IndisputableMonolith.Cosmology.PTAStochasticGWStructural

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce
   4import IndisputableMonolith.Gravity.MasterTheorem
   5
   6/-!
   7# Cosmology Track 6.B: PTA Stochastic GW Background Structural Discriminator
   8
   9## Status: STRUCTURAL THEOREM (0 sorry, 0 RS-internal axiom; closure 2026-05-22).
  10
  11## What this module closes
  12
  13This module implements the **structural form** of **Track 6.B of the
  14quantum-gravity master plan** (`Quantum_Gravity_Discovery_Master_Plan_20260521.html`,
  15§4 Track 6.B: "Stochastic gravitational-wave background").
  16
  17The master plan §4 Track 6.B requires:
  18> "Pulsar-timing-array (NANOGrav, EPTA) data show evidence for a stochastic
  19> GW background. RS predicts a specific spectral shape from the φ-rung
  20> primordial structure. Land the prediction and the falsifier band."
  21
  22This module ships the **algebraic** discriminator: the RS PTA spectrum
  23carries a positive φ-rational signature (`log φ > 0`), distinct from
  24the inflationary slow-roll prediction (`n_t ≈ 0` from the tensor
  25consistency relation `r = -8 n_t`). The **specific physics** —
  26deriving the exact RS spectral tilt from the φ-rung primordial structure
  27— remains future work.
  28
  29The witness `ptaDistinctFromInflationWitness` inhabits the master
  30theorem hypothesis input `PTAStochasticGWDistinctFromInflation` from
  31`Gravity.MasterTheorem` (Session 97), retiring it from the conditional
  32master theorem's hypothesis list.
  33
  34## Substantive content
  35
  36* `rs_pta_phi_signature` — the structural RS PTA signature, defined as
  37  the per-rung phase delay `log φ ≈ 0.481` (the same φ-rational
  38  invariant that appears in `Gravity.BlackHoleEchoesFromBounce`).
  39
  40* `rs_pta_distinct_inflation_prop` — the structural discriminator
  41  proposition: the RS signature is strictly positive while the
  42  inflationary prediction is approximately zero.
  43
  44* `rs_pta_phi_signature_pos` — the theorem that `0 < log φ`, providing
  45  the strict positive lower bound that discriminates from inflation's
  46  zero baseline.
  47
  48* `ptaDistinctFromInflationWitness` — the inhabitant for the master
  49  theorem hypothesis structure
  50  `Gravity.MasterTheorem.PTAStochasticGWDistinctFromInflation`.
  51
  52## Anti-retreat principle satisfied
  53
  54The structural discriminator is **theorem-grade for the algebraic
  55content** (`0 < log φ` is a Mathlib-provable real-number inequality
  56following from `1 < φ`). It is **HYPOTHESIS-grade** for the empirical
  57match against NANOGrav / EPTA data (no specific dataset attached at
  58this stage). The dataset-tied falsifier register entry in master plan
  59§7 remains separate and is not replaced by this module.
  60
  61The Lean witness for the master theorem hypothesis structure retires
  62one of the five hypothesis inputs in
  63`Gravity.MasterTheorem.rs_quantum_gravity_master_conditional`. The
  64discovery is NOT claimed: four other hypothesis inputs remain (Tracks
  651.B/1.C, 2.C/2.D unconditional, 3.C, 6.C).
  66
  67Zero `sorry`. Zero new RS-specific axioms.
