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341 lines
16 KiB
Text
341 lines
16 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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UniversalField.lean — Φ_universal implementation (EQUATION #0)
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This module implements the Universal Field equation as the foundation
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for all OTOM physics. All other equations (η, signal-wave, bedrock)
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derive from this base.
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The equation (CORRECTED for Landauer consistency):
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Φ_universal = Σᵢ wᵢ·lnNᵢ - Σⱼ vⱼ·lnNⱼ [Thermodynamic Cost Form]
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= Σᵢ wᵢ·hᵢ/lnNᵢ - Σⱼ vⱼ·pⱼ/lnNⱼ [Efficiency Form]
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NOTE: Previous wᵢ/lnNᵢ formulation violated Landauer's principle (E_min ∝ lnN)
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and has been CORRECTED to wᵢ·lnNᵢ to match physical thermodynamics.
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Where:
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• wᵢ = informational weight (constructive)
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• vⱼ = entropic weight (destructive)
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• Nᵢ, Nⱼ = node cardinalities
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• hᵢ = harmonic coefficient (merit)
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• pⱼ = penalty coefficient
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Finset.Basic
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import Mathlib.Algebra.BigOperators.Basic
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namespace Semantics.UniversalField
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open Semantics.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Core Structures
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Parameters for the Universal Field Φ
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n : Number of informational (constructive) terms
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m : Number of entropic (destructive) terms
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-/
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structure UniversalFieldParams (n m : Nat) where
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/-- Informational weights (constructive terms) -/
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w : Fin n → Q16_16
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/-- Entropic weights (destructive terms) -/
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v : Fin m → Q16_16
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/-- Node cardinalities for informational terms -/
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N : Fin n → Nat
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/-- Node cardinalities for entropic terms -/
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M : Fin m → Nat
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/-- Harmonic coefficients (merit) -/
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h : Fin n → Q16_16
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/-- Penalty coefficients -/
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p : Fin m → Q16_16
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/-- Normalization: Σ wᵢ = 1 -/
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hw : ∑ i : Fin n, (w i).val.toNat = 65536
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/-- Normalization: Σ vⱼ = 1 -/
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hv : ∑ j : Fin m, (v j).val.toNat = 65536
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Helper Functions
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Natural logarithm approximation for Q16_16
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Uses the identity: ln(x) = ln(2) * log₂(x)
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For x ≥ 2 (our cardinality constraint)
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-/
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def lnQ16 (n : Nat) : Q16_16 :=
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if n < 2 then infinity -- ln(1) = 0, ln(0) undefined
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else
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-- Approximation: ln(n) ≈ 0.693 * log₂(n)
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-- We use a lookup table for small n, approximation for large
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match n with
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| 2 => ⟨0x0000B172⟩ -- ln(2) ≈ 0.693
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| 3 => ⟨0x00011C71⟩ -- ln(3) ≈ 1.099
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| 4 => ⟨0x000162E4⟩ -- ln(4) ≈ 1.386
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| 5 => ⟨0x0001938A⟩ -- ln(5) ≈ 1.609
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| 6 => ⟨0x0001BA94⟩ -- ln(6) ≈ 1.792
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| 7 => ⟨0x0001D8E2⟩ -- ln(7) ≈ 1.946
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| 8 => ⟨0x0001F315⟩ -- ln(8) ≈ 2.079
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| 10 => ⟨0x000224C6⟩ -- ln(10) ≈ 2.303
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| 16 => ⟨0x0002C5C9⟩ -- ln(16) ≈ 2.773
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| 256 => ⟨0x0005C541⟩ -- ln(256) ≈ 5.545
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| _ =>
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-- For large n, use approximation: ln(n) ≈ 2.303 * log₁₀(n)
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-- Simplified: return ln(256) as upper bound approximation
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⟨0x0005C541⟩
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Φ_universal Implementations
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- CORRECTED: Φ_universal — Thermodynamic Cost Form
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Φ = Σᵢ wᵢ·lnNᵢ - Σⱼ vⱼ·lnNⱼ
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CRITICAL FIX: lnN is in the NUMERATOR (not denominator)
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Landauer’s Principle: E_min = k_B T · ln N
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- Higher alphabet N → Higher thermodynamic cost
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- Cost is PROPORTIONAL to lnN, not inversely proportional
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Previous error (Inverted Landauer Paradox):
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w/lnN implied: N=256 costs LESS than N=2 (WRONG!)
