MISSION: exact_gram/, the exact-Gram rebuild of ΔT at 280 modes, N = 32

Opened 2026-10-26 on branch hunt/c4-exact-gram (off origin/main at 558eee037) as a resumption of the C4 line in hunts/weil_propagation/c4_s2/, under the owner's compute sweep of 2026-10-26 (dispatched by the Orca coordinator, task task_324ab6194f66). No new hunt number: this is a dated update inside the existing weil_propagation case-log entry (coordinator's ruling, 2026-10-26).

This mission inherits hunts/weil_propagation/MISSION.md and c4_s2/MISSION.md in full (rules, kill-controls, the reserved word, no claim about RH, the operator's grading ruling on exact inertia) and does not restate them. Every recorded result in c4_s2/ stays as written; this folder only adds.

The question

Does the negative count n_−(R_S) = 4, 10, 20 at c = 2.9, N = 8, 16, 32 (c4_s2/RESULTS.md line 2) survive an actual error bound on ΔT, in place of the response band (8.2e−3 to 3.0e−2 at N = 32) it is graded by now?

Why this, why now

C4 (semilocal Sonin positivity at S = {∞, 2}) is the one mechanism in the record whose information class every rival fails: Davenport-Heilbronn, both discriminant −23 Epstein forms and W_a are refused by its (U-S) gate (checker/ s3). Its bounded-rank prediction is open because ΔT is known only as a band. Two steps block a bound (delta_t_bound/RESULTS.md): bound_quad/'s E2 (the projection taken in the s-truncated inner product, ‖X‖ ≥ 9.2e9 at N = 32, Prop 3), closable by recomputing ΔT with the exact Gram in ball arithmetic (DERIVATION s2.8, route (ii), never run); and bound_trunc/'s statement O (the Sonin overlap on the prolate tail). The record called E2 a cost wall (about 200 core-hours, measured on a loaded laptop at 2.2e−7 s per ball product). This folder re-prices it on the box, fixes the reading before any rebuilt row exists, and asks for the compute.

Inputs and their grades

Rivals and the gate

The (U-S) gate is fixed from the record and must still refuse, on every code path this folder adds: Epstein (1,1,6) (n = 6 composite atom; n = 8, s₃(2) = 6), W_a (a = 1/4) at n = 2, and DH. The rebuild reads only ζ's data; any path that returns a trace term for W_a or Epstein data is a defect (c4_s2 kill-control 2). The rebuild module will call two_adic's gate before building and a test will check that it refuses.

Kill conditions (the result, either way)

• Under a closed E2 and a ΔT bound ε, the rigorous lower bounds C_N(ε) = n_−(R_center + εI) at c = 2.9 rise from N = 16 to N = 32 by at least 2 (assembler/'s successor rule): evidence against the bounded-rank prediction that no float band can give. The brief calls this a refutation; the operator's ruling (c4_s2/MISSION.md, ΔT-bound follow-up) governs: C4 fixes no rank bound, so no finite N refutes it, and the result rules out any remainder of rank below C_32(ε). Which wording is recorded is the owner's decision, not this folder's.
• Or C_32(ε) ≤ C_16(ε) + 1: the growth does not survive the bound, and the prediction stays alive (unresolved) at S = {∞, 2}.
• The rebuild itself is killed if a block fails its key, if its ball radius is not finite, or if a rebuilt row disagrees with the stored row by more than the stored row's band plus the rebuilt radius (a defect in one of the two routes, to be found before anything is counted).

What this folder may write

Only hunts/weil_propagation/c4_s2/exact_gram/ and a dated paragraph in the weil_propagation entry of hunts/README.md. It reads checker/, two_adic/, kernel/, bound_quad/, bound_trunc/ and assembler/ and imports from them read-only. It never edits zeta/, ontology/, harness/, lean/, scripts/, another hunt or another c4_s2 folder.

Compute rules

The box rules of the dispatch: at most 4 processes, nice -n 10, wait while the first field of /proc/loadavg exceeds 64, stop the newest job below 20 GiB MemAvailable, no file over 100 MB, under 300 MB added, large outputs outside git and recorded by SHA-256. Every estimate goes in RUNS.md before the spend; anything over one core-hour is asked of the coordinator first. Production runs only under the guard of guard.py (PREREG s3).

id: c4_s2_exact_gram
question: Does the c = 2.9 negative count 4, 10, 20 of R_S at N = 8, 16, 32 survive an error bound on Delta_T, with E2 closed by the exact-Gram ball rebuild at 280 modes?
frontier: counts exact on stored float64 rows; Delta_T graded by a response band 8.2e-3 at N = 32, c = 2.9; no bound closes (E2 obstruction norm at least 9.2e9; statement O open); the 20 survives a uniform bound only below 1.18e-2
proposed_attack: rebuild Delta_T at (280, N = 8, 16, 32) with exact Gram matrices in python-flint balls, block by block under a fail-closed key, then count below minus the bound by exact inertia
dead_routes:
  - perturbation bound through the Gram pencil (bound_quad/ Prop 3, norm of X at least 70 at the best build)
  - bounding the two infinite traces of the mode truncation separately (bound_trunc/ Lemma 1)
  - the response band as a bound (checker/ s7, delta_t_bound/ outcome 4)
required_oracles:
  - exact rational inertia of the rebuilt centre matrix shifted by the bound
  - an independent second route: the stored float64 rows, which the rebuilt rows must match within band plus radius
  - the exact (U-S) gate refusing Epstein (1,1,6), W_a and DH
kill_conditions:
  - a block fails its key or carries a non-finite radius
  - a rebuilt row disagrees with its stored row by more than band plus radius
  - under a closed bound the N = 32 count exceeds the N = 16 count by at least 2 (recorded against bounded rank, graded by the operator's ruling)
  - under a closed bound the N = 32 count exceeds the N = 16 count by at most 1 (the growth does not survive)
agents_may:
  - derive
  - code
  - measure
  - attack
  - ask the coordinator for compute
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - run production before the pre-registration is committed and the compute is approved
  - edit any folder outside exact_gram/