IndisputableMonolith.Unification.FermionDOFGapBridge

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.BoltzmannConstant
   4import IndisputableMonolith.Cost
   5import IndisputableMonolith.Foundation.DimensionForcing
   6import IndisputableMonolith.Foundation.PhiForcing
   7import IndisputableMonolith.Unification.RecognitionBandwidth
   8
   9/-!
  10# Fermion DOF / Dimension-Gap Arithmetic Identities
  11
  12STATUS (re-scoped after the 2026-06-25 external review, Thapa, "Comment on
  13the RS Relativistic Degree Count"):
  14
  15This module proves **arithmetic identities** relating the imported Standard
  16Model degree-of-freedom counts to D = 3 combinatorial quantities. It does
  17NOT derive the Standard Model spectrum. Earlier language in this file
  18("pure D=3 formula", "zero empirical inputs", "master theorem",
  19"CONFIRMED in 2D conductors") was overclaiming and has been removed.
  20
  21## Derived vs imported (the honest split)
  22
  23- **IMPORTED (standard physics, not RS results)**: the SM matter
  24  representations, the minimal-neutrino convention (g_f = 90), the
  25  Fermi-Dirac thermal integral giving the 7/8 weight, and the
  26  high-temperature scope of g_star = 106.75. See
  27  `StandardModel.RelativisticDOF` for the full derived-vs-imported split
  28  and `Cosmology.GStarThresholds` for the temperature-dependent g_star(T).
  29- **RS-DERIVED (upstream theorems, cited not re-proved here)**: D = 3
  30  (Foundation.DimensionForcing / T8), the 8-tick period 2^D = 8
  31  (Foundation.EightTick), and the generation count 3 (Q₃ face pairs).
  32  The spin-statistics EXCHANGE SIGN (fermion −1 / boson +1 under 2π) is
  33  derived in `Foundation.SpinStatistics` (this repository); note
  34  that module proves the SIGN of the statistics, NOT the 7/8 thermal
  35  weight, which is the standard Fermi/Bose integral ratio and is imported.
  36- **PROVED HERE (exact kernel-checked arithmetic on the counts above)**:
  37  90 = 2 × dimensionGap(3), 7/8 = (2³−1)/2³, and the assembled identity
  38  28 + (7/8) × 90 = 106.75.
  39
  40## What these identities are, and are not
  41
  42The identities `fermionic_dof = 2 × D²(D+2)` and
  43`fermi_dirac_weight = (2^D − 1)/2^D` are re-expressions of already-known
  44numbers in D-flavored notation, verified by the kernel. They are exact and
  45machine-checked. But a re-expression obtained AFTER the target number is
  46known is not a derivation of that number. To make g_star = 106.75 an RS
  47derivation one would have to derive the gauge representations, the Higgs
  48doublet, the chiral neutrino content, and the spin-statistics thermal
  49integral from RS premises. None of that is done here or elsewhere in this
  50repository; only the gauge GROUP, the generation COUNT, and the exchange
  51SIGN have RS-side theorems.
  52
  53Whether the numerical coincidences recorded here (90 = 2 × 45 with 45 the
  54η_B-adjacent dimension gap; 7/8 = (8−1)/8 with 8 the tick period) reflect
  55structure or accident is an OPEN question. This module records the exact
  56arithmetic so the question is precisely posed; it does not answer it.
  57
  58## On the "g_star(D)" function (§5)
  59
  60`g_star_D d` varies ONLY the thermal weight (2^d − 1)/2^d. The bosonic
  61count 28 and the fermionic count 90 remain frozen at their D = 3 Standard
  62Model values. It is therefore NOT a variable-dimension physics formula and
  63no genuine g_star(D) theory is claimed. A real one would require deriving
  64the matter content at each D, which is not available.
  65
  66## On the D = 2 comparison (§8)
  67
  68An earlier version claimed the D = 2 weight 3/4 is "CONFIRMED in 2D
  69conductors". That was a category error: condensed-matter 2D electron gases
  70are nonrelativistic Fermi systems with a chemical potential; the g_star
  71count is a relativistic thermal-plasma object. The D = 2 and D = 4 rows
  72below are counterfactual arithmetic evaluations of the same expressions,
  73kept because they make the D-dependence of the FORMULAS explicit. They are
  74not experimental confirmations, and no clean falsifier via 2D materials is
  75claimed.
  76
  77## Epistemic status summary
  78
  79- 90 = 2 × dimensionGap(3), 7/8 = (2³−1)/2³, 106.75 assembly: THEOREM
  80  (exact arithmetic on imported counts; the counts themselves are inputs).
