Architecture v6.0 — Final¶

Date: 2026-03-15 Kill count: 10 Status: CLOSED. Theorem A complete. RH bridge broken. Manuscript ready.

Kill Log¶

What Survived: The Complete Theorem A¶

Theorem A1: Existence and Uniqueness of the Adelic GL Vacuum¶

Statement. The renormalized GL functional S_ren[W] on the constrained sector {W ∈ v + H¹(C_Q)} has a unique global minimizer W* = v (constant vacuum), modulo the global U(1) gauge.

Proof components (all verified):

Theorem A2: L∞ Regularity (Lemma M) — via Stampacchia Truncation¶

Statement. The global minimizer satisfies ‖W*‖_{L∞(A)} ≤ K = √(−α/β).

Proof (Jean-Paul's Stampacchia argument):

1. Diamagnetic reduction. |W| is also a minimizer (kinetic forms satisfy diamagnetic inequality). WLOG W ≥ 0 real.

2. Test function. φ = (W* − K)₊ is in the Dirichlet form domain (normal contraction, Beurling-Deny).

3. Euler-Lagrange. E_loc(W, φ) + E_glue(W, φ) + ∫(αW + β(W)³)φ dμ = 0.

4. Kinetic ≥ 0. E_loc(W*, φ) ≥ 0 by the Markov property: (W(x)−W(y))(φ(x)−φ(y)) ≥ 0 pointwise, because φ = g∘W with g monotone non-decreasing.

5. Glue ≥ 0. Same monotonicity argument for the jump-type glue Dirichlet form.

6. Potential > 0 on excess. On {W > K}: αW + β(W)³ = βW((W*)² − K²) > 0.

7. Conclusion. Three non-negative terms summing to zero ⟹ each is zero. Potential integral = 0 ⟹ meas({W* > K}) = 0.

Numerical verification: The true minimizer (E = −4.000, constant W* = K) satisfies the bound exactly. All component signs verified: 0 monotonicity violations in 16,320 tested pairs (kinetic + glue).

Hölder upgrade. Once ‖W‖_∞ ≤ K, the RHS of the Euler-Lagrange PDE is L∞. Standard fractional elliptic regularity (Vladimirov + archimedean) gives W ∈ C^{0,γ} locally.

What Died: The RH Bridge (Kill #10)¶

The Structural Beauty¶

The glue Hessian's cross-terms, when evaluated against Hecke character fluctuations η_f = ∫ f̂(t) χ_t dt, reproduce the prime sums of the Weil explicit formula:

−(1/2π) Σ_p Σ_m log(p) p^{−m/2} [F(m log p) + F(−m log p)]

This is verified numerically and algebraically (Checks 1–3 in the bridge audit).

The Two Fatal Gaps¶

Gap A (Categorical). The archimedean Hessian -(xd/dx)² gives eigenvalue 1/4 + t² on χ_t. The Weil formula needs Re Ψ(1/4 + it/2) ≈ log(t). These differ by t²/log(t) — a mismatch between second-order differential operators (polynomial spectrum) and pseudodifferential operators (logarithmic spectrum). No change of vacuum (including the Gaussian e^{−πx²}) fixes this: V_eff adds a t-independent constant.

Gap B (Divergence). The diagonal Σ_p log(p) · 2I diverges. In the Weil formula, this cancels against the archimedean Gamma integral. But operator-level cancellation IS the explicit formula — making the positivity claim circular.

No-Go Theorem (Discussion Section)¶

Any attempt to identify a local GL Hessian (second-order, polynomial spectrum) with the Weil distribution (pseudodifferential, logarithmic spectrum) must fail at the archimedean place. This is an obstruction theorem for "Hilbert-Pólya via physics" programs using local differential operators.

Speculative Extensions (Documented, Not Pursued)¶

Pivot 1: Pseudodifferential Redesign¶

Replace -(xd/dx)² with (1/2)log(-(xd/dx)²), giving eigenvalues (1/2)log(1/4+t²) ≈ Ψ(1/4+it/2). This would match Weil but abandons the local GL framework. Convexity and uniqueness would need entirely new proofs.

Pivot 2: Connes Spectral Triple¶

Embed the GL vacuum state into an adelic spectral triple (A, H, D) with GL mass as natural regularization replacing Connes' δ-cutoff. Requires reformulating the GL vacuum as a KMS state (temperature interpretation absent).

Manuscript Structure¶

Title: Variational Selection and L∞ Regularity of Adelic Ginzburg-Landau Vacua

Section 1: Introduction. Define the GL functional on C_Q with Vladimirov kinetic, Mexican hat potential, arithmetic glue, and log-norm penalty. State main theorems.

Section 2: Theorem A1 — Coercivity and Unique Vacuum. Direct method existence. Log-norm convexity. Strict convexity proof. Spectral gap computation.

Section 3: Theorem A2 — L∞ Regularity. Stampacchia truncation proof (5 steps). Hölder upgrade via fractional elliptic theory. No dimensional constants needed.

Section 4: Numerical Verification. Moser iteration numerics (chain rule C_k ≤ 1, product Sobolev bounded, spectral gap stable). Stampacchia bound confirmed for 2–7 primes.

Section 5: The Weil Connection and Spectral Obstructions. Cross-term correspondence with Weil prime sums. Archimedean mismatch (t² vs log t). Diagonal divergence. No-go theorem for local-operator Hilbert-Pólya programs.

Section 6: Conclusion. Standalone value for adelic QFT and non-archimedean functional analysis. Open problems: Lemma M for general Hecke L-functions, pseudodifferential extensions.

Computational Artifacts¶

Verdict¶

Architecture v6.0 is closed. The Riemann Hypothesis remains open. Theorem A — existence, uniqueness, and L∞ regularity of the adelic GL vacuum — is complete and publishable. The spectral bridge to RH is broken by two categorical obstructions documented as a no-go theorem.

Kill count: 10. Survival count: 1 (Theorem A, standing).