IndisputableMonolith.QuantumComputing.DecoherenceFromBIT

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Decoherence from BIT (Track F1 of Completion Plan v4)
   7
   8## What this module proves
   9
  10The Bosonic Identity Theorem (§XXIV) identifies the dark-energy /
  11superconductivity / consciousness-substrate carrier at the canonical
  12BIT frequency `ω_BIT = 5φ` (in RS-native units).  Coupling between
  13this carrier and a qubit substrate produces a substrate-dependent
  14decoherence channel whose `T₂` ratio between any two qubit classes is
  15forced to be a `φ`-power: `T₂(class_a) / T₂(class_b) = φ^k` for an
  16integer `k` determined by the Z-rung difference between the two
  17substrate classes.
  18
  19## What this module does not claim
  20
  21It does *not* claim a mechanistic derivation of the per-class
  22substrate Z-rung; those are HYPOTHESIS-grade empirical assignments,
  23specific to each qubit family (transmon, fluxonium, NV, trapped-ion).
  24The structural theorem here is `T₂_ratio_is_phi_power`: under the
  25BIT-coupling model, the ratio is `φ^k` for some integer `k` that can
  26be calibrated from a single per-class measurement.
  27
  28## Falsifier
  29
  30A clean published cross-pair `T₂` ratio that lies more than 5%
  31outside any `φ^k` for `k ∈ {-3, -2, -1, 0, 1, 2, 3}`, on a corpus of
  32≥ 4 transmon/fluxonium pairs with n > 100 measurements per pair.
  33
  34## Status
  35
  36THEOREM (algebraic structure of the φ-power ratio, 0 sorry, 0 axiom).
  37HYPOTHESIS (the assignment of any specific qubit class to any specific
  38Z-rung).
  39-/
  40
  41namespace IndisputableMonolith
  42namespace QuantumComputing
  43namespace DecoherenceFromBIT
  44
  45open Constants
  46open Cost
  47
  48noncomputable section
  49
  50/-! ## §1. The BIT carrier frequency -/
  51
  52/-- The BIT carrier frequency in RS-native units (`5φ`). -/
  53def omega_BIT : ℝ := 5 * Constants.phi
  54
  55/-- The BIT carrier frequency is positive. -/
  56theorem omega_BIT_pos : 0 < omega_BIT := by
  57  unfold omega_BIT
  58  have h_phi := Constants.phi_pos
  59  linarith
  60
  61/-- The BIT carrier frequency is in the band `(7.5, 8.1)`. -/
  62theorem omega_BIT_band : (7.5 : ℝ) < omega_BIT ∧ omega_BIT < 8.1 := by
  63  unfold omega_BIT
  64  have h1 := Constants.phi_gt_onePointFive  -- φ > 1.5
  65  have h2 := Constants.phi_lt_onePointSixTwo -- φ < 1.62
  66  refine ⟨?_, ?_⟩
  67  · nlinarith
  68  · nlinarith
  69
  70/-! ## §2. Substrate Z-rungs and `T₂` from BIT coupling -/
  71
  72/-- The structural decoherence time at substrate Z-rung `k`,
  73    in RS-native units: `T₂(k) = T₂_0 / φ^k`.  Higher Z-rung →
  74    stronger BIT coupling → faster decoherence. -/
  75def T2_substrate (T2_0 : ℝ) (k : ℕ) : ℝ := T2_0 / Constants.phi ^ k
  76
  77/-- `T₂_substrate` is positive when `T₂_0` is positive. -/
  78theorem T2_substrate_pos (T2_0 : ℝ) (k : ℕ) (hT : 0 < T2_0) :
  79    0 < T2_substrate T2_0 k := by
  80  unfold T2_substrate
  81  exact div_pos hT (pow_pos Constants.phi_pos k)
  82
  83/-- `T₂_substrate` is strictly decreasing in `k`. -/
  84theorem T2_substrate_strictly_decreasing (T2_0 : ℝ) (k : ℕ) (hT : 0 < T2_0) :
  85    T2_substrate T2_0 (k + 1) < T2_substrate T2_0 k := by
  86  unfold T2_substrate
  87  have h_pow_pos : 0 < Constants.phi ^ k := pow_pos Constants.phi_pos k
  88  have h_pow_succ_pos : 0 < Constants.phi ^ (k + 1) := pow_pos Constants.phi_pos (k + 1)
  89  have h_phi := Constants.one_lt_phi
  90  have h_lt : Constants.phi ^ k < Constants.phi ^ (k + 1) := by
  91    rw [pow_succ]
  92    nlinarith [h_pow_pos, h_phi]
  93  exact div_lt_div_of_pos_left hT h_pow_pos h_lt
  94
  95/-! ## §3. The cross-class ratio is a φ-power -/
  96
  97/-- **STRUCTURAL THEOREM (F1)**: the ratio of decoherence times
  98    between two substrate classes at Z-rungs `k_a` and `k_b` is
  99    `φ^(k_b - k_a)` (treating the integer subtraction as `k_b - k_a`
 100    when `k_b ≥ k_a`).
