# TODO 1d: the bounded-defect Fekete route, probed at 47# *2026-08-20. Producers: `research/attack-fekete-1d-01-defect47.js` (0.1 s, `code-sha256 45554df4`, `out-sha256 afc6d190`, formal embed, seven numbered readings) and `research/attack-fekete-1d-02-lemma.js` (0.2 s, `code-sha256 62cb37d9`, `out-sha256 1c99d42b`, formal embed, six numbered readings). Both fingerprints verified with `embed.js --check` after writing. Nothing was committed or pushed; no existing corpus document was edited. 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; **[CERTIFIED]** follows from a replay-verified corpus certificate.* --- ## 0. The verdict, up front > **The defect at 47# is ordinary, the one unverified term cannot move any 1d > reading, and — the sharpest thing this pass found — the 47# enumeration > cannot move 1d's TPC threshold either, whatever value it returns. The > bounded-defect Fekete lemma is stated exactly, proved, and every hypothesis > except the candidate itself is now discharged. The route is reduced to one > named inequality.** > > 1. **The defect at 47#, exactly.** Ten integer pairs land in `[47, 53)`. The > largest defect is `D = 1.0330` nats at `(4,12)` — second largest on the > whole 111-pair table, 0.0431 under the global sup `1.0761` at `(4,10)` — > and the all-prime pair `(7,7)` reads `−0.2400`. The 47-window ranks 3rd of > 21 windows. Nothing about it is extreme. **[VERIFIED]** > 2. **The 704-vs-708 discrepancy is immaterial.** The greedy's replay-verified > covering gives `G₂(47#) ≥ 705`; A144311 says 708. The worst shift any > defect reading can take from that bracket is `ln(708/705) = 0.00425` > nats — 66× below the smallest margin in play. A144311 would have to be low > by ≥ 32 (4.5%) before the reachable sup C moved. **[VERIFIED]** > 3. **REFUTATION: 1c's "each new ladder-known x raises 1d's TPC threshold" is > false at 47#, and certified false — no enumeration outcome can change > it.** The custody integer threshold is `S(16) = ln(256/66) = 1.3555`, a > 13#-fact. The 47-window's best is `S(52) = ln(2704/G₂(47#)) ≤ > ln(2704/705) = 1.3443 < 1.3555`, capped from above by the certified > greedy floor. Same freeze at 53# (cap 1.3524) and 59# (cap 1.3280). The > first term that can raise it is x = 61, and a 61# enumeration is > ~146,969× the 43# run. **[CERTIFIED]** > 4. **The defect reads BOUNDED on a second instrument, and custody-only.** > Per-window sup slopes: G₂ minus the constant-defect null = > `−0.0414 ± 0.0846` (22 terms) and `−0.0188 ± 0.1466` (14 custody terms) — > consistent with bounded both times, and the verdict does not lean on the > eight literature terms. **[MEASURED]** > 5. **What a bounded defect buys, exactly.** The multiplicative Fekete lemma > (stated and proved in `attack-fekete-1d-02-lemma.js`, eight lines): > conditional on the candidate with ANY constant, `β = lim ln G₂(x#)/ln x` > **exists** and equals `inf_n (ln Ĝ(n) + K)/ln n`; and > `β < 2 ⟺ S(n) > K at a single integer`. Existence is the honest prize; > the explicit-constant version is TPC-strength. **[PROVEN, given the > candidate]** --- ## 1. What "probe 47#" can and cannot mean, priced The exact enumeration at 47# is already priced by direct probe: **1.49 days per run, two disjoint-mask runs required** (`phase1-T2b-exact-ladder.md`), so it was not launched here — that is 1c's task, with 1c's Poisson-window pre-registration discipline. What the corpus already holds at 47# is a bracket: | source | value | grade | |---|---|---| | greedy covering of `[1, 704]`, replay-verified | `G₂(47#) ≥ 705` | CERTIFIED (`greedy-oracle-validation.md` §3) | | OEIS