mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
300 lines
15 KiB
Text
300 lines
15 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Research Stack Team
|
||
|
||
UniversalField.lean — Φ_universal implementation (EQUATION #0)
|
||
|
||
This module implements the Universal Field equation as the foundation
|
||
for all OTOM physics. All other equations (η, signal-wave, bedrock)
|
||
derive from this base.
|
||
|
||
The equation (CORRECTED for Landauer consistency):
|
||
Φ_universal = Σᵢ wᵢ·lnNᵢ - Σⱼ vⱼ·lnNⱼ [Thermodynamic Cost Form]
|
||
= Σᵢ wᵢ·hᵢ/lnNᵢ - Σⱼ vⱼ·pⱼ/lnNⱼ [Efficiency Form]
|
||
|
||
NOTE: Previous wᵢ/lnNᵢ formulation violated Landauer's principle (E_min ∝ lnN)
|
||
and has been CORRECTED to wᵢ·lnNᵢ to match physical thermodynamics.
|
||
|
||
Where:
|
||
• wᵢ = informational weight (constructive)
|
||
• vⱼ = entropic weight (destructive)
|
||
• Nᵢ, Nⱼ = node cardinalities
|
||
• hᵢ = harmonic coefficient (merit)
|
||
• pⱼ = penalty coefficient
|
||
-/
|
||
|
||
import Semantics.FixedPoint
|
||
import Mathlib.Data.Fin.Basic
|
||
import Mathlib.Data.Finset.Basic
|
||
|
||
namespace Semantics.UniversalField
|
||
|
||
open Semantics.Q16_16
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §1 Core Structures
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Parameters for the Universal Field Φ
|
||
|
||
n : Number of informational (constructive) terms
|
||
m : Number of entropic (destructive) terms
|
||
|
||
Normalization is expressed as a separate validity predicate to avoid
|
||
requiring AddCommMonoid on Q16_16.
|
||
-/
|
||
structure UniversalFieldParams (n m : Nat) where
|
||
/-- Informational weights (constructive terms) -/
|
||
w : Fin n → Q16_16
|
||
/-- Entropic weights (destructive terms) -/
|
||
v : Fin m → Q16_16
|
||
/-- Node cardinalities for informational terms -/
|
||
N : Fin n → Nat
|
||
/-- Node cardinalities for entropic terms -/
|
||
M : Fin m → Nat
|
||
/-- Harmonic coefficients (merit) -/
|
||
h : Fin n → Q16_16
|
||
/-- Penalty coefficients -/
|
||
p : Fin m → Q16_16
|
||
|
||
/-- Sum Q16_16 values over Fin n via List.foldl — avoids AddCommMonoid/CommFold. -/
|
||
def finSum {n : Nat} (f : Fin n → Q16_16) : Q16_16 :=
|
||
(List.ofFn f).foldl add zero
|
||
|
||
/-- Normalization predicate: Σ wᵢ = 1.0 in Q16_16 (raw value 65536). -/
|
||
def weightsNormalized {n m : Nat} (params : UniversalFieldParams n m) : Prop :=
|
||
finSum params.w = one ∧ finSum params.v = one
|
||
|
||
/-- All informational weights are non-negative; all cardinalities ≥ 2. -/
|
||
def weightsNonNeg {n m : Nat} (params : UniversalFieldParams n m) : Prop :=
|
||
(∀ i : Fin n, params.w i ≥ zero) ∧ (∀ j : Fin m, params.v j ≥ zero)
|
||
|
||
/-- Cardinality validity: all N_i, M_j ≥ 2 (prevents ln singularity at N=1). -/
|
||
def cardinalityConstraint {n m : Nat} (params : UniversalFieldParams n m) : Prop :=
|
||
(∀ i : Fin n, params.N i ≥ 2) ∧ (∀ j : Fin m, params.M j ≥ 2)
|
||
|
||
/-- Bounded alphabet: N_i, M_j ≤ 256 (hardware representability). -/
|
||
def alphabetBounded {n m : Nat} (params : UniversalFieldParams n m) : Prop :=
|
||
(∀ i : Fin n, params.N i ≤ 256) ∧ (∀ j : Fin m, params.M j ≤ 256)
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §2 Helper Functions
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Natural logarithm approximation for Q16_16
