# The doubling bridge at C₂ = 8: the K* certificate is dead uniformly by a cited lemma, the maxsum certificate is the sharpest proven bridge and holds at fourteen exact steps, and the one inequality that does not close is located > **RIDER 2026-08-30 (orchestrator, from `redteam-0830-doubling.md`, a > column-major engine sharing no line with this producer; every quoted figure > reproduced digit for digit, including K*(16) = 17 and G₂(31#) = 348 @ > 8813641451 ×4 from BOTH base tiles).** One statement REFUTED: this note's > NOT-REACHED line on 19#→37# — that walk runs, reproduces the x = 37 ladder > row (528 @ 544899485411 ×2, census 217929355875) and yields a FIFTEENTH > step at s = 19, 20: K* = 13, C₂ = 3.5200, certificate 3.8000. (M8) holds at > fifteen steps, and K*/N reads 3.25 there, so the "rising" reading does not > continue. `REFUTED.md` row 94 stands and in fact under-claims: Lemma 1 at > s = 128 alone clears the whole legal band. *2026-08-29 night wave. Producer: `research/history/staging/attack-0829n-doubling-bridge.js` (formal embed; the tail carries the hashes). Nothing committed; no git command run; no existing corpus document edited; publication moratorium respected. HELD, awaiting the standing one-pass adversarial review. Legend as in `research/sift-limit-attack.md`: **[PROVEN]** proof given here or in a cited corpus note; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[INFERRED]** deduction from sourced facts; **[CITED]** taken from a corpus artifact, not recomputed; **[OPEN]** not proven, not refuted.* --- ## 0. The verdict, up front > **No uniform bridge is proven. The one bridge the corpus had, the > K*-product certificate `Ĝ(2s) ≤ (K*+1)·Ĝ(s)`, is dead at `C₂ = 8` on the > chain itself: at `s = 16` the exact walk gives `K* = 17`, so the > certificate reads 18 against the target 8 while the true ratio is > 5.2727, and by the cited run floor `K* ≥ π(2s) − π(s)` it reads at least > 14 at `s = 64` and at least 24 at `s = 128`, past the whole legal band. > The sharper bridge `Ĝ(2s) ≤ maxsum_{K*+1}(T_s)` (the corpus's own > `G₂(new) ≤ maxsum_{1+L}(old)` shape) is a per-step theorem, stays under 8 > at all fourteen enumerable steps with sup 6.6364 at `s = 16`, and is the > tightest proven object here: slack 1.00 to 1.33 over the truth where the > product certificate's runs to 3.27. Its all-s form is the inequality that > does not close, and it is located exactly: the longest run of consecutive > level-`s` slots the primes in `(s, 2s]` can kill, weighted by the ground > it stands on, must stay within 8 records. Nothing proven bounds that run > from above; the counting bound needs `θ < 1` and `θ → ∞` (cited); the > run's proven floor grows like `s/ln s`. The doubling inequality itself is > untouched in both directions. The telescoped chain is a Cesàro mean of > `log₂ C₂`, so no rung gains anything and the per-step inequality at every > large `s` is the whole content; it is not the closed fold recursion, whose > per-block target was exponent 2, and it lands on no refuted row.** Numbers below are quoted from the producer's bound OUTPUT block (`attack-0829n-doubling-bridge.js`, lines 464 to 652) or from cited artifacts by file and line; none is typed from a run. ## 1. (a) The statement, quantifiers fixed **(D8).** For every integer `s ≥ 2`: `Ĝ(2s) ≤ 8·Ĝ(s)`, where `Ĝ(t) = G₂(P(t)#)`, `P(t)` the largest prime `≤ t`, and `G₂(p#)` the maximal cyclic gap between consecutive twin-admissible slots modulo `p#` (`attack-doubling-01.md` §1). Chain form: for every `k ≥ 1`, `Ĝ(2^{k+1}) ≤ 8·Ĝ(2^k)`. - **Quantifier.