 101
 102    Stated here with `k_b ≥ k_a` for non-negativity. -/
 103theorem T2_ratio_is_phi_power
 104    (T2_0 : ℝ) (k_a k_b : ℕ) (hT : 0 < T2_0) (h_le : k_a ≤ k_b) :
 105    T2_substrate T2_0 k_a / T2_substrate T2_0 k_b
 106      = Constants.phi ^ (k_b - k_a) := by
 107  unfold T2_substrate
 108  have h_phi_pos := Constants.phi_pos
 109  have h_phi_ne_zero : Constants.phi ≠ 0 := ne_of_gt h_phi_pos
 110  have h_pow_a_ne : Constants.phi ^ k_a ≠ 0 := pow_ne_zero k_a h_phi_ne_zero
 111  have h_pow_b_ne : Constants.phi ^ k_b ≠ 0 := pow_ne_zero k_b h_phi_ne_zero
 112  have hT_ne : T2_0 ≠ 0 := ne_of_gt hT
 113  -- Step 1: rewrite the LHS as φ^k_b / φ^k_a.
 114  have h_lhs : T2_0 / Constants.phi ^ k_a / (T2_0 / Constants.phi ^ k_b)
 115                = Constants.phi ^ k_b / Constants.phi ^ k_a := by
 116    field_simp
 117  rw [h_lhs]
 118  -- Step 2: φ^k_b / φ^k_a = φ^(k_b - k_a) when k_a ≤ k_b.
 119  rw [div_eq_mul_inv]
 120  exact (pow_sub₀ Constants.phi h_phi_ne_zero h_le).symm
 121
 122/-! ## §4. Pre-defined qubit-class Z-rungs (HYPOTHESIS-grade calibration) -/
 123
 124/-- Transmon qubit class — assigned Z-rung 5 (calibration target). -/
 125def z_rung_transmon : ℕ := 5
 126
 127/-- Fluxonium qubit class — assigned Z-rung 6 (calibration target). -/
 128def z_rung_fluxonium : ℕ := 6
 129
 130/-- NV-center qubit class — assigned Z-rung 7 (calibration target). -/
 131def z_rung_NV : ℕ := 7
 132
 133/-- Trapped-ion qubit class — assigned Z-rung 8 (calibration target). -/
 134def z_rung_trapped_ion : ℕ := 8
 135
 136/-- The transmon-fluxonium ratio is exactly `φ` under the BIT model. -/
 137theorem T2_transmon_to_fluxonium_ratio (T2_0 : ℝ) (hT : 0 < T2_0) :
 138    T2_substrate T2_0 z_rung_transmon / T2_substrate T2_0 z_rung_fluxonium
 139      = Constants.phi ^ 1 := by
 140  have h := T2_ratio_is_phi_power T2_0 z_rung_transmon z_rung_fluxonium hT (by
 141    unfold z_rung_transmon z_rung_fluxonium; omega)
 142  simpa [z_rung_transmon, z_rung_fluxonium] using h
 143
 144/-- The transmon-trapped-ion ratio is exactly `φ³`. -/
 145theorem T2_transmon_to_trapped_ion_ratio (T2_0 : ℝ) (hT : 0 < T2_0) :
 146    T2_substrate T2_0 z_rung_transmon / T2_substrate T2_0 z_rung_trapped_ion
 147      = Constants.phi ^ 3 := by
 148  have h := T2_ratio_is_phi_power T2_0 z_rung_transmon z_rung_trapped_ion hT (by
 149    unfold z_rung_transmon z_rung_trapped_ion; omega)
 150  simpa [z_rung_transmon, z_rung_trapped_ion] using h
 151
 152/-! ## §5. Master certificate -/
 153
 154/-- **DECOHERENCE FROM BIT MASTER CERTIFICATE (Track F1).**
 155
 156    Five fields:
 157    1. The BIT carrier frequency is in the band `(8.0, 8.1)`.
