IndisputableMonolith.RSBridge.Anchor

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# RSBridge Anchor: Fermion Species, Z-Map, and Gap Function
   6
   7This module defines the core bridge between the recognition framework and particle physics:
   8
   91. **Fermion**: The 12 Standard Model fermions (6 quarks + 3 leptons + 3 neutrinos).
  102. **ZOf**: The charge-indexed integer Z_i = q̃² + q̃⁴ (+ 4 for quarks).
  113. **gap (F)**: The display function F(Z) = ln(1 + Z/φ) / ln(φ).
  124. **massAtAnchor**: The mass at the anchor scale μ⋆.
  135. **anchor_ratio**: Family mass ratios are pure φ-powers for equal-Z species.
  14
  15## Relation to RG Transport
  16
  17The `gap` function F(Z) is the **closed form** that the integrated RG residue
  18`f_i(μ⋆)` is claimed to equal at the anchor scale. See:
  19- `Physics/RGTransport.lean` for the mathematical framework of γ and ∫γ.
  20- `Physics/AnchorPolicy.lean` for the axiom `display_identity_at_anchor`.
  21
  22The claim of Single Anchor Phenomenology is:
  23  `(1/ln φ) ∫_{ln μ⋆}^{ln m_phys} γ_i(μ) d(ln μ) ≈ gap(ZOf i)`
  24
  25with tolerance ~1e-6.
  26-/
  27
  28namespace IndisputableMonolith
  29namespace RSBridge
  30
  31inductive Sector | up | down | lepton | neutrino
  32deriving DecidableEq, Repr
  33
  34inductive Fermion
  35| d | s | b | u | c | t | e | mu | tau | nu1 | nu2 | nu3
  36deriving DecidableEq, Repr, Inhabited
  37
  38def sectorOf : Fermion → Sector
  39| .d | .s | .b => .down
  40| .u | .c | .t => .up
  41| .e | .mu | .tau => .lepton
  42| .nu1 | .nu2 | .nu3 => .neutrino
  43
  44def tildeQ : Fermion → ℤ
  45| .u | .c | .t => 4
  46| .d | .s | .b => -2
  47| .e | .mu | .tau => -6
  48| .nu1 | .nu2 | .nu3 => 0
  49
  50def ZOf (f : Fermion) : ℤ :=
  51  let q := tildeQ f
  52  match sectorOf f with
  53  | .up | .down => 4 + q*q + q*q*q*q
  54  | .lepton     =>     q*q + q*q*q*q
  55  | .neutrino   => 0
  56
  57/-- The display function F(Z) = ln(1 + Z/φ) / ln(φ).
  58
  59    This is the **closed form** that the integrated RG residue is claimed to equal
  60    at the anchor scale μ⋆. The claim is:
  61      `f_i(μ⋆) = gap(ZOf i)`
  62    where `f_i(μ⋆)` is the integral of the mass anomalous dimension.
  63
  64    **Properties**:
  65    - F(0) = 0
  66    - F is monotonically increasing for Z > -φ
  67    - For large Z: F(Z) ≈ log_φ(Z)
  68
  69    **Canonical values** (charged fermions):
  70    - F(24)   ≈ 5.74   (down quarks, q̃ = -2)
  71    - F(276)  ≈ 10.69  (up quarks, q̃ = +4)
  72    - F(1332) ≈ 13.95  (leptons, q̃ = -6) -/
  73noncomputable def gap (Z : ℤ) : ℝ :=
  74  (Real.log (1 + (Z : ℝ) / (Constants.phi))) / (Real.log (Constants.phi))
  75
  76notation "𝓕(" Z ")" => gap Z
  77
  78/-- The residue at the anchor for a fermion species.
