RESULTS: frontier math after the 10 August 2026 paper

Status: CLEAN KILL. The unconditional candidates 0.6725124, 0.672529, and 0.6725318 are withdrawn. Gate 0 failed: the zero-side summand uses transpose, not conjugate transpose, so the asserted on/off cross-block sign is false. The measure-level collapse and quantified walls below are unaffected.

Everything reproduces from the repo root:

.venv/bin/python hunts/frontier_math/configuration_lp.py .venv/bin/python hunts/frontier_math/cg_transplant.py .venv/bin/python -c "import sys; sys.path.insert(0,'hunts/frontier_math'); \ import blockpos; blockpos.__name__; exec(open('hunts/frontier_math/blockpos.py').read())"

Sources: the paper (URL in ../wide_search/HANDOFF.md; section/line cites follow its pdftotext rendering), Cheer–Goldston PAMS 118 (1993) 365–372, Chirre–Gonçalves–de Laat Adv. Math. 361 (2020) (arXiv:1810.08843), Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh (arXiv:2306.04799).

0. Closure: the candidate is withdrawn

The exact failure, pinned dependencies, smallest Gaussian-integer obstruction, formal obstruction, and reproduction commands are in CLEAN-KILL-REPORT.md. The first false statement was tr(P1 Q') >= 0. The old blockpos.py constructed u u*; upstream uses u u^T. For the one-dimensional vectors 1, i, and -i, the correct interaction is -2. A five-on-line-point witness makes the proposed additive inequality demand 9 >= 13.

The follow-up audit is closed in INTERACTION-CONTROL-REPORT.md. The paper's existing rank, trace, positive-index, and prime-side moment inputs do not bound the negative on/off interaction in the required direction. An exact family forces every universal recovery coefficient to zero, and a direct-sum version simultaneously matches tr(A)=N and approaches the paper's printed Frobenius ratio while leaving less slack than the full old gap floor. The missing invariant is a signed joint on/off incidence law, not a richer unmarked gap distribution.

1. THREAD 1 answered: the measure-level LP collapses

wide_search left one open route to the interval (0.6725007, 0.68185): a joint formulation keeping cross-window information. Formulated here (configuration_lp.py): minimise the density of simple on-line points over (multiplicity types p_m, off-line pair density q, off-diagonal pair measure ρ ≥ 0) subject to the bandwidth-one data R̂₂(α) = δ(α)+|α| on [−1,1].

• The type structure reduces out exactly: eliminating (p_m, q) against the density constraint gives p₁ = 2 − D + Σ_{m≥3}(m²−2m)p_m (the m ≥ 3 and off-line types enter with nonnegative coefficients m²−2m and 0). So the LP value is the dual of the Montgomery–Taylor extremal problem, and integrality devices beyond (m−1)(m−2) buy nothing at the measure level.
• Measured: the (X, J, ε) ladder descends monotonically 0.6794 → 0.6776 → 0.6765 → 0.6756 → 0.6750823 (X = 40, 80, 160, 320, 640 with J = 5X and ε = 0.4/X at the truncation floor ~1/(π²X); the last rung costs ~78 min), consistent with convergence to the paper's 0.6725007 from above and with nothing in between, the residual 2.6e-3 at X = 640 tracks the ε-band rather than any gap, since D climbs 1.3206 → 1.324918 toward the MT constant 1.3274993. Two independent corroborations: the paper scopes Theorem D's optimality to "the values of F on [−1,1] only" (§1.2), and Cheer–Goldston's closing remark records 1993 numerics that the pure-frequency problem attains the MT constant.
• Consequence: the ceiling gap (0.6725007, 0.68185) is not about the pair measure at all, it measures what configuration realizability (ordered real sequences) adds beyond measure positivity. CG's 1993 improvement (§3 below) is the constructive half of that statement.

2. The λ > 1 sieve wall, quantified

The paper's §4 machinery is support-agnostic; only the prime-side second moment caps λ at 1 (its §7.5(a)). The tempting unconditional route, bound the off-diagonal prime sums by the Selberg-sieve upper bound Σ_{n≤N} Λ(n)Λ(n+h) ≤ C·𝔖(h)·N with classical C, fails structurally: for X = T^λ, λ > 1, the off-diagonal and expected-value terms are each of scale x/T · N = T^{λ−1}·N and cancel to O(N) only under Hardy–Littlewood with error; a sieve constant C multiplies the x-scale term, so the loss is (C−1)·T^{λ−1}·N/log T, unbounded relative to N for any fixed C > 1. Only C = 1 + o(1), HL itself, closes it. This makes Remark 1.1's wall ("0.70 needs support ≈ 1.04") mechanism-explicit: no constant-factor upper bound on prime pair correlations, however sharp, opens the band.

3. WITHDRAWN: the Cheer–Goldston transplant

Withdrawn claim shape: the paper's Theorem D constant for simple zeros on the critical line was claimed to improve unconditionally,

N₀ˢ(T,2T) ≥ (0.6725124 + o(1)) N(T,2T) [paper: 0.6725007…]

Historical progression, now withdrawn: §5's gap-distribution LP raised the same arithmetic floor to 0.672529, which was the hunt's headline candidate before Gate 0 failed. The 0.6725124 below is the five-bucket version, kept because it is calibrated directly against Cheer–Goldston's printed 1993 constants. Neither value now has a zeta implication.

by transplanting Cheer–Goldston's 1993 gap-rigidity floor into the paper's Frobenius counting. The floor: consecutive gaps of on-line zeros cannot all sit at zeros λ_k of the MT kernel, because abutting near-λ₁ gaps force next-to-consecutive gaps near 2λ₁, and λ₂ = 2.03007 ≠ 2λ₁ = 2.11455.