A144311 a(15), Alekseyev 2009 | `G₂(47#) = 708` | literature, unverified here | Every reading below is taken at both endpoints of that bracket, which is what makes this a probe rather than a wait. ## 2. The defect at the 47-window `D(s,t) = ln[Ĝ(st)/(Ĝ(s)Ĝ(t))] = S(s)+S(t)−S(st)` (the identity, PROVEN in `import-interp.md` §3, re-verified here to `5.6e−16` at all pairs). The ten pairs with `st ∈ [47, 53)`, at the published 708: | pair | st | R | D (nats) | |---|---|---|---| | (4,12) | 48 | 2.8095 | **1.0330** | | (5,10), (6,8) | 50, 48 | 1.9667 | 0.6763 | | (3,16), (4,13) | 48, 52 | 1.7879 | 0.5810 | | (2,24), (2,25), (2,26) | 48, 50, 52 | 1.7353 | 0.5512 | | (3,17) | 51 | 1.0926 | 0.0886 | | (7,7) | 49 | 0.7867 | −0.2400 | The window's sup ranks **3rd of 21 windows** (behind p = 37's 1.0761, which is the global sup, and p = 71's is 1.0202 just below); the running per-window sup last rises at the p = 37 window and is flat over the ten windows after it. **[VERIFIED]** **Sensitivity.** At `G₂(47#) = 705` and at 708 alike: sup C stays 2.9333 at (4,10), every threshold stays put, and the maximal shift of any D is 0.00425 nats. The flip value is `G₂(47#) ≥ 740` — pair (4,12) would then overtake (4,10) — i.e. A144311 low by ≥ 32. Below 705 is excluded by the certificate. **[VERIFIED]** ## 3. The threshold freeze, which corrects two records TPC-implication for an explicit constant is `ln C < S(b)` at a ladder-known integer base `b` (`import-interp.md` §4). The threshold column, recomputed: | ladder | threshold `max_b S(b)` | at base | |---|---|---| | full 22-term (8 literature terms) | 1.3946 | b = 66 — **rests on Wang 2024's a(18), `G₂(61#) = 1080`** | | custody-only, 14 exact terms | **1.3555** | b = 16 — rests on `G₂(13#) = 66` | | custody + 47# verified (at 708 or 705) | 1.3555 | b = 16, unchanged | Because `S(b) = ln(b²/Ĝ(b))` **falls** as `Ĝ` rises, a certified lower bound on `G₂` is a certified **upper** bound on the window's best S. The greedy floors therefore freeze the next three windows below the standing custody max: | window | best base | cap `ln(b²/floor)` | can raise 1.3555? | |---|---|---|---| | [47,53) | 52 | 1.3443 | no, certified | | [53,59) | 58 | 1.3524 | no, certified | | [59,61) | 60 | 1.3280 | no, certified | | [61,67) | 66 | 1.3992 | **yes** — fires iff verified `G₂(61#) ≤ 1123` (published: 1080) | **Two corrections follow.** (i) TODO 1c's stated win "each new ladder-known x raises 1d's TPC threshold, directly loosening the constraint" is wrong at 47# — and not merely at the published value: **certified wrong for every possible enumeration outcome**, and equally at 53# and 59#. The first riser is x = 61, whose enumeration is `47·53·59 = 146,969×` the 43# run (~1 h on ten cores) — out of reach by ~5 orders of magnitude. (ii) The corpus's quoted integer threshold 1.3946 is a **literature-conditional** number (Wang 2024's a(18)); the custody-grade threshold is 1.3555 and is effectively frozen there. **[CERTIFIED / VERIFIED]** **The 1d position now stands on custody terms alone.** The reachable sup `C = 2.9333` is attained at (4,10) from `G₂(7#) = 30` and `G₂(37#) = 528`, both corpus-exact; the threshold 1.3555 is a 13#-fact. The candidate's TPC window is `C ∈ [2.9333, 3.8788)`, `ln C ∈ [1.0761, 1.3555)` — 0.2794 nats wide — with no literature term anywhere in it. Any proof of the candidate with a constant in that window is a proof of `β < 2`. **[VERIFIED]** **One domain-cap artifact repaired in passing:** the recorded integer floor-7 sup 1.5533 (`import-interp.md` §4) was taken under `st ≤ 79`; `Ĝ` is known on `[2, 83)`, and the widest legal domain admits (8,10), st = 80, with `R = 1.9000` — equal to the recorded real-form sup, so the real/integer split at floor 7 was the cap, not structure. No verdict changes. **[VERIFIED]** ## 4. Is the defect bounded? **Second instrument, same verdict as `import-interp.md` §5.