IndisputableMonolith.Gap45.ShimmerFactor

   1import Mathlib
   2import IndisputableMonolith.Gap45.RecognitionBarrier
   3
   4/-!
   5# The Shimmer Subjective-Time Factor: Pure Gap-45 Arithmetic
   6
   7## The claim being formalized
   8
   9> "The Shimmer subjective-time factor ~9.73 (i.e., 360/37) is a numerical
  10>  observation from the Gap-45 arithmetic, not a named theorem."
  11
  12## What this module establishes
  13
  14The Shimmer factor is **defined** from the two Gap-45 primitives:
  15
  16* `gapLCM := lcm(8, 45) = 360` — the minimal window in which both the
  17  8-tick body clock and the 45-tick consciousness window realign.
  18* `gapDiff := 45 - 8 = 37` — the arithmetic beat between the two
  19  periods, which happens to be prime (and therefore coprime with 360).
  20
  21The factor itself is their ratio:
  22
  23  `shimmerFactor := gapLCM / gapDiff = 360 / 37 ≈ 9.7297…`
  24
  25Every claim in this file reduces, after unfolding, to `native_decide`
  26on ℕ-valued facts or `norm_num` on ℝ-valued arithmetic. There is no
  27hidden physical input, no calibration, and no empirical step: the
  28factor is whatever the two Gap-45 primitives force it to be.
  29
  30The theorem `shimmer_is_gap45_arithmetic` packages the whole claim so
  31that any downstream caller can cite a single statement witnessing
  32"this number is pure arithmetic on `(8, 45)`".
  33
  34## Epistemic status
  35
  36- **THEOREM (trivial arithmetic).** The identity `shimmerFactor = 360/37`
  37  and the bounds `9.72 < shimmerFactor < 9.73` are proved here by
  38  `native_decide` / `norm_num`. Zero sorry, zero axiom.
  39- **Not a named physics theorem.** The non-trivial mathematical content
  40  of the Gap-45 structure lives in `RecognitionBarrier.lean` (coprimality,
  41  window insufficiency, 37 prime). This file only observes that once
  42  those structural facts are proved, the approximate value `~9.73`
  43  follows by counting.
  44- **Upstream dependence.** The physical interpretation — that this ratio
  45  is what "subjective time" feels like — is a HYPOTHESIS living in
  46  `Consciousness/DecoupledTickRate.lean` and related files. This module
  47  says nothing about that interpretation; it only checks the number.
  48
  49## Status: 0 sorry, 0 axiom.
  50-/
  51
  52namespace IndisputableMonolith
  53namespace Gap45
  54namespace ShimmerFactor
  55
  56noncomputable section
  57
  58open RecognitionBarrier
  59
  60/-! ## Gap-45 primitives -/
  61
  62/-- The 8-tick body-clock period (from T6: `2^D = 2^3 = 8`). -/
  63def bodyPeriod : ℕ := 8
  64
  65/-- The 45-tick consciousness window (Gap-45). -/
  66def gapPeriod : ℕ := 45
  67
  68/-- The minimal joint realignment window, `lcm(8, 45) = 360`. -/
  69def gapLCM : ℕ := Nat.lcm bodyPeriod gapPeriod
  70
  71/-- The arithmetic beat between the two periods, `45 - 8 = 37`. -/
  72def gapDiff : ℕ := gapPeriod - bodyPeriod
  73
  74theorem gapLCM_eq : gapLCM = 360 := by native_decide
  75
  76theorem gapDiff_eq : gapDiff = 37 := by native_decide
  77
  78/-- The beat is prime, and therefore coprime with every proper
  79    divisor of the 360-tick window. -/
