IndisputableMonolith.Gravity.NoGraviton.UnitBridge

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Gravity.NoGraviton
   4import IndisputableMonolith.Gravity.QuantumChannel.BMVPositive
   5
   6/-!
   7# Gravity IV: Unit Bridge from κ_rs to SI BMV Phase Rate (Theorem 4)
   8
   9This module formalizes the fourth load-bearing theorem of *Gravity from
  10Recognition IV: The Quantum Channel*: the dimensionless RS coupling
  11`κ_rs = 8 φ⁵`, with the band `(85.6, 90.4)` from
  12`ZeroParameterGravity.kappa_bounds`, converts to the dimensionful BMV
  13entangling phase rate via an explicit RS-native-to-SI bridge.
  14
  15The mathematical content is:
  16
  17* In RS-native units, `ℏ = φ⁻⁵`, `G = φ⁵/π`, hence
  18  `G/ℏ = φ¹⁰/π`, a closed-form quantity fixed by `φ` alone.
  19* The BMV entangling-phase rate is
  20  `dΦ/dT = (G m₁ m₂ / ℏ) · g(r_LL, r_LR, r_RL, r_RR)`
  21  where `g` is the geometry-dependent inverse-distance combination
  22  appearing in T3 (`branchPhaseInvariant`), divided by `T`.
  23* Therefore in RS-native units, the BMV phase rate is
  24  `(φ¹⁰/π) · m₁ m₂ · g`, and the SI value is obtained by composing
  25  with the canonical RS-native-to-SI calibration.
  26
  27The unit bridge to a tabletop observable in SI is parameterized by an
  28inhabitant of `Constants.RSNativeUnits.ExternalCalibration`, the named
  29open frontier of the dimensional bridge (conversion map closed in
  30`Foundation.SIBridgeClosure`; the anchor is external by dimensional
  31analysis). Until an anchor is supplied, T4 is a CONDITIONAL THEOREM
  32with the calibration as input.
  33
  34## What is proved here
  35
  36* `BMVPhaseRateNative`: the BMV entangling phase rate in RS-native
  37  units, a closed-form `φ`-rational quantity for fixed `(m₁, m₂, {r_ab})`.
  38* `bmv_phase_rate_native_eq` : closed-form expression
  39  `(φ¹⁰/π) · m₁ m₂ · g`.
  40* `bmv_phase_rate_native_in_kappa_band` : the band on `κ_rs` propagates
  41  linearly to the BMV phase rate in RS-native units.
  42-/
  43
  44namespace IndisputableMonolith
  45namespace Gravity
  46namespace NoGraviton
  47namespace UnitBridge
  48
  49open Constants
  50open Real
  51
  52noncomputable section
  53
  54/-! ## RS-native BMV phase rate -/
  55
  56/-- Geometric factor of the BMV protocol: the entangling
  57inverse-distance combination
  58`1/r_LL + 1/r_RR − 1/r_LR − 1/r_RL`. -/
  59def bmvGeometryFactor (r_LL r_LR r_RL r_RR : ℝ) : ℝ :=
  60  1 / r_LL + 1 / r_RR - 1 / r_LR - 1 / r_RL
  61
  62/-- BMV entangling phase rate in RS-native units. By T3 the entangling
  63invariant is `(G m₁ m₂ T / ℏ) · g`; the per-time rate is
  64`(G m₁ m₂ / ℏ) · g`. -/
  65noncomputable def BMVPhaseRateNative
  66    (m1 m2 r_LL r_LR r_RL r_RR : ℝ) : ℝ :=
  67  (G * m1 * m2 / hbar) * bmvGeometryFactor r_LL r_LR r_RL r_RR
  68
  69/-- **Helper: G/ℏ in RS-native units.**
  70We compute `G/ℏ = (φ⁵/π) · φ⁵ = φ¹⁰/π` directly by unfolding the
  71RS-native definitions:
  72* `G = λ_rec² c³ / (π ℏ)` with `λ_rec = c = 1` and `ℏ = φ⁻⁵`,
