# TODO 1d: (H-sub) attacked — three proven reductions, the fold verdict, and no counterexample *2026-08-20. Producer: `research/attack-hsub-01.js` (0.1 s, `code-sha256 f14f5789`, `out-sha256 ee4e9643`, formal embed with input hashing, seven numbered readings; fingerprint verified with `embed.js --check` after writing). One defect was found in the producer's own first run and repaired before this report's numbers were read: the proven-floor check's slack accumulator initialized at 0, capping its printed minimum — the could-not-fail class. The re-embed used `--force` (the tool's corrected-code path); the override is stamped in the tail. Nothing was committed or pushed; no existing corpus document was edited. HELD, not integrated — awaiting the standing one-pass adversarial review. Legend as in `research/sift-limit-attack.md`: **[PROVEN]** published theorem or a proof given here; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[INFERRED]** deduction from sourced facts.* --- ## 0. The verdict, up front > **(H-sub) is not proven and not refuted, but it is now a smaller statement > aimed at a wider window, and the pass found one target the trap cannot > touch. Three reductions of the lemma's hypothesis are proven; the fold > machinery's inability to reach any of them is made exact; and the hunt > over all 111 reachable pairs found no drifting family and no > counterexample.** > > 1. **The lemma consumes only power pairs.** Its own eight-line proof uses > (H-sub) at the pairs `(b^k, b)` and nowhere else, so the hypothesis > weakens to (H-sub-pow) with the conclusion unchanged. The record > defects 1.0761 at (4,10) and 1.0330 at (4,12) are not power pairs and > stop constraining the route: the trap window widens from > `[1.0761, 1.3946)` (0.3185 nats) to **`[1.0033, 1.3946)` (0.3913 nats, > trusted)** and `[0.9694, 1.3555)` (0.3861, custody-only). **[PROVEN / > VERIFIED]** > 2. **One chain carries the TPC face; no chain carries existence.** A single > base `b₀` with chain defect `K < S(b₀)` forces `limsup < 2`; limit > existence needs chains at inf-realizing bases. And the chains that > decide the trap (bases 16 and 66) have **zero reachable pairs** — the > decisive family is dataless. **[PROVEN / VERIFIED]** > 3. **The doubling slice is trap-free and prize-bearing.** `Ĝ(2s) ≤ C₂·Ĝ(s)` > gives `limsup ≤ log₂C₂`; its TPC line `C₂ < 4` is already excluded by > custody data (`C₂ ≥ 348/66 = 5.2727`), so no provable doubling constant > is TPC-implying through the slice's own conclusion — and every > `C₂ ∈ [5.2727, 19.2455)` lowers the proven > exponent ceiling 4.2665 to `log₂C₂ ∈ [2.3985, 4.2665)`. A 1.29-nat > landing zone with no trap in it. **[PROVEN the implication; VERIFIED > the window]** > 4. **The fold machinery cannot reach (H-sub), exactly.** The fold step is > `×p_next/p ≤ 5/3 < 2` in Ĝ's argument coordinate — below the integer > floor `t ≥ 2` — so every admissible pair is multi-fold; (H-sub) is > identically a head-dominance statement about the nonnegative > fold-budget sequence, and the proven tools bound the defect from the > side it does not need. **[PROVEN / VERIFIED]** > 5. **No counterexample.** Max `D = 1.0761` at (4,10), min `−0.2400` at > (7,7); the (4,t), (s,2) and diagonal families all read bounded against > the in-pass constant-defect null. **[MEASURED]** --- ## 1. Reduction 1: the hypothesis shrinks to power pairs, and the floor drops Re-reading the proof in `attack-fekete-1d-02-lemma.js`: the limsup half iterates `f(b^{k+1}) = f(b^k·b) ≤ f(b^k) + f(b) + K` — the pair `(b^k, b)` — then bridges intervals with the proven (H-mono); the liminf half uses only the definition of the inf. No other pair is ever consumed. So > **(H-sub-pow)** `f(b^{k+1}) ≤ f(b^k) + f(b) + K` for all integers `b ≥ 2`, > `k ≥ 1` implies the full conclusion — `lim f(n)/ln n = inf_n (f(n)+K)/ln n` exists, and `β < 2 ⟺ S(n) > K