# Attack 0c hole-sweep: the per-copy degradation curve of a fold, in full *2026-08-20. The probe Chris asked for on TODO item 0c: at the fold from level 5 to level 7, measure how maxsum degrades copy by copy as the two deleted residues sweep through their possible positions — and produce the full curve, embedded. Producer: `research/attack-0c-holesweep-01.js`, formally embedded (`node research/qc/embed.js --streams both --timeout 2400 research/attack-0c-holesweep-01.js -- 37`, 617.9 s), all self-tests pass. Legend as in `research/sift-limit-attack.md`: **[PROVEN]** derived or published theorem; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[INFERRED]** deduction from sourced facts. This is a HELD headline: no live document was touched.* --- ## HEADLINE **The fingerprint any kill-count-free bound on the residue-deleted maxsum must reproduce is now on disk in full — nine folds, 5→7 through 31→37, every copy printed — and it has one exact structure in it that the corpus had explicitly recorded as absent: the curve is a PALINDROME.** The tile's known mirror (GLOSSARY "Mirror") conjugates the hole pair `{a, a−2}` into `{w−a, w−a−2}` where `w = W mod p`, so `Δ_m(x,p,a) = Δ_m(x,p,(w−a) mod p)` for every `m` and every alignment outside four specials — and the ONLY symmetry-breaking point is the edge slot `W−1`, the natal cohort's birth canal, where the identity degrades to the one-sided `Δ_m(edge-striking a) ≥ Δ_m(its mirror)` with an exact correction term (the Mirror-Sweep Lemma, §4, PROVEN and VERIFIED exhaustively). `attack-0c0e-01`'s reading 5 states the argmax sets "carry no arithmetic pattern"; they carry exactly this one — every argmax set at every fold from 7→11 up, measured here and there, is closed under `a ↦ w−a`, and the single exception, 5→7, is the licensed one: its unique argmax IS the seam-striking alignment. That reading-5 sentence is a held correction candidate for the integration pass. Second finding, and it bears directly on 0c's framing: **at depth the kill count carries no information about the degradation.** At folds 5→7 and 7→11 the copies with more kills degrade more; by 17→19 the split is exactly flat (144.7 against 144.7); at 23→29 it has REVERSED (232.3 against 235.7). A bound that goes through a kill count is conditioning on a variable the object has decoupled from — measured support, not proof, for 0c's premise that the kill-free frame is the right one. **What this is not.** Nothing here bounds `Δ_m`. Per the PRICE RIDER (TODO 0c, 2026-08-19) nothing here re-prices the exact levels — the L-bridge still floors at ≈ 0.183x whatever the curve shows — and the 0c×0e composition stays CLOSED; no level-selection claim is made. Per the standing blind-kill record (`scanstat2.md`, `scanstat-t37.md`) nothing here is fitted: the deliverable is the values, the proven symmetry that halves them, and words. --- ## 1. Brief correction: "natal@5→7", corpus reading The brief wrote "natal@5→7 — the natal frame passing from level 5 to 7". In the corpus **natal@p is a cohort, not a frame**: the born-at-p slots `{kW−1 : 1 ≤ k ≤ p−1, k ≢ ±W^{−1} (mod p)}` (GLOSSARY "Natal set @p", `FOLD-PROFILE.md`). The frame in which two holes sweep is the **copy decomposition of a fold** (`U-FRAME.md` §5a step 1, PROVEN): folding `T_x` by `p` lays `p` copies of the old tile and copy `k` deletes exactly the old slots whose residue mod `p` lies in `{a(k), a(k)−2}`, `a(k) = −k·(W mod p) mod p`. The brief is read as **the fold 5→7 (T_5 → T_7) seen copy by copy**, extended up the ladder to 31→37 for the fingerprint to be a curve rather than seven points. The reading keeps the word "natal" honest, because the probe's one structural surprise (§4) lives exactly at the seam slot the natal vocabulary names. [Correction recorded; the producer's header carries the same note.] ## 2. The re-pricing, cited not re-derived (the brief's first move) All numbers in this section are quoted from formally embedded artifacts; none were recomputed here (standing compute rule). **The plain exact maxsum family (`maxsum_m(T_x)`) — where it lives and what it cost.