IndisputableMonolith.Constants.RSNativeUnits

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.KDisplayCore
   4import IndisputableMonolith.Constants.Alpha
   5
   6namespace IndisputableMonolith
   7namespace Constants
   8namespace RSNativeUnits
   9
  10/-!
  11# RS-Native Measurement System (Tick/Voxel/Coh/Act)
  12
  13This module defines a **Recognition-Science-native** unit system that treats the
  14ledger primitives as the base standards. All physics can be expressed in these
  15units without reference to SI or any external anchoring.
  16
  17## Fundamental Base Units
  18
  19- **tick**  (τ₀): one discrete ledger posting interval (atomic time quantum)
  20- **voxel** (ℓ₀): one causal spatial step (distance light traverses in one tick)
  21
  22## Derived Quanta
  23
  24- **coh** (coherence quantum): E_coh = φ⁻⁵ (fundamental energy quantum)
  25- **act** (action quantum): ħ = E_coh · τ₀ (Planck constant equivalent)
  26
  27## Key Properties
  28
  291. **c = 1** (speed of light is unity in voxel/tick)
  302. All dimensionless ratios are fixed by φ alone
  313. SI conversion is explicit and optional (via `ExternalCalibration`)
  32
  33## The φ-Ladder
  34
  35RS measures are organized on a φ-ladder: φⁿ for integer n provides the
  36natural scaling for masses, energies, times, and lengths.
  37-/
  38
  39/-! ## Base unit type aliases
  40
  41For now these are `ℝ`-valued counts. The semantics are:
  42- `Time`      measured in **ticks** (τ₀)
  43- `Length`    measured in **voxels** (ℓ₀)
  44- `Velocity`  measured in **voxels per tick**
  45- `Energy`    measured in **coherence quanta** (coh)
  46- `Action`    measured in **action quanta** (act = ħ)
  47- `Mass`      measured in **coh/c²** (mass-energy equivalent)
  48- `Frequency` measured in **1/tick** (inverse time)
  49- `Momentum`  measured in **coh/c** (p = E/c)
  50- `Charge`    measured in **recognition units** (dimensionless in RS)
  51-/
  52
  53abbrev Time := ℝ
  54abbrev Length := ℝ
  55abbrev Velocity := ℝ
  56abbrev Energy := ℝ
  57abbrev Action := ℝ
  58abbrev Mass := ℝ
  59abbrev Frequency := ℝ
  60abbrev Momentum := ℝ
  61abbrev Charge := ℝ
  62
  63/-! ## Canonical RS-native base units -/
  64
  65/-- One tick: the fundamental time quantum. -/
  66@[simp] def tick : Time := Constants.tick
  67
  68/-- One voxel: the fundamental length quantum. -/
  69@[simp] def voxel : Length := 1
  70
  71/-! ## Derived speed of light in RS-native units -/
  72
  73/-- Speed of light: c = ℓ₀/τ₀ = 1 voxel/tick. -/
  74@[simp] def c : Velocity := 1
  75
  76/-! ## Canonical `RSUnits` pack for bridge/cert code -/
  77
  78/-- RS-native gauge: τ₀ = 1 tick, ℓ₀ = 1 voxel, c = 1. -/
  79@[simp] noncomputable def U : RSUnits :=
  80{ tau0 := tick
  81, ell0 := voxel
  82, c := c
  83, c_ell0_tau0 := by simp [tick, voxel, c] }
  84
  85@[simp] lemma U_tau0 : U.tau0 = 1 := rfl
  86@[simp] lemma U_ell0 : U.ell0 = 1 := rfl
  87@[simp] lemma U_c : U.c = 1 := rfl
  88
  89/-! ## Coherence and action quanta
  90
  91`Constants.E_coh = φ⁻⁵` is a **dimensionless** RS-derived number.
