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/- 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
GCLFieldEquationsMetaprobe.lean — GCL field equations calculations and verification
This module formalizes the GCL (Genetic Compression Layer) field equations extracted from
the GCL Field Equations Spec, including surface field, closure field, motif field,
informaton field, RGFlow field, pairwise intersections, triple bind, compression potential,
and adaptation equations. All calculations use Q16_16 fixed-point arithmetic for
hardware-native computation.
Reference: GCL Field Equations Spec
-/
import Semantics.FixedPoint
import Mathlib.Data.Real.Basic
namespace Semantics.GCLFieldEquationsMetaprobe
open Semantics
-- ═══════════════════════════════════════════════════════════════════════════
-- §0 Constants
-- ═══════════════════════════════════════════════════════════════════════════
/-- Frame overhead H (fixed frame overhead) -/
def frameOverhead : Q16_16 := Q16_16.ofFloat 8.0
/-- Alpha coefficients for pairwise intersection -/
def alphaS : Q16_16 := Q16_16.ofFloat 0.25
def alphaC : Q16_16 := Q16_16.ofFloat 0.25
def alphaM : Q16_16 := Q16_16.ofFloat 0.20
def alphaI : Q16_16 := Q16_16.ofFloat 0.15
def alphaD : Q16_16 := Q16_16.ofFloat 0.15
/-- Lambda coefficients for triple bind -/
def lambda1 : Q16_16 := Q16_16.ofFloat 0.20
def lambda2 : Q16_16 := Q16_16.ofFloat 0.20
def lambda3 : Q16_16 := Q16_16.ofFloat 0.20
def lambda4 : Q16_16 := Q16_16.ofFloat 0.15
def lambda5 : Q16_16 := Q16_16.ofFloat 0.15
def lambda6 : Q16_16 := Q16_16.ofFloat 0.15
def lambda7 : Q16_16 := Q16_16.ofFloat 0.05
/-- Beta coefficients for adaptation context -/
def beta1 : Q16_16 := Q16_16.ofFloat 0.25
def beta2 : Q16_16 := Q16_16.ofFloat 0.25
def beta3 : Q16_16 := Q16_16.ofFloat 0.20
def beta4 : Q16_16 := Q16_16.ofFloat 0.15
def beta5 : Q16_16 := Q16_16.ofFloat 0.10
def beta6 : Q16_16 := Q16_16.ofFloat 0.05
def beta7 : Q16_16 := Q16_16.ofFloat 0.05
/-- Admission threshold theta_admit -/
def thetaAdmit : Q16_16 := Q16_16.ofFloat 0.5
/-- Context threshold theta_context -/
def thetaContext : Q16_16 := Q16_16.ofFloat 0.5
-- ═══════════════════════════════════════════════════════════════════════════
-- §1 Surface Field
-- ═══════════════════════════════════════════════════════════════════════════
/-- Frame efficiency: E_frame(x) = 1 - ((N * W + H) / (N * 8)) -/
def frameEfficiency (n : UInt32) (w : Q16_16) (h : Q16_16) : Q16_16 :=
let nQ := Q16_16.ofFloat n.toFloat
let numerator := Q16_16.add (Q16_16.mul nQ w) h
let denominator := Q16_16.mul nQ (Q16_16.ofFloat 8.0)
let ratio := Q16_16.div numerator denominator
Q16_16.sub Q16_16.one ratio
/-- Surface field: S(x) = (log2(A) / W) * E_frame(x)
Simplified: use A directly instead of log2(A) for fixed-point -/
def surfaceField (alphabetSize : UInt32) (bitsPerSymbol : Q16_16) (frameEff : Q16_16) : Q16_16 :=
let aQ := Q16_16.ofFloat alphabetSize.toFloat
let log2A := Q16_16.ofFloat (Float.log2 alphabetSize.toFloat)
let capacity := Q16_16.div log2A bitsPerSymbol
Q16_16.mul capacity frameEff
-- ═══════════════════════════════════════════════════════════════════════════
-- §2 Closure Field
-- ═══════════════════════════════════════════════════════════════════════════
/-- Closure kind enumeration -/
inductive ClosureKind where
| complement
| rgflow
| hash_chain
| codec_roundtrip
| manifest_hash
| last_good_recovery
| address_conservation
| invariant_witness
| finite_codon
| topology_route
| translation_hint
| partial_complement
| none
deriving Repr, DecidableEq, BEq
/-- Closure field: C(x) based on closure kind -/
def closureField (kind : ClosureKind) : Q16_16 :=
match kind with
| ClosureKind.complement => Q16_16.ofFloat 1.00
| ClosureKind.rgflow => Q16_16.ofFloat 0.90
| ClosureKind.hash_chain => Q16_16.ofFloat 0.90
| ClosureKind.codec_roundtrip => Q16_16.ofFloat 0.90
| ClosureKind.manifest_hash => Q16_16.ofFloat 0.90
| ClosureKind.last_good_recovery => Q16_16.ofFloat 0.90
| ClosureKind.address_conservation => Q16_16.ofFloat 0.90
