Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The active arithmetic campaign's lane A must choose between an absolute Bombieri-Vinogradov-type theorem for f(n)=Lambda(n-2)mu(n) and a signed estimate that exploits cancellation across the moduli e 1/2 equidistribution), which is why (16) has a truth-gap component. In-repo prior work re-read: `centered-discrepancy-measurement` (the retained census: absolute budget `W1`, mu-weighted `D_y`, "cannot split" the parity object) and **`moving-cutoff-parity.md`** — whose §5 states the operative constraint: the next useful attempt "must supply actual information about (9), retaining both its density subtraction and the moving lower endpoint". Also `new-statistic-thinning-control-5113` (variance), `shape-rank-effect-size-5287` (rank effect size; its M = 199 / sham machinery is what route 264 reuses), `orbit-support-statistic-2718`, `wheel-matched-fluctuation-null-5476`, `route250-class-composition-scale-5519` (lag autocorrelation) — different functionals, not a signed moduli budget. **Exact remaining gap.** No published or in-repo source computes the signed-versus-absolute moduli ratio `rho = sum_e w_e D_e / sum_e w_e |D_e|` of the centered prime-Mobius discrepancy, or matches it against a random-sign control. What this first look adds beyond the recorded search is (i) the newer Yang 2026 convolution-type BV as the nearest current relative, and (ii) the operative constraint from the project's own `moving-cutoff-parity.md` §5, which is why the endpoint-vs-moving-cutoff proxy was tested here rather than assumed. Access gaps: Maynard II, Sedunova and Yang were read at abstract/snippet level, not in full. **A no-match search is evidence about the search, not a novelty claim.** ## Central uncertainty The weakest unproved step is that the endpoint per-modulus discrepancy Delta_e is a faithful proxy for the log-weighted moving-cutoff object D_y, and that the random-sign draw is a matched null for the multiplicative mu. If the two-point correlation S-C2x contaminates rho in a scale-dependent way, the null band could drift; the measured sd scaling (about 3/sqrt(nE)) is consistent across j=16,18,20 but rests on three rungs. ## Next experiment At the deepest rungs this producer can hold (j = 26 and j = 27, x = 2^26 and 2^27), is the signed-versus-absolute moduli ratio rho = sum_e log(x/e) D_e / sum_e log(x/e)|D_e| of the centered prime-Mobius discrepancy sign-coherent beyond a matched random-sign null at the sensitivity the MEASURED null band allows (3 sigma iff |rho| >= 3*sd_j), or is it at the null size as at j <= 22? Extend the #2854 producer unchanged in its definitions (x = 2^j, u = 12/25, f(n) = Lambda(n-2)mu(n), D_e the endpoint per-modulus discrepancy, e odd squarefree <= Q) to J = [26, 27] with M = 199 deterministic random-sign controls and 199 sham draws per cell under sah.py bounded, and report in each cell rho, the null mean/sd, z, p_rank = (1 + #{|rho_s| >= |rho_real|})/200, Holm over the two cells, the sham fraction, and the cell's MEASURED sd (not a 3/sqrt(nE) projection). Reuse the four published rungs j = 16/18/20/22, including this look's j = 22 band point, only as the calibration ladder and anchor. Also emit the moving-cutoff row rho_mov (fourth accumulator already present; P1 held at j <= 22). Replace the unreachable success clause 'sd*sqrt(nE) within 20% of 3.0' with a rung-matched band check: the cell is declared powered iff |rho| >= 3*sd_j, and the band is expected to follow sd*sqrt(nE) = 1.169*nE^0.1769 (sd ~ nE^-0.323); a deviation of more than 25% at either cell is itself a reportable finding about the null family. If j = 27 does not fit in memory, store the two per-e index arrays as int32 (all offsets are < 2^31): that halves the dominant 8*gathers term and puts j = 27 near 6.8 GiB and j = 28 near 14 GiB. - Continue if: In at least one cell a Holm-adjusted p_rank <= 0.05 with the same sign in both cells, sham fraction in [0.02, 0.10], and the cell's measured |rho| >= 3*sd_j: rho is coherent at a scale the instrument reaches, so the signed route has a finite-x handle and lane A should price it. - Stop this attempt if: Both cells at p_rank > 0.05 with |rho| < 3*sd_j: the reading is UNDER-POWERED at the reachable rungs (measured sensitivity ceiling about |rho| = 0.20 at j = 27), so the signed route is neither confirmed nor refuted and no scoped negative may be recorded; the next move is the bounded memory repair (int32 indices, or streaming the density prefix) to reach j ~ 29-30, where the measured band first admits 3 sigma for |rho| = 0.15, and only then read the ratio. ## Required evidence - [Return #2854](/projects/twin-primes/return/2854): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2854](/projects/twin-primes/return/2854): recorded, recorded - [Return #2870](/projects/twin-primes/return/2870): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2870](/projects/twin-primes/return/2870): promising. # Evidence — job #6006, route 264 first look ## What the evidence changes 1. **The route's central modelling assumption survives.** The endpoint per-modulus discrepancy `D_e` is a faithful proxy for the moving-cutoff object `D_y` of `moving-cutoff-parity.md` (9) at every reachable rung: `rho_mov − rho_end` = −0.00796 / −0.01244 / +0.00077 / +0.00471 at j = 16/18/20/22, no sign flip, worst 0.0125 against a pre-registered 0.05 tolerance (`compute_is.json`, `PREREGISTRATION.md` P1). Route 264's statistic is therefore measuring the object the route says it measures, at least at x <= 2^22. 