|
||
|
||
Uses the identity: ln(x) = ln(2) * log₂(x)
|
||
For x ≥ 2 (our cardinality constraint)
|
||
-/
|
||
def lnQ16 (n : Nat) : Q16_16 :=
|
||
if n < 2 then infinity -- ln(1)=0 and ln(0) undefined; return sentinel
|
||
else
|
||
-- Q16_16 lookup: ln(n) × 65536, values accurate to ±1 ULP
|
||
match n with
|
||
| 2 => ofRawInt 0x0000B172 -- ln(2) ≈ 0.6931
|
||
| 3 => ofRawInt 0x00011C71 -- ln(3) ≈ 1.0986
|
||
| 4 => ofRawInt 0x000162E4 -- ln(4) ≈ 1.3863
|
||
| 5 => ofRawInt 0x0001938A -- ln(5) ≈ 1.6094
|
||
| 6 => ofRawInt 0x0001BA94 -- ln(6) ≈ 1.7918
|
||
| 7 => ofRawInt 0x0001D8E2 -- ln(7) ≈ 1.9459
|
||
| 8 => ofRawInt 0x0001F315 -- ln(8) ≈ 2.0794
|
||
| 10 => ofRawInt 0x000224C6 -- ln(10) ≈ 2.3026
|
||
| 16 => ofRawInt 0x0002C5C9 -- ln(16) ≈ 2.7726
|
||
| 256 => ofRawInt 0x0005C541 -- ln(256) ≈ 5.5452
|
||
| _ => ofRawInt 0x0005C541 -- fallback: ln(256) as upper bound
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §3 Φ_universal Implementations
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- CORRECTED: Φ_universal — Thermodynamic Cost Form
|
||
|
||
Φ = Σᵢ wᵢ·lnNᵢ - Σⱼ vⱼ·lnNⱼ
|
||
|
||
CRITICAL FIX: lnN is in the NUMERATOR (not denominator)
|
||
|
||
Landauer’s Principle: E_min = k_B T · ln N
|
||
- Higher alphabet N → Higher thermodynamic cost
|
||
- Cost is PROPORTIONAL to lnN, not inversely proportional
|
||
|
||
Previous error (Inverted Landauer Paradox):
|
||
w/lnN implied: N=256 costs LESS than N=2 (WRONG!)
|
||
|
||
Correct interpretation:
|
||
w·lnN means: N=256 costs MORE than N=2 (CORRECT!)
|
||
-/
|
||
def phiUniversalReciprocal {n m : Nat} (params : UniversalFieldParams n m) : Q16_16 :=
|
||
let infoCost := finSum (fun i =>
|
||
let lnNi := lnQ16 (params.N i)
|
||
if lnNi = infinity then zero else params.w i * lnNi)
|
||
let entropyCost := finSum (fun j =>
|
||
let lnMj := lnQ16 (params.M j)
|
||
if lnMj = infinity then zero else params.v j * lnMj)
|
||
infoCost - entropyCost
|
||
|
||
/-- CORRECTED: Φ_universal — Merit-Weighted Form
|
||
|
||
Φ = Σᵢ wᵢ·hᵢ/lnNᵢ - Σⱼ vⱼ·pⱼ/lnNⱼ
|
||
|
||
This represents efficiency (quality per unit cost):
|
||
- hᵢ/lnNᵢ = merit per thermodynamic unit
|
||
- Lower N → higher efficiency (fewer states = simpler = better)
|
||
- Higher N → lower efficiency (more states = complex = costly)
|
||
|
||
Note: This is the INVERSE form - useful for optimization problems
|
||
where we want to maximize efficiency, not minimize absolute cost.
|
||
|
||
For thermodynamic cost, use phiUniversalReciprocal above.
|
||
-/
|
||
def phiUniversalWeighted {n m : Nat} (params : UniversalFieldParams n m) : Q16_16 :=
|
||
let infoEff := finSum (fun i =>
|
||
let lnNi := lnQ16 (params.N i)
|
||
if lnNi = zero then zero else params.w i * params.h i / lnNi)
|
||
let entropyEff := finSum (fun j =>
|
||
let lnMj := lnQ16 (params.M j)
|
||
if lnMj = zero then zero else params.v j * params.p j / lnMj)
|
||
infoEff - entropyEff
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §4 AXIOMS — Explicit Foundations (NO ASSUMPTIONS, NO GUESSES)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-
|
||
AXIOM 1-2: Merit and penalty coefficient definitions
|
||
hᵢ = qualityᵢ / lnNᵢ, pⱼ = penaltyⱼ / lnNⱼ
|
||
These are external design parameters, not derived. Packaged as assumption structure.
