Cosmology Ptastochastic Gwstructural Rs Pta Distinct Inflation Prop

The structural discriminator

Pulsar timing arrays like NANOGrav and EPTA have gathered evidence for a stochastic gravitational wave background, a faint sea of ripples in spacetime. In standard cosmology, the leading explanation is inflation, a brief period of exponential expansion in the early universe. Inflation makes a specific, testable prediction: the spectrum of these gravitational waves should be nearly flat, with a tilt very close to zero.

Recognition Science (RS) proposes a different origin for this background. Its framework models the early universe as a discrete structure built on the golden ratio, φ ≈ 1.618. The key quantity is the per-rung phase delay, defined as the natural logarithm of φ, which is approximately 0.481. The declaration rs_pta_distinct_inflation_prop is a formal statement that this RS signature is strictly greater than zero. The accompanying theorem proves this is true, since φ is greater than 1.

This positivity is the entire content of the discriminator. It establishes a structural difference: the RS signature is a positive number, while the inflationary prediction is approximately zero. The formal theorem does not, however, claim that this number matches any observed data. The empirical match against NANOGrav or EPTA datasets is explicitly a separate, future obligation. The declaration is a statement about the framework's own internal consistency, not a measurement of the universe.

In the framework's larger project, this theorem serves a specific purpose. It provides a formal witness that satisfies one of the hypotheses in a conditional master theorem about quantum gravity. By proving this structural point, the framework retires one of the five open hypotheses, but it explicitly does not claim the full discovery. Four other hypotheses remain unproven.

/-- The structural discriminator proposition: the RS PTA signature is
strictly positive, distinct from the inflationary slow-roll prediction
of approximately zero. -/
def rs_pta_distinct_inflation_prop : Prop :=
  0 < rs_pta_phi_signature

theorem rs_pta_phi_signature_pos : 0 < rs_pta_phi_signature := by
  unfold rs_pta_phi_signature
  exact Real.log_pos one_lt_phi

theorem rs_pta_distinct_inflation_prop_holds :
    rs_pta_distinct_inflation_prop :=
  rs_pta_phi_signature_pos

What this page does not claim

The theorem does not claim any empirical match with pulsar timing array data. The theorem does not derive the exact spectral tilt of the RS gravitational wave background. The theorem does not prove that inflation is wrong, only that the RS signature is positive.

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/Cosmology/PTAStochasticGWStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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