  80theorem gapDiff_prime : Nat.Prime gapDiff := by
  81  rw [gapDiff_eq]; exact beat_is_prime
  82
  83/-- The two Gap-45 primitives are coprime, which is the hypothesis
  84    that generates the whole barrier. -/
  85theorem gap_coprime : Nat.Coprime bodyPeriod gapPeriod := coprime_barrier
  86
  87/-! ## The Shimmer factor -/
  88
  89/-- The Shimmer subjective-time factor. Defined directly from the two
  90    Gap-45 primitives as `lcm(8,45) / (45 - 8)`. -/
  91def shimmerFactor : ℝ := (gapLCM : ℝ) / (gapDiff : ℝ)
  92
  93/-- The factor reduces to the bare arithmetic ratio `360 / 37`. -/
  94theorem shimmerFactor_eq_360_div_37 : shimmerFactor = (360 : ℝ) / 37 := by
  95  unfold shimmerFactor
  96  have h1 : (gapLCM : ℝ) = 360 := by exact_mod_cast gapLCM_eq
  97  have h2 : (gapDiff : ℝ) = 37 := by exact_mod_cast gapDiff_eq
  98  rw [h1, h2]
  99
 100theorem shimmerFactor_pos : 0 < shimmerFactor := by
 101  rw [shimmerFactor_eq_360_div_37]; norm_num
 102
 103theorem shimmerFactor_gt_one : 1 < shimmerFactor := by
 104  rw [shimmerFactor_eq_360_div_37]; norm_num
 105
 106/-! ## Numerical bounds: `~ 9.73`
 107
 108The approximation `9.73` is accurate to the displayed precision:
 109`360/37 = 9.72972972…`, so the factor sits strictly between
 110`9.72` and `9.73`. -/
 111
 112theorem shimmerFactor_gt_9_72 : (9.72 : ℝ) < shimmerFactor := by
 113  rw [shimmerFactor_eq_360_div_37]; norm_num
 114
 115theorem shimmerFactor_lt_9_73 : shimmerFactor < (9.73 : ℝ) := by
 116  rw [shimmerFactor_eq_360_div_37]; norm_num
 117
 118theorem shimmerFactor_approx :
 119    (9.72 : ℝ) < shimmerFactor ∧ shimmerFactor < (9.73 : ℝ) :=
 120  ⟨shimmerFactor_gt_9_72, shimmerFactor_lt_9_73⟩
 121
 122/-- Strictly-tighter three-decimal bound: `9.729 < 360/37 < 9.730`.
 123    Confirms that `9.73` is the correct rounding at two decimals. -/
 124theorem shimmerFactor_three_decimal :
 125    (9.729 : ℝ) < shimmerFactor ∧ shimmerFactor < (9.730 : ℝ) := by
 126  rw [shimmerFactor_eq_360_div_37]
 127  refine ⟨?_, ?_⟩ <;> norm_num
 128
 129/-- Absolute-error bound: the factor is within `0.001` of `9.73`.
 130    In other words, `9.73` is correct to the displayed precision. -/
 131theorem shimmerFactor_error_bound :
 132    |shimmerFactor - (9.73 : ℝ)| < (1 / 1000 : ℝ) := by
 133  rw [shimmerFactor_eq_360_div_37]
 134  rw [abs_lt]
 135  refine ⟨?_, ?_⟩ <;> norm_num
 136
 137/-! ## The meta-claim as a single packaged theorem
 138
 139The full content of the user's statement — that the Shimmer
 140subjective-time factor `~9.73` is just `lcm(8,45) / (45 - 8)` with no
 141extra physics — is packaged below. Nothing in the proof uses anything
 142beyond:
 143
 144* `native_decide` on ℕ-valued identities (lcm, subtraction);
 145* `norm_num` on ℝ-valued inequalities;
 146* the already-proved Gap-45 primitives (coprimality, `37` prime).
 147
 148The proof is deliberately short; its brevity is the content of the
 149claim. -/
 150
 151/-- **Master statement** of the meta-claim:
 152
 153> The Shimmer subjective-time factor `~9.73` (i.e., `360/37`) is a
 154> numerical observation from the Gap-45 arithmetic, not a named theorem.
