IndisputableMonolith.Cognition.AnimalZComplexityBound

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Animal Z-Complexity Bound
   7
   8## Arc 10c (cognition seed): the φ-ladder of animal cognition
   9
  10This is the entry-point module for the animal-cognition domain in
  11§XXIII.C.  It establishes the structural framework for placing
  12non-human animals on the Z-complexity ladder forced by Recognition
  13Science.
  14
  15## What this module proves
  16
  17The Z-complexity ladder is a φ-geometric sequence indexed by an
  18integer rung `k`.  The structurally significant rungs for cognition
  19are:
  20
  21* **Bond rung** (`k = 8`): the threshold for sustained molecular
  22  recognition (chemistry).  Below this no recognition dynamics
  23  persists.
  24* **Life rung** (`k = 19`): the ignition threshold from
  25  `Biology.IgnitionThreshold`; this is where biological recognition
  26  becomes self-sustaining.
  27* **Counterfactual-floor rung** (`k = 5`): the floor for counterfactual
  28  reasoning (`Z_cf = φ^5` from
  29  `Consciousness.CounterfactualBoundary`).
  30* **Vertebrate-cognition rung** (`k = 12`): empirical fit to mammals
  31  and birds with documented tool-use.
  32* **Cetacean / corvid / great-ape rung** (`k = 15`): documented
  33  multi-step planning.
  34* **Octopus rung** (`k = 14`): documented distributed-substrate
  35  cognition (uncommonly high for an invertebrate).
  36* **Human rung** (`k = 17`): documented compositional language.
  37
  38## What this module does not claim
  39
  40It does *not* claim a mechanistic forcing of any specific empirical
  41rung assignment.  The structural theorems below are restricted to:
  42
  431. The ladder is well-defined (φ^k is positive for all integer k).
  442. The ladder is strictly increasing in k.
  453. The named rungs above are strictly ordered: bond < cf-floor <
  46   vertebrate < octopus < cetacean < human < life.
  474. Adjacent-rung gaps are exactly one factor of `φ`.
  485. The cognition bound `Z_cog` is the integer rung above which
  49   counterfactual reasoning is structurally possible (= 5, the
  50   counterfactual-floor rung).
  51
  52The empirical claim that any specific animal occupies any specific
  53rung is HYPOTHESIS-grade and given a falsifier in §6 below.
  54
  55## Falsifiers
  56
  571. A non-human animal that demonstrably solves multi-step
  58   counterfactual problems indistinguishable from a 5-year-old human
  59   (would falsify the Z = φ^5 floor for non-human cognition).
  602. An animal at structural rung > 17 (would exceed the human ceiling).
  613. An octopus that fails the basic recognition tasks documented in
  62   the published literature (would falsify the octopus rung
  63   assignment of k = 14).
  64
  65## Status
  66
  67THEOREM (algebraic structure on the Z-ladder, 0 sorry, 0 axiom).
  68HYPOTHESIS (the empirical rung assignments per species; sharper
  69falsifiers in the comparative-cognition literature).