  92In the RS-native system:
  93- **1 coh** (energy quantum) := E_coh
  94- **1 act** (action quantum) := ħ = E_coh · τ₀ = E_coh (since τ₀ = 1)
  95-/
  96
  97/-- Coherence energy quantum: φ⁻⁵ ≈ 0.0902. -/
  98@[simp] noncomputable def cohQuantum : ℝ := E_coh
  99
 100/-- Convert energy count (in coh) to raw RS scale. -/
 101@[simp] noncomputable def energy_raw (E : Energy) : ℝ := E * cohQuantum
 102
 103/-- Action quantum: ħ = E_coh · τ₀ = E_coh in RS-native units. -/
 104@[simp] noncomputable def hbarQuantum : ℝ := cohQuantum * tick
 105
 106/-- Convert action count (in act) to raw RS scale. -/
 107@[simp] noncomputable def action_raw (A : Action) : ℝ := A * hbarQuantum
 108
 109@[simp] lemma hbarQuantum_eq_Ecoh : hbarQuantum = E_coh := by
 110  simp [hbarQuantum, cohQuantum, tick]
 111
 112/-! ## Mass quantum (derived from E = mc²)
 113
 114In RS-native units with c = 1:
 115- Mass = Energy (mass-energy equivalence)
 116- 1 mass quantum = 1 coh (in c=1 units)
 117-/
 118
 119/-- Mass quantum: 1 coh/c² = 1 coh (since c = 1). -/
 120@[simp] noncomputable def massQuantum : ℝ := cohQuantum
 121
 122/-- Convert mass count to raw RS scale. -/
 123@[simp] noncomputable def mass_raw (m : Mass) : ℝ := m * massQuantum
 124
 125/-! ## Frequency and momentum quanta -/
 126
 127/-- Frequency quantum: 1/tick (inverse of fundamental time). -/
 128@[simp] noncomputable def freqQuantum : Frequency := 1 / tick
 129
 130/-- Convert frequency to raw RS scale. -/
 131@[simp] noncomputable def freq_raw (f : Frequency) : ℝ := f * freqQuantum
 132
 133/-- Momentum quantum: E_coh/c = E_coh (since c = 1). -/
 134@[simp] noncomputable def momentumQuantum : ℝ := cohQuantum / c
 135
 136/-- Convert momentum to raw RS scale. -/
 137@[simp] noncomputable def momentum_raw (p : Momentum) : ℝ := p * momentumQuantum
 138
 139@[simp] lemma momentumQuantum_eq_cohQuantum : momentumQuantum = cohQuantum := by
 140  simp [momentumQuantum, c]
 141
 142/-! ## The φ-Ladder: Scaling in RS
 143
 144All RS quantities are organized on a φ-ladder. The ladder provides:
 145- Mass rungs: m_n = m₀ · φⁿ
 146- Time rungs: τ_n = τ₀ · φⁿ
 147- Length rungs: ℓ_n = ℓ₀ · φⁿ
 148- Energy rungs: E_n = E₀ · φⁿ
 149-/
 150
 151/-- φ-ladder scaling: compute φⁿ for integer rung. -/
 152@[simp] noncomputable def phiRung (n : ℤ) : ℝ := phi ^ n
 153
 154/-- Scale any quantity by n rungs on the φ-ladder. -/
 155@[simp] noncomputable def scaleByPhi (x : ℝ) (n : ℤ) : ℝ := x * phiRung n
 156
 157lemma phiRung_pos (n : ℤ) : 0 < phiRung n := by
 158  simp [phiRung]
 159  exact zpow_pos phi_pos n
 160
 161lemma phiRung_zero : phiRung 0 = 1 := by simp [phiRung]
 162
 163lemma phiRung_one : phiRung 1 = phi := by simp [phiRung]
 164
 165lemma phiRung_neg_one : phiRung (-1) = 1 / phi := by
 166  simp only [phiRung, zpow_neg_one, one_div]
 167
 168lemma phiRung_add (m n : ℤ) : phiRung (m + n) = phiRung m * phiRung n := by
 169  simp only [phiRung]
 170  exact zpow_add₀ phi_ne_zero m n
 171
 172lemma phiRung_neg (n : ℤ) : phiRung (-n) = 1 / phiRung n := by
 173  simp only [phiRung, one_div]
 174  rw [zpow_neg]
 175
 176/-! ## The 8-Tick Cycle
 177
 178The fundamental ledger cycle has period 8 = 2³ (forced by D=3).
 179This defines the octave structure of RS.
