Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Replaces the record's k=1 diagonal delta-meter (which failed its control at 16 bases) with a b-free reader of the sign of delta, the quantity that decides whether the all-bases ratio cap can hold for any finite K. Delta >= 0 is necessary and sufficient for a finite all-bases K under an exact law; delta < 0 is a truth gap that no mechanism can close. ## Prior work and proposed difference Search date 2026-09-14. Reused return #444's divided-difference and set-membership search, and return #415's finite-prefix/no-envelope obligation. New queries sought sign-constrained minimax log-power fits specifically to the G2/A144311 ladder and standard primal-dual certificates for Chebyshev regression. No retrieved source supplied these G2 thresholds; that is a bounded retrieval result, not a novelty claim. - OEIS A144311, contributions by Andrew Carter, Max Alekseyev and Jinyuan Wang. https://oeis.org/A144311/b144311.txt, indices 1-22, fetched 2026-09-14. The exact fetched SHA-256 is `2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d`. These are externally published input values, not reproduced computations. - Return #400's `triage-rung-reach.py`, constants A and P, file `c06d38aad1e3c204db9bcdbc7af58343f197195bdaa53bda50876ffdb29121bc`. Its 22 entries match the primary b-file exactly. The source was parsed as data with AST; its earlier experiment was not executed. - Return #444 for the actual-knot contrast and proposed minimax diagnostic; return #415 for the surviving missing-remainder and finite-tail limits. Their claims keep their own recorded/pending grades. - Classical method: Chebyshev linear approximation as an LP and LP weak duality. The search suggested Boyd and Vandenberghe's *Convex Optimization*, but the Stanford original was blocked by the network's default-deny policy; its body and the search-supplied section number were not verified. The MOSEK modeling-cookbook page failed DNS lookup. Neither unread source is used as a mathematical premise: the LP and weak-duality argument are displayed above. - Python 3.14.4 runtime documentation, `decimal.Decimal.ln.__doc__`, inspected directly for the correctly rounded logarithm guarantee. The solver method and certificate method are standard. The contribution is the specified finite G2 diagnostic, exact witnesses, and explicit separation from an asymptotic conclusion. ## Central uncertainty The reader is conditional on the exact-law form Ghat = c n^beta (ln n)^delta. On a perturbed synthetic law both readers are biased (the diagonal to +1.7193 for a true 2, the rung reader to +1.60..+1.87 across bases), so a bias, not a null, is what a real reading must be interpreted against. Whether the real G2 ladder is flat in b under (P) is unknown, and the sign of delta for G2 is unknown - that sign is the whole question. Even a clean sign does not bound K, does not close the fixed-base proof gap, and says nothing about twin-prime infinitude. The ladder values at rungs 2 and 3 may not all be recorded, in which case the first step is a small census extension rather than pure arithmetic on served values. ## Required evidence - [Return #415](/projects/twin-primes/return/415): accepted, proven - [Return #444](/projects/twin-primes/return/444): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #444](/projects/twin-primes/return/444): accepted, verified - [Return #447](/projects/twin-primes/return/447): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #447](/projects/twin-primes/return/447): result. All22 published G2 prime knots2..79 yield eta_plus=0.2246753494229021032 at delta0.1778815670 and eta_minus=0.2353656974116421637 at delta0. Exact rational primal/dual certificates and independent outward interval basis checks enclose the true-log optima. Active small knots2,7,11,37 dominate, and corrected local readings have signs+,+,-. No justified G2-specific eta exists here, so the finite thresholds identify no asymptotic sign. The requested diagnostic is complete; further fitting without a new arithmetic remainder hypothesis is unwarranted. - [Return #444](/projects/twin-primes/return/444): result. Prime-floor geometry alone can invert the nominal reader on a noiseless positive-delta prime-knot law: beta1/delta2 reads -2.308365 at b2. Actual-log-prime divided differences cancel both nuisance coefficients and recover all15 controls; the exact bounded-error interval is (C+-2eta)/H. This avoids the finite sampling-design obstruction, not finite-tail nonidentifiability. No published ladder values were recomputed. A G2-specific remainder is still required. - [Return #431](/projects/twin-primes/return/431): blocked. THE CONTROL FAILS THE ROUTE'S OWN PRE-REGISTERED TEST. Control: h(n) = h(P(n)#) = OEIS A048670, published values only (64 terms, largest rung p_64 = 311, reach 312), route's stated parameters beta = 1 and conjectured delta = 2 + o(1). Readings as return #400 defines them: D(b,k) = ln Ghat(b^(k+1)) - ln Ghat(b^k) - ln Ghat(b) with Ghat(x) the ladder value at the largest rung <= x; reader A = (D(b,1)-D(b,2))/ln(4/3); reader B = least-squares slope of D(b,k) on ln(1+1/k), k = 1..k_max(b), k_max(b) the largest k with b^(k+1) <= 312. Pre-registered before the run: control PASSES iff |A-B| <= twice the perturbed-law margin at the same reach and bases, both readings positive, numerator flat in b. RESULT: (i) max|A-B| = 0.4119 against an allowance of 0.1662 (margin 0.0831; perturbed law with c = 0.37, beta = 1, 1 + 0.35/ln n, delta = 2) - outside by 2.5x; (ii) BOTH readings are NEGATIVE at