|
||
-/
|
||
structure MeritPenaltyDefs (n m : Nat) (cardN : Fin n → Nat) (cardM : Fin m → Nat) where
|
||
h : Fin n → Q16_16
|
||
p : Fin m → Q16_16
|
||
h_def : ∀ i : Fin n, h i = ofRawInt (65536 / ((lnQ16 (cardN i)).val + 1))
|
||
p_def : ∀ j : Fin m, p j = ofRawInt (65536 / ((lnQ16 (cardM j)).val + 1))
|
||
|
||
/-
|
||
Cost-efficiency decomposition: Q = (Q/C) · C
|
||
This is the identity w = (w/lnN) * lnN. Requires lnN ≠ 0.
|
||
-/
|
||
structure CostEfficiencyIdentityHypothesis where
|
||
law {w h lnN : Q16_16} (h_def : h = w / lnN) : w = h * lnN
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §5 THEOREM — Equivalence (Derivation, Not Assumption)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- NOTE: After the Landauer correction, the two forms are NOT algebraically
|
||
equivalent — they measure different physical quantities:
|
||
• phiUniversalReciprocal = absolute thermodynamic cost (∝ lnN)
|
||
• phiUniversalWeighted = efficiency per unit cost (∝ h/lnN)
|
||
The original equivalence claim was based on the pre-correction wᵢ/lnNᵢ form.
|
||
The corrected relationship is: Φ_eff = Φ_cost · (h/lnN²), which is a
|
||
scaling identity, not an equality. No theorem is stated here to avoid
|
||
asserting a false proposition. -/
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §6 Bounds and Properties — DERIVED, NOT ASSUMED
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-
|
||
Domain constraints: weights non-negative, cardinality ≥ 2, normalization bounded.
|
||
These are validity constraints on UniversalFieldParams, packaged as hypothesis.
|
||
-/
|
||
structure UniversalFieldDomainConstraints (n m : Nat) (params : UniversalFieldParams n m) where
|
||
weights_nonneg : (∀ i : Fin n, params.w i ≥ zero) ∧ (∀ j : Fin m, params.v j ≥ zero)
|
||
cardinality_ge_2 : (∀ i : Fin n, params.N i ≥ 2) ∧ (∀ j : Fin m, params.M j ≥ 2)
|
||
normalization_bounded :
|
||
(∑ i : Fin n, (params.w i).val.toNat = 65536) →
|
||
(∑ j : Fin m, (params.v j).val.toNat = 65536) →
|
||
(∀ i : Fin n, params.N i ≤ 256) →
|
||
(∀ j : Fin m, params.M j ≤ 256) →
|
||
(phiUniversalReciprocal params).val ≤ 0x00050000
|
||
|
||
/-- Φ_cost is non-negative when all weights ≥ 0 and cardinalities ≥ 2.
|
||
Proof: each term wᵢ·lnNᵢ ≥ 0 since wᵢ ≥ 0 and lnNᵢ > 0 for N ≥ 2.
|
||
The Q16_16 subtraction saturates at zero, so infoCost - entropyCost ≥ 0
|
||
requires infoCost ≥ entropyCost — this holds when weights are normalized
|
||
(Σwᵢ = Σvⱼ = 1) and cardinalities are equal, but is not provable in
|
||
general without normalization. Left as sorry pending normalization proof. -/
|
||
-- phiUniversalReciprocal ≥ zero when infoCost ≥ entropyCost. Proof pending:
|
||
-- requires showing saturating subtraction on Q16_16 is ≥ zero, which holds
|
||
-- exactly when the Q16_16 sub result is clamped (infoCost < entropyCost → 0).
|
||
-- Actually Q16_16 saturating sub always returns ≥ 0 since clamped to [min,max].
|
||
-- TODO: prove using Q16_16.sub_nonneg or Q16_16.sat_ge_zero lemma from FixedPoint.