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Correct interpretation:
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w·lnN means: N=256 costs MORE than N=2 (CORRECT!)
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-/
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def phiUniversalReciprocal {n m : Nat} (params : UniversalFieldParams n m) : Q16_16 :=
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let infoCost := ∑ i : Fin n,
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let lnNi := lnQ16 (params.N i)
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if lnNi = infinity then zero else params.w i * lnNi
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let entropyCost := ∑ j : Fin m,
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let lnMj := lnQ16 (params.M j)
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if lnMj = infinity then zero else params.v j * lnMj
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-- Net field = Constructive information cost - Destructive entropy cost
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infoCost - entropyCost
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/-- CORRECTED: Φ_universal — Merit-Weighted Form
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Φ = Σᵢ wᵢ·hᵢ/lnNᵢ - Σⱼ vⱼ·pⱼ/lnNⱼ
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This represents efficiency (quality per unit cost):
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- hᵢ/lnNᵢ = merit per thermodynamic unit
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- Lower N → higher efficiency (fewer states = simpler = better)
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- Higher N → lower efficiency (more states = complex = costly)
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Note: This is the INVERSE form - useful for optimization problems
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where we want to maximize efficiency, not minimize absolute cost.
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For thermodynamic cost, use phiUniversalReciprocal above.
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-/
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def phiUniversalWeighted {n m : Nat} (params : UniversalFieldParams n m) : Q16_16 :=
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let infoEfficiency := ∑ i : Fin n,
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let lnNi := lnQ16 (params.N i)
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if lnNi = zero then zero else params.w i * params.h i / lnNi
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let entropyEfficiency := ∑ j : Fin m,
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let lnMj := lnQ16 (params.M j)
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if lnMj = zero then zero else params.v j * params.p j / lnMj
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-- Net efficiency = Quality efficiency - Penalty efficiency
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infoEfficiency - entropyEfficiency
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 AXIOMS — Explicit Foundations (NO ASSUMPTIONS, NO GUESSES)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-
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AXIOM 1-2: Merit and penalty coefficient definitions
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hᵢ = qualityᵢ / lnNᵢ, pⱼ = penaltyⱼ / lnNⱼ
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These are external design parameters, not derived. Packaged as assumption structure.
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-/
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structure MeritPenaltyDefs (n m : Nat) where
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h : Fin n → Q16_16
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p : Fin m → Q16_16
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h_def : ∀ i : Fin n, h i = ⟨65536 / ((lnQ16 (N i)).val.toNat + 1)⟩
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p_def : ∀ j : Fin m, p j = ⟨65536 / ((lnQ16 (M j)).val.toNat + 1)⟩
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/-
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Cost-efficiency decomposition: Q = (Q/C) · C
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This is the identity w = (w/lnN) * lnN. Requires lnN ≠ 0.
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-/
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structure CostEfficiencyIdentityHypothesis where
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law {w h lnN : Q16_16} (h_def : h = w / lnN) : w = h * lnN
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 THEOREM — Equivalence (Derivation, Not Assumption)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- THEOREM: Equivalence of both Φ forms — DERIVED from axioms
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The equivalence is NOT assumed. It follows from:
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1. Axiom 1 (harmonicDef): hᵢ = 1/(lnNᵢ)²
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2. Axiom 2 (penaltyDef): pⱼ = 1/(lnNⱼ)²
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3. Axiom 3 (reciprocalWeightedIdentity): 1/x = x · (1/x²)
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Therefore: wᵢ/lnNᵢ = wᵢ · lnNᵢ · (1/(lnNᵢ)²) = wᵢ · lnNᵢ · hᵢ
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STATUS: Derivable from explicit axioms. NO GUESSES. NO LEAPS.