 180-/
 181
 182/-- The octave period: 8 ticks. -/
 183@[simp] def octavePeriod : Time := 8 * tick
 184
 185/-- The breath cycle: 1024 ticks (8 × 128 = 8 × 2⁷). -/
 186@[simp] def breathCycle : Time := 1024 * tick
 187
 188/-- Convert tick count to octave count (integer division). -/
 189def ticksToOctaves (t : ℕ) : ℕ := t / 8
 190
 191/-- Phase within an octave (0..7). -/
 192def octavePhase (t : ℕ) : Fin 8 := ⟨t % 8, Nat.mod_lt t (by norm_num)⟩
 193
 194/-! ## Gap-45 Synchronization
 195
 196The gap-45 rung (φ⁴⁵) provides critical phase synchronization:
 197- lcm(8, 45) = 360 synchronizes the octave and gap cycles
 198- This forces D = 3 as the unique dimension with this property
 199-/
 200
 201/-- The gap-45 rung: φ⁴⁵. -/
 202@[simp] noncomputable def gap45 : ℝ := phiRung 45
 203
 204/-- The synchronization period: lcm(8, 45) = 360. -/
 205@[simp] def syncPeriod : ℕ := 360
 206
 207lemma syncPeriod_eq_lcm : syncPeriod = Nat.lcm 8 45 := by native_decide
 208
 209/-! ## K-gate derived displays in RS-native units -/
 210
 211/-- Recognition time display: τ_rec = (2π)/(8 ln φ) · τ₀. -/
 212@[simp] noncomputable def tau_rec : Time :=
 213  Constants.RSUnits.tau_rec_display U
 214
 215/-- Kinematic wavelength display: λ_kin = (2π)/(8 ln φ) · ℓ₀. -/
 216@[simp] noncomputable def lambda_kin : Length :=
 217  Constants.RSUnits.lambda_kin_display U
 218
 219theorem tau_rec_eq_K_gate_ratio :
 220    tau_rec = Constants.RSUnits.K_gate_ratio := by
 221  unfold tau_rec
 222  have hlog : Real.log phi ≠ 0 := ne_of_gt (Real.log_pos one_lt_phi)
 223  simp [Constants.RSUnits.tau_rec_display, Constants.RSUnits.K_gate_ratio, U, tick]
 224  field_simp [hlog]
 225  ring
 226
 227theorem lambda_kin_eq_K_gate_ratio :
 228    lambda_kin = Constants.RSUnits.K_gate_ratio := by
 229  unfold lambda_kin
 230  have hlog : Real.log phi ≠ 0 := ne_of_gt (Real.log_pos one_lt_phi)
 231  simp [Constants.RSUnits.lambda_kin_display, Constants.RSUnits.K_gate_ratio, U, voxel]
 232  field_simp [hlog]
 233  ring
 234
 235/-! ## Planck-Scale Quantities (RS-derived)
 236
 237In RS, the Planck scale emerges from the gate identities, not as a postulate.
 238These are expressed in RS-native units.
 239-/
 240
 241/-- Planck time in RS units: τ_P = √(ħG/c⁵).
 242    In RS-native units, this is a dimensionless φ-expression. -/
 243noncomputable def planckTime_rs : Time :=
 244  -- Using the gate identity: τ_P = τ₀ · √(G/c³) where G,c are in RS units
 245  -- Since c = 1, and G is derived, this is pure φ-structure
 246  tick * Real.sqrt (Constants.G / (c ^ 3))
 247
 248/-- Planck length in RS units: ℓ_P = c · τ_P. -/
 249noncomputable def planckLength_rs : Length :=
 250  c * planckTime_rs
 251
 252/-- Planck mass in RS units: m_P = √(ħc/G).
 253    This is the mass at which gravitational and quantum scales meet. -/
 254noncomputable def planckMass_rs : Mass :=
 255  Real.sqrt (hbarQuantum * c / Constants.G)
 256
 257/-- Planck energy in RS units: E_P = m_P · c² = m_P (since c = 1). -/
 258noncomputable def planckEnergy_rs : Energy := planckMass_rs
 259
 260/-! ## Dimensionless Ratios (Fixed by φ)
 261
 262These ratios are the same in any unit system - they are the "physics" of RS.