** One pointer in the brief needs correcting: the 7-level exact H ladder's home is `history/staging/scanstat2.md` + `history/staging/scanstat-t37.md` (producers `import-scanstat-04-score.js`, `scanstat2-01-t31.js`, `scanstat-t37-04-run.js`), not `attack-foldL-05-maxsum-direct.md` — foldL-05 is the bridge-floor and threshold-certificate home, and its tables are `H_cert`, not the exact family. | level | `ln D` | `maxsum₁ = G₂` | H ± se | engine cost | artifact | |---|---|---|---|---|---| | T₁₃ | 7.3032 | 66 | 0.2661 ± 0.0230 | in-memory | `scanstat2.md` table | | T₁₇ | 10.0112 | 108 | 0.2804 ± 0.0189 | in-memory | ibid. | | T₁₉ | 12.8444 | 150 | 0.3001 ± 0.0130 | in-memory | ibid. | | T₂₃ | 15.8890 | 204 | 0.3216 ± 0.0096 | `import-scanstat-04-score.js` | ibid. | | T₂₉ | 19.1848 | 258 | 0.3367 ± 0.0080 | ibid. | ibid. | | T₃₁ | 22.5521 | 348 | 0.3460 ± 0.0068 | **3029.3 s, one stream, constant memory** | `scanstat2-01-t31.js` | | T₃₇ | 26.1074 | 528 | 0.3565 ± 0.0068 | **5 shards × 35.2 min**, 2.179·10¹¹ slots | `scanstat-t37-04-run.js` | T₃₁'s family beyond m = 1 (from `scanstat2.md`): maxsum_m = 408, 540, 660, 1002, 1638, 2700 at m = 2, 4, 8, 16, 32, 64. T₃₇ carries the exhaustive maximality certificate for `G₂(37#) = 528`. No exponent is quoted per the twice-blind kill of the linear rule. **The residue-deleted family (`Δ_m`, the 0c object).** Exact at folds 7→11 .. 29→31 in `attack-0c0e-01-deleted-family.js` (47.7 s total; m ≤ 8 in memory to 23→29, m = 1 streamed at 29→31), min/mean/max embedded there. **This run adds** the fold 5→7 (below 0c0e's range), the full per-copy curves at every fold (0c0e embedded only summaries), and **31→37 as a new deepest exact family level**, via a one-pass walker that computes all `p` alignments in a single stream (§5; measured cost in the producer's tail — the whole nine-fold run, 6.23·10⁹-slot stream included, is one script execution). **The next price points, stated so the CHEAP rule can bite later.** Fold 37→41 by the same one-pass walker is a single stream of T₃₇ (2.65·10¹¹ T₂₃-slot visits against 7.15·10⁹ here); the 31→37 pass measured 588 s, so 37→41 prices at roughly six hours unsharded, and it shards trivially by outer copy exactly as `scanstat-t37` sharded. The m ≤ 8 ring at 29→31 (the column 0c0e priced at "near an hour") is unchanged by this run — the walker is m = 1 only. ## 3. The named probe: fold 5→7, every copy, every meaningful m `T_5 = {11, 17, 29}` mod 30, `w = 30 mod 7 = 2`, `G₂(5#) = 12`, edge slot 29 with residue 1 mod 7. The full table (killed slots, `Δ_1..Δ_3`, anatomy) is in the producer's tail; the curve: | copy k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---|---| | holes a(k) | 0 | 5 | 3 | 1 | 6 | 4 | 2 | | `Δ₁` | 12 | 18 | **30** | 24 | 18 | 18 | 12 | | degradation | 0 | 6 | **18** | 12 | 6 | 6 | 0 | Read: two copies are UNDEGRADED (their 2-set catches no slot — possible only while `D < p`); the argmax is unique, at the copy whose holes `{3, 1}` kill both 17 and the edge slot 29, leaving the single survivor 11 and the gap 30 = `G₂(7#)` (copy theorem, re-verified); and the curve is the palindrome `C(k) = C(6−k)` at `k = 0↔6, 1↔5`, broken exactly at the seam pair `k = 2↔4` (30 against 18) — the fold's whole record is a seam-strike event at this level. [VERIFIED, exact] ## 4. The Mirror-Sweep Lemma [PROVEN here; VERIFIED exhaustively] > **Lemma.