Gravity Ptastructural Pta Structural One Statement

The structural discriminator

Pulsar timing arrays are networks of millisecond pulsars whose radio pulses arrive with extraordinary regularity. A passing gravitational wave, a ripple in spacetime, changes the arrival times by a few tens of nanoseconds in a correlated pattern across the sky. Astronomers search these timing residuals for a stochastic background, a random sea of gravitational waves from many sources, and they compare what they see against the faint hiss predicted from inflation, the rapid expansion of the early universe.

The Recognition Science framework contributes a structural discriminator, a way to tell the two possibilities apart before any data is even examined. The framework models the background's strength with a specific positive number, the golden ratio raised to the power minus 44, a tiny but nonzero value. Its library, a machine-checked collection of formal theorems, proves this number is greater than zero. It also proves the number is not equal to zero, which is the framework's proxy for a pure inflation baseline with no stochastic signal at all. The single theorem pta_structural_one_statement bundles these facts: the signature is positive, it is distinct from the zero baseline, and the framework's master theorem for a distinct gravitational wave background is satisfied.

The theorem goes one step further and strengthens the separation into an observable band. The framework defines a range of values, from half to one and a half times its predicted signature, and proves the signature sits inside that band. It also proves that the pure inflation baseline, represented by zero, lies outside the band. This is a structural claim about the mathematics, not a measurement of the sky.

What the declaration does not claim is equally precise. It attaches no pulsar timing array dataset, and it makes no claim that current observations already separate the signals. Dataset sensitivity and the spectral fitting of individual channels remain empirical work, the domain of falsifiable predictions, not of this theorem. The declaration is an algebraic, theorem-grade inhabitant for a master theorem input, a piece of the framework's internal structure, not a report from any telescope.

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

/-- Track 6.B structural one-statement.  The PTA stochastic signature is
positive and therefore distinct from the zero inflation-baseline proxy; the
master theorem PTA input is inhabited. -/
theorem pta_structural_one_statement :
    (0 < rs_pta_stochastic_phi_signature) ∧
    (rs_pta_stochastic_phi_signature ≠ inflation_zero_stochastic_baseline) ∧
    rs_pta_distinct_inflation_prop ∧
    Nonempty MasterTheorem.PTAStochasticGWDistinctFromInflation :=
  ⟨rs_pta_stochastic_phi_signature_pos,
   rs_pta_stochastic_phi_signature_ne_inflation_zero,
   rs_pta_distinct_inflation_prop_holds,
   ⟨ptaStochasticGWDistinctFromInflationWitness⟩⟩

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 pulsar timing array dataset is attached to this theorem. No claim is made that current observations already separate the signals. The zero baseline is a proxy for inflation, not a full model of inflationary gravitational wave production.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: