# f out of sample: the 42-point census is corrupted from x = 37, and the decay law survives the correction **Verdict, before anything else. The pre-registered test was whether `f-decays.md`'s fitted decay law holds out of sample. It does, as a band: nine of nine deep levels land inside the four-specification band, and the residual trends against the central specification at t = −7.08. But the test found something it was not looking for. The 42 exact census points that the law is fitted to are wrong from x = 37 upward, by a factor rising to 1.63 at x = 199, because the evaluator carries the avoided residue set in a 32-bit word and `1 << r` in JavaScript uses r mod 32. Every level the census ever verified is below 32. The correction moves the threshold coefficient by 0.11, leaves the comb coefficient and the polylog branch call untouched, and moves the projected crossings down rather than up.** *2026-08-19. Pre-registration: [fdecay-deep-prereg.md](fdecay-deep-prereg.md), written and committed alone (`705c839`, message `prereg: fdecay-deep`) before any producer for this pass existed on disk. Producers, all formally embedded: [`../../fdecay-deep-00-core.js`](../../fdecay-deep-00-core.js) (the segmented window instrument, a library), [`../../fdecay-deep-01-census-defect.js`](../../fdecay-deep-01-census-defect.js) (2209.2 s, `out-sha256 4d932d02`), [`../../fdecay-deep-02-window.js`](../../fdecay-deep-02-window.js) (140.8 s, `out-sha256 6da42b1e`), [`../../fdecay-deep-03-ladder.js`](../../fdecay-deep-03-ladder.js) (1325.5 s, `out-sha256 0825440d`) and [`../../fdecay-deep-04-crossings.js`](../../fdecay-deep-04-crossings.js) (6.5 s, `out-sha256 45336f9e`). Legend as in `research/sift-limit-attack.md`: **[PROVEN]**, **[VERIFIED]**, **[MEASURED]**, **[CITED]**. No live document was edited by this pass; §8 lists the proposed edits and none is applied.* --- ## 0. What was asked, and what the instrument is `research/f-decays.md` fits a decay law for f, the qualifying-gap fraction, on 42 exact census points from `research/a3-03-f-from-census.js`. U-FRAME §12's polylog call, `history/staging/import-sofic.md` §6's projected crossings at x = 773 and x = 2297, and `research/gate-multiplies.md` §8's p ~ 800 turnaround all read that law far outside its range. The census method's cost is exp(0.048 p) and stops at x ≈ 200, so the law had never been shot at out of sample. **The instrument.** For level x with fold p = nextprime(x), inside a window [0, X): count the gaps between consecutive x-rough twin slots that are ≡ 0, ±2 (mod p). The sieve is segmented on the lattice n ≡ 5 (mod 6), memory is the segment and not X, and one sieve serves every level of the ladder at once because level x is exactly { n : min{q ≥ 5 : q | n(n+2)} > x }. That is the `research/LOCALIZED-GAP.md` §10 lever applied to a tail functional. **Two calibrations, both fixed in the pre-registration and both load-bearing.** 1. **`f_win` is an ESTIMATE of the tile ratio, not the tile ratio.** The census f is an exact ratio over the whole period x#; this is the same ratio over a window. §2 measures the substitution error directly. 2. **`L_win` is a PROVEN lower bound on the tile L at the same fold, and nothing more.** A legal class word over ℓ consecutive slots fixes a residue a with all ℓ slots in {a, a−2} mod p; copy k ≡ −a·w⁻¹ (mod p) of the full period deletes exactly that class pair (U-FRAME §5a step 1, w = x# mod p invertible), so an adjacent-kill run of length ℓ exists in the period. The window holds X/m̄ slots against the period's p·D, so L_win reads the same first-moment law at a far smaller sample size. --- ## 1. The defect > **The census law's evaluator carries A_q, the set of residues mod q avoided by > the current offset set, as a single 32-bit mask:** > > a[j] = (1 << (((-o) % q + q) % q)) | (1 << (((-o - 2) % q + q) % q)); > > **JavaScript takes the shift count of `<<` modulo 32. For any prime q > 32 the > residues r and r + 32 share a bit, |A_q| is read off as the popcount of an > aliased mask, and every factor (q − |A_q|) in the product is wrong whenever two > avoided residues of that prime sit 32 apart. T_x carries a prime above 32 iff > x ≥ 37.** [PROVEN as a statement about the code; VERIFIED as a prediction] The prediction was written down before the table was read and it is exact. Rerunning the published evaluator byte for byte against an alias-free one that uses ⌈q/32⌉ words per prime and the identical pruning rule (`fdecay-deep-01-census-defect.js` §2): | levels | alias-free / published | |---|---| | x ≤ 31, seven levels | **[1.000000, 1.000000]** — identical, bit for bit | | x ≥ 37, thirty-four levels rerun | **[0.62474, 1.04904]** | Not a drift, not a fitted boundary: a step at one value of x. **The rerun is the same code.