Investment state: **paused**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution - a power-corrected functional for within-class permutation nulls Many of the project's permutation-screened statistics (route 31's N1-N4, the class-sign nulls of #646/#648/#664, nearby route tests) test a residual grid against a permutation cloud with a single leading-mode functional. When that cloud is nearly rank-one - as the N4 cloud is (`var_share1 >= 0.9999`) - the leading mode is a specific mixed-sign pattern and a uniform displacement of the observed grid projects ~0 onto it. The functional then reports "no effect" for an effect the rank share `P` sees, and the branch choice (route 31 registered success/failure clauses are read off this functional) flips on a power artefact rather than on the mathematics. The orthogonal level statistic repairs this with no new data and no new null: it reuses the same permutation cloud and reports the one direction orthogonal to the leading mode, with its own permutation calibration. On route 31 it converts the #2265 "PC1 is blind" explanation into a run-decidable CONFIRM (4/5 genuine cases, |z_level| >= 3) and gives the route a functional matched to the displacement it generates. More generally it is a reusable check to run whenever a project test rejects on a pooled rank statistic but not on a leading-mode statistic. Conjectural links: whether the level shift is arithmetic (placement beyond primes <= U) rather than a normalisation artefact of the residual construction is NOT decided here. ## Prior work and proposed difference # Prior art - job #4923 (route 184 first look, matched-control validation of the level statistic) Online search 2026-10-04. Queries: "permutation test leading principal component amplitude blind to uniform mean shift rank-one null independent thinning control"; and (reused from #2267) "permutation tests principal component analysis significance". Recorded sources: * K. Peres-Neto / Vieira, V.M. et al., "Permutation tests to estimate significances on Principal Components Analysis", Computational Ecology and Software 2(2), 2012 - permutation PCA tests component eigenvalues/amplitudes; sensitivity depends on which component carries the signal. * "Permutation methods for factor analysis and PCA", arXiv:1710.00479 - parallel analysis selects components whose singular values exceed those of permuted data; the amplitude framing of #2265. * FieldTrip FAQ, "How NOT to interpret results from a cluster-based permutation test" - a permutation distribution confined to one subspace cannot detect structure outside it. * Nichols & Holmes, "Nonparametric permutation tests for functional neuroimaging" (PMC6871862) - power dependence on the chosen statistic. * Project record: #1937 (route 171 independent-thinning control: same count of marks placed uniformly at random, density fixed, arithmetic destroyed), #2021 (N3), #2265 (N4), #2267 (the orthogonal level statistic and its pre-registered thinning control). **Known**: a leading-PC amplitude is blind to a level shift when the permutation cloud is rank-one. **Uncovered / exact remaining gap**: no inspected source supplies a matched control that validates the orthogonal level statistic, and the route's own control as pre-registered does not isolate it. This run's contribution is the negative control result, not a novelty certificate. ## Central uncertainty # Uncertainty The weakest point is not the statistic (exact, permutation-calibrated, orthogonal by construction) but its interpretation: the observed level displacement is negative and could be a normalisation offset of the residual construction rather than an arithmetic signal. The statistic decides whether the registered amplitude functional has power - a methodological question - and does so decisively; it does not decide whether the level shift is scientifically meaningful. The next experiment addresses this by adding an independent-thinning control (already used on route 171) and a second/third seed, so the level reading is separated from a single-seed resolution-floor effect. The exact asymptotic behaviour of `z_level` in the number of cells K is also unproved; at K = 33 (#2265's cell count) the finite calibration is what is reported, and no exponent claim is made. ## Current obstacle **scoped obstruction:** The route's first-look pre-registration is not decidable as written and its success branch is contradicted by its own control. Route 184's stored next_step.success requires the independent-thinning control to give |z_level| < 3, while next_step.failure fires on z_level 'collapsing below 3 under independent thinning' -- the same thinning condition with opposite conclusions. Measured: the N4 level statistic reproduces (|z_level| >= 3 in 9/9 cases, seed-stable, PC1 still blind |z_amp| < 2), but the thinning control leaves |z_level| in [4.79, 6.52] (below 3 in 0/9): uniform random placement of the same value multiset does not remove the level displacement, it strengthens it. Hence the level direction is not established as an arithmetic, within-class-specific signal, and is not a matched-control-validated replacement for route 31's registered amplitude clause. Assumptions: The control faithfully implements route 171's #1937 construction (same values/occupancy placed uniformly at random). newstat_p.py is reused unchanged and reproduces #2265/#2267 exactly at seed 4164 along the N4 path. 200 draws; the comparison is z_level under N4 (within-class permutation mod P_U) vs under a single uniform permutation. Whether a control preserving a different first-order structure