Gravity Ptastructural Rs Pta Stochastic Phi Signature Ne Inflation Zero

The structural discriminator

Pulsar timing arrays search for a stochastic gravitational-wave background, a faint hum of gravitational waves from many supermassive black hole mergers. A key question is whether any detected signal could be confused with a primordial background from cosmic inflation. The Recognition Science framework contributes a structural discriminator: a theorem-grade algebraic statement that its proposed signature, a positive number, cannot equal zero, the proxy baseline for a pure inflationary background.

The signature itself is defined as φ^(-44), where φ is the golden ratio, approximately 1.618. This is a positive real number because any positive number raised to any integer power remains positive. The theorem rs_pta_stochastic_phi_signature_ne_inflation_zero proves this signature is not equal to zero, the defined inflation baseline. The proof is direct: since the signature is positive, it cannot be zero. This is not a claim about real data; it is a purely mathematical statement about the framework's own definitions.

The framework's library, a machine-checked collection of formal theorems, extends this to a stronger statement. It defines an observable band around the signature, from half the signature to one and a half times it. The theorem rs_pta_stochastic_phi_signature_in_observable_band proves the signature lies inside this band. A separate theorem, inflationary_pta_family_baseline_not_in_rs_band, proves that the zero baseline lies outside this band. Together, these establish a structural separation: the proposed signature occupies a positive band, and the inflationary zero baseline does not.

In Recognition Science, this provides a witness for the master theorem input PTAStochasticGWDistinctFromInflation, meaning it gives a formal inhabitant for that hypothesis. The framework models the distinction as a structural property, not an observational one. The docstring is explicit: this does not attach a PTA dataset or claim current observational separation. Dataset sensitivity and spectral fitting remain empirical falsifier work.

What this changes is the nature of the claim. The framework offers a theorem that its signature is structurally distinct from zero, a formal guarantee within its own model. Whether that signature corresponds to a real signal is a separate question, one that only data can answer. The theorem is a piece of internal consistency, not a prediction that has been tested against the sky.

theorem rs_pta_stochastic_phi_signature_ne_inflation_zero :
    rs_pta_stochastic_phi_signature ≠ inflation_zero_stochastic_baseline := by
  intro h
  have hpos := rs_pta_stochastic_phi_signature_pos
  unfold inflation_zero_stochastic_baseline at h
  rw [h] at hpos
  linarith

theorem rs_pta_stochastic_phi_signature_in_observable_band :
    rs_pta_observable_band rs_pta_stochastic_phi_signature := by
  have hpos := rs_pta_stochastic_phi_signature_pos
  unfold rs_pta_observable_band
  constructor <;> nlinarith

theorem inflationary_pta_family_baseline_not_in_rs_band
    (x : ℝ) (hx : inflationary_pta_family_baseline x) :
    ¬ rs_pta_observable_band x := by
  intro hband
  rcases hband with ⟨hlow, _⟩
  unfold inflationary_pta_family_baseline inflation_zero_stochastic_baseline at hx
  subst x
  have hpos := rs_pta_stochastic_phi_signature_pos
  nlinarith

What this page does not claim

No observational claim: the theorem does not assert that any pulsar timing array has detected this signature. No empirical separation: the theorem does not state that current data can distinguish the signature from a zero baseline. No physical derivation: the theorem does not derive the φ^(-44) scale from first principles within this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/PTAStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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