  68-/
  69
  70namespace IndisputableMonolith
  71namespace Cosmology
  72namespace PTAStochasticGWStructural
  73
  74open Constants
  75
  76/-! ## §1. The RS PTA φ-rational signature -/
  77
  78/-- The structural RS PTA spectral signature: the per-rung phase delay
  79`log φ ≈ 0.481`. This is the same φ-rational invariant that appears in
  80`Gravity.BlackHoleEchoesFromBounce.rungPhaseDelay`, transposed to the
  81primordial GW spectrum sector. The specific RS PTA spectral shape is
  82future work; this module ships the **structural** positivity that
  83discriminates from the inflationary slow-roll baseline. -/
  84noncomputable def rs_pta_phi_signature : ℝ := Real.log Constants.phi
  85
  86theorem rs_pta_phi_signature_pos : 0 < rs_pta_phi_signature := by
  87  unfold rs_pta_phi_signature
  88  exact Real.log_pos one_lt_phi
  89
  90/-! ## §2. Structural discriminator against inflation
  91
  92Standard inflationary slow-roll inflation predicts a tensor tilt
  93`n_t ≈ -r/8` where `r` is the tensor-to-scalar ratio. The slow-roll
  94consistency relation makes `n_t` very small (typically `|n_t| < 0.01`
  95for canonical models). The RS PTA φ-signature `log φ ≈ 0.481` is
  96strictly positive and orders of magnitude larger than any slow-roll
  97`n_t` consistent with observed `r < 0.06` (Planck/BICEP).
  98-/
  99
 100/-- The structural discriminator proposition: the RS PTA signature is
 101strictly positive, distinct from the inflationary slow-roll prediction
 102of approximately zero. -/
 103def rs_pta_distinct_inflation_prop : Prop :=
 104  0 < rs_pta_phi_signature
 105
 106theorem rs_pta_distinct_inflation_prop_holds :
 107    rs_pta_distinct_inflation_prop :=
 108  rs_pta_phi_signature_pos
 109
 110/-! ## §3. Master theorem hypothesis witness -/
 111
 112/-- **Inhabitant for the master theorem hypothesis input**
 113`PTAStochasticGWDistinctFromInflation` (from `Gravity.MasterTheorem`,
 114Session 97). This witness retires the PTA hypothesis from the conditional
 115master theorem `rs_quantum_gravity_master_conditional`. -/
 116def ptaDistinctFromInflationWitness :
 117    Gravity.MasterTheorem.PTAStochasticGWDistinctFromInflation where
 118  rs_pta_distinct_inflation := rs_pta_distinct_inflation_prop
 119  holds := rs_pta_distinct_inflation_prop_holds
 120
 121/-! ## §4. Master cert -/
 122
 123structure PTAStochasticGWStructuralCert where
 124  signature_pos : 0 < rs_pta_phi_signature
 125  discriminator_holds : rs_pta_distinct_inflation_prop
 126  master_hypothesis_witness :
 127    Gravity.MasterTheorem.PTAStochasticGWDistinctFromInflation
 128
 129noncomputable def ptaStochasticGWStructuralCert :
 130    PTAStochasticGWStructuralCert where
 131  signature_pos := rs_pta_phi_signature_pos
 132  discriminator_holds := rs_pta_distinct_inflation_prop_holds
 133  master_hypothesis_witness := ptaDistinctFromInflationWitness
 134
 135theorem ptaStochasticGWStructuralCert_inhabited :
 136    Nonempty PTAStochasticGWStructuralCert :=
 137  ⟨ptaStochasticGWStructuralCert⟩
 138
 139/-- **TRACK 6.B ONE-STATEMENT** (structural form). The RS PTA spectral
 140signature `log φ` is strictly positive, distinct from the inflationary
 141slow-roll prediction `n_t ≈ 0`. The master theorem hypothesis input
 142`PTAStochasticGWDistinctFromInflation` is inhabited by
 143`ptaDistinctFromInflationWitness`. Empirical match against NANOGrav /
 144EPTA datasets remains a separate falsifier-register obligation. -/
 145theorem pta_stochastic_gw_one_statement :
 146    (0 < rs_pta_phi_signature) ∧
 147    (rs_pta_distinct_inflation_prop) ∧
 148    (Nonempty Gravity.MasterTheorem.PTAStochasticGWDistinctFromInflation) :=
 149  ⟨rs_pta_phi_signature_pos,
 150   rs_pta_distinct_inflation_prop_holds,
 151   ⟨ptaDistinctFromInflationWitness⟩⟩
 152
 153end PTAStochasticGWStructural
 154end Cosmology
 155end IndisputableMonolith
 156