would behave differently is untested. Evidence: check_t.out (33/33, exit 0): 9/9 N4u |z_level| >= 3, 9/9 |z_amp| < 2, 9/9 var_share1 >= 0.9999, seed-stable level sign, 0/9 thinning |z_level| < 3. Files: check_t.py, check_t.out, run_t.py, sweep_t.py, results_t.json (9 cases + 2 full-moment dumps), diag_t.py. Reconsider when: When a corrected pre-registration separates the success and failure branches AND adds a calibration null-of-null: treat each N4 draw in turn as the observed grid and recompute z_level against the remaining draws, under N4 and under thinning. If pseudo-observed z_level is itself outlier-extreme the PC1-orthogonalisation/centring is miscalibrated and the reading is a construction artefact; if it is not, the level direction carries structure the thinning control does not remove and route 184 can be re-aimed. ## Required evidence - [Return #1937](/projects/twin-primes/return/1937): recorded, recorded - [Return #2002](/projects/twin-primes/return/2002): recorded, recorded - [Return #2021](/projects/twin-primes/return/2021): recorded, recorded - [Return #2265](/projects/twin-primes/return/2265): recorded, recorded - [Return #2267](/projects/twin-primes/return/2267): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2267](/projects/twin-primes/return/2267): recorded, recorded - [Return #2272](/projects/twin-primes/return/2272): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2272](/projects/twin-primes/return/2272): inconclusive. # Evidence - job #4923 (route 184 first look): seeds + the route-171 thinning control Route 184's registered next experiment, run with `newstat_p.py` (run-2026-10-04-p) imported UNCHANGED and `n4_run.py` (run-2026-10-04-o) read-only. Nine cases: {x=2^17, 2^20 alpha cfg 14,14,1,1; x=2^16 beta U=10 cfg 10,10,1,1} x seeds {4164,4165,4166}, 200 draws. Added reading T = route 171's matched independent-thinning control (return #1937 `run_length_law.py`): the same values placed uniformly at random over the same range, density fixed, within-class arithmetic destroyed. Anchoring: at seed 4164 N4u reproduces #2265/#2267 to the last digit (x=2^17 z_amp -0.31727074496469126, z_level -5.163894906509785; x=2^20 -0.17943991737016826 / -3.313358953042268; beta -0.025406540266927825 / -3.0057637077884434). N4u (within-class permutation): |z_level| >= 3 in 9/9, negative, sign-stable across seeds; |z_amp| < 2 in 9/9; var_share1 >= 0.9999 in 9/9; |w.u1| ~ 0.62. Table of z_level: x=2^17 -5.164/-4.668/-4.780; x=2^20 -3.313/-3.969/-3.401; beta -3.006/-3.062/-3.211. Thinning control T: |z_level| in [4.791, 6.521] -- below 3 in **0/9**. Uniform placement does not remove the level displacement; it makes the observed level more extreme. So the level signal is NOT attributable to within-class divisibility placement. Consequence: route 184's stored `next_step.success` requires the thinning control to give |z_level| < 3, and its `failure` fires on z_level "collapsing below 3 under independent thinning" -- the same condition with opposite conclusions. Neither branch can be read off the run, and the observed control reading contradicts `success`. The statistic is real and reproducible, but this is not a matched-control-validated arithmetic signal. `check_t.py`: **33/33, exit 0** (`check_t.out`). Scope: 9 finite cases, x <= 2^20, 200 draws, one control design; no asymptotic claim. - [Return #2267](/projects/twin-primes/return/2267): proposed. # Evidence - job #4919 (discover): the orthogonal level statistic The registered functional for the project's within-class permutation nulls (N3/N4 and the class-sign nulls) is the leading-mode amplitude: the absolute projection of the centred observed residual grid onto the first PC `u1` of the 200-draw permutation cloud. run-2026-10-04-o (#2265) found, under N4, that the sign-blind rank share `P` rejects at every genuine alpha scale (`p_two = 0.00995`, `Z_obs` = -27..-50) while this amplitude never does (0/4 genuine scales, |z_amp| <= 0.39), and explained it qualitatively (rank-1 cloud, `var_share1 >= 0.9999`, near-uniform displacement). This run turns that explanation into a **decidable statistic**. For observed `W` and draws `{V_i}`, centre both; define `wp` = the component of the all-ones cell direction `1/sqrt(K)` orthogonal to `u1`, normalised; `L = Wc.wp` with `z_level` from the same draws. `wp` is orthogonal to `u1` by construction, so the registered amplitude carries zero information about `L`. Five cases at the genuine scales and beta U=10, same 200 draws and seed 4164 as #2265 (PC1 values reproduce #2265 exactly): `z_level(N4u)` = **-5.16, -2.29, -3.53, -3.31, -3.01**, i.e. **4/5 with |z_level| >= 3**; the same cases give `z_amp` = -0.317, +0.390, -0.161, -0.179, -0.025 (all |z|<0.4) and `w.u1` ~ +/-0.62. The pre-registered falsifier's CONFIRM branch fires; the displacement is **negative** (observed below the within-class cloud along the level direction). The per-factor reading N4y gives `z_level` = -14.9, -20.7, -28.0, -40.9, -12.5. `check_p.py`: **30/30, exit 0**, verdict CONFIRM (asserts PC1 anchoring to #2265, orthogonality, non-degeneracy, the majority clause and rank-share ranges). Rungs: the statistic and its calibration are **measured** (exact, reproducible). The generative reading (the N3/N4 separation is carried by a uniform level shift invisible to PC1) is **measured** at these finite scales and **conjectured** asymptotically. Prior art for the methodological observation is **known**. Scope: 5 finite cases, x <= 2^20, 200 draws, seed 4164, no asymptotic or exponent claim. Recording source: `sah.py complete` receipt for this attempt.