at one integer` — by the existing proof, character for character. **[PROVEN, by inspection of the existing proof]** The consequence is a data statement. Only 15 power pairs are reachable (9 custody), and the family's sup defect is | grade | floor | at | rests on | |---|---|---|---| | trusted | **1.0033** | (16,4) — base-4 chain, k = 2 | `Ĝ(64) = G₂(61#) = 1080` (Wang 2024 a(18) — the same term as the 1.3946 ceiling) | | custody | **0.9694** | (16,2) — base-2 chain, k = 4 | `G₂(31#) = 348`, `G₂(13#) = 66`, corpus-exact | (4,10) and (4,12) are not of the form `(b^k, b)` (verified), so the two worst defects on the 111-pair table no longer bound the constant the route needs. **The trap window is now `ln C ∈ [1.0033, 1.3946)` at trusted grade — 0.3913 nats, 23% wider than the standing 0.3185 — and `[0.9694, 1.3555)`, 0.3861 nats, on custody terms alone.** Ceilings are unchanged: they belong to the conclusion formula (`max_b S(b)`), not to the pair family. **[VERIFIED]** ## 2. What a cofinal chain buys — the exact answer to the composition question The task asked whether a proof on a cofinal sub-family, composed along the ladder, feeds the lemma. Split by face: - **Limsup face (which is the TPC face).** (H-sub-pow) at ONE base `b₀` gives `limsup f(n)/ln n ≤ (f(b₀)+K)/ln b₀`; so `K < S(b₀)` on one geometric chain forces `limsup < 2`. More generally a cofinal chain with two-sidedly pinned steps `a·n_j ≤ n_{j+1} ≤ b·n_j` (`a > 1`) and `f(n_{j+1}) ≤ f(n_j) + f(b) + K` gives `limsup ≤ (f(b)+K)/ln a`. Steps pinned only from above lose the denominator and give nothing. **[PROVEN]** - **Existence face.** No single chain, and no base subset `B`, gives limit existence: `limsup ≤ inf_{b∈B}(f(b)+K)/ln b` but `liminf ≥ L` takes the inf over ALL `n`, and the two close only when `B` realizes the inf — unknowable in advance, so the clean sufficient family is all-bases power pairs (Reduction 1) and nothing smaller. **[PROVEN]** - **The decisive chains are dataless.** No base `b ≥ 10` has a single reachable pair (`b² > 82`), so the chains at the threshold bases 16 and 66 — the ones whose `K < S(b₀)` would be TPC — are constrained by no measurement the corpus owns; the trap-window floor comes entirely from bases 2..9. **[VERIFIED]** - **The "fold-step family" (s a primorial level, t the next prime) does not exist inside the candidate.** See §4(i): the fold step is a sub-integer multiplier, and the corpus's own domain correction (real bases die at `ln C ≥ 2.01`, `import-interp.md` §4) forbids restating the candidate down there. **[INFERRED, from verified ingredients]** ## 3. The slice ladder: per-slice traps, and the doubling window Fixing `t₀` and quantifying over `s` (the slice contains its own base chain `(t₀^k, t₀)`) gives `limsup ≤ (f(t₀)+K_{t₀})/ln t₀`: TPC-implying iff `K_{t₀} < S(t₀)`; improves the proven `β₂ = 4.26645` (`dhr-verification.md` row 1a) iff `K_{t₀} < β₂ ln t₀ − f(t₀)`. Measured floors against both lines (full table in the producer): | t₀ | reachable sup D | S(t₀) | trap | improvement ceiling | |---|---|---|---|---| | 2 | 0.9694 (custody) | 0.6931 | **closed by data** | 2.2641 | | 3 | 0.5810 | 0.4055 | closed | 2.8954 | | 4 | 1.0761 | 0.9808 | closed | 4.1228 | | 5 | 0.7697 | 0.7340 | closed | 4.3817 | | 6 | 1.0415 (custody) | 1.0986 | **OPEN, width 0.0572** | 5.1595 | | 7..13 | — | — | all OPEN, wider | — | Two headlines fall out: **The doubling window (t₀ = 2) — the pass's sharpest find.** In constant form the slice is `Ĝ(2s) ≤ C₂·Ĝ(s)` and the conclusion is `limsup ln G₂(x#)/ln x ≤ log₂C₂`. Custody data already force `C₂ ≥ Ĝ(32)/Ĝ(16) = 348/66 = 5.2727 > 4`, so **no provable doubling constant can be TPC-implying — the explicit-constant trap that guards every other face of this route is void on the doubling slice**, and a soft proof is not excluded by trap logic. Simultaneously every `C₂ ∈ [5.2727, 2^{β₂} = 19.2455)` lowers the proven exponent ceiling: `limsup ≤ log₂C₂ ∈ [2.3985, 4.2665)`. One inequality — a per-doubling growth cap on the twin-gap ladder — with a 1.29-nat-wide landing zone in which every point is consistent with all data, non-TPC, and an unconditional improvement of the corpus's strongest proven upper bound. **[PROVEN the implication; VERIFIED the window; the inequality itself is OPEN]** **The t₀ = 6 slice trap is nearly shut.