 155
 156Concretely, the factor equals `lcm(8,45) / (45 - 8)`, reduces to
 157`360 / 37`, is bounded by `9.72 < · < 9.73`, is irreducible because the
 158denominator `37` is prime, and sits inside the Gap-45 barrier whose
 159coprimality is `gcd(8, 45) = 1`. No extra input is used. -/
 160theorem shimmer_is_gap45_arithmetic :
 161    shimmerFactor = (Nat.lcm 8 45 : ℝ) / ((45 - 8 : ℕ) : ℝ) ∧
 162    shimmerFactor = (360 : ℝ) / 37 ∧
 163    Nat.lcm 8 45 = 360 ∧
 164    (45 - 8 : ℕ) = 37 ∧
 165    Nat.Prime (45 - 8) ∧
 166    Nat.Coprime 8 45 ∧
 167    (9.72 : ℝ) < shimmerFactor ∧ shimmerFactor < (9.73 : ℝ) := by
 168  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
 169  · -- shimmerFactor = lcm(8,45) / (45 - 8)
 170    unfold shimmerFactor gapLCM gapDiff bodyPeriod gapPeriod; rfl
 171  · exact shimmerFactor_eq_360_div_37
 172  · native_decide
 173  · native_decide
 174  · show Nat.Prime (45 - 8); native_decide
 175  · exact coprime_barrier
 176  · exact shimmerFactor_gt_9_72
 177  · exact shimmerFactor_lt_9_73
 178
 179/-! ## Certificate: structural witness that nothing else is needed -/
 180
 181/-- Certificate bundling the Shimmer-factor claim. Any module citing
 182    "the Shimmer factor is Gap-45 arithmetic" can cite this structure
 183    without having to rebuild the proof. -/
 184structure ShimmerCert where
 185  /-- The factor has closed form `360 / 37`. -/
 186  value : shimmerFactor = (360 : ℝ) / 37
 187  /-- The factor is `lcm(bodyPeriod, gapPeriod)` divided by their difference. -/
 188  from_gap45 : shimmerFactor = (Nat.lcm 8 45 : ℝ) / ((45 - 8 : ℕ) : ℝ)
 189  /-- Numerator arithmetic. -/
 190  lcm_eq : Nat.lcm 8 45 = 360
 191  /-- Denominator arithmetic. -/
 192  diff_eq : (45 - 8 : ℕ) = 37
 193  /-- Denominator is prime (so `360/37` is already in lowest terms). -/
 194  diff_prime : Nat.Prime (45 - 8)
 195  /-- Structural hypothesis: the two periods are coprime. -/
 196  coprime : Nat.Coprime 8 45
 197  /-- Two-decimal bound confirming `~9.73`. -/
 198  lower : (9.72 : ℝ) < shimmerFactor
 199  upper : shimmerFactor < (9.73 : ℝ)
 200
 201/-- The canonical Shimmer-factor certificate. -/
 202def shimmer_cert : ShimmerCert where
 203  value := shimmerFactor_eq_360_div_37
 204  from_gap45 := by unfold shimmerFactor gapLCM gapDiff bodyPeriod gapPeriod; rfl
 205  lcm_eq := by native_decide
 206  diff_eq := by native_decide
 207  diff_prime := by show Nat.Prime (45 - 8); native_decide
 208  coprime := coprime_barrier
 209  lower := shimmerFactor_gt_9_72
 210  upper := shimmerFactor_lt_9_73
 211
 212/-! ## Consistency with existing names
 213
 214The existing `Gap45.RecognitionBarrier.time_dilation_factor` states the
 215loose bound `9 < 360/37 < 10`. The bounds below tighten that to `9.73`. -/
 216
 217/-- Tight bound implies the loose bound in `RecognitionBarrier`. -/
 218theorem tight_implies_loose :
 219    shimmerFactor = (360 : ℝ) / 37 ∧
 220    (9 : ℝ) < shimmerFactor ∧ shimmerFactor < (10 : ℝ) := by
 221  refine ⟨shimmerFactor_eq_360_div_37, ?_, ?_⟩
 222  · have := shimmerFactor_gt_9_72; linarith
 223  · have := shimmerFactor_lt_9_73; linarith
 224
 225end
 226
 227end ShimmerFactor
 228end Gap45
 229end IndisputableMonolith
 230