The failed step. The old instrument replaced the upstream transpose summand u u^T by the Hermitian summand u u*. That changed correct on/off blocks 2 Re(B(γ,z)^2), which can be negative, into 2|B(γ,z)|^2. The following advertised chain therefore breaks at its middle line:

‖Â‖²_F = tr P₁² + 2 tr(P₁Q′) + tr Q′² tr P₁² = s₁ + cross₁₁ (exact: simple on-line, B(γ,γ)→1) tr(P₁Q′) ≥ 0 (blockwise nonnegativity) tr Q′² ≥ 4 tr Q′ − 4(s₂+p) (per-eigenvalue (x−2)² ≥ 0; n₊(Q′) ≤ s₂+p is the paper's Prop 4.1)

The middle line is false. Hence the upstream inequality does not gain +cross11, and the computed ordered-gap floors below do not imply a zeta-zero bound.

Computed (cg_transplant.py):

The floor is 22× smaller than CG's because the unconditional census is thinner (ν_on = 0.6725 vs 0.83625: the adversary has fewer gaps to place, mean gap 1.49 vs 1.20, so the length constraint pinches less, but all-gaps-beyond-d is still infeasible: 1.99782 × 0.6725 = 1.336 > 1).

Taper, truncation, census, bootstrap, and exact-LP work cannot repair the failed algebraic implication. They are closed as moot for this candidate.

4. The CGdL transplant reduces to one named obstruction

Chirre–Gonçalves–de Laat's RH-conditional 0.6792 uses two zero-side facts: (i) F(α) ≥ 0 for all α, (ii) the g ≥ 0 diagonal-isolation drop. For (i), no RH is needed: BGSTB 2023 (arXiv:2306.04799, Theorem 1) show the weighted form factor is nonnegative unconditionally, the conjugate-closure of the zero multiset makes it an integral of |Σ_ρ x^ρ/(1−(ρ−(½+it))²)|²; the Cauchy weight's strip (poles at ±2i) covers every pair of strip zeros (total imaginary displacement < 1 < 2). We re-derived this before finding it; it is known, and recorded here so the next session does not re-derive it either. What does not transfer is (ii): for the paper's machinery the analogue would be running its inertia counting against a kernel with ĝ ≤ 0 outside [−1,1], which is not the Gram matrix of any window family (autocorrelations are ≥ 0), so its §4 does not apply as written. That is the single obstruction. The prize if it falls, measured by the LP with the out-of-band constraint added (configuration_lp.py, BGSTB positivity as data): the class value at (X=80, J=320) is 0.6863 and still descending with X, consistent with landing near CGdL's 0.6792 for ζ unconditionally.

5. WITHDRAWN AS A ZETA INPUT: the gap-distribution LP

gap_lp.py replaces CG's five hand-tuned buckets with the full projection of the configuration LP onto gap statistics: a fine-binned distribution of consecutive distinct on-line gap lengths, chain levels 2 and 3 (pairs j apart in the ordering are disjoint classes, so per-class floors add with no double counting), a conservative total-internal-length constraint with boundary slack, and the census bootstrap iterated to its fixed point (it converges in two rounds).

A control earned its keep here, twice. The first implementation assigned bins to partition cells by bin midpoint; a bin straddling a cell edge was credited wholesale to one cell, so the chain count n_I could exceed reality and the floor came out invalidly high, including a conditional value of 0.6728294 that would have "beaten" Cheer–Goldston. The bin-width ladder caught it (the floor fell under refinement; a real quantity rises toward a tight relaxation). After snapping cell edges onto the bin grid the ladder is monotone as it must be (0.69 → 1.02 → 1.44 → 1.47 ×1e-5 for h = 0.02 … 0.0025), and the inflated claim is withdrawn. Recorded because the defect is the instructive part: a chain-count credit is a claim about which cell a gap is in, and any discretisation that answers optimistically manufactures floor.

Historical discovery numbers (fixed snapped edges and monotone in the reported refinement ladder; sampled kernel minima are not exact continuum lower objects):

These values are ordered-real-configuration optimization outputs only. They do not yield an unconditional zeta bound. The former unconditional candidate 0.672529 is withdrawn. The grid-locked edge optimisation had completed for the unconditional census: at edges (1.035, 1.085, 1.900) the floor is monotone under fixed-cell refinement: 1.4371e-5 (h = 0.005), 1.4710e-5 (0.0025), 1.4988e-5 (0.00125), so the trend value is the arithmetic trend was 0.6725307, but no bound survives. The matching conditional search was interrupted mid-run (HANDOFF has the resume note). The conditional lane's verdict is honest: the generalised LP has not so far beaten the 1993 constant once the discretisation is done correctly: CG's hand-tuned edges were good, and the inflation that briefly suggested otherwise was an artifact. The floor's extreme sensitivity to the cell edge nearest λ₂ (a shift of 0.005 moves the level-2 kernel minimum by a factor ~1.4) is itself a finding: the binding structure is the distance from 2·(cell edge) to λ₂, which is where a sharper argument or a better kernel should focus.

6. Lanes deliberately left

• Conditional CG improvement: our coarse edge grid reaches 0.6727450, below CG's hand-tuned 0.6727534, their 1993 optimisation stands; a joint kernel-and-buckets optimisation (not just edges) is the open lane for "the best result attainable from Montgomery's theorem", which CG pose explicitly and which remains open.
• λ_k arithmetic structure: the whole floor exists because {λ_k} is not an arithmetic progression; nobody has characterised the best kernel for the floor rather than for the main term.

Controls ledger