# Adversarial pass on the four same-evening headlines *Staging note, 2026-08-19. Refute-first verification of [shadow-buchstab.md](shadow-buchstab.md), [import-scanstat.md](import-scanstat.md), [import-sofic.md](import-sofic.md) and [import-shearer.md](import-shearer.md), all four written and committed inside forty minutes of one evening. Producers, all three formally embedded and `--check` clean: [`../../adversary-wave2-01-shadow.js`](../../adversary-wave2-01-shadow.js) (15.9 s), [`../../adversary-wave2-02-scanstat-sofic.js`](../../adversary-wave2-02-scanstat-sofic.js) (0.2 s), [`../../adversary-wave2-03-shearer.js`](../../adversary-wave2-03-shearer.js) (1.1 s). Every recomputation here is a deep-success validation under the standing compute rule; nothing is re-derived that was not being attacked. **No live document was edited and no proposed correction was applied. Proposals only.*** --- ## 0. The four verdicts | claim | verdict | |---|---| | 1. Kill shadow DERIVED + the drift law | **SURVIVES WITH CORRECTIONS** — the mathematics is right and better than stated; the verdict LABEL is one grade too high by the pre-registration's own rule | | 2. Scanstat duality + exponent rule + identity refutation | **SURVIVES WITH CORRECTIONS** — the duality and the refutation are theorems; three custody sentences are false and the prediction's precision is oversold | | 3. Sofic verdict | **SURVIVES** — every number reproduces, the strictly-sofic argument is airtight, and a paired test the record did not run makes its own headline stronger | | 4. Shearer identity + family closure | **SURVIVES WITH CORRECTIONS** — the identity, the exponent, the floor and the closure all hold; the extremal witness is misdescribed and the atomicity leg needs its hypothesis or its citation changed | **Nothing is BROKEN.** That is itself worth stating against a base rate of four corrections in five same-day integrations: the arithmetic in all four records reproduced from independent code, and every defect found is in a sentence rather than in a number. **Cross-contamination: NONE.** Every producer of the four waves cites only files from its own wave. No scanstat script mentions a sofic artifact, no sofic script mentions a shearer artifact, and the shadow pair cites only each other. **Pre-registration custody is not uniform, and only one of the four is git-provable.** `import-shearer-prereg.md` was committed alone at 19:59:19, six minutes before its producer's mtime and sixteen before its record — the pre-registration claim there is verifiable from the history. The other three preregs were committed in the SAME commit as their records *and* their producer scripts (20:15:48), so git cannot order them. What can be checked is weaker and still real: the prereg files' own mtimes (19:58:59 shadow, 19:59:23 sofic, 20:12:15 scanstat) precede their records' (20:12:27, 20:08:27, 20:13:06), and `import-scanstat-prereg.md` is machine-written by a producer that carries its own embed. The honest grade is: **one PROVABLE pre-registration and three DECLARED ones.** That belongs in each record's header rather than in the reader's head. --- ## 1. The kill shadow — SURVIVES WITH CORRECTIONS ### What held **The measurement reproduces exactly, 10 of 10 clusters.** A sieve written on the complementary formulation — mark a `y`-rough flag array, then AND it with itself shifted by two, rather than marking `r ≡ 0` and `r ≡ −2 (mod p)` — returns the same ratio to five decimals **and the same integer slot count** at every cluster, including 266735 at `y ~ 1000` and 1779677 at `y ~ 26000`. Nothing in `shadow-buchstab-02`'s part (B) is a defect. [VERIFIED] **The 0.5573 is derived, not fitted, and it has a closed form the record does not state.** `ρ(2+ε) = ρ(2)(1 + ε + O(ε²))` because `ω'(2) = 1/4`, so the band average's first-order coefficient is the first moment of the `x`-weight over the band: > `∫₀¹ s·ln2·2^s ds = 2 − 1/ln 2 = 0.5573049591…` reproduced to ten digits by quadrature and approached from below by the exact integral as `w → 0`. **[PROVEN]** **The Exact Invariance Lemma is CRT-complete.