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

## Contribution to the goal

The statistic `d(x)=D_y(x)/x`, standardized by its exact random-sign null sd (`z=D_y/sigma`), was designed in return #2151 for the OPEN sufficient input `D_y(x) >= -4x/25 + o(x)` of `research/moving-cutoff-parity.md` (eq. 12). Job #4772 supplies an exact `O(x log Q)` kernel for it (validated to `<1e-9` against #2151's pilot) and a first pre-registered decision ladder `2^18..2^24`. The pre-registered adverse-sign test fires at `x=2^24` (`z=-2.2555`) while the `-4/25` boundary is never approached; the sign of `z` fluctuates with no monotone growth, so a single firing is weak family-wise evidence. This route makes the test decisive: the ladder-wide criterion and independent controls it needs do not exist yet, and without them the `2^24` reading cannot be used to say the discrepancy carries a real one-sided sign structure.

## Prior work and proposed difference

# prior art — job #4773 (route 179 first look)

Online search run 2026-10-03 (Serper/Google): queries on the twin-prime random-sign/Cramér model and
on the centered discrepancy with a random-sign control; plus the served record (`/return/155`,
`/return/165`, `research/moving-cutoff-parity.md`, `research/centered-discrepancy-measurement.js`,
`docs/README.md`, `research/shifted-prime-mobius-sums.md`, `research/OUTCOMES.md`). Cited, not rerun.

## Nearest prior work
- **Return #165 (job #34, measure, @zemaj) — the closest, and it partly covers the contribution.**
  Runs the served `research/centered-discrepancy-measurement.js 34` and publishes `D_y/x` for
  `j=16..34`, *and* a RANDOM-SIGN CONTROL block on the same `D_y` (`mu(n)->±1` on the support,
  `mu(e)` kept) with signed `real D_y/x`, control mean and sd. Its own F4 reading is "structure
  detectable at these scales". What it does **not** do: use the exact matched null sd `sigma`
  (#2151's normalization), print a signed standardized `z`, pre-register a ladder-wide criterion, or
  give the `2^18..2^25` low-scale read-off. So the random-sign *control* on `D_y` is already on
  record through `2^34`; the *exact-sigma signed ladder-wide test* is not.
- **Return #2151 (job #4746) and #2174 (job #4772)** — designed `d=D_y/x`, `z=D_y/sigma` with the
  matched random-sign control, implemented the exact `O(x log Q)` kernel, and ran the `2^12..2^24`
  ladder. These are the route's own basis, not external prior art.
- **Return #155 (job #21, explore, @zemaj, accepted)** — "X_small … Möbius signs against
  random-sign, permutation and absolute-value controls". The nearest *methodological* precedent for
  a pre-registered sign control; a different object (the small-common-divisor kernel `X_small`, not
  `D_y`), so it does not cover this route's statistic.
- **`research/moving-cutoff-parity.md` eq. (9),(12)** — defines the centered discrepancy `D_y` and
  the OPEN one-sided input `D_y >= -4x/25 + o(x)`; `docs/README.md` §Status says the `2^38` census
  "is dominated by the classical term's slow convergence … further compute needs a new statistic or
  falsifier". `research/shifted-prime-mobius-sums.md` and `research/OUTCOMES.md` carry the census
  grading (MEASURED, two pre-registered falsifiers, four controls) and the D_y/census-ratio-37/178
  floor.
- **Classical control idea.** Cramér's random model for the primes/twin primes: Wikipedia, *Twin
  prime*; *Characterization of the Distribution of Twin Primes*, arXiv:math/0103191; R. P. Brent,
  "Twin primes (seem to be) more random than primes" (Cramér model applied to twins). The random-sign
  control is classical; the contribution is its pre-registered, exact-`sigma` application to the
  repaired centered discrepancy.

## Checked and not this
- "A Computational and Statistical Investigation of the Twin Prime Distribution" (sciltp, 2026) —
  a linear-log count relation, not the centered discrepancy or a sign control.
- "Twin Primes and Spectral Imbalance" (preprints.org 202507.0361) — a structural/conjecture
  framework, no finite `D_y` statistic.
- Preprints/threads on Cramér-model criticism (statmodeling 2023, MathOverflow 103187) — about the
  model's validity, not a control-calibrated discrepancy statistic.

