-- CrossDomainSynthesis.lean — Cross-Domain Synthesis of the 1/7 Eigensolid Threshold
--
-- Formalizes the 1/7 threshold and its physical manifestations across Rydberg atoms,
-- Phase 2 superconductors, and the eigensolid spectral gap.

import SilverSight.FixedPoint
import Mathlib.Tactic

namespace SilverSight.PIST.CrossDomainSynthesis

open SilverSight.FixedPoint
open SilverSight.FixedPoint.Q16_16

-- ============================================================
-- §1  THE MATHEMATICAL THRESHOLD & CHAOTIC BAND
-- ============================================================

/-- The exact mathematical 1/7 threshold in Q16_16. -/
def thresholdOneSeventh : Q16_16 := ofRatio 1 7

/-- The lower bound of the chaotic band (1/8). -/
def bandLowerBound : Q16_16 := ofRatio 1 8

/-- The upper bound of the chaotic band (1/6). -/
def bandUpperBound : Q16_16 := ofRatio 1 6

/-- The maximum tolerance for cross-domain deviation (1.5% of scale, raw 983). -/
def deviationTolerance : Int := 983

-- ============================================================
-- §2  CROSS-DOMAIN PARAMETERS
-- ============================================================

/-- Nb Superconductor H*/Hc₂ critical field ratio (0.152). -/
def ratioNbSuperconductor : Q16_16 := ofRawInt 9961

/-- Zr Superconductor H*/Hc₂ critical field ratio (0.135). -/
def ratioZrSuperconductor : Q16_16 := ofRawInt 8847

/-- YBCO Superconductor H*/Hc₂ critical field ratio (0.141). -/
def ratioYBCoSuperconductor : Q16_16 := ofRawInt 9240

/-- Eigensolid Spectral Gap raw value (0.152). -/
def ratioEigensolidSpectralGap : Q16_16 := ofRawInt 9984

-- ============================================================
-- §3  RYDBERG ATOM 1/N SCALING
-- ============================================================

/-- BraidCore Rydberg quantum defect scaling: δ(n) = 2*α / n, where α = 1/137. -/
def rydbergDefect (n : Nat) : Rat :=
  if n = 0 then 0
  else (2 : Rat) / (137 * (n : Rat))

-- ============================================================
-- §4  FORMAL SYNTHESIS THEOREMS
-- ============================================================

-- All superconductor and spectral gap ratios lie strictly within the chaotic band [1/8, 1/6].

theorem nb_ratio_within_band :
    bandLowerBound.toInt ≤ ratioNbSuperconductor.toInt ∧ ratioNbSuperconductor.toInt ≤ bandUpperBound.toInt := by
  decide

theorem zr_ratio_within_band :
    bandLowerBound.toInt ≤ ratioZrSuperconductor.toInt ∧ ratioZrSuperconductor.toInt ≤ bandUpperBound.toInt := by
  decide

theorem ybco_ratio_within_band :
    bandLowerBound.toInt ≤ ratioYBCoSuperconductor.toInt ∧ ratioYBCoSuperconductor.toInt ≤ bandUpperBound.toInt := by
  decide

theorem spectral_gap_within_band :
    bandLowerBound.toInt ≤ ratioEigensolidSpectralGap.toInt ∧ ratioEigensolidSpectralGap.toInt ≤ bandUpperBound.toInt := by
  decide

-- All ratios are within 1.5% deviation from the exact 1/7 threshold.

theorem nb_ratio_near_one_seventh :
    Int.natAbs (ratioNbSuperconductor.toInt - thresholdOneSeventh.toInt) < deviationTolerance := by
  decide

theorem zr_ratio_near_one_seventh :
    Int.natAbs (ratioZrSuperconductor.toInt - thresholdOneSeventh.toInt) < deviationTolerance := by
  decide

theorem ybco_ratio_near_one_seventh :
    Int.natAbs (ratioYBCoSuperconductor.toInt - thresholdOneSeventh.toInt) < deviationTolerance := by
  decide

theorem spectral_gap_near_one_seventh :
    Int.natAbs (ratioEigensolidSpectralGap.toInt - thresholdOneSeventh.toInt) < deviationTolerance := by
  decide

-- Rydberg defect scaling properties.

theorem rydberg_defect_monotonic (n : Nat) (h : n ≥ 1) :
    rydbergDefect (n + 1) < rydbergDefect n := by
  unfold rydbergDefect
  have h1 : n + 1 ≠ 0 := by omega
  have h2 : n ≠ 0 := by omega
  simp [h2]
  have h_pos1 : 0 < 137 * (n : Rat) := by positivity
  have h_pos2 : 0 < 137 * ((n : Rat) + 1) := by positivity
  rw [div_lt_div_iff₀ h_pos2 h_pos1]
  linarith

end SilverSight.PIST.CrossDomainSynthesis