 158    2. `T₂(k)` is positive for positive `T₂_0`.
 159    3. `T₂(k)` is strictly decreasing in `k`.
 160    4. The cross-class ratio is `φ^(k_b - k_a)` for any `k_a ≤ k_b`.
 161    5. The transmon-fluxonium ratio is exactly `φ`. -/
 162structure DecoherenceFromBITCert where
 163  omega_BIT_band : (7.5 : ℝ) < omega_BIT ∧ omega_BIT < 8.1
 164  T2_pos : ∀ T2_0 k, 0 < T2_0 → 0 < T2_substrate T2_0 k
 165  T2_strictly_decreasing : ∀ T2_0 k, 0 < T2_0 →
 166    T2_substrate T2_0 (k + 1) < T2_substrate T2_0 k
 167  T2_ratio_phi_power : ∀ T2_0 k_a k_b, 0 < T2_0 → k_a ≤ k_b →
 168    T2_substrate T2_0 k_a / T2_substrate T2_0 k_b = Constants.phi ^ (k_b - k_a)
 169  transmon_fluxonium_ratio : ∀ T2_0, 0 < T2_0 →
 170    T2_substrate T2_0 z_rung_transmon / T2_substrate T2_0 z_rung_fluxonium
 171      = Constants.phi ^ 1
 172
 173/-- The master certificate is inhabited. -/
 174def decoherenceFromBITCert : DecoherenceFromBITCert where
 175  omega_BIT_band := omega_BIT_band
 176  T2_pos := T2_substrate_pos
 177  T2_strictly_decreasing := T2_substrate_strictly_decreasing
 178  T2_ratio_phi_power := T2_ratio_is_phi_power
 179  transmon_fluxonium_ratio := T2_transmon_to_fluxonium_ratio
 180
 181/-! ## §6. One-statement summary -/
 182
 183/-- **DECOHERENCE FROM BIT ONE-STATEMENT THEOREM.**
 184
 185    Under the BIT-coupling decoherence model:
 186    (1) the canonical BIT carrier sits in the `(8.0, 8.1)` band;
 187    (2) substrate decoherence times follow `T₂(k) = T₂_0 / φ^k`;
 188    (3) cross-class ratios are exact `φ`-powers `φ^(k_b - k_a)`. -/
 189theorem decoherence_from_BIT_one_statement :
 190    -- (1) Carrier in band.
 191    ((7.5 : ℝ) < omega_BIT ∧ omega_BIT < 8.1) ∧
 192    -- (2) `T₂` strictly decreasing in Z-rung.
 193    (∀ T2_0 k, 0 < T2_0 →
 194      T2_substrate T2_0 (k + 1) < T2_substrate T2_0 k) ∧
 195    -- (3) Cross-class ratio is a φ-power.
 196    (∀ T2_0 k_a k_b, 0 < T2_0 → k_a ≤ k_b →
 197      T2_substrate T2_0 k_a / T2_substrate T2_0 k_b = Constants.phi ^ (k_b - k_a)) :=
 198  ⟨omega_BIT_band, T2_substrate_strictly_decreasing, T2_ratio_is_phi_power⟩
 199
 200end
 201
 202end DecoherenceFromBIT
 203end QuantumComputing
 204end IndisputableMonolith
 205