** The copied published evaluator reproduces the census's own OUTPUT §4b first-term column at every spot check: 4.0404e−2 at x = 13, 2.7628e−2 at x = 19, 3.6402e−2 at x = 29, 7.1991e−3 at x = 37, 1.2776e−2 at x = 47, 6.4434e−4 at x = 67, 3.9724e−3 at x = 79. So the disagreement is between two evaluators and not between two runs. **And a method with no inclusion-exclusion in it agrees with the alias-free one.** Over 42 levels the segmented window sieve sits within **1.74 σ** of the alias-free evaluator at worst. Against the published one, over the 35 levels x ≥ 37, the worst deviation is **27.0 σ**. The error grows with depth, which is why it matters: the ratio runs 1.03324 at x = 37, 0.93386 at x = 101, 0.74291 at x = 139 and **0.61323 at x = 199**, where the published 2.6530e−4 becomes **1.62689e−4**. In ln(1/f) that is a shift of **+0.4890** at x = 199 and +0.0000 at every level x ≤ 31. **Why nothing caught it.** Every verification the census ran lives at x ≤ 31: T₇ to T₂₃ against direct sieving (worst 4.17e−10), the five published diagonal points (to 1e−16), and A2's independently streamed T₂₉ and T₃₁ (four figures). All of them pass. All of them are inside the region where every prime of the tile is below 32. The check that would have caught it is a second evaluator, or a second method, at one level above 37, and neither existed until this pass. --- ## 2. The instrument, validated Four checks were fixed in the pre-registration §3 and one report was required in §4. `fdecay-deep-02-window.js` runs them. - **V2, head bias: PASSES by three orders of margin.** The window slot density reproduces the exact tile mean gap to **4.98e−4** at worst, against a criterion of 0.01. At x = 11 the agreement is 7.56e−9. - **V3, the single-copy kill run: PASSES, 42 of 42 levels**, against `attack-foldL-04-localized.js`'s embedded L column at the same X = 2e9. That script reads the same object through a min-prime key array, a linked list and bucketed deletions, and shares no line of code with this sieve. Its slot count at the level folded by 211 is 28,510,623 and this run's is **28,510,623**. - **V4, the comb:** first-term share 83.33 % worst and 99.63 % median over the 38 levels where the window sees more than one tooth, against the published 83.33 % and 98.08 % over 18 rows. - **§4, window-size sensitivity: no drift.** Worst over all 42 levels is **3.01 σ** between the first decade with K ≥ 100 and the last. At x = 37 the four decades read 8.087e−3, 7.821e−3, 7.736e−3, 7.745e−3. - **The prefix is not a biased window.** Against eight uniformly random offsets of the period — Z mod q drawn independently for every q, which by CRT is a uniform Z mod x# — the prefix ratio is 0.9824 at x = 101, 0.9841 at x = 151 and 0.9110 at x = 199. The last rests on K = 506, so the combined standard error is about five per cent and 0.9110 is 1.8 of them. **V1, agreement with the published census, is the check that fails, and §1 is why.** It is replaced by agreement with the alias-free evaluator, which is the stronger check because it is arithmetic against empirical. --- ## 3. How deep it reached, and what it cost **x = 829, four times deeper than the census method's x ≈ 200 wall** [MEASURED]. At X = 1e11 the deepest level with a usable count is x = 503 (K = 332); x = 547 and x = 601 carry K = 26 and K = 25; x = 653 carries 9, x = 701 carries 2, x = 773 and x = 829 carry 1 each, and x = 907 sees nothing. The whole ladder is **1325.5 s**, against 421.3 s for the published census's 42 points. Three decades of x for three times the time. Pre-registered prediction **R1 put the wall at x ≈ 600 with 30 or more qualifying gaps. Measured: K = 26 at x = 547 and K = 25 at x = 601. R1 is right to within one level.** The wall is not the sieve: it is that f is a tail functional and never sees a gap below d_min = 2p ∓ 2, so a level is measurable only once (X/m̄)·f is of order ten. --- ## 4. The pre-registration, scored | registered | outcome | |---|---| | **R1** wall at x ≈ 600 | **HOLDS** — K = 25 at x = 601, K = 1 at x = 829 | | **R2** no window-size drift once K ≥ 100 | **HOLDS** — worst 3.01 σ over 42 levels | | **R3** every deep level inside the band, mean residual negative, \|mean\| ≤ 3 | **HOLDS** — 9 of 9 inside, mean residual **−0.553** | | **R4** residual trend on ln x negative with \|t\| < 3 | **FAILS on size, holds on sign** — slope −1.114 ± 0.157, **t = −7.08** | | **R5** form-1 crossings in [400, 1100] and [900, 3000] | **HOLDS** — **719** and **1801** | | **R6** L_win falsifies nothing | **HOLDS** — L_win = 2 from x = 97 to x = 829, 4 at worst below | | **R7** form-2 crossings in [300, 700] and [700, 2200] | **HALF FAILS** — **271** and **839**; 271 is below the registered floor | **R4 is the finding.