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Rigged-Space Theorem for the Weil Form

What it establishes

On the Gelfand triple C_c^\infty(R) ⊂ L²(R) ⊂ D', the pseudodifferential Weil form is well-defined, symmetric, and exact:

q_Weil[f] = W(ff̃, ff̃)

Finiteness results

  • Archimedean term: finite on compact-support tests because f̂ is Schwartz and multiplier grows only like log(1+|t|)
  • Prime translation term: finite on compact-support tests because only finitely many shifts k·log(p) overlap two fixed compact supports

Non-closability

The same form is NOT semibounded/closable on ordinary L²(R): the bump-sequence argument drives the prime part to −∞.

Classification

  1. It gives the correct exact Weil object
  2. It does NOT yield a Hilbert-space Hamiltonian
  3. Positivity of this form is still exactly Weil positivity, hence RH-equivalent, not a new proof mechanism

Recursive tree update

  • Standalone paper path: complete and stabilizing
  • Rigged-space Weil path: exact but equivalent
  • Remaining non-dead spectral path: find a genuinely new trace/noncommutative mechanism that does more than restate Weil positivity

Next move

Draft compact-support exactness proposition + non-closability proposition into manuscript no-go section.