** ALL `s` (all `k`), including `s = 16`. The eventual form `∀k ≥ k₀` is excluded on purpose: it drops the `s = 16` witness and past it the chain carries one datum, `1080/348 = 3.1034` at literature grade, so every `C₂ ∈ (3.1034, 4)` would be data-consistent and TPC-implying (`attack-wrongdirection-audit.md` §3.3). Every statement in this note is all-`s`. - **Constant.** `C₂ = 8`, inside the legal band `[4, 19.2455)`: the trap is `C₂ < 4` (`ln C₂ ∈ [1.1325, 1.3863)`), and 8 sits above the exact sup `348/66 = 5.2727` by a factor the data leave open. The exact table has every `C₂(s) ≤ 8` for `s = 2..41` (producer, A). **[VERIFIED]** - **Direction.** (D8) bounds the level-`2s` record ABOVE by the level-`s` record. That is the legal direction: an upper bound on `Ĝ` lowers the exponent ceiling and certifies no twin prime; the payout is `limsup ln Ĝ(x)/ln x ≤ log₂ 8 = 3.0000 < β₂ = 4.26645` through the Cesàro identity of §4 and (H-mono) (`fekete-1d.md` §5, cited). Nothing in this note bounds `Ĝ` below except where the bound is the adversary's own construction (§3 step 3), which runs in the opposite direction and is used only to measure slack. ## 2. (b) The finite-level mechanism, re-derived, and why it is per-level **The Bridging Lemma [PROVEN, re-derived].** A twin-admissible slot at level `2s` is an integer `r` with `r` and `r+2` coprime to `P(2s)#`; that condition contains the level-`s` one, so the level-`2s` slots are a sub-sequence of the (periodic) level-`s` slots. A level-`2s` gap `[a, b]` therefore contains `k ≥ 0` level-`s` slots, each divisible (itself or plus 2) by a prime `q ∈ (s, 2s]`, and `b − a` is the sum of the `k+1` consecutive level-`s` gaps it spans. Each of those is `≤ Ĝ(s)` by the definition of the record, so `Ĝ(2s) ≤ (k+1)·Ĝ(s) ≤ (K*(s)+1)·Ĝ(s)` with `K*(s)` the longest run of consecutive level-`s` slots all killed by the entering primes, anywhere in the level-`2s` period. This is the certificate `C₂ ≤ K*+1` (`attack-doubling-01.md` §3); the producer re-derives it at the eleven enumerable steps digit for digit against that file's own table (§5) and adds three steps. **Why it is per-level and not uniform.** `K*(s)` is defined by a full walk of the level-`2s` period, `P(s)#` times the product of the entering primes; the lemma converts one computed quantity into another and carries no statement about `sup_s K*(s)`. A uniform (D8) through it needs `K*(s) ≤ 7` for every `s`. That is false, twice over: - **On the chain, by exact data.** `K*(16) = 17` (13#→31#, walked here for the first time; producer line 536), so the certificate is 18 at the very rung that carries the sup. The record window there bridges `k = 14` killed level-13 slots and does not realize the longest run (`k = 14 < K* = 17`, line 618). **[VERIFIED, custody grade]** - **For all large `s`, by proof.** Lemma 1 of `hsubpow-explicit-K.md` §2b (three lines of CRT; red-team CONFIRMED, `redteam-0828-closures.md` row 1b) gives `K*(s) ≥ N(s) := π(2s) − π(s)` whenever `D_s ≥ N`. The producer checks `K* ≥ N` at all fourteen walked steps (`K*/N` from 1.00 to 3.40, line 547) and the hypothesis at the rungs that matter. `N ≥ 8` first at `s = 34`; on the chain `N(64) = 13`, `N(128) = 23`, and `N ≥ 8` at every chain rung from 64 to 1024 (lines 548 to 550). So the certificate exceeds 8 at `s = 64` and exceeds the entire band `[4, 19.2455)` at `s = 128`, and since `N(s) → ∞` it exceeds every constant from some `s` on. **[PROVEN, given the cited lemma; VERIFIED the rungs]** The lemma's second and larger reason for looseness is visible at the sup step: five entering primes deliver 15 strikes on 14 slots inside the record window (line 613), so a prime kills about three copies, not one, and the floor `N` explains under half of `K*` there. ## 3. (c) The bridge attempted at C₂ = 8, step by step Notation: `T_s` the level-`s` tile (the slot residues modulo `P(s)#`, `D_s = ∏_{3≤p≤s}(p−2)` of them), `ḡ(s) = P(s)#/D_s` its mean gap, `maxsum_m(T_s)` the largest sum of `m` cyclically consecutive level-`s` gaps (`a3-05-bound-L.md` §5, same definition), `N(s) = π(2s) − π(s)`. **Step 1 [PROVEN].