  81- D = 3, 8-tick, 3 generations, exchange sign: THEOREM upstream (cited).
  82- The "identity tick" reading of 7/8 (§7) and the matter/φ⁴⁵ duality gloss
  83  (§6): HYPOTHESIS — interpretive narratives attached to the arithmetic,
  84  with no theorem forcing them.
  85- g_star as an RS prediction: NOT CLAIMED. The number is standard SM
  86  bookkeeping; see `StandardModel.RelativisticDOF`.
  87-/
  88
  89namespace IndisputableMonolith
  90namespace Unification
  91namespace FermionDOFGapBridge
  92
  93open Constants
  94
  95/-! ## §1. Dimension and Gap -/
  96
  97/-- The spatial dimension forced by T8. -/
  98def D : ℕ := 3
  99
 100/-- The 8-tick period: the fundamental cadence of R̂. -/
 101def eightTick : ℕ := 2 ^ D
 102
 103theorem eightTick_eq : eightTick = 8 := by native_decide
 104
 105/-- The dimension gap function: D²(D+2).
 106    At D=3: dimensionGap = 9 × 5 = 45. -/
 107def dimensionGap (d : ℕ) : ℕ := d ^ 2 * (d + 2)
 108
 109theorem dimensionGap_at_D3 : dimensionGap D = 45 := by native_decide
 110
 111theorem dimensionGap_positive (d : ℕ) (hd : 0 < d) : 0 < dimensionGap d := by
 112  unfold dimensionGap
 113  apply Nat.mul_pos
 114  · exact Nat.pos_of_ne_zero (by positivity)
 115  · omega
 116
 117/-! ## §2. Fermionic DOF = 2 × dimensionGap (arithmetic identity) -/
 118
 119/-- Fermionic DOF per generation: the IMPORTED Standard Model count.
 120
 121    Quarks: 2 flavors × 3 colors × 2 chiralities × 2 (particle+antiparticle)
 122      = 24.
 123    Leptons: charged lepton 2 chiralities × 2 = 4, plus the left-handed
 124      neutrino × 2 = 2 (minimal-neutrino convention, no thermalized
 125      right-handed states); total 6.
 126    Per generation: 24 + 6 = 30.
 127
 128    The re-expression 30 = 2·D·(D+2) at D = 3 is exact arithmetic on this
 129    imported count, not a derivation of the representation content. -/
 130def dof_per_gen : ℕ := 30
 131
 132theorem dof_per_gen_eq : dof_per_gen = 2 * D * (D + 2) := by native_decide
 133
 134/-- Number of generations = 3. The count matches D via the Q₃ face-pair
 135    argument (Foundation.ParticleGenerations in the parent repository). -/
 136def n_generations : ℕ := 3  -- = D
 137
 138theorem n_generations_eq_D : n_generations = D := rfl
 139
 140/-- Total fermionic DOF: 3 generations × 30 DOF/gen = 90.
 141    Standard Model bookkeeping (minimal-neutrino convention); with
 142    thermalized right-handed Dirac partners this would be 96, and
 143    g_star would be 112 — see StandardModel.RelativisticDOF (g_star_dirac). -/
 144def fermionic_dof : ℕ := n_generations * dof_per_gen
 145
 146theorem fermionic_dof_eq : fermionic_dof = 90 := by native_decide
 147
 148/-- **ARITHMETIC IDENTITY**: fermionic_dof = 2 × dimensionGap(3).
 149
 150    90 = 2 × 45, i.e. the (imported) SM fermionic DOF count equals twice
 151    D²(D+2) at D = 3:
 152
 153    fermionic_dof = n_gen × dof_per_gen = 3 × 30 = 90 = 2 × 45.
 154
 155    This is a kernel-checked re-expression of two known integers, recorded
 156    because it is exact. It is NOT a derivation of the SM fermion content
 157    from RS premises, and no such derivation is claimed. Whether the match
 158    is structural or accidental is OPEN. -/
 159theorem fermionic_dof_eq_twice_gap :
 160    fermionic_dof = 2 * dimensionGap D := by native_decide
 161
 162/-- Corollary of the arithmetic: 90 splits as 45 + 45. The particle /
 163    antiparticle reading of the two halves is interpretation (each sector
 164    does contribute half the count), but nothing here derives the split
 165    from the dimension gap. -/
 166theorem fermionic_matter_antimatter_split :
 167    fermionic_dof = dimensionGap D + dimensionGap D := by
 168  have := fermionic_dof_eq_twice_gap
 169  omega
 170
 171/-! ## §3. The 7/8 weight: imported thermal integral, exact arithmetic here -/
 172
 173/-- The Fermi-Dirac thermal weight expression (2^d − 1)/2^d.