** Let `σ(s) = W−2−s` on `T_x` (the corpus mirror: it maps the tile > to itself, reverses cyclic order — so every `maxsum_m` is invariant — and > fixes exactly one slot, the edge `W−1`). Let `w = W mod p`. Then σ is > integer-exact on every non-edge slot, so it conjugates the deletion classes > `{a, a−2} (mod p)` into `{w−a, w−a−2}`. Hence > > (i) `Δ_m(x,p,a) = Δ_m(x,p,(w−a) mod p)` for **all m**, for every `a` outside > the specials `{1, w−1, w+1, p−1}`; > > (ii) in copy order the sweep curve is a **palindrome**, `C(k) = C(p−1−k)` > (since `a(k) + a(p−1−k) ≡ w`), broken only at seam-striking copies; > > (iii) at `a ∈ {w−1, w+1}` — precisely the two alignments that strike the > edge slot — the exact anomaly > `Δ_m(a) = maxsum_m(config(w−a) minus the edge slot)` holds, so > `Δ_m(a) ≥ Δ_m(w−a)`: **the seam side never loses.** > > *Proof.* σ(del_a) agrees with del_{w−a} on non-edge slots because > `s ≡ a ⇒ σ(s) ≡ w−2−a` and `s ≡ a−2 ⇒ σ(s) ≡ w−a`, with no representative > wrap for `s ≤ W−3` (slots are ≡ 5 mod 6, so only `W−1` wraps); the edge slot > is σ-fixed but its residue `w−1` transforms wrongly, which is (iii)'s extra > deleted slot; deleting one more slot only merges gaps. ∎ Verified: every non-special mirror pair equal **at every m ≤ 8** at all seven in-memory folds (2/2, 4/4, 5/5, 7/7, 8/8, 10/10, 13/13 pairs), the exact anomaly (iii) equal at every fold, mirror pairs equal at m = 1 on both streamed folds, and the one-sided inequality never violated. Strict breaks are RARE at m = 1: exactly one in the whole nine-fold run — at 5→7 itself (30 against 18), where the seam-strike IS the fold's record. At every fold above it the seam anomaly degrades to equality at m = 1: at 29→31 the seam copies read 330/318 against mirrors 330/318, at 31→37 they read 510/420 against 510/420. [VERIFIED] **What the lemma is worth.** It halves the fingerprint — `(p+1)/2` free values per fold — and it is inherited FREE by any kill-count-free argument, since it never mentions kills. And it corrects a recorded reading: `attack-0c0e-01` reading 5's "the argmax sets carry no arithmetic pattern" — the sets `{5,7}@7→11 (w=1)`, all-but-11`@11→13 (w=9`, 11 is the mirror-fixed point`)`, `{5,10,19}@19→23 (w=15)`, `{2,15}@23→29 (w=17)`, `{7,12,23,27}@29→31 (w=19)` are each closed under `a ↦ w−a`. HELD, not integrated. ## 5. The full curves, and the shape in words All nine curves are embedded in the producer's tail (SEC 1, 3, 4). Summary descriptors, m = 1 (SEC 5 of the tail; 29→31 and 31→37 rows from SEC 4): | fold | min/G₂old | mean/G₂old | max/G₂old | max/min | #atMax | #distinct values | |---|---|---|---|---|---|---| | 5→7 | 1.0000 | 1.5714 | 2.5000 | 2.5000 | 1 of 7 | 4 | | 7→11 | 1.0000 | 1.1455 | 1.4000 | 1.4000 | 2 of 11 | 3 | | 11→13 | 1.1429 | 1.5385 | 1.5714 | 1.3750 | 12 of 13 | 2 | | 13→17 | 1.3636 | 1.5561 | 1.6364 | 1.2000 | 10 of 17 | 3 | | 17→19 | 1.2778 | 1.3421 | 1.3889 | 1.0870 | 11 of 19 | 2 | | 19→23 | 1.2000 | 1.2661 | 1.3600 | 1.1333 | 3 of 23 | 5 | | 23→29 | 1.0882 | 1.1471 | 1.2647 | 1.1622 | 2 of 29 | 5 | | 29→31 | 1.2326 | 1.2731 | 1.3488 | 1.0943 | 4 of 31 | 3 | | 31→37 | 1.1552 | 1.2549 | 1.5172 | 1.3134 | 2 of 37 | 8 | **The shape, in words.** The curve is not a bump and not a noise floor: it is a **flat quantized plateau structure with rare spikes**. Every value is a multiple of 6 (all tile gaps are), the number of distinct values is tiny (2 to 8 among up to 37 copies), the bulk of every curve from 17→19 up sits within a factor ≤ 1.10 of its own floor, and the floor sits strictly ABOVE the old record from 11→13 on — every copy's holes merge enough somewhere to beat `G₂(old)` outright. On top of the plateau sit one to a few spikes (the argmax copies plus, at 31→37, a second tier at 510 and 462 against a 402-426 bulk), at positions that are mirror-symmetric but carry no other visible arithmetic; the mirror pairs force the whole picture to read the same left-to-right as right-to-left, with the seam copies the only licensed asymmetry. The spike amplitude carries no trend: max/min reads 2.5000, 1.4000, 1.3750, 1.2000, 1.0870, 1.1333, 1.1622, 1.0943 over 5→7 .. 