** Level-`2s` slots are a sub-sequence of level-`s` slots (§2). **Step 2 [PROVEN, an identity].** Call a run of consecutive level-`s` slots *killed* if every slot in it is struck by an entering prime, and its *span* the distance from the surviving slot before it to the surviving slot after it. Every level-`2s` gap is the span of a killed run (possibly empty) and every killed run lies inside one level-`2s` gap, so `Ĝ(2s) = max over killed runs R of span(R)`. This is the object itself written in level-`s` words; it proves nothing and is where the bridge must start. **Step 3 [PROVEN, the sandwich].** A killed run of length `k` spans `k+1` consecutive level-`s` gaps, so `span(R) ≤ maxsum_{k+1}(T_s) ≤ maxsum_{K*+1}(T_s)` (`maxsum_m` is monotone in `m`; checked at `m ≤ 40`, producer B). And Lemma 1's construction places a killed run of length `N` on ANY `N` consecutive slots, so placed on the `maxsum_{N+1}` window it forces a level-`2s` gap of at least that span. Hence, direction by direction: maxsum_{N+1}(T_s) ≤ Ĝ(2s) ≤ maxsum_{K*+1}(T_s) ≤ (K*+1)·Ĝ(s). The left inequality is the adversary's (a lower bound on `Ĝ(2s)`; the classical CRT family of `two-class-lower-bounds.md` §1, not new); the middle one is the **maxsum certificate**, the corpus's `G₂(new) ≤ maxsum_{1+L}(old)` (`U-FRAME.md` §5a as quoted at `a3-09-histogram-operator.md` line 176; `a3-05-bound-L.md` §5 has the same shape for the single-fold `L`), here applied to the doubling block; the right one is the product certificate of §2. Verified at all fourteen steps, and the floor half at `s = 19..23` against the exact table (producer E, lines 556 to 586): | step | s | N | K* | floor | C₂ | msc | K*+1 | |---|---|---|---|---|---|---|---| | 13#→23# | 13,14 | 3 | 8 | 2.3636 | 3.0909 | 3.6364 | 9 | | 13#→29# | 15 | 4 | 10 | 2.5455 | 3.9091 | 4.5455 | 11 | | 13#→31# | 16 | 5 | 17 | 2.8182 | 5.2727 | 6.6364 | 18 | | 17#→31# | 17,18 | 4 | 13 | 1.9444 | 3.2222 | 4.2778 | 14 | (`floor = maxsum_{N+1}/Ĝ(s)`, `msc = maxsum_{K*+1}/Ĝ(s)`; the ten earlier rows are in the block, lines 562 to 571.) `msc/C₂` runs 1.00 to 1.33 across the fourteen steps and `(K*+1)/msc` 1.00 to 3.27: the product certificate loses most of its slack by replacing a sum of gaps with a multiple of the largest one. **[VERIFIED]** **Step 4 [PROVEN sufficient; OPEN as a statement].** (D8) follows from (M8) maxsum_{K*(s)+1}(T_s) ≤ 8·Ĝ(s) for every s ≥ 2. (M8) holds at all fourteen enumerable steps with sup `6.6364` at `s = 16` (line 578), and nothing is known about it past `s = 18`. It is sufficient and not necessary: the record window need not realize the longest run (`s = 16, 17`) and need not stand on the thickest ground (`ρ = 1.147` at `s = 16` against the tile mean, line 617). **[VERIFIED the range]** **Step 5, the split, and the single inequality that does not close.** Write `maxsum_{m}(T_s) = m·ḡ(s)·ρ(s, m)` with `ρ` the thick-ground factor (measured `ρ_run` at `m = K*+1`: 1.000 to 1.440 across the fourteen steps, lines 587 to 602). Then (M8) reads (R) K*(s) + 1 ≤ 8 · [Ĝ(s)/ḡ(s)] / ρ(s, K*+1) for every s. The left side is a covering quantity: the longest run the two residue classes of each prime in `(s, 2s]` can jointly cover on the slot sequence. The right side is a record quantity: how far the level-`s` record exceeds the tile's mean gap (`Ĝ/ḡ` measured 1.00 at `s = 2` to 4.71 at `s = 17`, same lines). (R) is the inequality the bridge needs and cannot supply, and each of its three pieces is open in the direction needed: - **Upper bound on `K*(s)`: OPEN.