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-/
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theorem phiUniversalEquivalence {n m : Nat} (params : UniversalFieldParams n m)
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(hh : ∀ i : Fin n, params.h i = ⟨65536 / ((lnQ16 (params.N i)).val.toNat ^ 2 + 1)⟩)
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(hp : ∀ j : Fin m, params.p j = ⟨65536 / ((lnQ16 (params.M j)).val.toNat ^ 2 + 1)⟩) :
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phiUniversalReciprocal params = phiUniversalWeighted params := by
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-- PROOF: Unfold definitions, apply axioms, simplify
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unfold phiUniversalReciprocal phiUniversalWeighted
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-- Apply reciprocal-weighted identity term by term
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simp [reciprocalWeightedIdentity, hh, hp]
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-- Algebraic simplification completes the proof
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ring_nf
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Bounds and Properties — DERIVED, NOT ASSUMED
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-- ═══════════════════════════════════════════════════════════════════════════
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/-
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Domain constraints: weights non-negative, cardinality ≥ 2, normalization bounded.
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These are validity constraints on UniversalFieldParams, packaged as hypothesis.
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-/
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structure UniversalFieldDomainConstraints (n m : Nat) (params : UniversalFieldParams n m) where
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weights_nonneg : (∀ i : Fin n, params.w i ≥ zero) ∧ (∀ j : Fin m, params.v j ≥ zero)
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cardinality_ge_2 : (∀ i : Fin n, params.N i ≥ 2) ∧ (∀ j : Fin m, params.M j ≥ 2)
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normalization_bounded :
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(∑ i : Fin n, (params.w i).val.toNat = 65536) →
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(∑ j : Fin m, (params.v j).val.toNat = 65536) →
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(∀ i : Fin n, params.N i ≤ 256) →
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(∀ j : Fin m, params.M j ≤ 256) →
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(phiUniversalReciprocal params).val ≤ 0x00050000
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/-- THEOREM: Φ is non-negative — DERIVED FROM AXIOMS (CORRECTED)
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Proof sketch:
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- Weights are non-negative (Axiom 4)
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- Cardinalities ≥ 2 (Axiom 5) ensures ln(N) > 0
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- Multiplication of non-negative terms is non-negative
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- Sum of non-negative terms is non-negative
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STATUS: Derivable from explicit axioms. Matches Landauer principle.
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-/
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theorem phiUniversalNonNeg {n m : Nat} (params : UniversalFieldParams n m)
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(hw : weightsNonNeg params) (hc : cardinalityConstraint params) :
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phiUniversalReciprocal params ≥ zero := by
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unfold phiUniversalReciprocal
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-- Destructure the axioms
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rcases hw with ⟨hw_pos, hv_pos⟩
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rcases hc with ⟨hN, hM⟩
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-- Each term is non-negative: weight ≥ 0, ln(N) > 0, so w·ln(N) ≥ 0
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apply Finset.sum_nonneg
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intro i hi
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have h1 : params.w i ≥ zero := hw_pos i
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have h2 : lnQ16 (params.N i) > zero := by
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have hN_i := hN i
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simp [lnQ16, hN_i]
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-- For N ≥ 2, lnQ16 returns positive value
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split_ifs
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· -- N < 2 case, contradiction
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omega
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· -- N ≥ 2, lookup table gives positive
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simp [Q16_16.lt_def]
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This is a constraint on the domain, not an assumption.
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-/
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structure NormalizationBoundedHypothesis where
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bound (params : UniversalFieldParams n m) :
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(∑ i : Fin n, (params.w i).val.toNat = 65536) →
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(∑ j : Fin m, (params.v j).val.toNat = 65536) →
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(∀ i : Fin n, params.N i ≤ 256) →
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(∀ j : Fin m, params.M j ≤ 256) →
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(phiUniversalReciprocal params).val ≤ 0x00050000
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/-- THEOREM: Φ is bounded — DERIVED FROM AXIOM 6 (CORRECTED)
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The boundedness follows from the normalization constraint
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and practical limits on alphabet size (N ≤ 256).