 174
 175    PROVENANCE: in 3+1 dimensions the fermion/boson thermal energy-density
 176    ratio is the standard Fermi/Bose integral result
 177      ∫ x³/(eˣ+1) dx / ∫ x³/(eˣ−1) dx = 1 − 2⁻³ = 7/8,
 178    i.e. (1 − 2^(−D)) at D = 3. That analytic integral is IMPORTED standard
 179    statistical mechanics; it is not proved in Lean here or elsewhere in
 180    this repository. What this module proves is the exact arithmetic of the
 181    expression (2^d − 1)/2^d at specific d.
 182
 183    The coincidence that the same expression can be read as
 184    (tick period − 1)/(tick period) at 2^D = 8 is recorded in §7 as a
 185    HYPOTHESIS-grade interpretation, not a derivation. -/
 186noncomputable def fermi_dirac_weight_D (d : ℕ) : ℝ :=
 187  ((2 : ℝ)^d - 1) / (2 : ℝ)^d
 188
 189/-- At D=3, the expression evaluates to 7/8 (exact arithmetic). -/
 190theorem fermi_dirac_weight_D3 : fermi_dirac_weight_D D = 7 / 8 := by
 191  unfold fermi_dirac_weight_D D
 192  norm_num
 193
 194/-- Numerical identity: (2^D − 1)/2^D = (eightTick − 1)/eightTick at D = 3.
 195    Both sides are the same number because eightTick := 2^D; this equation
 196    is bookkeeping, not new physics. -/
 197theorem fermi_dirac_from_eight_tick :
 198    fermi_dirac_weight_D D = ((eightTick - 1 : ℕ) : ℝ) / ((eightTick : ℕ) : ℝ) := by
 199  rw [fermi_dirac_weight_D3, eightTick_eq]
 200  norm_num
 201
 202/-- The expression at d = 2: 3/4. This matches the RELATIVISTIC thermal
 203    integral in 2+1 dimensions (1 − 2⁻²); it is NOT the nonrelativistic 2D
 204    electron-gas result from condensed matter, and no experimental
 205    confirmation via 2D conductors is claimed (see module header). -/
 206theorem fermi_weight_in_D2 : fermi_dirac_weight_D 2 = 3 / 4 := by
 207  unfold fermi_dirac_weight_D; norm_num
 208
 209/-- The expression at d = 4: 15/16 (counterfactual arithmetic). -/
 210theorem fermi_weight_in_D4 : fermi_dirac_weight_D 4 = 15 / 16 := by
 211  unfold fermi_dirac_weight_D; norm_num
 212
 213/-! ## §4. Bosonic DOF: imported SM content, D-flavored bookkeeping -/
 214
 215/-- Gluons: SU(3) adjoint = 8 generators × 2 polarizations = 16.
 216    The re-expression 2(D²−1) uses SU(D) at D = 3; the choice of SU(3)
 217    color is imported SM content (the gauge GROUP has an RS-side argument;
 218    the representation assignments do not). -/
 219def gluon_dof : ℕ := 2 * (D ^ 2 - 1)
 220
 221theorem gluon_dof_eq : gluon_dof = 16 := by native_decide
 222
 223/-- EW gauge bosons (T > T_EW, unbroken phase): SU(2) × U(1) = 4 generators
 224    × 2 transverse polarizations = 8. The re-expression 2(D−1)² is
 225    D-flavored bookkeeping of the imported content. -/
 226def ew_boson_dof : ℕ := 2 * ((D - 1) ^ 2)
 227
 228theorem ew_boson_dof_eq : ew_boson_dof = 8 := by native_decide
 229
 230/-- Higgs doublet: complex SU(2) doublet = 4 real DOF; re-expressed 2(D−1). -/
 231def higgs_dof : ℕ := 2 * (D - 1)
 232
 233theorem higgs_dof_eq : higgs_dof = 4 := by native_decide
 234
 235/-- Total bosonic DOF: 16 + 8 + 4 = 28 (standard SM high-T count). -/
 236def bosonic_dof : ℕ := gluon_dof + ew_boson_dof + higgs_dof
 237
 238theorem bosonic_dof_eq : bosonic_dof = 28 := by native_decide
 239
 240/-- The D-flavored polynomial re-expression: 28 = 4D² − 2D − 2 at D = 3.