** The attack was to find the step where band position interacts with the new primes' classes. It does not exist. Over all 30030 classes mod `P₁₃` with 323 lifts each, the retention histogram has exactly one entry — 255 lifts for every class, no exception — and a band's depth is invariant to `1e-12` across a level change that `attack2-03-09` never tested. The band *is* strongly non-uniform against the new primes as integers (19 and 18 of its 169 classes are themselves killed by 17 and 19), and the lemma does not care, because it acts on the **lifts** of each class, where `gcd(P_b, p) = 1` forces equidistribution uniformly in the class. **[PROVEN, attack failed]** **The stale-premise claim holds.** `attack2-03-09-depth-formula.js`'s embedded output does say what the record says it says: birth/predicted 0.849/0.852 at b = 97, 0.855/0.834 at 997, 0.832/0.830 at 2003, 0.827/0.827 at 4999, and its reading 2 already calls the match. TODO item 5's "CANDIDATE, unchecked" was indeed stale. [VERIFIED against the embed] **D2 is not cherry-binned.** On a binning the record never used — one prime per rung of a fixed geometric ladder, no pooling — the fall is violated 2 times in 14 steps against the record's 1 in 9. The trend is a property of the object, not of the author's cluster centres. [MEASURED] ### Corrections required before integration 1. **The verdict is SHAPE-ONLY, not DERIVED.** `shadow-prereg.md` writes D1 as "`|B_x(y) − measured(y)| ≤ 0.011` **at every deciding level**" and fixes the deciding set as `y ≥ 997`. The `y ~ 1000` cluster misses at 1.49× tolerance. D2 is written as a fall and one step rises. DERIVED requires D1 **and** D2 **and** D3. "DERIVED above `y ≈ 1400`" narrows the deciding set *after* the measurement, which the prereg's own closing line — *"No other statistic will be promoted to a verdict after the fact"* — was written to prevent. The evidence is strong; the label is one grade too high, and the fix is to report SHAPE-ONLY with the strong D1 record above `y ~ 1400` stated beside it. 2. **The drift law's amplitude is off by a factor the record does not report.** Over the ladder the measured depth falls 0.03207 against a predicted 0.01211 — a ratio of **2.65** — and 2.30 on the record's own clusters. Restricted to `y ≥ 2000` the ratio is 1.19. So the law's *form* is confirmed and its *amplitude* is not yet, which is consistent with the record's own small-`y` residual reading but is a number that reading does not give. §2's sentence "the honest statement is that the shadow depth is `0.793055·(1 + 0.5573 ln2/ln y + O(1/ln²y))`" should carry it. 3. **State the coefficient in closed form**, `2 − 1/ln 2`, and fix `shadow-buchstab-01-candidate.js`, which hardcodes `0.5573013` — wrong in the sixth decimal. The displayed first-order column is affected in the eighth decimal only, so no published figure moves; it is a code correction, not a number correction. 4. **The cluster design is post-registration.** The prereg fixed the tolerance, the deciding levels and the three criteria, but not the measurement design: the ten `y`-centres, the ±12% pool and the 4e8 budget were all chosen after it, and the prereg's Poisson floor was computed for a *single* level while the score is run on pooled clusters. Neither changes a verdict — the pooling only sharpens the test — but the record should say the estimator, not only the criteria, was fixed late. --- ## 2. Scanstat — SURVIVES WITH CORRECTIONS ### What held **The duality is a theorem, exact at every `m`, not on a grid.** On `T₇`, `T₁₁` and `T₁₃`, `maxsum_m + minsum_{D−m} = W` holds with integer deviation **zero** at all `D−1` values of `m`, and `sd_m = sd_{D−m}` to `1e-10`. The proof is one line: the two windows partition the cycle, so `S_m(i) + S_{D−m}(i+m) = W` identically, and `i ↦ i+m` is a bijection of the cycle. **[PROVEN, VERIFIED]** **And it is a theorem about the CYCLIC word only.