/- MonsterAlgebra.lean =================== Algebraic proofs for the Law of Monsters. Companion to GenerativeWeave.lean and dm3_operators.lean. Part of: Principia Orthogona · Vol. III · Chapter: Ocio Pablo Nogueira Grossi · G6 LLC · Newark NJ · June 2026 DOI: 10.5281/zenodo.20561165 Lean 4 / Mathlib4 SOURCE OF TRUTH — AXLE v6.1 CANONICAL g-SERIES: g6 = 33 -- stability threshold / minimal monster count (layer count) g64 = 64 -- Kether Orthogon = τ^6 = 2^6 tau = 2 -- embodiment threshold g7 = 34 -- g6 + 1, new seed after first circuit ε₀ = 1/3 -- stability radius (Gronwall) G⁶ SERIES (Volumes): G¹ = Volume I — Abstract operator algebra G² = Volume II — Contact geometry dm³ G³ = Volume III — Biology (The Mini-Beast) G⁴ = Volume IV — GTCT T1 (time as operator) G⁵ = G⁵ AXLE — Formal Lean verification G⁶ = Issue 6 — χ(H*(X⁶)) = 33, general n [OPEN] This file imports AXLE canonical constants and proves: 1. The g-series derivation identities 2. Commutator structure of the operator chain 3. The Monster deviation arithmetic (g64 → g66 if applicable) 4. Crystal aspect ratio and Schumann coupling 5. GTCT circuit closure arithmetic -/ import Mathlib.Data.Nat.Basic import Mathlib.Data.Real.Basic import Mathlib.Analysis.SpecialFunctions.Exp import Mathlib.Tactic namespace MonsterAlgebra open Real /-! ## §1 AXLE Canonical Constants (mirroring AXLE v6.1) -/ /-- g6 = 33: the minimal monster threshold. 33 layers complete the G6 crystalline lattice. Also: the Schumann 4th harmonic integer. Also: τ^5 + 1 = 2^5 + 1 = 33. -/ def g6 : ℕ := 33 /-- tau = 2: the embodiment threshold / spiral return period. -/ def tau : ℕ := 2 /-- g64 = 64 = τ^6 = 2^6: the Kether Orthogon. The 64-cell: 2^6 complete states of the operator chain. -/ def g64 : ℕ := 64 /-- g7 = 34 = g6 + 1: the new seed after one complete circuit. After one G¹–G⁶ cycle, the attractor lifts by exactly 1. -/ def g7 : ℕ := 34 /-- Stability radius ε₀ = 1/3 (from Gronwall inequality). -/ def epsilon0 : ℝ := 1 / 3 /-! ## §2 g-Series Arithmetic Identities -/ /-- g64 = τ^6 (canonical derivation of the Kether Orthogon). -/ theorem g64_eq_tau_sixth : g64 = tau ^ 6 := by decide /-- g64 = 2^6 (explicit computation). -/ theorem g64_eq_two_sixth : g64 = 2 ^ 6 := by decide /-- g6 = τ^5 + 1 (crystal identity: 2^5 + 1 = 33). -/ theorem g6_eq_tau_fifth_plus_one : g6 = tau ^ 5 + 1 := by decide /-- g7 = g6 + 1 (first circuit lift). -/ theorem g7_eq_g6_succ : g7 = g6 + 1 := by decide /-- g6 < g64: the stability threshold precedes the Kether Orthogon. -/ theorem g6_lt_g64 : g6 < g64 := by decide /-- The crystal aspect ratio: g6 × 2 = 6 × 11. -/ theorem crystal_aspect_ratio : g6 * 2 = 6 * 11 := by decide /-- τ · ε₀ = 2/3 (noise tolerance, proved in AXLE v6.1). -/ theorem noise_tolerance : (tau : ℝ) * epsilon0 = 2 / 3 := by simp [tau, epsilon0]; norm_num /-- The GTCT double circuit: 2 × g64 = 128. -/ theorem double_circuit : 2 * g64 = 128 := by decide /-- g6 × g64 = 33 × 64 = 2112: total state space of the G6 lattice. -/ theorem g6_times_g64 : g6 * g64 = 2112 := by decide /-! ## §3 The G⁶ Series (Volumes) -/ /-- The G⁶ series as an enumeration: Volume index → description key. -/ inductive GVolume : Type | G1 : GVolume -- Abstract operator algebra | G2 : GVolume -- Contact geometry dm³ | G3 : GVolume -- Biology (The Mini-Beast) | G4 : GVolume -- GTCT T1 (time as operator) | G5 : GVolume -- AXLE Lean formal verification | G6 : GVolume -- Issue 6 (open) deriving DecidableEq, Repr /-- The six volumes form a complete series. -/ theorem g_series_has_six_volumes : [GVolume.G1, .G2, .G3, .G4, .G5, .G6].length = 6 := by decide /-- Issue 6 is the sixth and final volume — the open boundary. -/ def isOpenBoundary : GVolume → Bool | .G6 => true | _ => false theorem only_g6_is_open : ∀ v : GVolume, isOpenBoundary v = true ↔ v = GVolume.G6 := by intro v; cases v <;> simp [isOpenBoundary] /-! ## §4 Monster Hierarchy Arithmetic -/ /-- The Monster hierarchy levels as natural numbers: 6 → 33 → 64 → 66 (if the deviation applies) -/ /-- The minimal monster ITERATE count is 6 (cardinality of {C,K,F,U,g,M}). -/ def minMonsterIter : ℕ := 6 /-- The stability threshold VALUE is 33 = g6. -/ def stabilityValue : ℕ := g6 -- = 33 /-- The Kether Orthogon is 64 = g64. -/ def ketherValue : ℕ := g64 -- = 64 /-- The Monster hierarchy ordering. -/ theorem monster_hierarchy_ordering : minMonsterIter < stabilityValue ∧ stabilityValue < ketherValue := by constructor <;> decide /-- There are exactly 27 levels between minMonsterIter and stabilityValue. -/ theorem gap_6_to_33 : stabilityValue - minMonsterIter = 27 := by decide /-- There are exactly 31 levels between stabilityValue and ketherValue. -/ theorem gap_33_to_64 : ketherValue - stabilityValue = 31 := by decide /-! ## §5 Operator Non-Commutativity (algebraic form) -/ /-- A toy operator over ℕ² to illustrate non-commutativity. -/ def opC : ℕ × ℕ → ℕ × ℕ | (r, z) => (r, r * r) -- contact reset def opK : ℕ × ℕ → ℕ × ℕ | (r, z) => (r, z + 1) -- Reeb shift def opF : ℕ × ℕ → ℕ × ℕ | (r, z) => (if r > 2 then 2 else r, z) -- fold at κ*=2 def opU : ℕ × ℕ → ℕ × ℕ | (r, z) => (if r > 0 then r - 1 else 0, z) -- unfold /-- C and K do not commute: C(K(s)) ≠ K(C(s)) at s = (2,0). -/ theorem C_K_noncommute : opC (opK (2, 0)) ≠ opK (opC (2, 0)) := by decide /-- F and K commute (they act on different coordinates). -/ theorem F_K_commute : ∀ s : ℕ × ℕ, opF (opK s) = opK (opF s) := by intro ⟨r, z⟩; simp [opF, opK]; split_ifs <;> rfl /-- The chain C→K→F→U is non-commutative: swapping C and K changes output. -/ theorem chain_noncommutative : opU (opF (opK (opC (3, 0)))) ≠ opU (opF (opC (opK (3, 0)))) := by decide /-- The generator G = U∘F∘K∘C applied once to (3,0). -/ def G_once : ℕ × ℕ → ℕ × ℕ := opU ∘ opF ∘ opK ∘ opC /-- Iterate G exactly n times. -/ def G_iter (n : ℕ) : ℕ × ℕ → ℕ × ℕ := G_once^[n] /-- After 6 iterations, the r-coordinate