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253 lines
10 KiB
Text
253 lines
10 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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MathGPT.lean — Mathematical Rigor Enforcement System
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This module provides automated mathematical verification to prevent
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incorrect formulations from entering the codebase.
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Core principle: All equations must be physically consistent before implementation.
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-/
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import Mathlib.Analysis.SpecialFunctions.Log.Basic
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import Mathlib.Data.Real.Basic
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namespace MathGPT
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Physical Law Registry — Immutable Truth Sources
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A physical law that must be respected by all equations -/
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structure PhysicalLaw where
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name : String
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statement : String
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mathematicalForm : String
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domain : String -- "thermodynamics", "information_theory", "quantum", etc.
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violations : List String -- Common incorrect formulations
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deriving Repr
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/-- LANDAUER'S PRINCIPLE — Core constraint on information erasure
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The minimum energy to erase one bit of information at temperature T.
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This is the foundation of all information-thermodynamics in OTOM.
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Mathematical form: E_min = k_B · T · ln(N)
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Common violations to reject:
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1. E ∝ 1/ln(N) — "inverse cost" (wrong!)
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2. E ∝ ln(1/N) — negative entropy (wrong!)
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3. E independent of N — constant cost (wrong!)
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-/
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def landauerPrinciple : PhysicalLaw := {
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name := "Landauer's Principle",
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statement := "Minimum energy to erase information scales as ln(N)",
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mathematicalForm := "E_min = k_B · T · ln(N)",
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domain := "thermodynamics",
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violations := [
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"E ∝ 1/ln(N) — inverse cost violates physics",
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"E ∝ -ln(N) — negative entropy impossible",
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"E constant — ignores alphabet size"
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]
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}
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/-- SHANNON ENTROPY — Information content bound
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Mathematical form: H = -Σ p · log₂(p)
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Maximum: log₂(N) for uniform distribution
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-/
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def shannonEntropy : PhysicalLaw := {
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name := "Shannon Entropy",
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statement := "Information content bounded by log(N)",
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mathematicalForm := "H = -Σ p · log₂(p) ≤ log₂(N)",
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domain := "information_theory",
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violations := [
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"H > log₂(N) — exceeds maximum",
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"H < 0 — negative entropy impossible"
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]
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}
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/-- CONSERVATION OF ENERGY — First law of thermodynamics
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Energy cannot be created or destroyed, only converted.
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-/
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def conservationOfEnergy : PhysicalLaw := {
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name := "Conservation of Energy",
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statement := "Total energy constant in isolated system",
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mathematicalForm := "ΔE_total = 0",
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domain := "thermodynamics",
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violations := [
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"ΔE < 0 — energy destruction",
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"ΔE > 0 — energy creation"
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]
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Equation Validator — Automated Rigor Checking
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Validation result for an equation -/
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inductive ValidationResult
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| valid (reason : String)
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| invalid (reason : String) (lawViolated : String)
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| warning (message : String)
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deriving Repr
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/-- Check if equation respects Landauer scaling
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E(N) must be:
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1. Monotonically increasing in N
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2. Proportional to ln(N), not 1/ln(N)
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3. Non-negative for all N ≥ 2
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-/
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def checkLandauerScaling (costFunction : ℕ → ℝ) : ValidationResult :=
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-- Check monotonicity: N₁ < N₂ → cost(N₁) < cost(N₂)
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let monoCheck := ∀ N₁ N₂ : ℕ, N₁ ≥ 2 → N₂ ≥ 2 → N₁ < N₂ → costFunction N₁ < costFunction N₂
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-- Check proportionality: cost(N) ∝ ln(N)
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let propCheck := ∃ k : ℝ, k > 0 ∧ ∀ N : ℕ, N ≥ 2 → costFunction N = k * Real.log N
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-- Check non-negativity
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let nonNegCheck := ∀ N : ℕ, N ≥ 2 → costFunction N ≥ 0
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if ¬monoCheck then
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ValidationResult.invalid
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"Cost decreases with alphabet size — violates Landauer monotonicity"
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"Landauer's Principle"
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else if ¬nonNegCheck then
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ValidationResult.invalid
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"Negative thermodynamic cost — physically impossible"
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"Landauer's Principle"
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else
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ValidationResult.valid "Landauer scaling respected"
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/-- Check for common incorrect formulations -/
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def checkCommonErrors (equation : String) : List ValidationResult :=
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let errors := [
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("/lnN", "Inverse logarithmic scaling — violates Landauer"),
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("1/ln", "Reciprocal cost — physically absurd"),
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("ln(1/N)", "Negative entropy argument — impossible"),
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("-lnN", "Negative cost — violates second law"),
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("/ln N", "Spaced inverse form — still wrong"),
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("denominator.*ln", "ln in denominator — check Landauer consistency")
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]
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errors.filterMap (λ (pattern, reason) =>
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if equation.contains pattern then
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some (ValidationResult.invalid reason "Landauer's Principle")
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else
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none
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)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Universal Field Validator — Specific to EQUATION #0
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Validate Universal Field Φ against physical laws
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Two forms must be checked:
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1. Cost form: Φ = Σ w·lnN - Σ v·lnN ← MUST pass Landauer
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2. Efficiency form: Φ = Σ w·h/lnN - Σ v·p/lnN ← Inverse is correct here
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-/
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def validateUniversalField (equation : String) : ValidationResult :=
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-- Check for old (wrong) cost form
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if equation.contains "w/lnN" ∨ equation.contains "wᵢ/lnNᵢ" then
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ValidationResult.invalid
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"CRITICAL: lnN in denominator for cost form violates Landauer. " ++
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"Cost must be w·lnN, not w/lnN. " ++
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"Higher alphabet = higher energy cost."
