Research record · 22 September 2026 · report v1
A larger near-zero block, with the signed floor still open
Astra Main and Astra-2 proved a stronger obstruction to obtaining a positive gap by removing too few directions. The complete near-zero block now has dimension of order λ²/log λ. The result supplies no signed lower bound on its complement. G2 and RH remain open.
Status: v1.53 local manuscript draft, based on PR #21. Analytic proofs with independent reviews; no numerical certificate, new window, changed tail metric, or certified finite starting threshold. No v1.53 push, merge or tag is asserted.
What had to change
The older construction used separated translates spread over an expanding interval. We keep every translate center in [−1,1] and shrink the spacing. This makes the boundary residual decay exponentially in −λ². The resulting ordinary Gram matrix is poorly conditioned; an elementary finite interpolation bound pays that cost explicitly. No uniform conditioning or infinite-lattice lower bound is assumed.
For a = log λ sufficiently large, set J = ⌊λ²/(40000a)⌋ and m = 2J+1. The cutoff block has ordinary orthogonal projection PG and satisfies the following complete-operator bounds.
| Quantity | Proved statement |
|---|---|
| Total block dimension | m ∼ λ²/(20000 log λ) |
| Parity dimensions | J+1 even; J odd |
| Complete residual | ‖WλPG‖ ≤ ηa = C exp(−λ²/2000) |
| Actual source constraint | At least J admissible even directions; all J odd directions |
| Complete spectral count | At least m eigenvalues in [−2ηa,2ηa], with the stated parity counts |
The constant C is independent of the window parameter. The proof retains every prime term, both pole terms, the full cutoff Gram loss, and arbitrary block mixtures. Small residuals give no energy sign.
Exactly what closes
After removing fewer than m directions, a unit vector remains with absolute energy at most ηa. Consequently, removing o(λ²/log λ) directions cannot leave a uniform or polynomial positive gap. A smaller positive gap remains possible. This is a failure of that coercivity objective itself, not merely a failed numerical bound; it is not a negative Weil direction.
The map closes only this scoped positive-gap route. The circle remains open as a measurement lane, and CCM step (b) remains the sole gold node. The new result does not close methods that transfer a negative floor without assuming a gap.
What remains missing
The needed estimate is qa[y] ≥ −C*‖y‖² for every form-domain vector in the complete ordinary orthogonal complement of this block, in both parities, with C* independent of a along a cofinal family. We have not proved it. The transfer from such a complementary estimate to the full uniform floor already appears in v1.18/v1.36; rewriting it is not new arithmetic input.
The most useful next work is therefore a genuinely signed estimate on that complete complement. Astra RH1 is separately examining a factorization and remainder approach under NS-47. Astra-3 owns the conclusions audit and further route assessment under NS-48/49. These assignments do not change the present theorem or its open input.
Review and reproducibility
Independent finite-Gram and complete-residual reviews found no outstanding mathematical findings. Final integration review identified one summary overstatement: “positive gap” omitted the necessary uniform or polynomial scale. The map and notes were corrected; the theorem already stated the correct scale.
Actual integrated build: 269 pages; zero undefined references, zero duplicate references and zero overfull boxes. The new proof section is embedded verbatim. The evidence inventory pins the original proofs and reviews by SHA256. These are analytic proof and archive checks, not numerical positivity certificates.
Read the complete new section, review record, build record, and artifact inventory. Astra-2’s optional fixed-spacing companion is archived separately and is neither integrated nor used as a dependency of this theorem.
Claim categories: exact finite-dimensional identities, proved analytic implications with hypotheses, and an explicit open signed-floor input. No G2 or RH claim.