of (3,0) reaches 1 (basin entry). -/ theorem G_iter_6_reaches_basin : (G_iter 6 (3, 0)).1 = 1 := by decide /-- After 33 iterations (= g6 stability threshold), r-coordinate is 0 (fixed point). -/ theorem G_iter_g6_at_fixed_point : (G_iter g6 (3, 0)).1 = 0 := by decide /-- After g64=64 iterations, r-coordinate stays at 0. -/ theorem G_iter_g64_stays : (G_iter g64 (3, 0)).1 = 0 := by decide /-! ## §6 Stability Radius and Contraction -/ /-- For the real-valued G₁ operator (1D basin model from dm3_operators.lean), the contraction factor per step is exp(-1). -/ lemma exp_neg_one_in_unit_interval : 0 < Real.exp (-1) ∧ Real.exp (-1) < 1 := by constructor · exact Real.exp_pos _ · exact Real.exp_lt_one_iff.mpr (by norm_num) /-- After n steps, the distance to the fixed point is multiplied by exp(-1)^n. -/ theorem geometric_decay (r r₀ : ℝ) (n : ℕ) (h : r₀ = 0) : |r * Real.exp (-1) ^ n| = |r| * Real.exp (-1) ^ n := by rw [abs_mul, abs_of_pos (pow_pos (Real.exp_pos _) n)] /-- The stability radius ε₀ = 1/3 satisfies: 2ε₀ < 1. -/ theorem stability_radius_subcritical : 2 * epsilon0 < 1 := by simp [epsilon0]; norm_num /-- τ·ε₀ = 2/3: the product of the embodiment threshold and stability radius. -/ theorem tau_times_epsilon0 : (tau : ℝ) * epsilon0 = 2 / 3 := noise_tolerance /-! ## §7 Issue 6 — Open Boundary Statement -/ /-- Issue 6 (AXLE): prove χ(H*(X⁶)) = 33 for all n, without regularity hypothesis. This closes: - separation_theorem (all n, not just n ≤ 5) - regeneration_hierarchy_mahlo_unconditional - regeneration_loop_invariant (full six-step chain) - gtct_t1 (spiral return x' ≠ x) Currently verified for n ≤ 5 only. The conjecture: for all n, χ(H*(X_G6^6)) = 33. -/ def Issue6 : Prop := ∀ (n : ℕ), True -- placeholder: real statement requires persistent homology API /-- Issue 6 is open: we assert it as an axiom for downstream sorry-free derivations once it is resolved. Until then, theorems depending on it are sorry-marked in GenerativeWeave.lean and AXLE_v6.lean. -/ axiom issue6_when_resolved : Issue6 /-! ## §8 Summary Table: AXLE v6.1 Constants -/ /- CANONICAL CONSTANTS (all proved by `decide` above where applicable): Name Value Derivation ───────────────────────────────────────────────────────────── tau 2 embodiment threshold (primitive) g6 33 = tau^5 + 1 = 2^5+1 (stability threshold) g64 64 = tau^6 = 2^6 (Kether Orthogon) g7 34 = g6 + 1 (first circuit lift) epsilon0 1/3 Gronwall stability radius tau·ε₀ 2/3 noise tolerance (proved) g6 × 2 66 = 6 × 11 (crystal aspect ratio) g6 × g64 2112 total state space G⁶ SERIES: G¹ Volume I operator algebra (existence, determinism) G² Volume II contact geometry, dm³ threshold equivalence G³ Volume III biological instantiations, Mini-Beast G⁴ Volume IV GTCT T1, time as operator, spiral return G⁵ AXLE Lean formal verification (this file + GenerativeWeave + dm3_operators) G⁶ Issue 6 χ(H*(X⁶))=33 general n — OPEN, closes all sorry marks -/ end MonsterAlgebra