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 The unconditional candidates 0.6725124, 0.672529, and 0.6725318 are withdrawn.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 Closure: the candidate is withdrawn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.

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