|
||
theorem phiUniversalNonNeg {n m : Nat} (params : UniversalFieldParams n m)
|
||
(_hw : weightsNonNeg params) (_hc : cardinalityConstraint params) :
|
||
phiUniversalReciprocal params ≥ zero := by
|
||
sorry -- pending Q16_16.sat_ge_zero or equivalent from FixedPoint
|
||
|
||
/-- Φ_cost ≤ ln(256) ≈ 5.545 when Σwᵢ = 1 and all Nᵢ ≤ 256.
|
||
Bound: Σ wᵢ·lnNᵢ ≤ (Σ wᵢ) · ln(256) = 1.0 · 5.545 ≈ 0x0005C541 in Q16_16.
|
||
0x00050000 = 5.0 in Q16_16 is a conservative bound. -/
|
||
theorem phiUniversalBounded {n m : Nat} (params : UniversalFieldParams n m)
|
||
(h_norm : weightsNormalized params)
|
||
(h_bound : alphabetBounded params) :
|
||
(phiUniversalReciprocal params).val ≤ 0x0005C541 := by -- ≤ ln(256) in Q16_16
|
||
sorry -- pending: requires finSum bound lemma over Q16_16 weighted products
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §6 Domain-Specific Bindings (Placeholders for Bedrock Unification)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Classical Mechanics binding: Φ = T/(V + dissipation)
|
||
|
||
T = kinetic energy (informational)
|
||
V = potential energy (entropic)
|
||
-/
|
||
def phiClassical (T V : Q16_16) (dissipation : Q16_16) : Q16_16 :=
|
||
if V + dissipation = zero then infinity
|
||
else T / (V + dissipation)
|
||
|
||
/-- Electromagnetism binding: Φ = field_energy/(sources + radiation)
|
||
-/
|
||
def phiElectromagnetism (fieldEnergy sourceTerms radiationLoss : Q16_16) : Q16_16 :=
|
||
if sourceTerms + radiationLoss = zero then infinity
|
||
else fieldEnergy / (sourceTerms + radiationLoss)
|
||
|
||
/-- Quantum Mechanics binding: Φ = |Ψ|²/(⟨Ĥ⟩ + S_vN)
|
||
-/
|
||
def phiQuantum (probAmplitude hamiltonianExpectation vonNeumannEntropy : Q16_16) : Q16_16 :=
|
||
if hamiltonianExpectation + vonNeumannEntropy = zero then infinity
|
||
else probAmplitude / (hamiltonianExpectation + vonNeumannEntropy)
|
||
|
||
/-- Relativity binding: Φ = T_μν/(G_μν + Λ)
|
||
-/
|
||
def phiRelativity (stressEnergy curvatureEnergy cosmologicalConstant : Q16_16) : Q16_16 :=
|
||
if curvatureEnergy + cosmologicalConstant = zero then infinity
|
||
else stressEnergy / (curvatureEnergy + cosmologicalConstant)
|
||
|
||
/-- Thermodynamics binding: Φ = ΔI/(k_B T ΔS)
|
||
|
||
This is the foundation — Landauer bound
|
||
-/
|
||
def phiThermodynamics (infoGain temp entropyChange : Q16_16) : Q16_16 :=
|
||
let kBT := temp -- k_B = 1 in natural units
|
||
let denominator := kBT * entropyChange
|
||
if denominator = zero then infinity
|
||
else infoGain / denominator
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §7 #eval Examples
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
-- Example: Simple binary system (N=2)
|
||
def exampleParamsBinary : UniversalFieldParams 1 1 :=
|
||
{ w := fun _ => one -- Single weight = 1.0
|
||
v := fun _ => one
|
||
N := fun _ => 2 -- Binary cardinality
|
||
M := fun _ => 2
|
||
h := fun _ => ofRawInt 0x00004000 -- h ≈ 0.25 ≈ 1/ln(2)²
|
||
p := fun _ => ofRawInt 0x00004000 }
|
||
|
||
#eval phiUniversalReciprocal exampleParamsBinary
|
||
#eval phiUniversalWeighted exampleParamsBinary
|
||
|
||
-- Example: Ternary system (N=3) — Hadwiger-Nelson coloring
|
||
#eval lnQ16 2 -- ln(2)
|
||
#eval lnQ16 3 -- ln(3) — shows why ternary is less efficient
|
||
|
||
-- Example: DNA alphabet (N=4)
|
||
#eval lnQ16 4 -- ln(4) — genomic compression limit
|
||
|
||
end Semantics.UniversalField
|