  73* so `G = 1/(π · φ⁻⁵) = φ⁵/π`,
  74* and `G/ℏ = (φ⁵/π)/φ⁻⁵ = φ¹⁰/π`. -/
  75theorem G_over_hbar_RS_native :
  76    G / hbar = phi ^ (5 : ℝ) / Real.pi * phi ^ (5 : ℝ) := by
  77  unfold G hbar cLagLock lambda_rec ell0 c tau0 tick
  78  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  79  have hphi_ne : phi ^ (-(5 : ℝ)) ≠ 0 :=
  80    ne_of_gt (Real.rpow_pos_of_pos phi_pos _)
  81  -- After the unfold, we have (1^2 * 1^3) / (π * (φ⁻⁵ * 1)) / (φ⁻⁵ * 1)
  82  simp only [one_pow, mul_one, div_one]
  83  -- Goal: 1 / (π * φ⁻⁵) / φ⁻⁵ = φ⁵/π * φ⁵
  84  rw [Real.rpow_neg phi_pos.le]
  85  field_simp
  86
  87/-- **Closed form for the RS-native BMV phase rate.** -/
  88theorem bmv_phase_rate_native_eq
  89    (m1 m2 r_LL r_LR r_RL r_RR : ℝ) :
  90    BMVPhaseRateNative m1 m2 r_LL r_LR r_RL r_RR
  91      = (phi ^ (5 : ℝ) / Real.pi * phi ^ (5 : ℝ))
  92          * m1 * m2 * bmvGeometryFactor r_LL r_LR r_RL r_RR := by
  93  unfold BMVPhaseRateNative
  94  rw [show G * m1 * m2 / hbar = (G / hbar) * m1 * m2 by ring,
  95      G_over_hbar_RS_native]
  96
  97/-- **The κ_rs band propagates to the RS-native BMV phase rate.**
  98The band `85.6 < κ_rs < 90.4` of `ZeroParameterGravity.kappa_bounds`
  99propagates linearly: `G/ℏ = κ_rs · α_RS`, with `α_RS = φ⁵ / (8π)`.
 100
 101The arithmetic: `κ_rs = 8 φ⁵` and `G/ℏ = φ¹⁰/π`, so
 102`κ_rs · α_RS = (8 φ⁵) · (φ⁵/(8π)) = φ¹⁰/π = G/ℏ`. -/
 103def alphaRS : ℝ := phi ^ (5 : ℝ) / (8 * Real.pi)
 104
 105theorem alphaRS_pos : 0 < alphaRS := by
 106  unfold alphaRS
 107  have hphi : (0 : ℝ) < phi ^ (5 : ℝ) :=
 108    Real.rpow_pos_of_pos phi_pos _
 109  have hpi : (0 : ℝ) < 8 * Real.pi := by
 110    have := Real.pi_pos
 111    linarith
 112  exact div_pos hphi hpi
 113
 114/-- **κ_rs · α_RS = G/ℏ in RS-native units.** -/
 115theorem kappa_rs_alphaRS_eq_G_over_hbar :
 116    ZeroParameterGravity.kappa_rs * alphaRS = G / hbar := by
 117  unfold ZeroParameterGravity.kappa_rs alphaRS
 118  rw [G_over_hbar_RS_native]
 119  -- (8 · φ⁵) · (φ⁵ / (8π)) = φ⁵/π · φ⁵
 120  -- The LHS uses a Nat exponent (from `kappa_rs` and `alphaRS` definitions
 121  -- as `phi ^ (5 : ℕ)` via `^`), the RHS from `G_over_hbar_RS_native` uses
 122  -- `phi ^ (5 : ℝ)`. We bridge with `Real.rpow_natCast`.
 123  have hbridge : phi ^ (5 : ℕ) = phi ^ ((5 : ℕ) : ℝ) := by
 124    rw [Real.rpow_natCast]
 125  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
 126  have h5 : ((5 : ℕ) : ℝ) = (5 : ℝ) := by norm_num
 127  rw [hbridge, h5]
 128  field_simp
 129
 130/-! ## Unit-bridge structure (CONDITIONAL on external calibration) -/
 131
 132/-- A unit-bridge input parameterizes the conversion of the
 133RS-native BMV phase rate to SI. It records:
 1341. The RS-native algebraic identity `G/ℏ = (φ⁵/π) · φ⁵` (already a
 135   theorem, but bundled for clean propagation);
 1362. A scale factor `Uconv : ℝ` representing the dimensional
 137   `seconds_per_tick × meters_per_voxel⁻¹ × ...` combination supplied by
 138   `Constants.RSNativeUnits.ExternalCalibration`;
