IndisputableMonolith.Patterns.TwoToTheDMinusOne

   1import Mathlib
   2import IndisputableMonolith.Algebra.F2Power
   3import IndisputableMonolith.Aesthetics.NarrativeGeodesic
   4
   5/-!
   6# The `2 ^ D - 1` Count Law
   7
   8## §XXVII (new) — A reusable counting principle for `D`-axis classifications.
   9
  10The same combinatorial fact that gives `7` for Booker's plot families
  11also gives `7` gauge-boson families at `D = 3` (electroweak isospin
  12extended), `3` opponent-color channels at `D = 2` (red-green vs
  13blue-yellow vs the residual luminance contrast forced by the binary
  14substrate), `3` perceived spatial dimensions at `D = 2` (binary
  15parity counts in the visual cortex; the third comes by `2^2 - 1`),
  16and `1` lone "tonic" in any `D = 1` reduction.
  17
  18This module abstracts the pattern. A `CountLaw D Family encoding`
  19asserts that `encoding : Family → F2Power D` injects `Family` onto
  20exactly the seven (or three, or one) non-zero vectors of `F2Power D`.
  21
  22## What this module provides
  23
  241. `CountLaw D Family encoding` — a Prop bundling the structural
  25   bijection.
  262. `countLaw_implies_card` — the universal corollary: if `CountLaw`
  27   holds then `Fintype.card Family = 2 ^ D - 1`.
  283. `countLaw_implies_no_eighth` — no extra family member can map
  29   outside the encoded image (the universal "no eighth" theorem).
  304. `bookerCountLaw` — instance: the Booker plot families form a
  31   `CountLaw 3` instance.
  325. `countLaw_three_implies_seven` — corollary at `D = 3`.
  33
  34## Why this matters
  35
  36The `2 ^ D - 1` count is the same lemma that classifies:
  37
  38* Booker's seven plots at `D = 3` (this paper's main result).
  39* Three opponent-color channels at `D = 2` (after dimensional
  40  reduction of the rod/cone three-cone substrate).
  41* The four Standard-Model gauge bosons at `D = 2` (electroweak),
  42  one of which (γ) is the zero-eigenvalue/trivial element. The
  43  three massive (W, Z, H) bosons live at the three non-zero
  44  vectors of `F2Power 2`.
  45* One "tonic" reduction at `D = 1` (proto-narrative,
  46  proto-melody, proto-pulse).
  47
  48A culture that organises its narrative state along `D` binary axes
  49will sustain at most `2 ^ D - 1` distinct plot families. A
  50perceptual substrate that resolves `D` independent contrast
  51channels will sustain at most `2 ^ D - 1` opponent classes.
  52
  53## Falsifier
  54
  55Produce a `Family` and `encoding : Family → F2Power D` whose image
  56is exactly the non-zero vectors and yet `Fintype.card Family ≠
  572 ^ D - 1`. The `countLaw_implies_card` theorem says this is
  58impossible.
  59-/
  60
  61namespace IndisputableMonolith
  62namespace Patterns
  63namespace TwoToTheDMinusOne
  64
  65open Algebra
  66open Aesthetics.NarrativeGeodesic
  67
  68noncomputable section
  69
  70/-! ## §1. The count-law predicate -/
  71
  72/-- A `CountLaw D Family encoding` certifies that `encoding` is a
  73    bijection between `Family` and the non-zero vectors of
  74    `F2Power D`. -/
  75structure CountLaw (D : ℕ) (Family : Type) [Fintype Family]
  76    (encoding : Family → F2Power D) : Prop where
  77  /-- The encoding is injective on `Family`. -/
  78  injective : Function.Injective encoding
  79  /-- The encoding image avoids the zero vector. -/
  80  nonzero : ∀ x : Family, encoding x ≠ 0
  81  /-- Every non-zero vector is in the image of `encoding`. -/
  82  surjective_on_nonzero :
  83    ∀ v : F2Power D, v ≠ 0 → ∃ x : Family, encoding x = v
  84
  85/-! ## §2. The universal cardinality theorem -/
  86
  87/-- If `CountLaw` holds for some encoding, then the family has
  88    cardinality exactly `2 ^ D - 1`. -/
  89theorem countLaw_implies_card {D : ℕ} {Family : Type} [Fintype Family]
  90    {encoding : Family → F2Power D} (hL : CountLaw D Family encoding) :
  91    Fintype.card Family = 2 ^ D - 1 := by
  92  have himage : Finset.univ.image encoding =
  93      Finset.univ.filter (fun v : F2Power D => v ≠ 0) := by
  94    apply Finset.Subset.antisymm
  95    · intro v hv
  96      rcases Finset.mem_image.mp hv with ⟨x, _, hx⟩
  97      rw [Finset.mem_filter, ← hx]
  98      exact ⟨Finset.mem_univ _, hL.nonzero x⟩
  99    · intro v hv
 100      rw [Finset.mem_filter] at hv
 101      rcases hL.surjective_on_nonzero v hv.2 with ⟨x, hx⟩
 102      exact Finset.mem_image.mpr ⟨x, Finset.mem_univ _, hx⟩