  70-/
  71
  72namespace IndisputableMonolith
  73namespace Cognition
  74namespace AnimalZComplexityBound
  75
  76open Constants
  77open Cost
  78
  79noncomputable section
  80
  81/-! ## §1. The Z-complexity rung function -/
  82
  83/-- The Z-complexity at rung `k`: `Z_k := φ^k`. -/
  84def z_rung (k : ℕ) : ℝ := Constants.phi ^ k
  85
  86/-- Every rung is positive. -/
  87theorem z_rung_pos (k : ℕ) : 0 < z_rung k := by
  88  unfold z_rung
  89  exact pow_pos Constants.phi_pos k
  90
  91/-- Rung 0 is exactly 1. -/
  92theorem z_rung_zero : z_rung 0 = 1 := by
  93  unfold z_rung; simp
  94
  95/-- Adjacent rungs are related by one factor of `φ`. -/
  96theorem z_rung_succ (k : ℕ) :
  97    z_rung (k + 1) = Constants.phi * z_rung k := by
  98  unfold z_rung
  99  exact pow_succ' Constants.phi k
 100
 101/-- The ladder is strictly increasing. -/
 102theorem z_rung_strictly_increasing (k : ℕ) :
 103    z_rung k < z_rung (k + 1) := by
 104  rw [z_rung_succ]
 105  have h_pos := z_rung_pos k
 106  have h_phi : (1 : ℝ) < Constants.phi := Constants.one_lt_phi
 107  calc z_rung k = 1 * z_rung k := by ring
 108    _ < Constants.phi * z_rung k :=
 109        mul_lt_mul_of_pos_right h_phi h_pos
 110
 111/-! ## §2. Structural rung assignments -/
 112
 113/-- The bond rung: threshold for sustained molecular recognition. -/
 114def z_rung_bond : ℕ := 8
 115
 116/-- The counterfactual-floor rung: structural floor for counterfactual
 117    reasoning (matches `Z_cf = φ^5` from
 118    `Consciousness.CounterfactualBoundary`). -/
 119def z_rung_cf : ℕ := 5
 120
 121/-- The vertebrate-cognition rung. -/
 122def z_rung_vertebrate : ℕ := 12
 123
 124/-- The octopus rung (distributed-substrate cognition). -/
 125def z_rung_octopus : ℕ := 14
 126
 127/-- The cetacean / corvid / great-ape rung. -/
 128def z_rung_cetacean : ℕ := 15
 129
 130/-- The human rung (compositional language). -/
 131def z_rung_human : ℕ := 17
 132
 133/-- The life-ignition rung from `Biology.IgnitionThreshold`. -/
 134def z_rung_life : ℕ := 19
 135
 136/-! ## §3. Strict ordering of the named rungs -/
 137
 138/-- Counterfactual floor is below bond rung. -/
 139theorem cf_below_bond : z_rung_cf < z_rung_bond := by
 140  unfold z_rung_cf z_rung_bond; norm_num
 141
 142/-- Bond rung is below vertebrate cognition. -/
 143theorem bond_below_vertebrate : z_rung_bond < z_rung_vertebrate := by
 144  unfold z_rung_bond z_rung_vertebrate; norm_num
 145
 146/-- Vertebrate is below octopus. -/
 147theorem vertebrate_below_octopus : z_rung_vertebrate < z_rung_octopus := by
 148  unfold z_rung_vertebrate z_rung_octopus; norm_num
 149
 150/-- Octopus is below cetacean. -/
 151theorem octopus_below_cetacean : z_rung_octopus < z_rung_cetacean := by
 152  unfold z_rung_octopus z_rung_cetacean; norm_num
 153
 154/-- Cetacean is below human. -/
 155theorem cetacean_below_human : z_rung_cetacean < z_rung_human := by
 156  unfold z_rung_cetacean z_rung_human; norm_num
 157
 158/-- Human is below life-ignition (the structural ceiling). -/
 159theorem human_below_life : z_rung_human < z_rung_life := by
 160  unfold z_rung_human z_rung_life; norm_num
 161
 162/-- The full strict ordering. -/
 163theorem rung_ordering :
 164    z_rung_cf < z_rung_bond ∧
 165    z_rung_bond < z_rung_vertebrate ∧
 166    z_rung_vertebrate < z_rung_octopus ∧
 167    z_rung_octopus < z_rung_cetacean ∧
 168    z_rung_cetacean < z_rung_human ∧
 169    z_rung_human < z_rung_life :=
 170  ⟨cf_below_bond, bond_below_vertebrate, vertebrate_below_octopus,
 171   octopus_below_cetacean, cetacean_below_human, human_below_life⟩
 172
 173/-! ## §4. Z-complexity ordering implications -/
 174
 175/-- The φ-ladder respects rung ordering: lower rung → strictly smaller
 176    Z-complexity. -/
 177theorem z_rung_monotone (j k : ℕ) (h : j < k) :
 178    z_rung j < z_rung k := by
 179  unfold z_rung
 180  exact pow_lt_pow_right₀ Constants.one_lt_phi h
 181
 182/-- The cognition floor `Z_cog := φ^5` matches the counterfactual
 183    boundary `Z_cf` from `Consciousness.CounterfactualBoundary`. -/
 184def z_cognition_floor : ℝ := z_rung z_rung_cf
 185
 186/-- The cognition floor equals `φ^5`. -/
 187theorem z_cognition_floor_eq : z_cognition_floor = Constants.phi ^ 5 := rfl
 188
 189/-- The cognition floor is strictly less than the bond rung. -/
 190theorem cognition_floor_below_bond_z :
 191    z_cognition_floor < z_rung z_rung_bond :=
 192  z_rung_monotone z_rung_cf z_rung_bond cf_below_bond
 193
 194/-- The cognition floor is strictly less than the human rung Z. -/
 195theorem cognition_floor_below_human_z :
 196    z_cognition_floor < z_rung z_rung_human :=
 197  z_rung_monotone z_rung_cf z_rung_human (by
 198    have h1 := cf_below_bond
 199    have h2 := bond_below_vertebrate
 200    have h3 := vertebrate_below_octopus
 201    have h4 := octopus_below_cetacean
 202    have h5 := cetacean_below_human
 203    omega)
 204
 205/-! ## §5. Master certificate -/
 206
 207/-- **ANIMAL Z-COMPLEXITY BOUND MASTER CERTIFICATE (arc 10c).**
 208
 209    Five fields:
 210    1. The Z-rung function is geometric: each rung is `φ` times the
 211       previous.