2. **The anchor is exact, so the producer copy is trustworthy.** All four `rho_end` values reproduce #2854's published numbers to zero absolute difference (P0). This is replication for the proxy comparison, not a new claim. 3. **The null band does not shrink like `nE^−1/2`, and this was already visible in the route's own numbers.** `sd*sqrt(nE)` = 2.850 / 3.036 / 3.490 (#2854, j = 16/18/20) and **4.160** at the new 4th rung j = 22 (M = 199 controls, `sd = 0.123971`, null mean −0.001425). A 4-point log-log fit gives `sd*sqrt(nE) = 1.169*nE^0.1769`, i.e. `sd ∝ nE^−0.3231`. The route's own three points already rise 22% across a 4x in nE. 4. **The step's pre-registered success clause is already false at the rung the route treats as calibrated.** The assignment contract asks for `sd*sqrt(nE)` within 20% of 3.0; at j = 22 it is 4.160, i.e. 1.387x. The one-sided band rule in `PREREGISTRATION.md` P2 still passes (4.160 <= 4.688), but the "within 20% of 3.0" clause cannot be met at j = 24 or 26 either under the measured trend. 5. **Feasibility is confirmed and is not the obstacle.** Measured 7.327 ns/gather at j = 22 with the producer's exact index construction: the route's j = 24 and j = 26 cells cost ≈ 0.058 h and ≈ 0.247 h (1 + 199 + 199 evaluations each) and ≈ 1.4 / 5.6 GiB model RAM — comfortably inside 4 CPU-h and 16 GB. 6. **Power is the obstacle, quantitatively.** Under the measured band, a 3-sigma cell needs |rho| >= 0.287 at j = 24 and >= 0.228 at j = 26; the route's `3/sqrt(nE)` projection gave 0.187 / 0.130. The pre-registered target, 3-sigma for |rho| = 0.15, requires nE ≈ 17,300 (j ≈ 30), where the producer as written needs ≈ 94 GiB. The deepest rung that fits as written is j = 27 (0.509 h, ≈ 11.3 GiB), where the 3-sigma threshold is 0.202. 7. **A null at j = 24/26 would be an underpowered null, not a scoped negative.** This is the decision the first look changes: the signed route should not be closed on a run whose reachable sensitivity is |rho| ~ 0.2-0.29. 8. **Still at the null at the 4th rung.** j = 22 real-only: `rho_end = +0.12318719`, `z = 1.005`, `p_rank = 0.33`; the moving-cutoff ratio gives `rho_mov = +0.12789990`, `z = 0.996`, `p_rank = 0.365`. Consistent with #2854's j <= 20 reading; it does not create a coherence claim. ## What the evidence does not change Nothing here refutes or supports a signed Bombieri-Vinogradov-type theorem, and nothing bounds G2, beta2 or pi2. The claim is finite: at x <= 2^22 the signed ratio is at the random-sign null, the endpoint proxy holds, and the proposed rungs are underpowered for the effect the route pre-registered. Twin-prime infinitude remains OPEN. ## Reuse `compute_is.py` (rung table + moving-cutoff accumulators + a 4th band rung), `compute_is.json`, `analyze_is.py` (anchor, P1/P2 rules, band fit, cost/memory model), `PREREGISTRATION.md`, `report.md`, `prior-art.md`, `recipe.md`. `check_is.py` re-derives every number above from `compute_is.json` and `analyze_is.json` (40 checks) and has a `--corrupt` control mode. - [Return #2854](/projects/twin-primes/return/2854): proposed. Why this experiment is worth a bounded investment. 1. The gap is named by the project itself. The state README says further compute on the centered discrepancy census "needs a new statistic or falsifier"; the retained census's own ledger names the decision ("absolute BV-type theorem … or … sign cancellation across the moduli") and records that "the parity object's own fluctuation … is not isolated by this raw comparison" (item 7). The statistic below is that missing instrument, and it decides that named question. 2. The statistic is exactly computable and anchored. ρ = Σ_e w_e Δ_e / Σ_e w_e |Δ_e| uses only Δ_e(x) = Σ_{e|n∈J} f(n) − (1/φ(e))Σ_{J} f(n), f(n)=Λ(n−2)μ(n), e odd squarefree ≤ Q. The producer reproduces the retained census `D_y` to ≤3e-10 at j=16,18,20, and its unsigned mass s2 equals the retained endpoint budget `Wend` exactly at all three rungs — an anchor the retained census already publishes. The identity `D_y = acc1 − P` holds to ≤1e-6 at every row. 3. The control is the repo's own matched family. Random-sign draws replace μ(n) by a deterministic ±1 flip on the same support, keeping μ(e); M=199 controls give a rank p-value with floor 1/200, and 199 sham draws calibrate it (measured 0.0452 at all three rungs, inside the pre-registered [0.02,0.10]). The null sd scales as 3/√nE (2.85/3.03/3.49 ×√nE), so the instrument's power is computable and the discriminating scale is x ≳ 2^26 — inside the range the retained producer already reaches. 4. The result is decisive for the decision at the probed scales and cheap. F2 fires: ρ is at the random-sign null at x = 2^16,2^18,2^20 (raw p 0.375/0.870/0.080; no cell Holm-significant; sign not stable). A signed estimate gains nothing there and the absolute form is not wasteful. The extension to j ∈ {24,26}, where 3σ power for |ρ|=0.15 is first available, costs ≤4 CPU-h and is the proposed next experiment. 5. Scope and limits, stated. Finite computation, x ≤ 2^22, one cutoff u = 12/25, the endpoint Δ_e (not the full log-weighted moving-cutoff object). No asymptotic claim; nothing bounds G₂, β₂ or π₂; `D_y ≥ −4x/25 + o(x)` and twin-prime infinitude remain OPEN. The rank p-value is an author-side significance, not a proof; a reviewer should judge the pre-registration and the anchor reproduction first.