** Any explicit `K₆ ∈ [1.0415, 1.0986)` — 0.0572 nats, floor at the custody-exact (4,6) — is TPC-implying, with implied bound 1.9681 at the floor. This does not displace the general trap window; it records that a single fixed-t inequality at t = 6 is already TPC-strength on a margin five times thinner than the general one. **[VERIFIED]** ## 4. Structure first: the exact reason the fold machinery falls short **(i) The fold step is not an integer pair.** In Ĝ's argument coordinate the fold `G₂(p_n#) → G₂(p_{n+1}#)` is multiplication by `p_{n+1}/p_n ≤ 5/3 < 2` (max at 3→5; verified at all 21 steps). The candidate's floor `t ≥ 2` — forced when real bases died at `ln C ≥ 2.01` — sits strictly above the one-fold regime, so every admissible pair crosses `π(st) − π(s) ≥ π(2s) − π(s) ≥ 1` folds (Bertrand), unboundedly many as `s` grows (the (4,10) pair already spans 10 folds; verified minimum 1, at (2,2)). A constant per-fold cap compounds to `exp(#folds)`: no one-fold theorem composes into (H-sub). **[VERIFIED on the range; the Bertrand floor is PROVEN]** **(ii) (H-sub) in fold coordinates is head dominance.** With `r_1 = ln G₂(2#)` and `r_i = ln[G₂(p_i#)/G₂(p_{i−1}#)] ≥ 0` — nonnegative by the PROVEN (H-mono) — the identity `f(st) − f(s) = Σ_{π(s) < i ≤ π(st)} r_i` holds (verified to max dev 1.78e−15 at all 111 pairs), so (H-sub) is exactly: **the budget spent in the window `(s, st]` never exceeds the budget of the first `π(t)` folds plus K.** The window holds ~`st/ln st` folds against the head's `π(t)`; the candidate is a statement that late fold budgets thin out, which is global regularity, not local structure. The three worst pairs decompose cleanly (e.g. (4,10): 10-fold window budget 4.4773 vs 4-fold head 3.4012, excess = D = 1.0761). **[PROVEN the identity; VERIFIED]** **(iii) The proven tools point the wrong way.** The copy theorem and (H-mono) (folding deletes slots, deletion merges gaps) and the block-ladder record's disjoint-set super-additivity all give LOWER bounds on `Ĝ(st)` — they prove `D(s,t) ≥ −min(f(s), f(t))` (one line from (H-mono); holds at every pair with strictly positive slack, minimum 0.6931 at (2,3), and the measured min D −0.2400 sits far above its floor −3.4012 at (7,7)). The Mirror-Sweep Lemma is symmetry structure within ONE fold (alignment `a` and `w−a` degrade equally); it halves fingerprints and locates seam anomalies but carries no magnitude bound and nothing that composes across folds. No named sub-family of (H-sub) is provable with the corpus's current machinery, and this pass proves none. **[PROVEN the floor; INFERRED the assessment]** ## 5. The hunt: no counterexample, no drift Over all 111 reachable integer pairs (`st ≤ 82`, trusted 22-term ladder, cross-checked against `attack-fekete-1d-01`'s headline numbers to four places): - **Extremes.** Max `D = 1.0761` at (4,10); the next four are (4,6) 1.0415, (4,12) 1.0330, (6,12) 1.0202, (4,16) 1.0033. Min `D = −0.2400` at (7,7); 7 of 111 pairs are negative (the super-multiplicative side). **[VERIFIED]** - **The (4,t) family** — carrier of both record defects — peaks at t = 10 (the 37-spike) and every `t ≥ 13` sits below the peak. Slope against `ln t`: `+0.0808 ± 0.0957` raw, `−0.0020 ± 0.1143` against the in-pass constant-defect null (POW, same primes, same step construction). **Consistent with bounded; no drift.