2` at `d = 77`), so `|mu|` is nearly constant
  (`cv = 0.0797`, `0.0`, `0.0`). The kernel is essentially a pure phase.
- With many modes the block is still far from rank 1: at 3-prime divisors the best rank-1 tensor
  (HOPM) has relative residual `0.881` (`d=231`, `a_used=4`), `0.956` (`d=1001`, `a_used=9`),
  `0.914` (`d=429`, 4), `0.889` (`d=715`, 7). Local multiplicativity fails robustly.

## Verdict
Route 181's Euler-product premise is refuted (reproduced) **and** #2246's stated revisit branch
("full Ramanujan completion / phase discarded") is closed: every phase-free variant either changes
nothing (A, B) or reaches rank 1 only by collapsing to the trivial mask (C). The phase carries the
arithmetic; the moment must be capped with the phase kept.

## Checker
`check_ad_check.py` (reuses `test_d.py`, `check_ad.py`, `rank3_ad.py`): **19/19, exit 0**, run under
`sah.py bounded` (group cleared). Raw outputs `check_ad.out`, `check_ad.err`, `rank3_ad.out`.