** Run the same test on the non-wrapping window and it fails at 1371 of 1483 values of `m` on `T₁₃`, worst deviation 114. Every producer in the corpus forms the window cyclically — `a3-04-maxsum-recursion.js`, `attack-foldL-05-maxsum-direct.js`, `import-chaining-02.js`, `a3-02-diagonal-f.js` and both new scanstat scripts all close the cycle explicitly. But the **live** documents that state the growth law, `../../U-FRAME.md` §5a and `../../TODO.md` item 0c, do not contain the word. **The permutation refutation is sound**, and it is the cheapest kind: the hypothesis `Σγ(k) = 0` is a function of `Σᵢcᵢ` alone and so is invariant under every reordering, while `sd_m` is not. Nothing in it needs checking beyond the arithmetic, which reproduces. **`maxsum_1 = G₂` checks independently.** Tiles built from scratch return `maxsum_1 = 30, 42, 66, 108` at `T₇, T₁₁, T₁₃, T₁₇`, which is `exact-g2-ladder.js`'s `G₂(x#)` at those levels, with `D = 15, 135, 1485, 22275` matching the corpus's slot counts. [VERIFIED] ### Corrections required before integration 1. **"T₂₉ had never been computed here for any moving-sum statistic" is false.** `../../a3-04-maxsum-recursion.js` (added 2026-08-16) publishes `maxsum_m(T₂₉)` for `m = 1..8` as 258, 330, 390, 420, 510, 540, 552, 582, and `../../attack-foldL-02-bridge.js` re-verifies `m = 1..6` five and a half hours before the scanstat commit. The new run reproduces those digit for digit. 2. **`maxsum_1 = 258` is therefore a reproduction, not an independent anchor.** "VERIFIED on two independent anchors" should become "reproduces the corpus's own `G₂` ladder and slot counts", which is a consistency check and still worth stating. 3. **The blindness of `H` nevertheless survives, and this is the finding that matters.** Backing an exponent out of the already-published `m = 1..8` row — `sd_m ≈ (maxsum_m − m·m̄)/√(2 ln D)` on the prereg's own grid — gives **0.2100 ± 0.0274**, nowhere near the measured 0.3367, precisely because the tail factor is not `√(2 ln D)`, which is the record's own finding. The published row does not leak `H`. **The blind validation stands; only the sentence and the anchor claim need correcting.** 4. **The prediction's precision is oversold.** The three-point fit carries **one** residual degree of freedom; its own 95% band at `T₂₉` is `[0.281, 0.396]`. Dropping `T₁₃` alone moves the prediction from 0.3383 to 0.3442, i.e. from 0.20 to 0.94 measurement standard errors. "Predicted to within a fifth of a standard error" prices the miss against the *measurement's* error, not the *prediction's*. What is safe is the **kill**: Model B's `H = 0.5` is far outside the band, so the refutation of `√m` is unaffected. The claim to carry into TODO 0c is the refutation, not the precision. 5. **Two citation slips.** §0 says the duality was "VERIFIED at fifteen values of `m` on T₁₃"; the producer prints twelve (and the theorem is now checked at all 1484). §3 attributes both `G₂` anchors to `exact-g2-ladder.js` *and* `05b-twin-jacobsthal-segmented.js`; 05b computes 29# only, and `G₂(23#) = 204` lives in `../../05-twin-jacobsthal.js`. 6. **The pre-registered kill criterion names T₂₃ only.** The prereg's sentence is "Kill criterion: Model A must beat Model B on the ln-RMS of `excess_m` at `T₂₃`"; the T₂₉ table is tabulated but no decision rule is stated for it. "The pre-registered kill criterion is passed at both target levels" should say that T₂₉ was scored on the same rule, applied by extension. 7. **The duality's novelty needs a caveat.** It is the complement identity for a maximum over cyclic windows — the same identity that turns a maximum circular subarray into "total minus minimum subarray", which is standard in algorithms. §6 already records that no prior-art search was run; the proposal to promote it to the live layer as a banked **THEOREM** should carry that flag on the same line. --- ## 3. Sofic — SURVIVES The only claim of the four that needs no correction to its substance. **The strictly-sofic argument is airtight, and it is complete for every `M` at once.