** The only a-priori handle is residue counting, which closes iff `θ = 2Ĝ(s)Σ_{q∈(s,2s]} 1/q < 1`; `θ ≥ 1.3333` at the first step and `θ → ∞` at every base (`attack-doubling-01.md` §3; `hsubpow-explicit-K.md` §3). The proven floor `K* ≥ N ~ s/ln s` runs the wrong way. `K*` is the entering primes' two-class Jacobsthal problem on the slot sequence: self-similar (doubling-01 §3), and the block campaign's combined-`L` object at the doubling block (`attack-block-00-ADJUDICATION.md`; its counting bound is the same `θ` machinery). - **Upper bound on `ρ(s, m)` at `m ≈ K*`: OPEN.** A bound on the largest sum of `m` consecutive gaps for `m ~ s/ln s` is a generalized Jacobsthal statement; the trivial one, `ρ ≤ Ĝ/ḡ`, returns the product certificate and Lemma 1's divergence. - **Lower bound on `Ĝ(s)/ḡ(s)`: order proven, constant not in hand.** `ḡ(s) ≍ ln² s` (Mertens) and `G₂(x#) ≫ x ln x` from published ingredients (`G2-STATE.md` §PROVEN, NOT refereed), so the right side of (R) grows at least like `s/ln s`, the same order as the left side's floor. A race of constants at exponent one, with the constant on the left unproven in any form. **[CITED the bound; INFERRED the comparison]** So the bridge at `C₂ = 8` reduces to (R): a run-length upper bound of the same order as the run's proven lower bound, against a record lower bound of the same order. Random-model heuristics put `K*` near `s/ln ln s` and `Ĝ(s)/ḡ(s)` near `s ln s`, which would leave (R) room of order `ln s`; that is a heuristic and the fourteen points cannot test it. **[HEURISTIC]** **A shortcut that is not one.** Bridging through the intermediate primes one fold at a time composes `N(s)` single-fold certificates, one per entering prime, and `N(s) → ∞`: a constant per-fold cap compounds to `exp(#folds)`, which is `attack-hsub-01.md` §4(i)'s point about (H-sub) and applies unchanged to (D8). Not pursued. **[CITED]** ## 4. (d) The kill check against the fold recursion and the refuted registry **Against the fold recursion `β_{k+1} = (β_k + φ_k)/2` (CLOSED, `attack-block-00-ADJUDICATION.md` attack 10).** Telescoping (D8) along the chain gives `Ĝ(2^{k+1}) = Ĝ(2)·∏_{j≤k} C₂(2^j)`, hence β(2^{k+1}) := ln Ĝ(2^{k+1}) / ln 2^{k+1} = (log₂ Ĝ(2) + Σ_{j=1..k} log₂ C₂(2^j)) / (k+1), a Cesàro mean of `log₂ C₂` along the chain (checked at `k = 6`: `β(64) = 1.6795` both ways, line 637). It is an averaging and it forgets its start, at rate `1/k` rather than the squaring ladder's `2^{−k}`. The adjudication's lesson transfers in full: the limsup of `β` is at most the limsup of `log₂ C₂`, no rung contributes anything, and the whole content is the per-step inequality at every large `s`. What does NOT transfer is the closure: attack 10 closed because the per-block exponent to prove equalled the global exponent 2, a TPC-strength requirement. Here the per-step requirement is `C₂ ≤ 8`, that is exponent `log₂ 8 = 3.0000`, legal and finite. The bridge is not that row. **[VERIFIED the identity; INFERRED the scoping]** **Against the refuted registry.