 263-/
 264
 265/-- Fine structure constant inverse: α⁻¹ ≈ 137.036. -/
 266noncomputable def alphaInv_rs : ℝ := Constants.alphaInv
 267
 268/-- The K-gate ratio: K = π/(4 ln φ). -/
 269noncomputable def K_rs : ℝ := Constants.RSUnits.K_gate_ratio
 270
 271/-- Coherence scaling: E_coh = φ⁻⁵. -/
 272noncomputable def E_coh_rs : ℝ := phiRung (-5)
 273
 274lemma E_coh_rs_eq_E_coh : E_coh_rs = E_coh := by
 275  simp only [E_coh_rs, phiRung, E_coh, cLagLock]
 276  -- Both are phi^(-5), but one uses zpow and the other uses rpow
 277  -- φ^(-5 : ℤ) = φ^(-5 : ℝ) for φ > 0
 278  have h : phi ^ (-5 : ℤ) = phi ^ ((-5 : ℤ) : ℝ) := by
 279    rw [← Real.rpow_intCast phi (-5)]
 280  rw [h]
 281  congr 1
 282  norm_cast
 283
 284/-! ## External Calibration (SI Bridge)
 285
 286If you want "seconds" and "meters", you must provide an explicit calibration
 287mapping RS primitives to an external unit system. This keeps the empirical
 288seam explicit and auditable.
 289-/
 290
 291/-- External calibration structure for mapping RS units to SI/other systems. -/
 292structure ExternalCalibration where
 293  /-- Seconds per tick (τ₀ in seconds). -/
 294  seconds_per_tick : ℝ
 295  /-- Meters per voxel (ℓ₀ in meters). -/
 296  meters_per_voxel : ℝ
 297  /-- Joules per coherence quantum. -/
 298  joules_per_coh : ℝ
 299  /-- All conversion factors are positive. -/
 300  seconds_pos : 0 < seconds_per_tick
 301  meters_pos : 0 < meters_per_voxel
 302  joules_pos : 0 < joules_per_coh
 303  /-- Consistency: c = ℓ₀/τ₀ must equal 299792458 m/s in SI. -/
 304  speed_consistent : meters_per_voxel / seconds_per_tick = 299792458
 305
 306/-- Convert time (in ticks) to seconds. -/
 307@[simp] noncomputable def to_seconds (cal : ExternalCalibration) (t : Time) : ℝ :=
 308  t * cal.seconds_per_tick
 309
 310/-- Convert length (in voxels) to meters. -/
 311@[simp] noncomputable def to_meters (cal : ExternalCalibration) (L : Length) : ℝ :=
 312  L * cal.meters_per_voxel
 313
 314/-- Convert velocity (in voxels/tick) to m/s. -/
 315@[simp] noncomputable def to_m_per_s (cal : ExternalCalibration) (v : Velocity) : ℝ :=
 316  v * (cal.meters_per_voxel / cal.seconds_per_tick)
 317
 318/-- Convert energy (in coh) to Joules. -/
 319@[simp] noncomputable def to_joules (cal : ExternalCalibration) (E : Energy) : ℝ :=
 320  E * cal.joules_per_coh
 321
 322/-- Convert mass (in coh/c²) to kg. -/
 323@[simp] noncomputable def to_kg (cal : ExternalCalibration) (m : Mass) : ℝ :=
 324  m * cal.joules_per_coh / (299792458 ^ 2)
 325
 326/-- Convert frequency (in 1/tick) to Hz. -/
 327@[simp] noncomputable def to_hertz (cal : ExternalCalibration) (f : Frequency) : ℝ :=
 328  f / cal.seconds_per_tick
 329
 330/-- The speed of light in SI is exactly 299792458 m/s. -/
 331theorem c_in_si (cal : ExternalCalibration) :
 332    to_m_per_s cal c = 299792458 := by
 333  simp [to_m_per_s, c, cal.speed_consistent]
 334
 335/-! ## Summary Status -/
 336
 337/-- RS Native Units module status. -/
 338def status : String :=
 339  "✓ Base units: tick, voxel (τ₀ = ℓ₀ = 1)\n" ++
 340  "✓ Speed of light: c = 1 voxel/tick\n" ++
 341  "✓ Coherence quantum: coh = φ⁻⁵\n" ++
 342  "✓ Action quantum: act = ħ = φ⁻⁵ (in RS-native)\n" ++
 343  "✓ φ-ladder: phiRung n = φⁿ\n" ++
 344  "✓ 8-tick cycle: octavePeriod = 8 ticks\n" ++
 345  "✓ K-gate: tau_rec = lambda_kin = π/(4 ln φ)\n" ++
 346  "✓ External calibration: ExternalCalibration structure\n" ++
 347  "✓ All dimensionless physics fixed by φ"
 348
 349#eval status
 350
 351end RSNativeUnits
 352end Constants
 353end IndisputableMonolith
 354