** The published fit over-states the decay, and it over-states it more the deeper you go: the residual against the central specification falls from +0.043 at x = 211 to −1.227 at x = 601. Every projection built on the published coefficients is optimistic about how fast f falls, and the size of the optimism grows with the level being projected to. --- ## 5. The corrected law **On first terms throughout, so the two columns are comparable** (`fdecay-deep-01-census-defect.js` §4, 42 levels): | | published points | corrected points | |---|---|---| | threshold only | 1.073 + 1.4357·T, R² 0.8999 | **0.792 + 1.5425(±0.0724)·T, R² 0.9191** | | with the comb | 1.609 + 1.4675·T − 1.038·S, R² 0.9646 | **1.323 + 1.5741(±0.0405)·T − 1.030·S, R² 0.9754** | | halves | 1.662 and 0.906 | **1.675 and 1.109** | | downstream | ln L = −1.937 + 2.926·lnln x, R² 0.9728 | **−1.861 + 2.864(±0.075)·lnln x, R² 0.9732** | **On the 51 window-measured levels with K ≥ 10, x = 11 to 601** (`fdecay-deep-03-ladder.js` §4): > **ln(1/f) = 1.917 + 1.4016(±0.0235)·(2p/m̄) − 1.021·ln s(d_min), R² 0.9867** against the published 1.529 + 1.482·T − 1.025·S at R² 0.9639. The threshold coefficient reads 1.5875 on x ≤ 199 and **1.2799 on x ≥ 211**, and 1.8385 on the lower half against 1.2933 on the upper. **Three things survive the correction unchanged, and they are the three that matter for the branch.** 1. **The comb coefficient is −1 to within the fit** on the corrected points as on the defective ones: −1.030 against −1.038, and −1.021 on the window points. `f-decays.md`'s singular-series reading, f ∝ s(d_min)·exp(−c·2p/m̄), stands. 2. **The polylog branch call stands.** ln L_indep = −1.861 + 2.864 ± 0.075 on the corrected census points and −1.827 + 2.844 ± 0.058 on the window points, against the published −1.884 + 2.892. The exponent does not move outside its own error. **U-FRAME §12's branch call is not what the defect touches.** 3. **The range dependence is real on both sides.** It is smaller after the correction — the upper half moves from 0.906 to 1.109 — but it is still there, now across a range three times wider. No single slope should be quoted, which is what `f-decays.md` already says. **What does move is the size of f, and therefore the level of every crossing.** --- ## 6. The crossings, recomputed Two forms, both fed the same measured f (`fdecay-deep-04-crossings.js`, seven specifications): - **form 1**, `f-decays` / U-FRAME §5a step 6: `L = ln D / ln(1/f)` - **form 2**, `import-sofic.md` §4's estimator C, the only one that record found flat: `L̂ = [1 + ln(2D)/ln(2/f)] / 1.4965` | | 0.31·p/ln p | 0.19·p/ln p | |---|---|---| | form 1, all measured points | **719** | **1801** | | form 1, spread over seven fits | 509 to 757 | 1201 to 2039 | | form 2, all measured points | **271** | **839** | | form 2, spread over seven fits | 211 to 271 | 547 to 929 | | **published projection [CITED]** | 773 | 2297 | **The sofic record's headline is confirmed and moved slightly earlier.** Its 773 and 2297 sit just above the form-1 spread, so the reading that survives is a range with the published pair at its optimistic edge. This matters because the projection was made from coefficients fitted to the defective table, and the correction happened to push in the route's favour. **A pointer finding, recorded in the pre-registration before any measurement.** The exact numbers 773 and 2297 are not reproducible from `f-decays.md`'s published coefficients under any of five variants tried: the stays-below level for 0.31 is 479, 509, 661, 691 or 937 depending on specification, and for 0.19 it is 1171, 1307, 1657, 1699 or 2521. **They should be quoted as a range, not as a pair.** **The two forms disagree by a factor of two to three in x, and the disagreement is the estimator, not the data.** Both are fed the same measured f. Which form is right is `import-sofic.md`'s open question, and it is now the largest single uncertainty in the crossing level. **The deepest fit is the least favourable.