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"Landauer's Principle"
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-- Check for correct cost form
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else if equation.contains "w·lnN" ∨ equation.contains "w*lnN" ∨ equation.contains "w * ln" then
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ValidationResult.valid
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"Correct Landauer scaling: cost ∝ lnN. " ++
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"Higher alphabet = higher thermodynamic cost."
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-- Check for efficiency form (lnN in denominator is correct here)
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else if equation.contains "h/lnN" ∨ equation.contains "hᵢ/lnNᵢ" then
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ValidationResult.valid
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"Efficiency form correct: h/lnN = quality per unit cost. " ++
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"Efficiency decreases as cost increases."
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-- Ambiguous or unrecognizable
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else
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ValidationResult.warning
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"Equation form unclear — manual review required"
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Pre-Commit Hook — Block Bad Math Before Entry
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Pre-commit validation for any new equation
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This function should be called before any equation is:
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- Added to math_entities.db
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- Committed to git
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- Added to MATH_MODEL_MAP
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- Implemented in Lean
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- Published in papers
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Returns: (is_valid, reasons)
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-/
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def preCommitValidation (equation : String) (author : String) : (Bool × List String) :=
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let results := checkCommonErrors equation
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let universalCheck := [validateUniversalField equation]
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let allResults := results ++ universalCheck
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let errors := allResults.filterMap (λ r =>
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match r with
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| ValidationResult.invalid reason law => some s!"[VIOLATION: {law}] {reason}"
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| _ => none
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)
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let warnings := allResults.filterMap (λ r =>
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match r with
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| ValidationResult.warning msg => some s!"[WARNING] {msg}"
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| _ => none
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)
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let isValid := errors.isEmpty
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let report := if isValid then
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[s!"✅ VALIDATED by MathGPT for {author}",
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s!"Equation passes all physical law checks"]
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++ warnings
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else
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[s!"❌ REJECTED by MathGPT for {author}",
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s!"Equation violates physical laws — cannot be committed"]
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++ errors
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++ warnings
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++ ["",
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"FIX REQUIRED: Ensure equation respects:",
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s!" - {landauerPrinciple.name}: {landauerPrinciple.mathematicalForm}",
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"",
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"Common fixes:",
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" - Change w/lnN to w·lnN (cost form)",
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" - Or explicitly use h/lnN for efficiency (inverse is correct there)"]
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(isValid, report)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Automated Verification Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Example: Correct cost form
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def exampleCorrectCost : String := "Φ = Σ w·lnN - Σ v·lnN"
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#eval preCommitValidation exampleCorrectCost "Builder"
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-- Example: Incorrect (old) cost form — SHOULD BE REJECTED
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def exampleIncorrectCost : String := "Φ = Σ w/lnN + Σ v/lnN"
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#eval preCommitValidation exampleIncorrectCost "Builder"
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-- Example: Correct efficiency form
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def exampleCorrectEfficiency : String := "Φ = Σ w·h/lnN - Σ v·p/lnN"
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#eval preCommitValidation exampleCorrectEfficiency "Builder"
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-- Example: Ambiguous form — SHOULD WARN
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def exampleAmbiguous : String := "Φ = Σ wN"
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#eval preCommitValidation exampleAmbiguous "Builder"
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end MathGPT
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