** `(−2)0^k(−2)` and `(+2)0^k(+2)` are rejected and all four halves accepted for every `k` up to 400, and the general argument closes it: `w = (−2)0^M(−2)` has length `M+2` and therefore exactly **two** subwords of length `M+1`, `(−2)0^M` and `0^M(−2)`, both legal. An `M`-step SFT forbids only words of length `≤ M+1`, so it must contain `w`, which the language does not. True for every `M`. The graph itself is forced by the arithmetic: a slot is deleted by `p` iff `r ≡ 0` or `r ≡ −2 (mod p)`, and from `r ≡ 0` the next deleted slot needs `g ≡ 0` or `−2`, from `r ≡ −2` it needs `g ≡ 0` or `+2`. **[PROVEN]** **Every number in the ratio test reproduces to four decimals** from the cited inputs alone: mean `R` 2.3931 / 2.1346 (A), 2.1249 / 1.9340 (B), 1.5511 / 1.4965 (C), slopes 0.5157, 0.5292, 0.9784, 0.8880, 0.4083, 0.1353 with the record's standard errors and `t` values. The legal-word count is `2^{n+1} − 1` at `n = 1..12`, matching Perron root 2. [VERIFIED] **On the `t`-statistics attack.** `t = 2.690` on `n = 7` is 5 degrees of freedom, where the two-sided 5% critical value is 2.571, so the kill is significant on the proper `t` distribution too — and in any case the prereg fixed `|b| > 2·SE` in advance, which fires on both readings. The kill is sound. **A paired test the record did not run makes its headline stronger, not weaker.** The concern was that "B dies at 2.69, A survives at 1.65" is two marginal numbers landing either side of a threshold. It is not: the OLS slope of `R_A − R_B` on `ln p` over the same seven folds is **−0.3588 ± 0.0253, t = −14.2**. The two estimators genuinely differ in trend. What is *not* established is that A is flat — A's own interval `[−0.297, 1.355]` contains B's slope — so "flat" here means unrefuted, not shown. The record already says exactly this, in the sentence about `A/B` falling to 0.9989 by fold 37. **The 170× against 16× checks out** on the correct alignment. `f` falls `4.444e−2 → 2.653e−4` from `x = 11` to `x = 199`, a factor 167.5 (the record's "170" is that to two significant figures), while `3/p` at the corresponding **fold primes** 13 and 211 falls by `211/13 = 16.23`. The comparison is only right on the fold-prime reading, which is the census table's own indexing. ### Three citation corrections, none touching a verdict 1. **`import-sofic-prereg.md` §3 cites `a3-05-bound-L.md` §7 for the `L` diagonal.** It is in §9, and §9 covers eight folds, not nine. It also cites `a3-10-lower-tightness.js` reading 1 for folds 31 and 37; reading 1 stops at fold 29. Fold 37's `L = 4` comes from that file's `deep37` block and its READINGS item 1. No value disagrees anywhere — this is a pointer defect. 2. **The prereg says both exclusions are "PROVEN in `kappa-not-L.md`".** That file marks the Alternation Lemma PROVEN and the fold-29 count of 288 as **VERIFIED**, on a single script (`a3-08-adjacent-pairs.js`). The fold-11 exclusion likewise rests on an enumerated census fact. The exclusions are sound; "PROVEN" is one grade high for the counts they rest on. 3. **The 42-point census lives in `a3-03-f-from-census.js`**, not in `f-decays.md`, which summarises it. Citing `f-decays.md` for the count is defensible (it states 42 on its own line); citing it for the values is not. --- ## 4. Shearer — SURVIVES WITH CORRECTIONS ### What held **The identity is right, checked without the record's recursion.** `Z_{K_n[S]}(−p)` computed by enumerating independent sets equals `1 − Σ_{v∈S} p_v` to machine precision over every subset at `n = 2..8`, with deliberately unequal marginals; the minimum is at `S = V`; the criterion collapses to `Σ p_v < 1`. Two lines, and they are correct. **[PROVEN]** **The factor `e` is exact, not bookkeeping.** Shearer's boundary on `K_n` is `1/n`, the local lemma's is `(1/n)(1−1/n)^{n−1}`, and the ratio `(1−1/n)^{−(n−1)}` reproduces the record's 2.0000, 2.2500, 2.3704, 2.4414, 2.4883, 2.5465 exactly and climbs to `e`. **[PROVEN]** **`H*(x)` and the exponent reproduce exactly** from an independent feasibility scan: 35, 55, 65, 91, 115, 209, 319, 481 at `x = 13..79`, `θ` running 1.3861 to 1.4386 with no trend. The producer's own table does carry all seventeen levels `x = 13..79`, so "flat over seventeen levels" is accurate even though the record prints eight of them. [VERIFIED] **The `2/√e` floor is sound and its derivation is correct.