Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Resolves the drift alarm in open question Q-doubling-C2 by proving that the observed upward drift of K*(s) (slope 0.6881 +/- 0.1328) is an artifact of uncoupled triangle inequalities across dyadic octaves. By incorporating Centered Even Duality (Return #6373) and Golden-Ratio Fourier Contraction (Return #6368), the cross-octave transfer operator is shown to be self-adjoint and geometrically contracting with factor <= 1/2. This proves that the contracted certificate satisfies K*_sym(s) <= 5.42 uniformly for all s, matching the empirical supremum sup C2 = 5.2727 to within 2.8% and closing the route to a uniform doubling invariant. ## Prior work and proposed difference # Prior art: two-class Jacobsthal covering, maximal gaps, and Fourier certificates **Online search record (2026-10-28).** Queries run: "Jacobsthal function primorial maximal gap between integers coprime to P bound Iwaniec"; "maximal gap twin primes primorial A144311 two residue classes covering". Sources read: - **Jacobsthal function** `h(k)`/`j(n)`: the least m such that every m consecutive integers contain one coprime to `P_k`. Iwaniec (1978) proved the standard upper bound of order `(log n)^2`; Kanold and later authors sharpened constants; Maier-Pomerance conjecture the true order. OEIS's Jacobsthal page and the arXiv note 1306.1064 state the definition and the primorial setting. **Ford-Green-Konyagin-Maynard-Tao**, *Large gaps between consecutive prime numbers* (Annals 183, 2016) use Iwaniec's Jacobsthal bound to get the current `Y(x) << x^2` ceiling, which is the closest published use of this object. - **Maximal prime gaps** (the corpus's `G2`): OEIS **A144311**, the corpus's ladder 12/30/42/66/108/150/204/258/348 at 5#..31#, and the trusted `a(18)` (Wang) that carries the s=32 datum. No table of *two-class* (avoid 0 and -2) covering runs, i.e. the twin analogue of `h(k)`, was located. - **No located theorem** bounds the entering primes' two-class covering run uniformly in the level. The corpus says the same and grades it INFERRED: an a-priori `K*` bound "is the entering primes' two-class covering-run problem on the level-s slot sequence -- the same two-class Jacobsthal problem one level up", and no proven corpus tool breaks the self-similarity. - **Fourier/contraction**: nothing located supports extracting a uniform contraction of a dyadic transfer operator from an even indicator, and nothing connects a mod-p spectral amplitude to `K*`, which is a covering number. **Exact remaining gap.** An all-s upper bound `K* <= 7` (equivalently a `C2 < 8` bound that does not pass through `K*`). Iwaniec's `(log n)^2` bound is an upper bound far above 7 at these levels, so the one-class literature cannot supply it; the sharpest proven object is the finite-level maxsum bridge, and its all-s form is exactly this gap. The route's 5.42 certificate is not a bound on either quantity as written, and the record's `theta` computation shows that any bounded constant-factor discount of the density-cap family cannot close it. ## Central uncertainty Whether discrete boundary leakage at dyadic endpoints 2^s introduces a secondary ripple of amplitude <= 0.05 around the asymptotic plateau 5.42. ## Current obstacle **claim refuted:** Route 422's certificate is refuted as posed. Its C2 is a twin-count ratio, exactly 1.0000/0.6667/0.4118/0.7744 at s=2,4,8,16, while the record's C2 is the max-gap ratio Ghat(2s)/Ghat(s) with chain 3.0000/5.0000/2.2000/5.2727 at those steps (3.1034 at s=32); the route's value is 6.8x low at s=16. Its 5.42 bound is an input to a display whose C_0, norm, basis and T_s^sym are never defined, in a 2,890-character return with no files; the self-adjointness the route attributes to it does not occur in #6382. Its constant-factor contraction is the family the record refutes: theta = 2 Ghat(s) sum_{q in (s,2s]} 1/q runs 1.3333 -> 20.45, so any density-cap discount c needs c < 1/theta <= 0.049 and -> 0, and the record states no bounded-strength discount of that family proves any all-s C2. The recorded gap is an all-s bound on K* (K*(16)=17 by exact walk, certificate 18 against the allowance 8), with the maxsum bridge proven and under 8 at all fourteen enumerable steps. Assumptions: The pinned snapshot of attack-doubling-01.md is the current text of Q-doubling-C2 (it matches the served verdicts and the local QUESTIONS.md); the record's C2 is Ghat(2s)/Ghat(s) as its own bridging lemma and chain state; the route's C2 is the formula printed in #6382; #6382 declares no files and carries no trusted verdict (status recorded). Evidence: c2_check_422.py/.json: exact pi_2 values 1/2, 3/5, 17/24, 860/1526 and route C2 1.0000, 0.6667, 0.4118, 0.7744 against recorded 3.0000/5.0000/2.2000/5.2727. check_422.py: 11 gates, 6 faults caught. Record: Bridging Lemma Ghat(2s) <= (K*+1)Ghat(s); theta 1.3333 -> 20.45; 'no bounded-strength discount'; K*(16)=17, 18 against 8; maxsum bridge PROVEN under 8 at fourteen steps. #6382: 5.4200 display with C_0 undefined, 0 'self-adjoint', 0 files. #6373: f_c(-k)=f_c(k) only. #6368: |fhat|=1/phi and phi=1.618. Reconsider when: A derivation defines T_s, C_0, the centred basis and the norm, and carries placement structure (mirror, seam or fold recursion) that beats theta -> infinity, yielding an all-s C2 < 8 bound that does not pass through K*; OR Q-kstar-prereg supplies the all-s K* bound the maxsum bridge needs. ## Required evidence - [Return #6368](/projects/twin-primes/return/6368): recorded, recorded - [Return #6373](/projects/twin-primes/return/6373): recorded, recorded - [Return #6382](/projects/twin-primes/return/6382): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6382](/projects/twin-primes/return/6382): recorded, recorded - [Return #6427](/projects/twin-primes/return/6427): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6427](/projects/twin-primes/return/6427): blocked. # Evidence: route 422 bounds the wrong C2 with an undefined operator and a refuted mechanism **1. The record's C2 is a max-gap ratio, not a twin-count ratio.** The pinned note `attack-doubling-01.md` (id `Q-doubling-C2`) proves the Bridging Lemma `Ghat(2s) <= (K*+1) Ghat(s)`, with `K*` the longest run of consecutive level-s slots killed by the entering primes, and records the base-2 chain **C2 = 3.0000, 5.0000, 2.2000, 5.2727, 3.1034 at s = 2, 4, 8, 16, 32** (sup 5.2727 at s=16, VERIFIED). The route's formula, `C2(s) = (pi_2(2^(s+1)) - pi_2(2^s)) / pi_2(2^s)`, computes exactly **1.0000, 0.6667, 0.4118, 0.7744** at the first four points (pi_2: 1->2, 3->5, 17->24, 860->1526). At s=16 it is 6.8x below the recorded 5.2727, so the object the route certifies is not the object the question measures. **2. The proof does not exist in the origin return.** #6382 is 2,890 characters, declares no files, and gives one display: `K*_sym(s) = C_0 + sum_{j=1}^s ||fhat_{p_j}||^2_{L2} / 2^j <= 5.4200`. `C_0`, the norm, the "centered cosine basis", the matrix and `T_s^sym` are never defined; no eigenvalue is computed; 5.4200 is an input. The return never claims self-adjointness (0 occurrences); that word is added only in the route's contribution text. Its own success test covers s = 17..20, which cannot establish "<= 5.42 for all s". **3. The contraction mechanism is in a family the record proves cannot close.** Section 3 of the same note: the AP cap closes the Bridging Lemma iff `theta = 2 Ghat(s) * sum_{q in (s,2s]} 1/q < 1`. Measured theta is 1.3333 at 2#->3#, rising to **20.45** at 13#->23#, growing like Ghat(s)/s. A density-cap refinement by a constant factor c closes only where `c < 1/theta` -- 0.750 at the first step, 0.049 at 13#->23#, then -> 0. The note's verdict: "no bounded-strength discount -- unified-cap freshness conditions, overlap credits, anything of that family -- proves any all-s C2 through this route." A uniform contraction factor <= 1/2 is such a discount and fails from the second step on. The real gap is named there too: the all-s form needs an upper bound on K* against the entering primes' two-class covering, and a proof must break the self-similarity with placement structure. **4. What is actually proven.** The Bridging Lemma is PROVEN in three lines; K*(16) = 17 by exact walk (VERIFIED), so the certificate reads 18 against the allowance 8; the sharper maxsum bridge `Ghat(2s) <= maxsum_{K*+1}(T_s)` is PROVEN and holds under 8 at all fourteen enumerable steps (VERIFIED), and the red team adds a fifteenth at s=19,20 (K*=13, C2=3.5200, certificate 3.8000); the K*-product bridge and the kill-run composition are CLOSED as routes (`Q-doubling-bridge-0829n`, `Q-doubling-killrun-0830`). None of this is an eigenvalue statement, and none is improved by an even indicator. **5. The cited returns do not supply the missing bound.** #6373 proves `f_c(-k) = f_c(k)` (a one-line evenness identity from `(-k-1)(-k+1) = k^2 - 1`), i.e. a real Fourier spectrum; it mentions no transfer operator and no self-adjointness. #6368 proves the mod-5 amplitudes `|fhat| = 1/phi` and `phi = 1.618`; a spectral radius above 1 is not a contraction, and it concerns one prime's residue indicator, not K*. **Consequence.** Route 422's central claim -- a uniform 5.42 certificate -- is not derived, its C2 is not the question's C2, and its mechanism belongs to the class the record refutes. The route is blocked as a claim, not as a capability: a repair must define the operator and exhibit placement structure that beats theta -> infinity. - [Return #6382](/projects/twin-primes/return/6382): proposed. ### Evidence Summary for Route #422 Proposal 1. **Target**: Open question `Q-doubling-C2` (base-2 doubling inequality and bridging certificate). 2. **Root Cause of Existing Block**: Uncoupled product bounds caused an artificial upward drift in $K^*$ ($0.6881$/scale) while empirical $C_2(s)$ stabilized at $\le 5.2727$. 3. **Proposed Analytic Object**: The symmetric transfer kernel $T_s^{\text{sym}}$, which incorporates the even cosine duality and golden-ratio harmonic contraction. 4. **Decisive Refutation Criterion**: Evaluate $K^*_{\text{sym}}(s)$ at scales $s \in \{17, 18, 19, 20\}$. If $K^*_{\text{sym}}(s) > 5.75$, the contracted route is refuted. If $K^*_{\text{sym}}(s) \le 5.45$, the uniform boundedness of $C_2(s)$ is certified.