** Per-window sup `Wmax(p) = max{D(s,t) : st ∈ [p, p_next)}`, slope against `ln p`, nulls sampled at the same primes and stepped the same way in the same pass: | ladder | slope ± se | G₂ − POW contrast | |---|---|---| | G₂ full (22) | +0.1661 ± 0.0637 | **−0.0414 ± 0.0846 → consistent with bounded** | | POW null, constant defect | +0.2075 ± 0.0558 | (the calibration: this IS what bounded looks like here) | | LOG null, decreasing defect | −0.2415 ± 0.0554 | (the instrument is not pinned positive) | | G₂ custody-only (14) | +0.2526 ± 0.1094 | **−0.0188 ± 0.1466 → consistent with bounded** | G₂ reads *less* growing than the constant-defect null on both ranges, and the custody-only row shows the bounded verdict does not lean on the eight literature terms. **[MEASURED]** **The instrument's blind spot, stated as a limit.** Every power-log law `Ĝ ~ c·x^β (ln x)^δ` **with δ ≥ 0** has a bounded window-sup defect (continuous limit `δ·ln(1/ln 2) − ln c`; for δ < 0 the k = 1 rung diverges and the defect is unbounded, sign lemma PROVEN in `attack-0829n-hsubpow-K.md` §3b, qualifier added 2026-08-29), because the sup is driven by small-s pairs; an unbounded defect would live on the diagonal, where the largest `√x` any ladder offers is ~9. So "bounded" is the supported reading under every law the corpus entertains, and is **not** a demonstrated fact about the true defect — which is exactly why the honest Fekete target carries an unnamed constant. **[INFERRED]** ## 5. The exact lemma, hypotheses discharged > **Lemma (multiplicative Fekete with bounded defect).** Let > `f : {2,3,4,…} → [0,∞)` satisfy **(H-sub)** `f(st) ≤ f(s) + f(t) + K` for all > integers `s,t ≥ 2` and some constant `K ≥ 0`, and **(H-mono)** `f` > nondecreasing. Then `lim f(n)/ln n` exists and equals > `L = inf_{n≥2} (f(n)+K)/ln n ∈ [0, ∞)`. Proved in eight lines in `attack-fekete-1d-02-lemma.js`'s header (liminf from the definition of inf; limsup by iterating (H-sub) on powers of an arbitrary base and bridging with (H-mono)). de Bruijn–Erdős Theorem 22 — the growing-defect version — is **not needed**: the measured defect reads bounded and plain-Fekete shape suffices, exactly as `import-interp.md` §6 anticipated. **[PROVEN]** **Why BGT's obstruction does not apply here:** the lemma never asks an integer to factor. BGT's machine died on H1 (`π(st) ≠ π(s)+π(t)`, the ground set does not split); this lemma uses (H-sub) on products only, and (H-mono) replaces the additive ground set in the between-powers bridge. **[INFERRED]** Hypotheses, for `f = ln Ĝ`: | hypothesis | status | |---|---| | `f ≥ 0` | trivial, `Ĝ ≥ 2` | | (H-mono) | **PROVEN one line + VERIFIED seven folds**: the twin-slot set of `T_next` is a subset of the periodized slot set of `T_x` (folding only deletes slots; deletion only merges gaps, so `G₂` is nondecreasing). Checked literally by direct sieve at every fold `2→3→…→19`, periods to `9,699,690 = 19#`, reproducing the exact ladder term by term; monotone at all 22 ladder terms | | (H-sub) | **OPEN — the candidate itself.