 212    2. The ladder is strictly increasing in `k`.
 213    3. The named rungs satisfy the strict ordering
 214       cf < bond < vertebrate < octopus < cetacean < human < life.
 215    4. The cognition floor `Z_cog = z_rung 5 = φ^5`.
 216    5. The cognition floor is strictly below the human-rung Z. -/
 217structure AnimalZComplexityBoundCert where
 218  rung_zero : z_rung 0 = 1
 219  rung_succ : ∀ k : ℕ, z_rung (k + 1) = Constants.phi * z_rung k
 220  rung_strictly_increasing : ∀ k : ℕ, z_rung k < z_rung (k + 1)
 221  rung_ordering :
 222    z_rung_cf < z_rung_bond ∧
 223    z_rung_bond < z_rung_vertebrate ∧
 224    z_rung_vertebrate < z_rung_octopus ∧
 225    z_rung_octopus < z_rung_cetacean ∧
 226    z_rung_cetacean < z_rung_human ∧
 227    z_rung_human < z_rung_life
 228  cognition_floor_eq : z_cognition_floor = Constants.phi ^ 5
 229
 230/-- The master certificate is inhabited. -/
 231def animalZComplexityBoundCert : AnimalZComplexityBoundCert where
 232  rung_zero := z_rung_zero
 233  rung_succ := z_rung_succ
 234  rung_strictly_increasing := z_rung_strictly_increasing
 235  rung_ordering := rung_ordering
 236  cognition_floor_eq := z_cognition_floor_eq
 237
 238/-! ## §6. One-statement summary -/
 239
 240/-- **ANIMAL Z-COMPLEXITY BOUND ONE-STATEMENT THEOREM.**
 241
 242    The animal-cognition Z-complexity ladder is the φ-geometric
 243    sequence `z_rung k = φ^k` for `k ∈ ℕ`, strictly increasing in `k`,
 244    with the structural rung ordering
 245    cf < bond < vertebrate < octopus < cetacean < human < life. -/
 246theorem animal_z_complexity_one_statement :
 247    -- (1) Geometric structure.
 248    (∀ k : ℕ, z_rung (k + 1) = Constants.phi * z_rung k) ∧
 249    -- (2) Strict monotonicity.
 250    (∀ k : ℕ, z_rung k < z_rung (k + 1)) ∧
 251    -- (3) Rung ordering.
 252    (z_rung_cf < z_rung_bond ∧
 253     z_rung_bond < z_rung_vertebrate ∧
 254     z_rung_vertebrate < z_rung_octopus ∧
 255     z_rung_octopus < z_rung_cetacean ∧
 256     z_rung_cetacean < z_rung_human ∧
 257     z_rung_human < z_rung_life) ∧
 258    -- (4) Cognition floor = φ^5.
 259    z_cognition_floor = Constants.phi ^ 5 :=
 260  ⟨z_rung_succ, z_rung_strictly_increasing, rung_ordering, z_cognition_floor_eq⟩
 261
 262end
 263
 264end AnimalZComplexityBound
 265end Cognition
 266end IndisputableMonolith
 267