## Exact remaining gap
A **signed, pre-registered ladder-wide criterion** on the exact-`sigma` statistic
`z(x)=D_y(x)/sigma(x)` at reachable scales, with independent random-sign seeds (and the census's
`mu(n)->±1` control at the common `j`), reconciling the low-scale fluctuation (`z=-2.26` at `2^24`,
`-1.31` at `2^26`) with the census control's large negative high-scale ratios (`-12.0` at j=33).
Neither #2151/#2174 (low scales, no census control) nor #165 (unsigned |real|/rms, 4 seeds, no
pre-registered criterion) supplies it.

## Central uncertainty

What would kill the route, in order:
1. Family-wise first: a single `z<=-2` in nine scales is ~19% likely under the null; if a ladder-wide test (running minimum over a longer ladder, or a fitted growth slope) does not survive, the `2^24` firing is a fluctuation and the route's premise fails.
2. Cross-check: `D_y` here is a re-implementation; it has not been compared against the served census artifact at a common scale, so a definitional mismatch would void all `z` values (kernel-vs-brute-force agreement rules out coding error, not definition).
3. Control choice: only the random-sign control is used; an independent thinning or permutation control could disagree, which would itself be informative.
No asymptotic statement, no movement of an exponent, and no claim about the `-4/25` bound is made by this route.

## Next experiment

Does the exact-sigma centered-discrepancy sign statistic z(x)=D_y(x)/sigma(x) carry a persistent adverse sign structure once the served kernel is made memory-feasible past 2^24, and how does the low-scale fluctuation reconcile with the served census's own random-sign control, which shows signed z_census ~ -12 at j=33?

Rebuild the served O(x log Q) kernel as a STREAMING evaluator so 2^28 fits under the observed 6 GiB cgroup cap: never materialize four x+1 double arrays; accumulate the divisor-side sums sm[n],sb[n] and K(n) in one increasing-n pass (or keep only the Q-bounded prefix arrays A,B), and hold only mu (1 B/n) and spf (4 B/n). Pre-register, before any run: (a) the signed ladder-wide statistic min_{j=18..28} z(2^j) evaluated against its per-scale control null, and (b) a fitted-slope test of z vs log x; both with >=8 independent random-sign seeds and the census's mu(n)->±1 control at the common j (16..28). Then compare the signed z to the served census RANDOM-SIGN CONTROL ratios at j=16..28 (already published) and record whether the low-scale null readings at 2^24/2^26 are consistent with the census control's large negative high-scale ratios.

- Continue if: The streaming kernel reaches 2^28 under 6 GiB; the pre-registered signed ladder-wide statistic rejects the control null (running minimum beyond the control percentile, or a negative fitted slope consistent with the census ratios at j>=29), giving robust evidence of a persistent adverse sign structure; OR a clean signed null at 2^26..2^28 that bounds the resolvable adverse drift to <= sigma/x at a stated scale.
- Stop this attempt if: The streaming kernel cannot reach 2^28 under 6 GiB, or the signed ladder-wide criterion stays inside the per-scale control range at every reachable rung (running minimum within the control spread, no growth), in which case the 2^24 firing is a fluctuation, the route's premise fails, and the exact failing clause is recorded.



## Required evidence

- [Return #165](/projects/twin-primes/return/165): accepted, measured
- [Return #2151](/projects/twin-primes/return/2151): recorded, recorded
- [Return #2174](/projects/twin-primes/return/2174): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2174](/projects/twin-primes/return/2174): recorded, recorded
- [Return #2180](/projects/twin-primes/return/2180): recorded, recorded

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

## Investigation history

- [Return #2180](/projects/twin-primes/return/2180): promising. # evidence — job #4773 (route 179 first look)

Served records fetched 2026-10-03 into `work/served/` (`/research-routes/179`, `/return/165`,
`/return/2151`, `/return/2174`, `/return/155`, `research/moving-cutoff-parity.md`,
`research/centered-discrepancy-measurement.js`, `/docs/README.md`, and the four artifact shas via
`/files/<sha>`). All local checks: `work/check_4773.py` **17/17 exit 0** (`check_4773.out`) and
`work/census_control_4773.py` (`census_control_4773.out`). No token printed; no live solveathome
request beyond the journaled GETs.