 241    This packages the three imported counts above into one polynomial; it
 242    does not derive the SM boson content from RS premises. -/
 243theorem bosonic_dof_eq_poly :
 244    bosonic_dof = 4 * D ^ 2 - 2 * D - 2 := by native_decide
 245
 246/-! ## §5. The assembled g_star identity -/
 247
 248/-- g_star assembly with the thermal weight evaluated at `d`.
 249
 250    HONEST SCOPE (per the 2026-06-25 review): this function varies ONLY the
 251    weight (2^d − 1)/2^d. The bosonic count (28) and fermionic count (90)
 252    are frozen at their D = 3 Standard Model values, so `g_star_D` is NOT a
 253    variable-dimension physics formula and no g_star(D) theory is claimed.
 254    A genuine one would need the matter representations derived at each D,
 255    which this repository does not have.
 256
 257    At d = 3 it reproduces the standard high-temperature SM value 106.75
 258    (see StandardModel.RelativisticDOF for the derived-vs-imported split,
 259    and Cosmology.GStarThresholds for the temperature dependence g_star(T),
 260    which is the physically meaningful variation). -/
 261noncomputable def g_star_D (d : ℕ) : ℝ :=
 262  (bosonic_dof : ℝ) + fermi_dirac_weight_D d * (fermionic_dof : ℝ)
 263
 264/-- The assembled identity 28 + (7/8) × 90 = 106.75 (exact arithmetic on
 265    the imported counts; the standard high-T SM value, not a new number). -/
 266theorem g_star_D3_eq : g_star_D D = 106.75 := by
 267  unfold g_star_D
 268  rw [fermi_dirac_weight_D3, fermionic_dof_eq, bosonic_dof_eq]
 269  norm_num
 270
 271theorem g_star_D3_positive : 0 < g_star_D D := by
 272  rw [g_star_D3_eq]; norm_num
 273
 274/-- Decomposition through the gap identity: g_star = 28 + (7/8)(2 × 45).
 275    Follows from `fermionic_dof_eq_twice_gap`; same arithmetic, gap-flavored. -/
 276theorem g_star_via_gap :
 277    g_star_D D = (bosonic_dof : ℝ) +
 278                 fermi_dirac_weight_D D * (2 * dimensionGap D) := by
 279  unfold g_star_D
 280  have := fermionic_dof_eq_twice_gap
 281  push_cast [this]
 282  ring
 283
 284/-! ## §6. The 45 ↔ φ⁴⁵ numerology (HYPOTHESIS-grade gloss)
 285
 286The η_B rung −44 = 1 − 45 and the φ⁴⁵ saturation scale share the integer
 28745 = D²(D+2) with half the fermionic DOF count (90/2). Per the review:
 288these are bookkeeping re-expressions of the SAME integer, not independent
 289confirmations of the rung, and none of them is a mechanism that forces it.
 290The arithmetic below is exact; the physical gloss ("matter/antimatter
 291balance broken by η_B") is interpretation with no supporting theorem. -/
 292
 293/-- Arithmetic: half the fermionic DOF equals dimensionGap(3) (45 = 45).
 294    A re-expression of one integer, recorded exactly; not independent
 295    evidence for the η_B rung (see section header). -/
 296theorem matter_phi45_complementarity :
 297    fermionic_dof / 2 = dimensionGap D := by native_decide
 298
 299/-- Rung bookkeeping: (1 − 45) + 45 = 1. Trivial integer arithmetic linking
 300    the DEFINED rung assignments −44 and 45; proves no physical mechanism. -/
 301theorem rung_sum_equals_one :
 302    (1 : ℤ) - (dimensionGap D : ℤ) + dimensionGap D = 1 := by omega
 303
 304/-! ## §7. The identity-tick reading of 7/8 (HYPOTHESIS)
 305
 306Interpretive picture: in each 8-tick cycle one tick is the balanced
 307"identity tick" (σ = 0, J(1) = 0); bosons can occupy it, fermions (carrying
 308half-integer σ) cannot, so fermions access 7 of 8 ticks, matching the 7/8
 309thermal weight.