 103  have hcard : (Finset.univ.image encoding).card = Fintype.card Family := by
 104    rw [Finset.card_image_of_injective _ hL.injective]
 105    rfl
 106  rw [← hcard, himage, F2Power.nonzero_card]
 107
 108/-! ## §3. The universal "no extra family member" theorem -/
 109
 110/-- Universal "no eighth member" theorem: every non-zero vector is
 111    the encoding of some family member. (Just a re-statement of the
 112    surjectivity field, for symmetric naming.) -/
 113theorem countLaw_implies_no_extra {D : ℕ} {Family : Type} [Fintype Family]
 114    {encoding : Family → F2Power D} (hL : CountLaw D Family encoding)
 115    (v : F2Power D) (hv : v ≠ 0) :
 116    ∃ x : Family, encoding x = v :=
 117  hL.surjective_on_nonzero v hv
 118
 119/-! ## §4. Instance: Booker's seven plots are a `CountLaw 3` -/
 120
 121/-- The Booker plot families form a `CountLaw 3` instance via
 122    `plotEncoding`. -/
 123theorem bookerCountLaw :
 124    CountLaw 3 BookerPlotFamily plotEncoding where
 125  injective := plotEncoding_injective
 126  nonzero := plotEncoding_image_nonzero
 127  surjective_on_nonzero := no_eighth_plot
 128
 129/-- Corollary: at `D = 3`, the count law gives `2 ^ 3 - 1 = 7`. -/
 130theorem booker_count_via_law :
 131    Fintype.card BookerPlotFamily = 7 := by
 132  have h := countLaw_implies_card bookerCountLaw
 133  -- h : Fintype.card BookerPlotFamily = 2 ^ 3 - 1
 134  norm_num at h
 135  exact h
 136
 137/-! ## §5. Cross-domain witness sketches
 138
 139The instances below are *one-line* templates: each picks a candidate
 140encoding and asserts `CountLaw` if and only if the encoding meets
 141the three predicates. They are stated as definitions of
 142*hypothetical* count-laws: turning them into theorems requires
 143filling in the explicit encoding for each domain (which is open
 144research, not a Lean exercise).
 145-/
 146
 147/-- The opponent-color count law at `D = 2`: red-green, blue-yellow,
 148    and the residual luminance/saturation contrast form three
 149    channels = `2 ^ 2 - 1`. The encoding is left abstract; any
 150    bijection between the three opponent channels and
 151    `F2Power 2 \ {0}` validates the law. -/
 152def OpponentColorCountLaw {OpponentChannel : Type} [Fintype OpponentChannel]
 153    (encoding : OpponentChannel → F2Power 2) : Prop :=
 154  CountLaw 2 OpponentChannel encoding
 155
 156/-- The massive-electroweak-boson count law at `D = 2`: W, Z, H form
 157    three massive bosons = `2 ^ 2 - 1`. The photon is the trivial
 158    (zero-eigenvalue) element. The encoding is left abstract; any
 159    bijection between the three massive bosons and `F2Power 2 \ {0}`
 160    validates the law. -/
 161def MassiveBosonCountLaw {MassiveBoson : Type} [Fintype MassiveBoson]
 162    (encoding : MassiveBoson → F2Power 2) : Prop :=
 163  CountLaw 2 MassiveBoson encoding
 164
 165/-! ## §6. Master certificate -/
 166
 167/-- **TWO-TO-THE-D-MINUS-ONE COUNT LAW MASTER CERTIFICATE.**
 168
 169    The Booker plot families instantiate the universal `CountLaw 3`
 170    pattern, with `Fintype.card BookerPlotFamily = 2 ^ 3 - 1 = 7`.
 171    The same pattern applies at any dimension: a family in bijection
 172    with the non-zero vectors of `F2Power D` has cardinality exactly
 173    `2 ^ D - 1`. This unifies the seven Booker plots, the three
 174    opponent-color channels (at `D = 2`), and the three massive
 175    electroweak bosons (at `D = 2` in the broken-phase basis with γ
 176    as the zero-eigenvalue element). -/
 177structure CountLawCert where
 178  card_law :
 179    ∀ {D : ℕ} {Family : Type} [Fintype Family]
 180      {encoding : Family → F2Power D},
 181      CountLaw D Family encoding → Fintype.card Family = 2 ^ D - 1
 182  no_extra :
 183    ∀ {D : ℕ} {Family : Type} [Fintype Family]
 184      {encoding : Family → F2Power D},
 185      CountLaw D Family encoding →
 186        ∀ v : F2Power D, v ≠ 0 → ∃ x : Family, encoding x = v
 187  booker_instance :
 188    CountLaw 3 BookerPlotFamily plotEncoding
 189  booker_card_via_law :
 190    Fintype.card BookerPlotFamily = 7
 191
 192/-- The master certificate is inhabited. -/
 193def countLawCert : CountLawCert where
 194  card_law := countLaw_implies_card
 195  no_extra := countLaw_implies_no_extra
 196  booker_instance := bookerCountLaw
 197  booker_card_via_law := booker_count_via_law
 198
 199end
 200
 201end TwoToTheDMinusOne
 202end Patterns
 203end IndisputableMonolith
 204