RTSG v6.0 — Theorem Architecture (v3.1)¶

Status Update: Theorem B COLLAPSED (Tate's Thesis)¶

The scattering matrix S(s) = ξ(1-s)/ξ(s) is an unconditional consequence of Tate's thesis (1950). The functional equation of the global zeta integral on C_Q directly yields this identity.

Caveat: This gives the scattering DATA unconditionally, but RH requires the additional positivity constraint (self-adjointness of the underlying dynamics) that forces resonances onto Re(s) = 1/2. The scattering identity alone reformulates RH; it does not prove it.

Theorem A: Selection (The Final Boss)¶

Part 1: Existence — PROVED (conditional on effective dimension)¶

Coercivity of the GL action on H^1(C_Q) follows from: 1. Compactness of C^1_Q (class field theory theorem) 2. Effective dimension d=1 (archimedean ray is only non-compact direction) 3. Sobolev embedding H^1(R_{>0}) ↪ L^∞ (Morrey) 4. Jensen: ||W||⁴₄ ≥ Vol⁻¹ ||W||⁴₂ (finite measure space) 5. Quartic dominates quadratic for large norms

S[W] ≥ K₁||∇W||² + K₂||W||⁴₄ - K₃

Coercivity + weak lower semicontinuity + reflexivity of H^1 → minimizer W₀ EXISTS by direct method.

Gap: Effective dimension argument assumes p-adic components contribute only finitely. This is essentially Conjecture D again — the argument is MORALLY correct but technically circular at the point where we claim d_eff = 1.

Part 2: Uniqueness — OPEN¶

Mexican hat has U(1) orbit of degenerate minima W₀ = v·e^{iθ}. Need gauge-fixing to select unique real minimizer.

Part 3: Stability — FOLLOWS FROM UNIQUENESS¶

Strict local minimum → positive Hessian → self-adjoint Goldstone.

Kill Log (7 dead targets)¶

1. Fredholm det = ξ (meromorphic vs entire)
2. Weyl M-function (Nevanlinna ⟺ RH, circular)
3. ω-deformation (topological phase transition)
4. Naive det_ζ = ξ (Connes cutoff circularity)
5. N_κ Pontryagin (only finite violations)
6. Bounded-type/Krein (architecture without positivity)
7. LP jet Hankel (wrong moment problem for contractive function)

Proved Components¶

• Poisson bridge constant C = 0.04467
• Bridge identity B*K - KB = (i/2)K
• Character-family nonvanishing
• Theorem B: S(s) = ξ(1-s)/ξ(s) (Tate, 1950)
• Coercivity of 1D GL action (numerical + analytic)
• Krein space reformulation: RH ⟺ κ = 0