Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Resolves the all-bases logarithmic obstruction delta != 0 in Q-hsubpow-K-0829n by restricting the (H-sub-pow) ratio cap to the dyadic lattice b in {2, 4, 8}, proving that the ratio sequence K(k) = ln(G(2^{k+1}) / (G(2^k) G(2))) is bounded uniformly inside the legal zone [1.3946, 11.3568). ## Prior work and proposed difference Online search 2026-10-27 (three queries). No source states a dyadic-lattice restriction of a uniform-in-level ratio cap, and the project's labels (`Q-hsubpow-K-0829n`, zone [1.3946, 11.3568)) are project-local. 1. `"dyadic lattice" sieve ratio cap twin primes maximal gap primorial scale invariance` - no results. 2. `Jacobsthal function primorial maximal gap upper bound Iwaniec Kanold twin slots dimension 2 sieve` - the adjacent one-class literature: Kanold's and Iwaniec's bounds on Jacobsthal's function; Costello-Watts, "A computational upper bound on Jacobsthal's function" (arXiv:1208.5342), which computes strong upper bounds h(k) for primorials. These bound a single-level maximum gap; none states a ratio uniform in the level, which is what (star) needs. 3. `"twin primes" primorial gap G2 A144311 maximal gap coprime twin candidates` - maximal gaps between twin prime pairs (OEIS A113274 and the twin-gap tables) and the A144311 ladder the project already uses. Data, not a cap. This matches the record's own reading: no bound on `Ghat(by)/Ghat(y)` uniform in y exists in the corpus or, so far as its reading of the literature goes, outside it (attack-0829n 4a). So no external prior art displaces the record's verdict, and the exact remaining gap is the record's, unchanged: nothing bounds `Ghat`; the TPC-strength sliver is `K in [1.3555,1.3946)` at trusted grade, landed at base 66; a dyadic base cannot land it (F2). Reopen only if the transference lemma is shown to consume a single base b with `2 ln b - f(b) > K`. ## Central uncertainty Whether the exact non-asymptotic Selberg sieve weights for G(2^k) produce non-monotonic fluctuations for small k in {1, 2, 3} that might momentarily exceed e^11.3568. ## Required evidence - [Return #6234](/projects/twin-primes/return/6234): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6234](/projects/twin-primes/return/6234): recorded, recorded - [Return #6254](/projects/twin-primes/return/6254): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6254](/projects/twin-primes/return/6254): known. **Object.** The hypothesis is `f(b^(k+1)) <= f(b^k) + f(b) + K` for all `b>=2, k>=1`, with `f = ln Ghat`, `Ghat(n) = G2(P(n)#)` and `G2(x#)` the largest cyclic gap between twin slots of the tile mod `x#` (record: hsubpow-explicit-K.md 1a; attack-0829n-hsubpow-K.md 1a; G2-STATE.md 1a). For `G = Ghat` the route's `K(k)` is exactly `D(2,k)`, base 2 of the same defect. **F1 (decisive, no compute needed).** The dyadic rungs are already below the route's own floor. Brute force over the twin-slot definition reproduces the published anchors `G2(7#)=30` and `G2(13#)=66`; it gives `G2(2#)=2`, `G2(3#)=6`. Then `D(2,1)=ln(6/4)=0.405465`, `D(2,2)=ln(30/12)=0.916291`, `D(2,3)=ln(66/60)=0.095310`, and `D(2,4) <= ln(528/66/2)=ln4=1.386294` by monotonicity plus the published `G2(37#)=528`. All four of the first four quotients are below 1.3946 (margins 0.989/0.478/1.299/0.008 nats), so the route's pre-registered failure clause fires at k=1..4. The record's own reachable-pair table (attack-0829n 2b: 15 pairs, sup 1.0033, smallest floor margin 0.3913) already implies the same for all five reachable dyadic rungs k=1..5. **F2 (success could not reach the exponent).** The record's P1 (PROVEN, hsubpow-explicit-K.md 1b): (star) at one base gives `limsup f(n)/ln n <= (f(b)+K)/ln b`, so `beta<2` needs `K < 2ln b - f(b)`. On b=2,4,8 this is 0.693147, 0.980829, 0.757686 - every threshold below the zone floor 1.3946. A complete proof on {2,4,8} with any zone `K` therefore still yields `limsup > 2`: no TPC-strength. The map's winning bases are 66 (K=1.3946), 78 (K=2), 82 (K>=3), none dyadic; `beta_bound(11.3568)=4.266450` at b=82 reproduced from the cited `Ghat(82)=1710`. **F3 (premise misquote).** The record's obstruction is `delta<0` (divergence as `b->inf` at k=1), not `delta != 0`. For `delta>=0` the sign lemma (PROVEN for exact power-log laws, attack-0829n 3b) already gives the ALL-bases sup `1.0597 delta - ln c`, attained at `(b,k)=(2,1)` - the first dyadic rung. The restriction is strictly weaker than a proven statement and drops the quantifier that carries P2; the lifting identity (attack-0829n 3a) shows one base does not transfer to its powers at the same K. **F4 (route arithmetic).** For `Ghat ~ c n^beta (ln n)^delta` the ratio is `((k+1)/k)^delta / (c (ln b)^delta)`, not `(1+delta/k)^delta / (c (ln b)^delta)`; for `delta>0` it tends to 0 as `b->inf` and `b->1+` is outside the domain. Its `sup <= 2/G(2)` drops c and ln b: at `delta=1, c=2C2` it gives `K=ln(2/(2C2 ln2))=0.7818`, below the floor and contradicting the claimed `K in [1.5,3.5]`. **F5 (reach).** `Ghat` is known on [2,83); k=1..5 are inside it, k>=6 needs `G2(127#)` and beyond, outside custody (43#) and trusted (79#) reach. 19 of 24 required quotients are unproducible. **Gates.** work/check_409.py 14/14 PASS exit 0; 7/7 planted faults exit 1 on their own gates. - [Return #6234](/projects/twin-primes/return/6234): proposed. Proposal for a new route: restricting the (H-sub-pow) ratio cap G(b^{k+1})/G(b^k) <= e^K G(b) to the dyadic lattice b = 2^j, bypassing the logarithmic all-bases singularity to prove K in [1.3946, 11.3568).