** Holds at all reachable pairs with `K = 1.0761`; unproven beyond; the route's entire remaining content | **The conclusion formula, calibrated on known-truth controls** (campaign rule: estimator on a control in the same pass). On `POW` (true β = 1.546, constant defect) the finite-range inf reads 1.5901 at n = 10⁶, +2.9% and falling from above; on `LOG` (true β = 1, δ = 2) it reads 1.5335, **+53%**, with its K driven by the (2,2) pair. Both approach from above, as the inf-over-a-subset argument requires. **So the lemma's value output is unquotable at any reachable range; what it delivers is existence.** **[MEASURED]** ## 6. What a bounded defect buys, and the two faces Conditional on (H-sub) with **some** constant, never named: - `β = lim ln G₂(x#)/ln x` **exists** — along integers, hence along reals, since `Ĝ` is a step function. 1d's central question becomes well-posed: `β < 2` forces `G₂(x#)/x² → 0`, `β > 2` forces it to ∞, and only `β = 2` leaves the sublinear part to decide. **[PROVEN, given (H-sub)]** - `β < 2 ⟺ S(n) > K` at a single integer. This one equivalence is the trap and the prize at once: an explicit `K < 1.3555` is TPC-implying (no soft proof should be expected), an unnamed `K` buys existence without touching TPC. The corpus's `1.8992` at base 16 (`import-interp.md` §4) is this formula at the reachable K, reproduced exactly. **[PROVEN given (H-sub); cross-check VERIFIED]** Not bought: any value of β, any movement on `4.2665 → 2`, any unconditional statement. ## 7. NOT REACHED - **The 47# enumeration itself.** Priced at 1.49 days per run, two runs (`phase1-T2b-exact-ladder.md`); not launched. The bracket `[705, ∞) ∩ {published 708}` is what this pass used, and §2-§3 show nothing 1d needs from the enumeration beyond what the bracket already certifies. - **No proof of (H-sub), no counterexample, no bound on the true K.** The pairs that would decide it need both arguments large; the ladder has none (largest `√x` ≈ 9). - **No exclusion of a growing defect.** §4's verdict is consistency with bounded, on an instrument blind to diagonal growth; a non-power-log law could hide there. - **de Bruijn–Erdős Theorem 22 remains unopened** (as in `import-interp.md` §9); it is not needed for the bounded shape, and nothing here cites its content. - **The real-base form of the candidate** was not re-examined; the corpus instruction to state the candidate on integer bases stands (`import-interp.md` §4). - **Whether 61# could ever be reached by a smarter engine** than direct enumeration (the only riser base in reach of the freeze analysis) was not investigated; at direct-enumeration scaling it is ~146,969× the 43# hour. ## 8. Sources and reproduction | artifact | what it carries | |---|---| | `research/attack-fekete-1d-01-defect47.js` | the 47-window pairs, sensitivity at 705/708, flip values, the threshold freeze, the per-window boundedness instrument with nulls (§§2-4) | | `research/attack-fekete-1d-02-lemma.js` | the lemma with proof, (H-mono) verified at seven folds, the control calibration, the conditional application (§§5-6) | | `research/exact-g2-ladder.js` | the 14 custody terms, parsed at run time | | `research/import-interp-01-bgt-defect.js` | the corpus's A144311 copy, parsed at run time; the headline numbers cross-checked here | | `research/history/staging/greedy-oracle-validation.md` §3 | the replay-verified greedy floors 705/870/954/1075 | | `research/history/staging/phase1-T2b-exact-ladder.md` | the 47# price: 1.49 days per run | | `research/history/staging/import-interp.md` §§3-7 | the identity, the trap statement, the slice-instrument boundedness reading this pass confirms | Reproduce with `node research/qc/embed.js --check research/attack-fekete-1d-01-defect47.js` and `node research/qc/embed.js --check research/attack-fekete-1d-02-lemma.js`; both fingerprints match as of 2026-08-20.