Gravity Ptastructural Rs Pta Distinct Inflation Observable Band Prop Holds

The structural discriminator

Pulsar timing arrays (PTAs) listen for a background hum of gravitational waves by clocking the regular flashes of millisecond pulsars. A stochastic background, a random superposition of many waves, would show up as a correlated jitter in those arrival times. The Recognition Science framework, a system of formal theorems built from a single cost function, contributes a structural prediction about that background: its signature should be a positive number, specifically the golden ratio raised to the power minus 44, written φ^(-44).

The framework's library, a machine-checked collection of formal theorems, proves this signature is positive and therefore not equal to zero. It then defines a pure inflation baseline as exactly zero, the absence of any stochastic signal. The key theorem, rs_pta_distinct_inflation_observable_band_prop_holds, states that the φ^(-44) signature lies inside a band between half and one and a half times its own value, while the zero baseline lies entirely outside that band. This is a formal, algebraic separation: the predicted signal and the baseline are provably distinct in this band.

This is not a measurement. The theorem does not attach any PTA dataset, nor does it claim current observational separation of the two signals. The framework's own documentation states that dataset sensitivity and channel-specific spectral fitting remain empirical falsifier work. The theorem is the algebraic inhabitant for a master theorem input, a structural discriminator that says: if the framework's prediction is correct, then its signal must be positive and distinct from a zero inflation baseline.

What the theorem changes is the shape of a future test. It provides a precise, falsifiable target: look for a stochastic background whose amplitude corresponds to φ^(-44) in the framework's units, and check that it is not zero. The separation is proved, but the observation is open. The framework has set the stage; the pulsar timing arrays must now provide the data.

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 _

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

theorem rs_pta_distinct_inflation_observable_band_prop_holds :
    rs_pta_distinct_inflation_observable_band_prop :=
  ⟨rs_pta_stochastic_phi_signature_in_observable_band,
   inflationary_pta_family_baseline_not_in_rs_band⟩

What this page does not claim

No PTA dataset has been analyzed, and no current observational separation is claimed. The theorem does not prove that the framework's prediction is the correct description of the actual stochastic background. The zero baseline is a proxy for pure inflation, not a full model of all inflationary gravitational-wave spectra.

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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