  79
  80    This is the **definitional** (closed-form) residue: `F(Z_i) = gap(ZOf f)`.
  81
  82    **Relation to the axiom `f_residue`**: The physics claim (stated as an axiom in
  83    `AnchorPolicy.lean`) is that the RG-transport residue equals this value:
  84      `f_residue f μ⋆ = residueAtAnchor f`
  85    This is verified numerically by external tools. -/
  86noncomputable def residueAtAnchor (f : Fermion) : ℝ := gap (ZOf f)
  87
  88theorem anchorEquality (f : Fermion) : residueAtAnchor f = gap (ZOf f) := by rfl
  89
  90theorem equalZ_residue (f g : Fermion) (hZ : ZOf f = ZOf g) :
  91  residueAtAnchor f = residueAtAnchor g := by
  92  simp [residueAtAnchor, hZ]
  93
  94noncomputable def rung : Fermion → ℤ
  95| .e   => 2   | .mu  => 13  | .tau => 19
  96| .u   => 4   | .c   => 15  | .t   => 21
  97| .d   => 4   | .s   => 15  | .b   => 21
  98| .nu1 => 0   | .nu2 => 11  | .nu3 => 19
  99
 100def M0 : ℝ := 1
 101@[simp] theorem M0_pos : 0 < M0 := by
 102  dsimp [M0]; norm_num
 103
 104noncomputable def massAtAnchor (f : Fermion) : ℝ :=
 105  M0 * Real.exp (((rung f : ℝ) - 8 + gap (ZOf f)) * Real.log (Constants.phi))
 106
 107theorem anchor_ratio (f g : Fermion) (hZ : ZOf f = ZOf g) :
 108  massAtAnchor f / massAtAnchor g =
 109    Real.exp (((rung f : ℝ) - rung g) * Real.log (Constants.phi)) := by
 110  unfold massAtAnchor
 111  set Af := ((rung f : ℝ) - 8 + gap (ZOf f)) * Real.log (Constants.phi)
 112  set Ag := ((rung g : ℝ) - 8 + gap (ZOf g)) * Real.log (Constants.phi)
 113  -- Since M0=1, factor cancels directly
 114  calc
 115    (M0 * Real.exp Af) / (M0 * Real.exp Ag)
 116        = (Real.exp Af) / (Real.exp Ag) := by simpa [M0]
 117    _ = Real.exp (Af - Ag) := by
 118              simpa [Real.exp_sub] using (Real.exp_sub Af Ag).symm
 119    _ = Real.exp ((((rung f : ℝ) - 8 + gap (ZOf f)) - ((rung g : ℝ) - 8 + gap (ZOf g)))
 120                   * Real.log (Constants.phi)) := by
 121              have : Af - Ag
 122                    = (((rung f : ℝ) - 8 + gap (ZOf f)) - ((rung g : ℝ) - 8 + gap (ZOf g)))
 123                       * Real.log (Constants.phi) := by
 124                        simp [Af, Ag, sub_eq_add_neg, add_comm, add_left_comm, add_assoc,
 125                              mul_add, add_mul]
 126              have h' :
 127                ((rung f : ℝ) - 8 + gap (ZOf f)) - ((rung g : ℝ) - 8 + gap (ZOf g))
 128                = (rung f : ℝ) - rung g + (gap (ZOf f) - gap (ZOf g)) := by ring
 129              simpa [this, h']
 130    _ = Real.exp (((rung f : ℝ) - rung g) * Real.log (Constants.phi)) := by
 131              simpa [hZ, sub_self, add_zero, add_comm, add_left_comm, add_assoc, mul_add,
 132                     add_right_comm, mul_comm, mul_left_comm, mul_assoc]
 133
 134structure ResidueCert where
 135  f  : Fermion
 136  lo : ℚ
 137  hi : ℚ
 138  lo_le_hi : lo ≤ hi
 139
 140def ResidueCert.valid (c : ResidueCert) : Prop :=
 141  (c.lo : ℝ) ≤ gap (ZOf c.f) ∧ gap (ZOf c.f) ≤ (c.hi : ℝ)
 142
 143/-! ### Generation indexing (three disjoint families) -/
 144
 145/-- Generation index (0,1,2) assigned by rung/sector ordering. -/
 146def genOf : Fermion → Fin 3
 147| .e   => ⟨0, by decide⟩ | .mu  => ⟨1, by decide⟩ | .tau => ⟨2, by decide⟩
 148| .u   => ⟨0, by decide⟩ | .c   => ⟨1, by decide⟩ | .t   => ⟨2, by decide⟩
 149| .d   => ⟨0, by decide⟩ | .s   => ⟨1, by decide⟩ | .b   => ⟨2, by decide⟩
 150| .nu1 => ⟨0, by decide⟩ | .nu2 => ⟨1, by decide⟩ | .nu3 => ⟨2, by decide⟩
 151
 152/-- Surjectivity of the generation index: there are exactly three generations. -/
 153theorem genOf_surjective : Function.Surjective genOf := by
 154  intro i
 155  have h : i.val = 0 ∨ i.val = 1 ∨ i.val = 2 := by
 156    fin_cases i <;> simp
 157  rcases h with h0 | h12
 158  · -- i = 0
 159    refine ⟨Fermion.e, ?_⟩
 160    apply Fin.ext
 161    simp [genOf, h0]
 162  · rcases h12 with h1 | h2
 163    · -- i = 1
 164      refine ⟨Fermion.mu, ?_⟩
 165      apply Fin.ext
 166      simp [genOf, h1]
 167    · -- i = 2
 168      refine ⟨Fermion.tau, ?_⟩
 169      apply Fin.ext
 170      simp [genOf, h2]
 171
 172/-! ### Admissible family encoding via rung residue classes and equal‑Z -/
 173
 174/-- Rung residue class modulo 360 (the joint sync scale of 8‑beat and rung‑45). -/
 175def rungResidueClass (a : ℤ) : Set Fermion :=
 176  { f | Int.ModEq (360 : ℤ) (rung f) a }
 177
 178/-- An admissible family is a set of fermions that share a single Z value
 179    (equal‑Z degeneracy at μ*) and land on a common rung residue class
 180    modulo 360. This packages the “equal‑Z + rung‑offset” policy encoding. -/
 181structure AdmissibleFamily (S : Set Fermion) : Prop where
 182  equalZ_const : ∃ Z0 : ℤ, ∀ {f}, f ∈ S → ZOf f = Z0
 183  rung_residue : ∃ a : ℤ, ∀ {f}, f ∈ S → Int.ModEq (360 : ℤ) (rung f) a
 184
 185
 186end RSBridge
 187end IndisputableMonolith
 188