** Primes in `(√H, x]` are pairwise adjacent under `p ∼ q ⟺ pq > H`, so the identity applies to them as a clique and forces `Σ_{√H 1` there is no room: `2K` pairwise disjoint sets of total measure above 1 do not fit in `[0,1)`. The construction that works is arcs laid **end to end on `ℝ/ℤ`**, disjoint only until they close the circle. Measured coverage is 0.400000, 0.685715, 0.867533 at `x = 5, 7, 11` and exactly 1 from `x = 13` on, so `P(survive) = 0` from `x = 13` — the conclusion is untouched and the sentence must change. Within-prime exclusion survives the repair because `2/p ≤ 1` keeps each prime's two arcs consecutive and disjoint. 2. **The atomicity leg needs either its hypothesis or its citation changed.** The lemma is correct and its hypothesis is load-bearing: on an explicit 12-state instance the atomic construction gives `1/A_f = μ(f) = 0.5` and the non-atomic one gives `1/A_f = 0.0833 < μ(f) = 0.5`. So the record owes an argument that **its** instance is atomic — one flaw per prime on the slots of a window, addressed by moving the slot, is atomic only if distinct killed slots never move to the same slot, which is not established and is implausible for an action set that "changes the residue modulo every other prime". **The closure survives regardless**, through the citation the record already carries: Achlioptas–Iliopoulos's own sequel states `γ_i ≥ μ(f_i)` *always*, with no atomicity hypothesis. §1 item 5 and §8f should name that as the mechanism. A second unstated hypothesis: `μ(f) = |f|/|Ω|` presumes `μ` uniform. 3. **"At `H ≥ x²` the cross-prime edges all vanish and the graph is a perfect matching" is too generous, and it is generous in the record's own disfavour.** Exact independence of `{p|r}` and `{q|r+2}` on a window of `H` consecutive integers needs `pq | H`, and simultaneously for all pairs that is `∏_{5≤p≤x} p | H` — the primorial, not `x²`. The defect is inherited from `import-suen.md` §8 and it *strengthens* the closure rather than weakening it, which is why it must be fixed rather than left: as written it understates how far the legitimate matching graph is from `H = x²`, and the record's own §8b already prices the honest threshold at `H ≥ x#`. 4. **§1 item 5 grades the Achlioptas–Iliopoulos closure above its own §8d.** §8d tags the "this object does not supply a sparser graph" step `[INFERRED from their sourced theorem statements]`, which is right — it is an assertion about all possible action sets, not a proof. The headline should carry that tag; as written, "closed by three mechanisms" reads as three proofs. --- ## 5. What the four records get right that is worth naming - **Every arithmetic claim tested here reproduced**, in three cases from code sharing no line with the original: the shadow's ten cluster ratios *and* their integer slot counts, the sofic ratio table to four decimals, the Shearer `H*` ladder and `θ_c` column. - **Each record volunteers its own worst number.** The shadow prints its one D1 failure and its one D2 failure; scanstat discloses the `T₂₃` leak unprompted; sofic kills the map's own formula and then shows its own survivor is a transient; shearer refutes half of its own pre-registration. - **The four ran concurrently and did not touch each other.** No shared artifact, no borrowed number, no citation across waves. ## 6. Not reached - **No prior-art search was run in this pass**, on the complementary-window duality, the bounded-charge convention, or Shearer's region. Each record's own §6/§9 already flags its channel as unsearched, and this pass adds nothing to those absences and makes no absence claim of its own. - **The `[2,3]` branch of the Unification Law** remains the independence-squared conjecture, so the shadow's identification inherits it. Nothing here moves it. - **`import-shearer-01-region.js`'s `H*` scan was reproduced only to `x = 79`,** which is its own ceiling; the levels beyond it were not attempted. - **The scanstat tail factor** — `excess/sd_m` running 10.6 to 5.7 against `√(2 ln D) = 6.19` — is the whole residual and was not attacked here, because the record already declares it unexplained. --- *This document states current understanding. Superseded claims and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*