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Updated 08-17-2026

D-009 — Bandwidth-one ceiling ≈ 0.682: re-derivation + non-tightness inventory

Date: 2026-08-17 | Session: 001 | Task: 1.1 | Status: complete Sources: paper §7.2 (D-006); zeta-23-lean PairCeiling/ (Defs, Ceiling, Stability, Bridge, NearCUE, RowCert, LawN256, CeilingLaw256 — read directly).

1. The ceiling theorem, re-derived from the Lean source

Setup (Defs.lean). A test configuration is encoded by atom masses s_j at grid points j/N (j = 1..N; s_j = S(j)/N where S is the grid form factor). The reference law ("GUE datum") is the one with cumulative x²/2 (density x) — i.e. form factor S(x) = x on [0,1]. Define: C(x) = Σ_{j/N ≤ x} s_j (empirical cumulative) D(x) = C(x) − x²/2 (discrepancy vs GUE datum) E(x) = ∫₀ˣ D (integrated discrepancy)

Certificate language. A bandwidth-one certificate is a pair (c₀, r) — r a window of Fourier support in [−1,1] — whose VALUE against the true zero ensemble is v = c₀ + ∫₀¹ r(k) k dk (k dk = the sine-kernel/GUE form-factor datum on [0,1]). Validity requires: configuration by configuration, v(config) ≤ (simple-point fraction of config).

Ceiling theorem (ceiling_of_valid_at, Ceiling.lean). If (c₀, r) satisfies the certificate inequality at ONE configuration (masses s_j, simple fraction p₁): c₀ + Σ_j s_j r(j/N) ≤ p₁, then its value obeys v ≤ p₁ + |r(1)|·|D(1)| + |r′(1)|·|E(1)| + (sup|E|)·∫₀¹|r″|. Proof structure (Stability.lean): integration by parts twice against the discrepancy: ∫ r d(C − x²/2) = boundary terms |r(1)D(1)|, |r′(1)E(1)| + bulk term bounded by sup|E|·∫|r″|. [VERIFIED by reading the kernel-checked Lean statement; the analytic content is two integration-by-parts.]

The extremal law (LawN256.lean). The adversary is the optimal law of an exact-rational LINEAR PROGRAM over 256-periodic marked configurations: weights w_c ≥ 0 (Σ=1), rational positions x_{c,i} ∈ [0,256), marks m_{c,i} ∈ {1,2} (simple/double only), Σ_i m_{c,i} = 256. Its grid form factor S(j) = (1/256)Σ_c w_c |Σ_i m_{c,i} e^{2πi j x_{c,i}/256}|² satisfies the row certificate |256·S(j) − j| ≤ 3×10⁻⁴⁰ (0 < j < 256) [near-CUE rows], |D(1)| ≤ 0.82395317 [edge bound], and its simple-point fraction is the exact rational p₀ = 1 − a_N = 0.68182868746…

Assembly (paper §7.2). Any bandwidth-one certificate p must obey p ≤ N^s_0/N on EVERY configuration, in particular this one: v ≤ p₀ + stability errors. For band-limited r (r(±1) = 0 kills the |r(1)|·|D(1)| edge term) with |r′(1)| + ∫|r″| < 8.2: v ≤ 0.6818287 + 2.55×10⁻⁶·(|r′(1)| + ∫|r″|) < 0.6819. Both windows (2.6) qualify. ⟹ no bandwidth-one certificate exceeds ≈ 0.682. Theorem A(i) at 2/3 is within 0.016 of this ceiling.

Audit caveat (from D-006): the enclosures EnclOK (that the true S lies in the stated 70-digit intervals) are interval-arithmetic certified, NOT kernel-checked; everything downstream is kernel-checked (LawN256_check by decide). The LP certificate file (cert_N256_blk_b128m.json, sha256 cc3de…) is "available from the authors", not in the repo.

2. What the ceiling MEANS (the information-theoretic wall)

The extremal law's two-trace-moment data (bandwidth ≤ 1) matches the sine process to 10⁻⁴⁰ on every row — yet only 68.18% of its points are simple. So: two moments at bandwidth one cannot distinguish "all zeros simple on the line" from "68.18% simple". No cleverness within this data class can pass 0.682. This is not a weakness of one inequality; it is a fundamental identifiability limit of the data set.