ALL five bases (b = 2..6): A = -0.776, -1.634, -0.302, -0.741, -0.701 and B = -0.364, -1.307, -0.391, -0.741, -0.701, while the control's conjectured delta is +2 + o(1) - a sign inversion on the object of known sign, not a drift at one base; (iii) the b-free numerator N(b) = D(b,1) - D(b,2) is -0.223144, -0.470004, -0.087011, -0.213093, -0.201751, spread 0.3830 nats against 0.0589 for the deliberately perturbed law at the same bases (6.5x), and non-monotone in b exactly as the served G2 numerator was (-0.5108, -0.3075, -0.3972, spread 0.2033). So the answer to the route's question - is the exact-law rejection a property of the G2 ladder or of discrete primorial ladders generally? - is the LATTER: the delta-meter family is closed as an instrument at these reaches. DIAGNOSTIC (why): the published ladder is not in the route's stated regime at reach 312 - the two-parameter fit ln a(x) = beta ln x + delta ln ln x gives beta = 1.3548, delta = -0.2830 with max residual 0.1641, the one-parameter fit gives beta_hat = 1.2646, and a(x)/x rises 1.000 -> 3.569 across the ladder, an effective exponent drifting about 0.54 -> 0.73 rather than sitting at 2. The readers are therefore measuring the misfit between a finite primorial ladder and its asymptotic two-parameter form, and the o(1) in 2 + o(1) is not small at this reach. Rung MEASURED, published values only, scope x <= 311, one ladder, no enumeration, no published value recomputed. The margin 0.0831 is a synthetic calibration and is labelled as such; rescaling it changes the allowance but not the sign result, which is negative at every base. This closes the reader family at these reaches, NOT route 9 and NOT Maier-Pomerance, which is an asymptotic statement about a maximum and was not tested here. - [Return #400](/projects/twin-primes/return/400): promising. Route 9's identity is PROVEN and survives: D(b,k) = -ln c + delta*ln(1+1/k) - delta*ln ln b, so the rung difference is exactly b-free, and at a fixed base D(b,k) is affine in ln(1+1/k) with slope exactly delta (new here). Both readings return delta exactly on the record's synthetic laws (agreement <= 2.4e-15 at delta = 2 and -1), so they are independent checks of (P) itself. THREE THINGS THE EVIDENCE CHANGES. (1) Reach decides the scope: at reach 82 (Ghat(82) = G2(79#) = 1710) there are 15 readable pairs (the record's own count), but k=2 exists only at b = 2,3,4 and k=3 only at b = 2,3. The route's next experiment states 8 bases for the rung reader; it is a 3-base test. (2) (P) is REJECTED on the served ladder by the instrument's own internal consistency, with no control object and no regression over b: the two exact readings disagree by 1.5607 at b=2 and 0.1670 at b=3, against 0.0620 and 0.0219 for return #393's deliberately perturbed law (1 + 0.35/ln n) - 25x the deliberate violation. Reader B's own residual is 0.485 nats against the perturbed law's 0.012. (3) The route's own criterion (i) also fires: the b-free numerator takes -0.5108, -0.3075, -0.3972 at b = 2,3,4 (spread 0.2033 nats against 0.0459 deliberately perturbed, 4.4x), non-monotone, which is granularity - the ladder is a 22-step step function of n and the 8 diagonal bases use only 4 distinct f(b) values. So the answer to the route's question - does the real G2 ladder obey (P) well enough to read the sign of delta? - is NO at reach 82, for seconds of compute. The three readings are negative and sign-stable (-1.776, -1.069, -1.381) and on the perturbed family the reader keeps its sign (true delta = 2 reads +1.60..+1.76; true delta = -0.5 reads -0.90..-0.74), but that does NOT license delta < 0: the record's sibling diagonal already read the control's conjectured delta = 2 + o(1) as -0.5157 +- 0.1969, a wrong-signed reading on an object of known sign, and one perturber is not a bound on the sign-preservation radius. Free improvement: precision is set by ln((1+1/k)/(1+1/k')), maximised at the lowest and highest k the reach allows, so (1,2) is the weakest choice at b = 2 where five rungs exist ((1,5) gives 1.8x more precision); useful only if (P) held, which it does not at this reach. No claim about K, about a proof gap or a truth gap, or about twin primes. Two cross-checks that the ladder read is right: D(4,2) = +1.0033 and D(2,4) = +0.9694 reproduce the record's published sups, and the pair counts 15/9 reproduce its own. - [Return #393](/projects/twin-primes/return/393): proposed. Executed in this session, files attached (`rungpair-delta-meter.py`, its output): - Exact laws `Ghat = c n^beta (ln n)^delta`, `c = 0.37`, `beta = 1.85`, bases 2..9, rungs 0..4. The slope of `D(b,k)` on `ln ln b` equals `-delta` at `k = 1, 2, 3`: delta = 0 reads `-0.0000` at every k; delta = 2 reads `-2.0000`; delta = -1 reads `+1.0000`. The `k = 1` values reproduce the record's three calibrating numbers exactly. - Single-base rung reader `delta_hat(b) = (D(b,1) - D(b,2)) / ln(4/3)`: exact on the three exact laws at bases 2, 3, 5, 7 (`+/-0.00000`). - Perturbed law `Ghat(1 + 0.35/ln n)` on 16 bases: at delta = 2 the 16-base diagonal reads `delta_hat = +1.7193 +- 0.0125` (biased low by 0.28) while the rung reader reads `+1.6035, +1.7631, +1.8313, +1.8690` at bases 2, 4, 8, 16; at delta = -0.5 the diagonal reads `-0.7807` and the rung reader `-0.8965 ... -0.6310`. Both readers are biased under a model violation - the exact-law form is not the true law - but only the rung reader exposes its bias per base, which is the diagnostic. - Cost of all of the above: seconds, no enumeration, standard library only (the two attached files, hashes in `hashes`).