 310
 311STATUS: HYPOTHESIS. The 7/8 weight's actual provenance is the Fermi/Bose
 312thermal integral (imported; §3). The tick-fraction reading below reproduces
 313the same number by construction — (8−1)/8 — and the exchange-sign half of
 314the story (fermion −1 under 2π) IS derived upstream
 315(`Foundation.SpinStatistics`, this repository). But no theorem connects
 316tick-occupancy counting to the thermal integral, so the identification of
 317the two 7/8's is an interpretation, not a result. -/
 318
 319/-- One balanced tick per 8-tick cycle (definition used by the gloss). -/
 320def identity_tick_count : ℕ := 1
 321def available_ticks_boson : ℕ := eightTick       -- gloss: all 8 ticks
 322def available_ticks_fermion : ℕ := eightTick - 1  -- gloss: excluded from one
 323
 324theorem fermion_missing_identity_tick :
 325    available_ticks_fermion = eightTick - identity_tick_count := by
 326  unfold available_ticks_fermion identity_tick_count
 327  rfl
 328
 329/-- The tick fraction (8−1)/8 numerically equals the thermal weight 7/8.
 330    Both sides are the same rational by construction; the equation records
 331    the numerical coincidence the §7 gloss is built on, nothing more. -/
 332theorem fermi_weight_is_tick_fraction :
 333    (available_ticks_fermion : ℝ) / eightTick = fermi_dirac_weight_D D := by
 334  unfold available_ticks_fermion eightTick fermi_dirac_weight_D D
 335  norm_num
 336
 337/-! ## §8. Counterfactual D-evaluations (NOT confirmed falsifiers) -/
 338
 339/-- Counterfactual arithmetic at D = 2: dimensionGap(2) = 16 and the weight
 340    expression gives 3/4. The 3/4 matches the RELATIVISTIC 2+1-dimensional
 341    thermal integral; it is NOT confirmed by 2D conductors (nonrelativistic
 342    Fermi gases with a chemical potential are a different object — the
 343    earlier "CONFIRMED" claim was a category error and is withdrawn). -/
 344theorem D2_evaluation :
 345    dimensionGap 2 = 16 ∧ fermi_dirac_weight_D 2 = 3 / 4 :=
 346  ⟨by native_decide, by unfold fermi_dirac_weight_D; norm_num⟩
 347
 348/-- Counterfactual arithmetic at D = 4: gap 96, weight 15/16. No physical
 349    system is claimed to realize this; it displays the D-dependence of the
 350    expressions only. -/
 351theorem D4_evaluation :
 352    dimensionGap 4 = 96 ∧ fermi_dirac_weight_D 4 = 15 / 16 :=
 353  ⟨by native_decide, by unfold fermi_dirac_weight_D; norm_num⟩
 354
 355/-- Deprecated alias (old name overstated the epistemic status). -/
 356@[deprecated D2_evaluation (since := "2026-07-02")]
 357theorem D2_prediction :
 358    dimensionGap 2 = 16 ∧ fermi_dirac_weight_D 2 = 3 / 4 := D2_evaluation
 359
 360/-- Deprecated alias (old name overstated the epistemic status). -/
 361@[deprecated D4_evaluation (since := "2026-07-02")]
 362theorem D4_prediction :
 363    dimensionGap 4 = 96 ∧ fermi_dirac_weight_D 4 = 15 / 16 := D4_evaluation
 364
 365/-! ## §9. Certificate (arithmetic identities only) -/
 366
 367/-- **FERMION DOF / DIMENSION-GAP ARITHMETIC CERTIFICATE**
 368
 369    Kernel-checked arithmetic identities on the imported SM counts:
 370    1. dimensionGap(3) = 45
 371    2. fermionic_dof = 90 = 2 × dimensionGap(3)
 372    3. weight expression at D=3 = 7/8
 373    4. assembled g_star = 28 + (7/8)×90 = 106.75
 374    5. 90/2 = 45
 375    6. counterfactual D=2 evaluation (16, 3/4)
 376
 377    NOT certified (and not claimed): a derivation of the SM spectrum, a
 378    variable-D g_star theory, the thermal integral itself, the identity-tick
 379    mechanism, or independent evidence for the η_B rung. See module header
 380    for the full derived-vs-imported split. -/
 381theorem fermion_dof_gap_certificate :
 382    dimensionGap D = 45 ∧
 383    fermionic_dof = 2 * dimensionGap D ∧
 384    fermi_dirac_weight_D D = 7 / 8 ∧
 385    g_star_D D = 106.75 ∧
 386    fermionic_dof / 2 = dimensionGap D ∧
 387    (dimensionGap 2 = 16 ∧ fermi_dirac_weight_D 2 = 3 / 4) := by
 388  refine ⟨
 389    dimensionGap_at_D3,
 390    fermionic_dof_eq_twice_gap,
 391    fermi_dirac_weight_D3,
 392    g_star_D3_eq,
 393    matter_phi45_complementarity,
 394    D2_evaluation
 395  ⟩
 396
 397end FermionDOFGapBridge
 398end Unification
 399end IndisputableMonolith
 400