Gravity Ptastructural Inflationary Pta Family Baseline Not In Rs Band

The structural discriminator

Pulsar timing arrays search for a gravitational-wave background by timing an array of millisecond pulsars and looking for a common, correlated signal. The standard astrophysical interpretation is a background from supermassive black hole binaries. A competing cosmological source could be a stochastic background produced during inflation. In the Recognition Science framework, the question is whether the framework's predicted stochastic background can be distinguished, structurally, from a pure inflationary one.

The framework's prediction is a positive signal at a specific scale, the rung-44 scale, written as φ^(-44), where φ is the golden ratio. This is a recognition scale, one of the discrete steps the framework uses to structure physical quantities. The inflationary baseline is modeled as exactly zero. The theorem inflationary_pta_family_baseline_not_in_rs_band proves that this zero baseline cannot lie inside the framework's observable band, which is defined as any value between half and three-halves of the rung-44 signature. Since the signature is positive, and the band is an interval of positive numbers, zero is necessarily outside it.

This is a purely algebraic result. It establishes a structural separation: the framework's predicted signal is a positive number, and the inflationary baseline is zero, so they cannot be the same. The theorem is a formal statement in the framework's machine-checked library of formal theorems. It does not attach a pulsar timing array dataset, and it does not claim any current observational separation between these two possible backgrounds. The sensitivity of current datasets and the spectral fitting needed to distinguish them remain empirical work.

What the theorem does provide is a precise, checkable statement of one way the framework's prediction differs from a simple inflationary model. It is a structural discriminator, not an observational one. The framework's contribution is to make the distinction a matter of formal proof, given its starting definitions. Whether the real sky shows the rung-44 signal or not is a question for data.

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

theorem rs_pta_stochastic_phi_signature_pos :
    0 < rs_pta_stochastic_phi_signature := by
  unfold rs_pta_stochastic_phi_signature
  exact zpow_pos phi_pos _

What this page does not claim

The theorem does not claim any current observational separation between the framework's signal and an inflationary background. The theorem does not prove that the inflationary baseline is the only possible zero-signal model. The theorem does not derive the rung-44 scale from first principles; it is a definitional input.

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)

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