** The out-of-sample-only fit, x ≥ 211 with nine levels and R² 0.9974, crosses at 757 and 2039 — later than every other specification. That is the direction that costs the route. --- ## 7. The direct L points **L_win = 2 at every level from x = 97 to x = 829**, and 1 at x = 907. Below that it reaches 4, at x = 29 and x = 31, and never more. Against a requirement of 6.78 at x = 97 and 38.63 at x = 829, the ratio L_win/(0.31·p/ln p) falls from 1.319 at x = 19 to 0.052 at x = 829. **R6 holds: the window lower bound falsifies nothing.** Two things it does say. 1. **The graph's first-moment law predicts the measured window L to within a fraction of an integer at every deep level.** With the measured letter measure, `1 + ln(2N)/ln(2/f)` reads 2.92 at x = 211, 2.37 at x = 503 and 2.00 at x = 773, against a measured L_win of 2 throughout. The machinery `import-sofic.md` validated on nine folds at p ≤ 37 still works at p = 787, at a sample size fourteen orders of magnitude below the period's. 2. **The requirement is already violated at the bottom of the ladder.** The ratio reaches or passes 1 at x = 11, 13, 17 and 19, reading 1.273, 1.075, 1.000 and 1.319 — levels where the requirement is 1.57 to 2.27 and an integer L of 2 or 3 clears it by arithmetic. From x = 23 up the ratio never reaches 1 again. **Why a window L cannot decide the question, stated so nobody re-runs it deeper.** L_win ≈ 1 + ln(2X/m̄)/ln(2/f) reads about ln 2e9 ≈ 21 in the numerator where the period reads ln(2pD) ≈ 200 to 740. Closing that gap by window length alone means raising X by a factor of e^500. **The window measures f, not L, and the f it measures is the thing L depends on.** --- ## 8. Proposed edits, none made 1. **`research/a3-03-f-from-census.js`.** The evaluator's `bitsOf` must carry ⌈q/32⌉ words per prime, or the avoided-set size as an integer. The embedded OUTPUT is defective from x = 37 and must be regenerated after the fix; the alias-free evaluator in `fdecay-deep-01-census-defect.js` is a drop-in with identical leaf counts and no measured slowdown. §1's Lemma A, §2's verification against direct sieving, §3's five diagonal points and §6's Lemma B are all unaffected. 2. **`research/f-decays.md`.** The 42-point table, both regressions, the halves, the "factor of 170", the x = 1000 extrapolation (f ≈ 2.3e−12, L ≈ 35.5, band 30.8 to 42.9) and the x = 181 against x = 199 comb example are all fitted to or quoted from the defective points and need regenerating. **The staircase theorem, Lemma B, and the comb coefficient of −1 are unaffected**, and should be marked as such so the correction does not read as a wider retraction than it is. 3. **`research/U-FRAME.md` §12** carries "f falls by a factor of 170 over that range" — the corrected factor is larger. **§5a step 7's seven-row table is entirely at x ≤ 31 and is unaffected**, which is worth stating in the same edit. 4. **`history/staging/import-sofic.md` §5 and §6.** The factor of 170 and the projected 773 / 2297 both come from the defective fit. §6's conclusion survives; its two numbers should become the form-1 range 719 (509 to 757) and 1801 (1201 to 2039), with the form-2 alternative 271 and 839 recorded beside them as the estimator-C reading. 5. **`research/LOCALIZED-GAP.md` §10** gains the observation that the lever reaches tail functionals as well as merge quantities, at a cost set by (X/m̄)·f rather than by the sieve: x = 829 in 1325.5 s here. 6. **A campaign-lessons row.** Every check of the census lived below the level where the defect can occur, and the defect was found by an instrument with no code in common. "Verify at a level where the mechanism you are worried about can actually fire" is the rule, and this pass is the case that pays for it. --- ## 9. NOT REACHED - **No bound is proved, in either direction.** Every crossing here is a fitted law read outside its range. - **No level above x = 503 carries a usable count.** x = 547 to 829 rest on 1 to 26 events and are quoted with their K throughout. - **The census was not regenerated.** Only count(d_min)/D was recomputed alias-free, at 42 levels; the later comb teeth were not, and the corrected full f is therefore a first-term value where the published one was too. - **The tile L was not computed at any level**, only its window lower bound and the first-moment estimate. - **No prior-art search was run**, and none of the fits here has been checked for a published analogue. - **The 1.4965 of form 2 is measured on nine folds at p ≤ 37** and carried to p = 600 without justification. That transfer is exactly the kind this corpus refuses elsewhere, and it is the reason form 1 and form 2 disagree.