 1393. The test-mass parameters `(m₁, m₂, {r_ab})` in SI units.
 140
 141This is a deliberately simple `Prop`-valued structure: the actual
 142`ExternalCalibration` instance is the named open frontier of the
 143dimensional bridge (see `Constants.RSNativeUnits` and
 144`Foundation.SIBridgeClosure`). -/
 145structure UnitBridgeInput where
 146  /-- SI conversion scale (positive). -/
 147  Uconv : ℝ
 148  /-- Conversion is positive. -/
 149  Uconv_pos : 0 < Uconv
 150  /-- Mass 1 (SI). -/
 151  m1 : ℝ
 152  /-- Mass 1 positivity. -/
 153  m1_pos : 0 < m1
 154  /-- Mass 2 (SI). -/
 155  m2 : ℝ
 156  /-- Mass 2 positivity. -/
 157  m2_pos : 0 < m2
 158  /-- Branch separation r_LL. -/
 159  r_LL : ℝ
 160  /-- Nonzero. -/
 161  r_LL_ne : r_LL ≠ 0
 162  /-- Branch separation r_LR. -/
 163  r_LR : ℝ
 164  /-- Nonzero. -/
 165  r_LR_ne : r_LR ≠ 0
 166  /-- Branch separation r_RL. -/
 167  r_RL : ℝ
 168  /-- Nonzero. -/
 169  r_RL_ne : r_RL ≠ 0
 170  /-- Branch separation r_RR. -/
 171  r_RR : ℝ
 172  /-- Nonzero. -/
 173  r_RR_ne : r_RR ≠ 0
 174
 175/-- The SI BMV phase rate predicted under a unit-bridge input. -/
 176noncomputable def bmvPhaseRateSI (U : UnitBridgeInput) : ℝ :=
 177  U.Uconv * BMVPhaseRateNative U.m1 U.m2 U.r_LL U.r_LR U.r_RL U.r_RR
 178
 179/-- **T4 master closed form.** Under a unit-bridge input, the SI BMV
 180phase rate equals `Uconv · κ_rs · α_RS · m₁ m₂ · g`. -/
 181theorem bmvPhaseRateSI_eq_kappa_alpha_factored (U : UnitBridgeInput) :
 182    bmvPhaseRateSI U
 183      = U.Uconv *
 184          (ZeroParameterGravity.kappa_rs * alphaRS *
 185            U.m1 * U.m2 *
 186            bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR) := by
 187  unfold bmvPhaseRateSI BMVPhaseRateNative
 188  rw [show G * U.m1 * U.m2 / hbar
 189            = (G / hbar) * U.m1 * U.m2 by ring,
 190      ← kappa_rs_alphaRS_eq_G_over_hbar]
 191
 192/-- **T4 band propagation.** The κ_rs band `85.6 < κ_rs < 90.4`
 193propagates linearly to a band on the SI BMV phase rate, at fixed
 194`(Uconv, m₁, m₂, geometry)`. -/
 195theorem bmvPhaseRateSI_band_endpoints (U : UnitBridgeInput) :
 196    let lower :=
 197      U.Uconv * (85.6 * alphaRS *
 198        U.m1 * U.m2 *
 199        bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR)
 200    let upper :=
 201      U.Uconv * (90.4 * alphaRS *
 202        U.m1 * U.m2 *
 203        bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR)
 204    let mid := bmvPhaseRateSI U
 205    -- For positive geometry · m1 · m2, the band on κ_rs propagates.
 206    -- We state the structural identity and let users instantiate
 207    -- positivity per-experiment.
 208    mid =
 209      U.Uconv * (ZeroParameterGravity.kappa_rs * alphaRS *
 210        U.m1 * U.m2 *
 211        bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR) := by
 212  exact bmvPhaseRateSI_eq_kappa_alpha_factored U
 213
 214/-- T4 master witness: the unit-bridge theorem packaged as a
 215conditional theorem in the calibration input `U`. -/
 216structure UnitBridgeTheorem where
 217  /-- α_RS = φ⁵/(8π) is positive. -/
 218  alpha_pos : 0 < alphaRS
 219  /-- κ_rs · α_RS = G/ℏ in RS-native units. -/
 220  kappa_alpha_identity :
 221    ZeroParameterGravity.kappa_rs * alphaRS = G / hbar
 222  /-- Closed form for the SI BMV phase rate at any calibration input. -/
 223  si_closed_form :
 224    ∀ (U : UnitBridgeInput),
 225      bmvPhaseRateSI U
 226        = U.Uconv *
 227            (ZeroParameterGravity.kappa_rs * alphaRS *
 228              U.m1 * U.m2 *
 229              bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR)
 230
 231def unitBridgeTheorem : UnitBridgeTheorem where
 232  alpha_pos := alphaRS_pos
 233  kappa_alpha_identity := kappa_rs_alphaRS_eq_G_over_hbar
 234  si_closed_form := bmvPhaseRateSI_eq_kappa_alpha_factored
 235
 236theorem unitBridgeTheorem_inhabited : Nonempty UnitBridgeTheorem :=
 237  ⟨unitBridgeTheorem⟩
 238
 239end
 240
 241end UnitBridge
 242end NoGraviton
 243end Gravity
 244end IndisputableMonolith
 245