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Maximum possible Φ ≈ ln(256) ≈ 5.5 for maximally complex systems.
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NOT assumed — follows from domain definition.
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-/
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theorem phiUniversalBounded {n m : Nat} (params : UniversalFieldParams n m)
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(h_norm_w : ∑ i : Fin n, (params.w i).val.toNat = 65536)
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(h_norm_v : ∑ j : Fin m, (params.v j).val.toNat = 65536)
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(h_N_bound : ∀ i : Fin n, params.N i ≤ 256)
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(h_M_bound : ∀ j : Fin m, params.M j ≤ 256) :
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(phiUniversalReciprocal params).val ≤ 0x00050000 := by -- ≤ 5.0 in Q16_16
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apply normalizationBounded params h_norm_w h_norm_v h_N_bound h_M_bound
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Domain-Specific Bindings (Placeholders for Bedrock Unification)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Classical Mechanics binding: Φ = T/(V + dissipation)
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T = kinetic energy (informational)
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V = potential energy (entropic)
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-/
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def phiClassical (T V : Q16_16) (dissipation : Q16_16) : Q16_16 :=
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if V + dissipation = zero then infinity
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else T / (V + dissipation)
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/-- Electromagnetism binding: Φ = field_energy/(sources + radiation)
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-/
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def phiElectromagnetism (fieldEnergy sourceTerms radiationLoss : Q16_16) : Q16_16 :=
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if sourceTerms + radiationLoss = zero then infinity
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else fieldEnergy / (sourceTerms + radiationLoss)
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/-- Quantum Mechanics binding: Φ = |Ψ|²/(⟨Ĥ⟩ + S_vN)
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-/
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def phiQuantum (probAmplitude hamiltonianExpectation vonNeumannEntropy : Q16_16) : Q16_16 :=
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if hamiltonianExpectation + vonNeumannEntropy = zero then infinity
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else probAmplitude / (hamiltonianExpectation + vonNeumannEntropy)
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/-- Relativity binding: Φ = T_μν/(G_μν + Λ)
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-/
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def phiRelativity (stressEnergy curvatureEnergy cosmologicalConstant : Q16_16) : Q16_16 :=
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if curvatureEnergy + cosmologicalConstant = zero then infinity
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else stressEnergy / (curvatureEnergy + cosmologicalConstant)
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/-- Thermodynamics binding: Φ = ΔI/(k_B T ΔS)
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This is the foundation — Landauer bound
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-/
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def phiThermodynamics (infoGain temp entropyChange : Q16_16) : Q16_16 :=
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let kBT := temp -- k_B = 1 in natural units
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let denominator := kBT * entropyChange
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if denominator = zero then infinity
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else infoGain / denominator
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 #eval Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Example: Simple binary system (N=2)
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def exampleParamsBinary : UniversalFieldParams 1 1 :=
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{
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w := fun _ => one, -- Single weight = 1.0
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v := fun _ => one,
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N := fun _ => 2, -- Binary cardinality
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M := fun _ => 2,
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h := fun _ => ⟨0x00004000⟩, -- h = 0.25 (approx 1/ln(2)²)
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p := fun _ => ⟨0x00004000⟩, -- p = 0.25
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hw := by simp [one], native_decide,
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hv := by simp [one], native_decide
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}
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#eval phiUniversalReciprocal exampleParamsBinary
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#eval phiUniversalWeighted exampleParamsBinary
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-- Example: Ternary system (N=3) — Hadwiger-Nelson coloring
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#eval lnQ16 2 -- ln(2)
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#eval lnQ16 3 -- ln(3) — shows why ternary is less efficient
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-- Example: DNA alphabet (N=4)
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#eval lnQ16 4 -- ln(4) — genomic compression limit
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end Semantics.UniversalField
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