** The residue-density closure and every constant-factor sharpening (`attack-doubling-01.md` §3): not re-run; used here only as the reason the `K*` side of (R) has no handle. The block ladder rows (`REFUTED.md` rows on `L ≤ 62`, the A/B-coupling depth axis, the submodular certificate): the `K*` object is the block campaign's combined `L` at a different block; none of those rows bounds it uniformly, and this note reopens none. The localized merge chain and its block composition (`REFUTED.md` rows 27 and 30): a different object (the merge chain's `M`), not touched. The `θ` ladder (row 35): a different `θ`. **Verdict: the K*-product bridge is CLOSED at every `C₂ < 19.2455` (by exact data at `s = 16` for `C₂ < 18`, by Lemma 1 at `s = 128` for the rest); the maxsum bridge is on no refuted row and its all-s form (R) is OPEN; the question's status is PARTIAL.** ## 5. (e) Chain of custody: the producer against the exact table Nothing new was trusted before the old was reproduced, in this order (producer A to C, lines 464 to 543): - The two ladders are parsed from their keepers (`exact-g2-ladder.js`, `import-interp-01-bgt-defect.js`), never retyped, and agree on the 14 shared terms. - The `C₂` table `s = 2..41` is rebuilt and compared field by field (`P(s)`, `P(2s)`, `Ĝ(s)`, `Ĝ(2s)`, exact fraction, four-decimal ratio, grade) against the table bound inside `attack-doubling-01.js`'s own OUTPUT block, parsed from that file at run time: 40 of 40 rows identical (line 469). **[VERIFIED]** - Tiles to 23# by fold recursion match the ladder in census, record, least argmax and multiplicity at every `p ≤ 23` (lines 474 to 485). - A walk engine written fresh for this pass (bitmask over copy residues, no shift operator; `qc/widths.js` guards on every wide quantity) re-derives at the eleven enumerable steps the record, least argmax, multiplicity, census, `k`, `K*` and the certificate, each compared against `attack-doubling-01.js`'s F-table and walk rows: all equal (lines 487 to 532). **[VERIFIED]** - Only then the three new steps: 13#→29#, 13#→31#, 17#→31#. They re-derive `G₂(29#) = 258 @ 1205437109 (×2)` and `G₂(31#) = 348 @ 8813641451 (×4)` against the exact ladder, the 31# row by two independent base tiles (lines 533 to 543). Wall 0.7 s, 26.3 s and 17.6 s. **[VERIFIED]** - Self-test failures: 0 (line 652). Fingerprint: `code-sha256 a12d2919`, `out-sha256 217204ab`, 190 body lines; written by `embed.js`, no `--force`. ## 6. How low can C₂ be pushed while staying ≥ 4 - **The K*-product bridge**: nothing below 18 on the chain (`s = 16`, exact), and no constant at all for all `s` (Lemma 1). Lowering `C₂` is moot. - **The maxsum bridge**: on the enumerable range it certifies exactly `6.6364` (its sup, `s = 16`, line 578) and cannot certify anything lower there, since at `s = 16` the certificate is itself 6.6364 while the truth is 5.2727. For every `C ∈ [6.6364, 8]` the all-`s` statement is (R) with 8 replaced by `C`, the same open inequality with less room; for `C ∈ [5.2727, 6.6364)` the bridge is already refuted at `s = 16` as a method, though (D8) with that `C` may still be true. Nothing here approaches 4, and nothing here is stated for any `C₂ < 4`. The number 8 was chosen for margin, and the margin bought no structural relief: (R) at 8 and (R) at 19.2455 have the same three open pieces. **[VERIFIED the range; INFERRED the rest]** ## 7. NOT REACHED, falsifiers, trap grading **NOT REACHED.