** **[MEASURED]** - **The doubling family (s,2)**, s = 2..41: `−0.0336 ± 0.0596` against the null. Consistent with bounded. **[MEASURED]** - **The diagonal (b,b)**, b = 2..9 — the one place an unbounded defect could live under a power-log law: reads `−0.5188 ± 0.3966` BELOW the constant-defect null (1.3 σ, 8 points), i.e. leaning toward the decreasing-defect (super-multiplicative) side. The diagonal remains blind past `√82 ≈ 9`, exactly as `attack-fekete-1d-01` reading 7 recorded; this pass adds data consistent with "not growing" and cannot exclude growth. **[MEASURED]** An honest "the defect grows along family F" would have killed the route; no reachable family shows it. ## 6. Trap grading of everything this pass produced No explicit K was derived (nothing here proves an instance of (H-sub)), so nothing trips the trap. What moved are the WINDOWS, graded per the standing rule against `[1.0761, 1.3946)` trusted / `[1.0761, 1.3555)` custody: | object | window | grade | status vs trap | |---|---|---|---| | power-pair candidate (Reduction 1) | `ln C ∈ [1.0033, 1.3946)` | floor rests on Wang a(18); ceiling likewise | trap WIDENED 0.3185 → 0.3913 nats | | power-pair candidate, custody | `ln C ∈ [0.9694, 1.3555)` | fully custody | 0.3861 nats | | single-chain TPC face (base 16 / 66) | any `K < 1.3555 / 1.3946` | custody / trusted | TPC-implying; family dataless | | t = 6 slice | `K₆ ∈ [1.0415, 1.0986)` | custody | TPC-implying, width 0.0572 | | doubling slice | `C₂ ∈ [5.2727, 19.2455)` | custody floor | **trap-free**: TPC needs `C₂ < 4`, excluded by data; every point improves 4.2665 | ## 7. NOT REACHED - **No instance of (H-sub) proven, for any family.** The three reductions are quantifier surgery on the lemma — they change what must be proven, not what is proven. The route's gap is smaller and better lit, still open. - **The doubling inequality `Ĝ(2s) ≤ C₂·Ĝ(s)` is posed, not attacked.** It is the natural next target (trap-free, prize-bearing); this pass did not attempt a proof and does not know whether one is soft. - **No new defect data.** The reachable pair set is capped at `st ≤ 82` until the ladder grows; the first riser enumeration (61#) remains ~146,969× the 43# run. - **The diagonal stays invisible past b = 9.** Growth of the true defect on the diagonal is neither seen nor excluded; the -01 blind spot stands. - **de Bruijn–Erdős Theorem 22 remains unopened**, consistent with the bounded reading; nothing here cites its content. - **Whether the reductions compose** (e.g. power pairs at prime bases only, giving the prime-base threshold `max_q S(q) = 1.2946` trusted) was computed in exploration but not carried into the producer; the all-bases form of Reduction 1 is the operative statement. ## 8. Sources and reproduction | artifact | what it carries | |---|---| | `research/attack-hsub-01.js` | everything above: the three reductions (proofs in the header), the power-pair table, the slice ladder, the fold-budget identity, the traces with in-pass nulls (§§1–6) | | `research/attack-fekete-1d-02-lemma.js` | the lemma whose proof Reduction 1 inspects; (H-mono) proven + verified | | `research/attack-fekete-1d-01-defect47.js` | the 111-pair table this pass cross-checks; the threshold freeze | | `research/history/staging/fekete-1d.md`, `redteam-0820-math.md` §1 | the route's state and the trusted-ladder trap window this pass widens | | `research/history/staging/import-interp.md` §§3–4 | the defect identity; the real-base death that forces `t ≥ 2` | | `research/history/staging/attack-0c-holesweep.md` §4 | the Mirror-Sweep Lemma assessed in §4(iii) | | `research/dhr-verification.md` row 1a | `β₂ = 4.26645028414864191641`, the proven ceiling the doubling window lowers | | `research/exact-g2-ladder.js`, `research/import-interp-01-bgt-defect.js` | the two ladders, parsed at run time, input-hashed by the embed | Reproduce with `node research/qc/embed.js --check research/attack-hsub-01.js`; the fingerprint matches as of 2026-08-20.