Foundations.lean

import TauLib.BookI.Kernel.Signature
import TauLib.BookI.Kernel.Axioms
import TauLib.BookI.Kernel.Diagonal
import TauLib.BookI.Orbit.Generation
import TauLib.BookI.Orbit.Rigidity
import TauLib.BookI.Denotation.TauIdx
import TauLib.BookI.Denotation.Arithmetic
import TauLib.BookI.Boundary.Iota
import TauLib.BookI.Polarity.Spectral

/-!
# Tour 01: Foundations of Category τ

A 10-minute interactive introduction. Step through this file in VS Code
with the Lean 4 extension — hover over `#check` and `#eval` to see results.

Category τ begins with 5 generators, 6 structural axioms, and a single
primitive operator ρ. From these alone, TauLib derives arithmetic, algebra,
analysis, physics, biology, and philosophy — with zero imports from Mathlib's
mathematical content.

**Prerequisites:** None. This tour is self-contained.
-/

open Tau.Kernel Generator Tau.Orbit Tau.Denotation Tau.Boundary

-- ================================================================
-- PART 1: THE FIVE GENERATORS
-- ================================================================

-- Category τ starts with exactly five generators in strict total order:
--   α < π < γ < η < ω
-- This is the entire alphabet. Everything else is derived.

#check Generator          -- The 5-element type: α, π, γ, η, ω
#check Generator.alpha    -- α: radial seed (becomes the natural numbers)
#check Generator.pi       -- π: prime base / multiplicative spine
#check Generator.gamma    -- γ: exponent channel
#check Generator.eta      -- η: tetration channel
#check Generator.omega    -- ω: fixed-point absorber / closure beacon

-- Each generator has a canonical index (0–4):
#eval Generator.alpha.toNat  -- 0
#eval Generator.pi.toNat     -- 1
#eval Generator.gamma.toNat  -- 2
#eval Generator.eta.toNat    -- 3
#eval Generator.omega.toNat  -- 4

-- There are exactly five. No more, no fewer.
-- (The `Generator` inductive has exactly 5 constructors.)

-- ================================================================
-- PART 2: THE SIX AXIOMS (K1–K6)
-- ================================================================

-- K0 (Universe Postulate) is implicit in Lean's type system:
-- declaring `Generator : Type` and `TauObj : Type` postulates
-- the universe of discourse.

-- K1: Strict Order — the five generators are strictly ordered.
#check @K1_strict_order

-- K2: Omega Fixed Point — ρ(ω) = ω at every depth. ω is the
-- unique element that the iterator cannot move.
#check @K2_omega_fixed

-- K3: Orbit-Seeded Generation — applying ρ to any non-ω generator g
-- produces an object seeded by g. This generates the orbit rays.
#check @K3_orbit_seeded

-- K4: No-Jump (Cover) — ρ advances depth by exactly 1. No skipping.
#check @K4_no_jump

-- K5: Beacon Non-Successor — ω is never reached by iterating ρ.
-- It stands outside all orbit rays as the fixed-point beacon.
#check @K5_beacon_non_succ

-- K6: Object Closure — every TauObj is either a generator or ρ-generated.
-- Nothing exists outside the axioms.
#check @K6_object_closure

-- ================================================================
-- PART 3: THE ρ OPERATOR AND ORBIT RAYS
-- ================================================================

-- ρ is the sole primitive operator. It maps TauObj → TauObj.
-- On non-ω objects, it advances depth by 1 (K4).
-- On ω, it returns ω (K2).

#check @rho  -- TauObj → TauObj

-- Starting from generator α and iterating ρ, we get the orbit ray O_α:
--   α, ρ(α), ρ²(α), ρ³(α), ...
-- This infinite sequence becomes τ-Idx — the internal natural numbers.

-- The four orbit rays (from α, π, γ, η) are pairwise disjoint:
#check @orbit_disjoint

-- ω sits alone as a fixed point, outside all rays.

-- ================================================================
-- PART 4: τ-IDX — INTERNAL NATURAL NUMBERS
-- ================================================================

-- TauIdx is Nat — but earned from the orbit ray O_α, not postulated.
-- The natural numbers are a *consequence* of the axioms, not an input.

#check TauIdx  -- = Nat (but with internal meaning)

-- Arithmetic operates on TauIdx just as on Nat — but the meaning
-- is grounded in orbit structure, not Peano axioms.
#eval (3 : TauIdx) + 5      -- 8
#eval (4 : TauIdx) * 7      -- 28

-- ================================================================
-- PART 5: THE MASTER CONSTANT ι_τ
-- ================================================================

-- The master constant ι_τ = 2/(π + e) ≈ 0.341304 emerges from
-- the asymptotic ratio of B-polarity to C-polarity primes.
-- It is NOT a free parameter — it is derived from the structure.

#eval iota_tau_float       -- 0.341304
#eval iota_tau_numer       -- 341304 (numerator of rational approx)
#eval iota_tau_denom       -- 1000000 (denominator)

-- κ_D = 1 − ι_τ ≈ 0.658696 (the complementary constant)
-- κ_ω = ι_τ/(1 + ι_τ) ≈ 0.254485

-- These two constants, together with ι_τ, govern all quantitative
-- predictions across physics (Books IV–V).

-- ================================================================
-- PART 6: RIGIDITY — Aut(τ) = {id}
-- ================================================================

-- The most striking structural theorem in Book I:
-- Category τ admits NO non-trivial automorphisms.
-- Any structure-preserving map must be the identity.

-- This means τ is *categorically unique* — there is exactly one
-- model satisfying the axioms, up to isomorphism.

#check @rigidity_non_omega
-- For any automorphism φ and non-ω generator g:
-- φ maps g's orbit ray to itself, preserving all depths.

-- ================================================================
-- PART 7: WHAT COMES NEXT
-- ================================================================

-- From these foundations, TauLib builds:
--
-- Book I:  Coordinates, Polarity, Boundary ring, Set theory,
--          Logic (Truth₄), Holomorphy, Topos, Meta-Logic
--
-- Book II: The Central Theorem O(τ³) ≅ A_spec(L)
--          → See Tour/CentralTheorem.lean
--
-- Books IV–V: All of physics from ι_τ + one anchor (neutron mass)
--          → See Tour/Physics.lean
--
-- Book VII: Ethics as categorical theorem (the Categorical Imperative)
--
-- The entire 127,000-line library, 0 sorry across all seven books,
-- traces back to these five generators and six axioms.