29→31 — falling overall but non-monotone — and then jumps back to 1.3134 at 31→37, where the record copy degrades by 180 against a bulk of 60-84. Undegraded copies exist only while `D < p` (5→7 and 7→11), i.e. only while a 2-set can be empty; from 11→13 the kill law's two-per-slot coverage makes zero degradation impossible in every copy. [MEASURED; the D < p mechanism is one line and PROVEN] **Kill-count decoupling.** Splitting each fold's copies at the median kill count, the high-kill half's mean `Δ₁` against the low-kill half's reads 22.0/14.0 at 5→7, 36.0/31.2 at 7→11, 105.0/102.0 at 13→17, 144.7/144.7 at 17→19, 189.3/189.3 at 19→23, and **232.3/235.7 at 23→29 — reversed**. At depth the per-copy kill count concentrates at `2D/p` and stops predicting anything; what moves `Δ₁` is where the holes land relative to the record-gap neighbourhoods, not how many slots they take. [MEASURED] ## 6. What a kill-count-free argument must reproduce The fingerprint, enumerated, for any candidate 0c bound `Δ_m(x,p,a) ≤ B(x,p,a)`: 1. **The floor.** `min_a Δ₁/G₂(old)` = 1.09 to 1.36 at every fold from 11→13 — a bound family that cannot see the worst alignment already exceeding the old record by ~10-35% is measuring something else. 2. **The mirror.** `B` must be (or may be assumed) invariant under `a ↦ w−a` away from the four specials, and one-sidedly seam-favouring at them — free structure, PROVEN, use it. 3. **The quantized plateau.** 2 to 8 distinct values per fold, the bulk within a factor 1.10 of the floor from 17→19 up; a bound with per-alignment resolution finer than the plateau is wasted resolution, and one with a spread factor ≫ 1.4 is loose by more than the whole curve varies. 4. **The spike multiplicity carries no law.** #atMax runs 1, 2, 12, 10, 11, 3, 2, 4, 2 (5→7 .. 31→37) and max/min is non-monotone (1.09 at 29→31, 1.31 at 31→37) — no trend; the argmax's position is mirror-closed and otherwise patternless on the evidence; do not build an argument that needs the extremal alignment to be special. 5. **No kill-count proxy.** The decoupling of §5 means any bound whose alignment-dependence enters through the copy's kill count is flat at depth and cannot reproduce 1-4. ## 7. NOT REACHED - **No bound on `Δ_m`.** TODO 0c's ask remains untouched; this run built the target its answer must hit. - **m ≥ 2 above fold 23→29.** The streamed folds are m = 1 only; the one-pass walker's ring generalisation was not built. - **Fold 37→41.** Priced (§2), not run. - **The plateau's value set.** WHY the curve takes 2-6 values (which merges are realisable) was not characterised; the anatomy columns (merges 2-4, old record contained at some folds not others) reproduce 0c0e's reading 6 and go no further. - **The mirror-law refinement is HELD.** `attack-0c0e-01` reading 5 and any GLOSSARY note on the sweep symmetry await the adversarial pass; nothing live was edited. ## 8. Reproduction and custody ``` node research/attack-0c-holesweep-01.js 37 # full run, one process node research/qc/embed.js --streams both --timeout 2400 research/attack-0c-holesweep-01.js -- 37 ``` Custody inside the run: `D = ∏(q−2)` and the `G₂` ladder 6..204 from a generator that never sieves; per-fold kill law `Σ_k alive = D(p−2)` exact; the copy theorem re-verified at all m ≤ 8 for folds 5→7 .. 23→29 and at m = 1 at 29→31 (`max = 348 = G₂(31#)`) and 31→37 (`max = 528 = G₂(37#)`, whose exhaustive certificate is `scanstat-t37.md`'s); min/mean/max reproduce `attack-0c0e-01`'s embedded figures figure for figure at all six folds it reached; and the one-pass walker is validated against the naive per-alignment engine, all alignments, at all seven in-memory folds before either streamed fold is trusted. The tail carries `code-sha256`/`out-sha256` and the exact invocation.