** - (D8) itself: not proven, not refuted, for any `s > 41`; on the chain the next datum is `Ĝ(128) = G₂(127#)`, out of reach of any walk. - (R), equivalently (M8), past `s = 18`: `K*` at 19#→37# and beyond needs walks over `D_19 = 378675` slots times four or more entering primes, hours in this engine rather than seconds; `s = 32` (31#→61#) has `D_31 = 6226553025` base slots (line 536) times seven entering primes and is beyond any walk. - No a-priori bound on `K*(s)`, on `ρ(s, m)`, or on `maxsum_{K*+1}(T_s)` is derived. (R) is TODO item 0c's shape at the doubling block (a residue-deleted maxsum bounded without a kill count, `U-FRAME.md` §5a), which that item records as untried in this repository; this pass did not try it either. - The constant in `G₂(x#) ≫ x ln x` was not extracted and compared with `N(s)·ḡ(s)`; the comparison in §3 step 5 is at the level of orders. - Whether `K*/N` (1.00 to 3.40, rising) keeps rising: fourteen points at `P(2s) ≤ 31`. **What would falsify each claim, and whether the check has run.** | claim | falsifier | has it run | |---|---|---| | the K*-product bridge fails at `C₂ = 8` on the chain | `K*(16) ≤ 7` | yes: `K*(16) = 17` by an exact walk that also reproduces `G₂(31#)` and its argmax from two base tiles | | it fails for all large `s` | a proof that `K*` is bounded | refuted by the cited Lemma 1, checked here at 14 steps and at the chain rungs 64 to 1024 | | the sandwich `maxsum_{N+1} ≤ Ĝ(2s) ≤ maxsum_{K*+1}` | one step violating either side | yes: 14 walked steps, plus the floor side at `s = 19..23` against the exact table; none | | (M8) holds on the enumerable range | one step with `msc > 8` | yes, 14 steps, sup 6.6364 | | (R) is open | a proven upper bound on `K*(s)` of order `s/ln s` with an explicit constant, plus a bound on `ρ` | none exists in the corpus; the counting bound needs `θ < 1` and `θ → ∞` (cited) | | the bridge is not the closed fold recursion | a derivation in which the per-step target is exponent 2 | the per-step target here is exponent 3 (`log₂ 8`); the Cesàro identity is verified at `k = 6` | | the C₂ table reproduces doubling-01 | any differing field | yes, 40 of 40 rows, 8 fields | **Trap grading.** `C₂ = 8 ≥ 4` throughout; the all-`s` quantifier is kept in (D8), (M8) and (R); no constant below 4 is derived or implied; the only lower bounds on `Ĝ` used are the adversary's, in their own direction, for slack measurement. Nothing here is TPC-implying, and nothing here is a result: one route is closed by data and a cited lemma, one route is sharpened and left open at a named inequality. ## 8. Sources and reproduction | artifact | what it carries | |---|---| | `research/history/staging/attack-0829n-doubling-bridge.js` | custody (A), tiles (B), fourteen walks (C), the run floor at the rungs (D), the maxsum sandwich and ground table (E), the sup-step anatomy (F), the Cesàro identity (G) | | `research/attack-doubling-01.js`, `attack-doubling-01.md` | the C₂ table, the eleven certificates, the residue-density closure, the mechanism; reproduced here field by field | | `research/history/staging/hsubpow-explicit-K.md` §2b, `redteam-0828-closures.md` row 1b | Lemma 1, `K* ≥ π(y′) − π(y)`, proven and red-team confirmed | | `research/history/staging/attack-wrongdirection-audit.md` §3.3 | the all-`s` quantifier discipline and the trap window | | `research/history/staging/attack-block-00-ADJUDICATION.md` attack 10 | the closed averaging recursion the kill check is run against | | `research/a3-05-bound-L.md` §5, `research/U-FRAME.md` §5a, `research/a3-09-histogram-operator.md` line 176 | the maxsum object and the `G₂(new) ≤ maxsum_{1+L}(old)` shape | | `research/two-class-lower-bounds.md` §1; `research/G2-STATE.md` §PROVEN | the CRT adversary and the `x ln x` lower bound | | `research/history/staging/attack-hsub-01.md` §4(i) | folds compound to `exp(#folds)` | | `research/exact-g2-ladder.js`, `research/import-interp-01-bgt-defect.js` | the two ladders, parsed at run time | Reproduce with `node research/qc/embed.js --check --timeout 600 research/history/staging/attack-0829n-doubling-bridge.js` (about 45 s).