3. NON-TIGHTNESS INVENTORY (task 1.1 second half)

Level 1 — inside Lemma 3.2 (the chain's losses; from D-005)

L1. Dropped term 2tr(PQ⁺) ≥ 0 — tight only if P ⊥ Q⁺. L2. Von Neumann alignment tr(PQ⁻) ≤ Σpᵢnᵢ — tight only under simultaneous ordered diagonalizability. L3. x² ≥ 2x−1 — tight only at pᵢ−nᵢ = 1. L4. x² ≥ 4x−4 — tight only at qⱼ = 2 (flat charge). All four tight simultaneously ⟺ extremal {1}_P/{2}_Q pattern. The REAL zero configuration is never exactly extremal — but the method cannot perceive the deviation with 2 moments.

Level 2 — inside the ceiling bound itself

C1. Edge term |r(1)|·|D(1)| with |D(1)| ≤ 0.82395317: the near-CUE rows are 10⁻⁴⁰-tight but the EDGE leaks 0.82 — the extremal law concentrates its deviation at the boundary. For band-limited r this term is zeroed (r(±1)=0), so it doesn't help the attacker — but it flags WHERE the adversarial law hides its non-GUE-ness. C2. sup|E|·∫|r″|: triangle-inequality loss; sign information of D/E discarded. Lossy for oscillating r, mild for smooth windows. C3. The LP restriction m_{c,i} ∈ {1,2}: the adversary may use only simple/double points. Allowing marks ≥ 3 could push the ceiling LOWER (stronger adversary) — i.e. 0.682 is not proven optimal among adversaries, only optimal for the proved class. (Worth a numeric LP experiment: N larger, marks ≤ 3 — E-00X candidate.) C4. Grid N = 256 finite: N→∞ refinements might lower p₀ slightly. (0.6818287 is the 256-grid optimum; the paper quotes ceiling ≈ 0.682.)

Level 3 — structural (the real gaps)

S1. The certificate uses 2 moments; zeta's true law has ALL sine-process moments. CMS/Christoffel upgrade path (§7.2d) needs tr G̃ᵏ, k ≥ 3 — walled by Rudnick–Sarnak Xᵏ ≤ T^{2−ε} (D-004, D-006). S2. Bandwidth 1 ⟺ X ≤ T: off-diagonal prime pairs bounded away by MV, never evaluated (D-004 §5 O₁). A support-(1+δ) form-factor fragment enters exactly there. S3. The method is insensitive to o(N) off-line zeros (D–H/Epstein gate). Any RH-proof needs input that IS sensitive — a different data class entirely.

4. Escape-route checks (adversarial review of obvious ideas)

Idea A: zero-density estimates. Unconditional N(σ,T) ≪ T^{A(1−σ)+ε} for σ > ½ implies all but o(N) zeros lie within any fixed δ of the line. Does this constrain the extremal law? NO: the extremal adversary uses off-line pairs of depth → 0 (spectrally = doubles), which zero-density cannot see. [RULED OUT — logged; do not retest without new structure.]

Idea B: larger grid / marks ≤ 3 LP. Only lowers the ceiling (stronger adversary). Does not help exceed it. [RULED OUT as an attack; useful as sharpening — candidate numerics E-00X, low priority.]

Idea C: Levinson-style multiplicities input (thread T5). The certificate's worst case is on-line doubles (cost 2, return 0 rank). Classical bounds on multiple-zero density (via ξ′, ξ″...) are unconditional and orthogonal to the 2-moment data — NOT covered by the ceiling theorem (which constrains only certificates of the 2-moment class!). ⟹ candidate: feed an unconditional multiple-zero upper bound into the chain as an extra constraint on s₂. Next: quantify what bound on s₂/N would move 2/3 → seen in the chain: N^s_0 ≥ 4tr G̃ − 2N − ‖G̃‖² used tr P₁ + 2(s₂+p) ≤ N; an a priori s₂/N ≤ κ strengthens to ... (computation deferred to task 1.3 sub-attack — OPEN, promising).

Idea D (the two sanctioned escapes, per D-006): support > 1 form-factor fragment (task 1.4) or unconditional 3rd moment (task 1.3). Both remain the primary tracks.

5. Domain sweep (§4A.5) — fields shaped like this wall

6. OUTPUTS of task 1.1

  1. Ceiling re-derived from kernel-checked Lean + paper. [VERIFIED]
  2. Non-tightness inventory: 4 chain losses (L1–L4), 4 ceiling losses (C1–C4), 3 structural gaps (S1–S3).
  3. Two ideas ruled out WITH REASONS (A: zero-density; B: bigger LP).
  4. One new candidate attack opened (Idea C: multiple-zero bounds as extra chain constraint — cross-track, not ceiling-constrained).
  5. DECISION-GATE INPUT (task 1.5): beating 2/3 inside bandwidth one has ≤ 0.009 room (0.6725→0.682); real escape = S1/S2 or Idea C.