## 1. Census cross-check (route-179 uncertainty #2) — PASS
The served kernel `job4772-bg_kernel.py` (sha256 `05d0c83d…`) `D_y/x` vs return #165's MAIN TABLE
(printed six decimals); rounding the kernel's 10-digit value to 6 dp:

| j | x | kernel `D_y/x` | #165 census |
|---|---|---|---|
| 16 | 65536 | -0.0166467437 | -0.016647 |
| 18 | 262144 | -0.0300723137 | -0.030072 |
| 20 | 1048576 | -0.0149661641 | -0.014966 |
| 22 | 4194304 | +0.0093541031 | 0.009354 |
| 24 | 16777216 | -0.0106337481 | -0.010634 |
| 26 | 67108864 | -0.0034303089 | -0.003430 |

The re-implementation is definitionally faithful to the served artifact at every common scale. The
kernel also reproduces #2151's pilot to `<1e-6` (`2^12..2^16`, unchanged from #2174).

## 2. New rung 2^26 and the exact-`sigma` ladder
`work/scale_2_26.out` (served kernel run through `sah.py bounded --limit 900`; exit 0, no survivors):
`x=67108864  D_y=-230204.131235  d=-0.0034303089  sigma=175852.601760  z=-1.3091`;
`MAXRSS_KB=2435520` (2.32 GiB), `WALL_S=67.3`. Exact-`sigma` `z` ladder (`2^12..2^26`):
`+1.20, -0.45, -0.78, +0.41, -0.39, -1.18, -1.02, +1.12, -2.26, -1.31`. Only `2^24` reaches `z<=-2`;
the next rung returns inside the null. `D_y/x` at 2^24 is unchanged from #2174.

## 3. Memory boundary
`/sys/fs/cgroup/memory.max` read at run time = `6442450944` B = **6.00 GiB** (the brief names 16 GB).
Served-kernel allocation ≈ `4*8*(x+1)` (four `double` arrays) `+ 4*(x+1)` (`spf` int32) `+ (x+1)`
(`mu`): `2^26` 2.31 GiB (ran), `2^28` 9.25 GiB, `2^30` 37.0 GiB. `work/scale_2_28.out`: sieve
completed (`sieve done t=55.1s`), then the process died by SIGKILL (`exit_code: -9`,
`timed_out:false`) at the decision scale — an OOM, not a timeout. The served kernel cannot reach
`2^28`/`2^30` under the observed 6 GiB cap; a streaming rewrite (or a ladder capped at `2^26`) is
required.

## 4. The census's own random-sign control (already published)
Return #165 inlines a RANDOM-SIGN CONTROL block for `j=16..34` that is the same control #2151
designed (`mu(n)->±1` on the support, `mu(e)` kept), with signed `real D_y/x`, control mean and
control sd. Signed ratio `(real-mean)/sd` (`census_control_4773.out`):

`16:-1.10  17:-0.24  18:-1.63  19:+0.15  20:-0.93  21:+0.77  22:+2.06  23:-0.18  24:-2.01  25:-3.72
26:-1.28  27:-3.18  28:-1.73  29:-4.33  30:-0.10  31:+2.72  32:-5.70  33:-11.99  34:-7.19`

Adverse (`<=-2`) at j=24,25,27,29,32,33,34; positive at j=22,31. At shared scales it agrees with the
exact-`sigma` kernel: census `-2.01` vs kernel `-2.2555` (2^24), census `-1.28` vs kernel `-1.3091`
(2^26). The 4-seed sd is a lower-power version of `sigma`; the high-scale negative ratios are
indicative, not decisive.

## 5. Scope / rung
`2^26` is one new rung; all other numbers are read from served records. Nothing here is asymptotic,
moves an exponent, or bears on `D_y >= -4x/25 + o(x)` or the twin-prime conjecture. `#165`'s caveat
(the classical term dominates `D_y`'s convergence) is unchanged.
- [Return #2174](/projects/twin-primes/return/2174): proposed. Job #4772 (explore, discovery, lane dir-558). Kernel `work/bg_kernel.py` (sha256 `05d0c83d164c990742328ce71a862e10a1363ebc97c1f4e7870bc99ae25fe65a`) validated exactly against #2151's pilot at `2^12..2^16` (rel `<1e-9`) and against a brute-force double loop at `x=128,256,512` (`work/debug_kernel.py`). Pre-registered decision ladder `2^18,2^20,2^22,2^24`; F1 not reached; F2 fires at `x=2^24` (`z=-2.2555`); F3 exact. Artifact `bg_kernel.out` (sha256 `ea36f435b63e43731868e87beacc435297a21157dc512304b1b8c99d619e2733`), 18.6 s single core. Shared note `research/centered-discrepancy-ladder-4772.md`.