D-052 — Phase 4 synthesis: strongest results, best-theorem statements, classification

Date: 2026-08-19 | Session: 008 | Tasks 4.1/4.2 Status: complete — with D-054 (session 009, task 4.3b) corrections applied: T-2 tag split (⇒ proved / ⇐ skeleton), T-3 re-tagged (upper bounds proved-conditional; lower bounds sketch + repair 4.5c), T-1 tag downgraded (R-bound tracing open, repair 4.5a), G-001/(A) label fixed, D-045 supersession of D-042 cited, criterion-3 wording now cites W-3, T-3 detector moved to criterion 4, Track-C wording narrowed to the Li route.

1. The result inventory after one year-equivalent (sessions 001–008)

Ledger: R-000 (inherited 2/3) through R-004 + the spectral program (D-040–D-051). Tracks: A closed (monitoring), B mapped to exhaustion at current technology, C complete (maximal deliverable), D pending.

2. The best theorems (final forms + classifications)

T-1 (R-004). Finite-height Li positivity. λ_n > 0 for all 1 ≤ n ≤ N₁ = (T₀−1)² = 8 999 999 999 994 000 000 000 001, T₀ = 3×10¹² (Platt–Trudgian). Tag: [PROVED-CONDITIONAL(computational RH-to-T₀; computational finite verification; R-bound constants E-021-measured, analytic tracing open — task 4.5a)]. Erratum D-054: D-039's digamma proof and D-036 §1's far-sum tracing are broken (bounds true, proofs being repaired); margins absorb any plausible constant error. Finite legs computational (ours to 3000, Misguich–Voros to 5×10⁵, 10× overlap). Maximal for the Li route (Voros: no partial infinite-range version).

T-2 (R-003 + D-041). The reduction. W(a,θ;z) = F(z)(R₀ + R_τ) exactly, F = z²ξ/ξ′, R₀ explicit; hence Suzuki's limit formula (1.12) ⟺ (★) tail cancellation; and in the Krein frame (Theorem D1) "(1.12) ⟹ the limit operator D has real spectrum", with the converse a named conditional skeleton (L3/L4; D-051: L4 collapses to (★)). Tag: factorization + (★) ⟺ (1.12): [PROVED (modulo tempered-FT bookkeeping — punctured-compact formulation of (1.12); R_τ's distributional representation; L1 write-up) — task 4.5d]. Spectral converse: [SKELETON]. Criterion-3 content: the explicit (★) form + the deficiency-convergence theorem D1 are the new reduction; the paywall is W-2 (= RH), with W-3 circularity confining the converse to conditional use (D-051). This is a new equivalence/reduction (criterion 3): it converts Cor 1.6 into an explicit remainder statement with a spectral form.

T-3 (D-049/D-050). The valley theorem. Under RH, the model-space kernel diagonal satisfies: plateau K_a(z,z) ≤ 2a (exact, proved), valley floors K_a(γ,γ) ≤ 1/m_γ (exact, proved). The LOWER bounds (floors 1 − O(1/a), sinc² profile envelope, Δ(a) = log 2a + o(1)) are supported by the written skeletons but the D-049 proofs are broken as written (D-054 F9/F10: Lemma-D 1/g² term dropped; sinc² argument factor 2) — repair task 4.5c; the measured laws (E-022 family) stand. Off-line pairs with δ > π/(2a) produce negative form directions (existence proved; the converse threshold claim is open) — so the valleys are a positivity detector. Tag: upper bounds [PROVED-CONDITIONAL(RH)]; lower bounds [SKETCH + NUMERICAL-VALIDATED]; detector [one-direction PROVED, converse open].

3. The obstruction map (final, criterion 4)

Track A: >2/3 impossible in the certificate data class (D-013 six walls; moment ladders capped, G-002; s₂/p substitution symmetry, G-001; zero-density invisible, graveyard entry (A)).
Track C (Li route): "λ_n > 0 ∀n ≥ N₀" = RH (Voros) — the route's maximal deliverable is T-1. (3.1's Weil-positivity horizon remains resolution-limited, D-017.)
Track B (★): consolidation obstruction = the linear-K equilibrium (residual → Z/(2π); D-045's K → 2a law superseding D-042's Z/6-from-1.9a fit); θ-channel dead (E-024); λ-channel machinery-blocked (G-004); inverse-spectral representation diffuse (D-047); the open content is spectral purity of the limit measure (Eichinger–Lukić–Woracek dichotomy) = (★).
Graveyard: G-001…G-004.
Repair map (4.3b, D-054): 4.5a (R-bound tracing), 4.5b (margin concavity), 4.5c (valley lower bounds + detector converse), 4.5d (L1 + form-domain + bookkeeping).

4. Success-criteria accounting (NGUYEN.md §9)

Full proof of RH: not claimed; the paywall is exactly located (W-2: real-spectrum persistence), with W-3 (factorization circularity) confining the converse direction to conditional use.
Unconditional improvement past 2/3: not achieved; proved impossible in the certificate framework (D-013).
New equivalence/reduction: ACHIEVED — T-2: Cor 1.6 as the explicit remainder statement (★) (R-003, new) + the spectral form via Theorem D1 (⇒ proved; ⇐ named skeleton).
Documented obstruction map: ACHIEVED — §3 + graveyard + repair map. T-3's off-line detector (δ_crit = π/(2a), existence direction) belongs here as a conditional detection statement. Plus T-1: a new explicit theorem (finite-height Li), conditional on accepted computation (tag per D-054 erratum; repair 4.5a/4.5b open).

5. What would change the picture (revival conditions, collected)

A λ_a lower-bound law (opens G-004 machinery).
A symbol-space (discretization-free) ground-state formulation (D-046 revival ii).
A spectral-purity theorem for the limit canonical system (opens (★)).
Prime-pair input beyond Montgomery (would reopen Track A's W1).