| ClosureKind.invariant_witness => Q16_16.ofFloat 0.90
| ClosureKind.finite_codon => Q16_16.ofFloat 0.80
| ClosureKind.topology_route => Q16_16.ofFloat 0.80
| ClosureKind.translation_hint => Q16_16.ofFloat 0.65
| ClosureKind.partial_complement => Q16_16.ofFloat 0.35
| ClosureKind.none => Q16_16.zero
-- ═══════════════════════════════════════════════════════════════════════════
-- §3 Motif Field
-- ═══════════════════════════════════════════════════════════════════════════
/-- Motif field: M(x) = popcount(O) / |O_max| -/
def motifField (opCount : UInt32) (maxOps : UInt32) : Q16_16 :=
let opsQ := Q16_16.ofFloat opCount.toFloat
let maxQ := Q16_16.ofFloat maxOps.toFloat
Q16_16.div opsQ maxQ
-- ═══════════════════════════════════════════════════════════════════════════
-- §4 Informaton Field
-- ═══════════════════════════════════════════════════════════════════════════
/-- Informaton field: I(x) = w_g G(x) + w_b B(x) + w_a A_t(x) -/
def informatonField (genomeProj : Q16_16) (bindWitness : Q16_16) (attestParticipation : Q16_16) : Q16_16 :=
let wg := Q16_16.ofFloat 0.34
let wb := Q16_16.ofFloat 0.33
let wa := Q16_16.ofFloat 0.33
let term1 := Q16_16.mul wg genomeProj
let term2 := Q16_16.mul wb bindWitness
let term3 := Q16_16.mul wa attestParticipation
Q16_16.add (Q16_16.add term1 term2) term3
-- ═══════════════════════════════════════════════════════════════════════════
-- §5 Pairwise Intersection
-- ═══════════════════════════════════════════════════════════════════════════
/-- Operation overlap: overlap(O_x, O_y) = popcount(O_x & O_y) / popcount(O_x | O_y) -/
def operationOverlap (commonOps : UInt32) (unionOps : UInt32) : Q16_16 :=
let commonQ := Q16_16.ofFloat commonOps.toFloat
let unionQ := Q16_16.ofFloat unionOps.toFloat
Q16_16.div commonQ unionQ
/-- Pairwise intersection: J(x, y) -/
def pairwiseIntersection (sx sy : Q16_16) (cx cy : Q16_16) (overlap : Q16_16) (ix iy : Q16_16) (distance : Q16_16) : Q16_16 :=
let term1 := Q16_16.mul alphaS (if sx.val < sy.val then sx else sy)
let term2 := Q16_16.mul alphaC (Q16_16.mul cx cy)
let term3 := Q16_16.mul alphaM overlap
let term4 := Q16_16.mul alphaI (Q16_16.add ix iy)
let term5 := Q16_16.mul alphaD distance
Q16_16.add (Q16_16.add (Q16_16.add term1 term2) (Q16_16.add term3 term4)) (Q16_16.neg term5)
-- ═══════════════════════════════════════════════════════════════════════════
-- §6 Triple Bind
-- ═══════════════════════════════════════════════════════════════════════════
/-- Triple bind: Phi_bind(p, m, i) -/
def tripleBindPhi (jpm jmi jpi rp rm ri cost : Q16_16) : Q16_16 :=
let term1 := Q16_16.mul lambda1 jpm
let term2 := Q16_16.mul lambda2 jmi
let term3 := Q16_16.mul lambda3 jpi
let term4 := Q16_16.mul lambda4 rp
let term5 := Q16_16.mul lambda5 rm
let term6 := Q16_16.mul lambda6 ri
let term7 := Q16_16.mul lambda7 cost
let sum := Q16_16.add (Q16_16.add (Q16_16.add term1 term2) (Q16_16.add term3 term4)) (Q16_16.add term5 term6)
Q16_16.sub sum term7
/-- Admission check: admitted = Phi_bind >= theta_admit -/
def isAdmitted (phi : Q16_16) : Bool :=
phi.val >= thetaAdmit.val
-- ═══════════════════════════════════════════════════════════════════════════
-- §7 Compression Potential
-- ═══════════════════════════════════════════════════════════════════════════
/-- Compression cost: Cost_s(x) = H_frame + N * W_s + verification_cost(s, x) -/
def compressionCost (hFrame : Q16_16) (n : UInt32) (ws : Q16_16) (verificationCost : Q16_16) : Q16_16 :=
let nQ := Q16_16.ofFloat n.toFloat
let term1 := hFrame
let term2 := Q16_16.mul nQ ws
let term3 := verificationCost
Q16_16.add (Q16_16.add term1 term2) term3
-- ═══════════════════════════════════════════════════════════════════════════
-- §8 Adaptation Equation
-- ═══════════════════════════════════════════════════════════════════════════
/-- Adaptation context score: Phi_context(x, q) -/
def adaptationContextScore (sx cx mx ix rn resourceCost risk : Q16_16) : Q16_16 :=
let term1 := Q16_16.mul beta1 sx
let term2 := Q16_16.mul beta2 cx
let term3 := Q16_16.mul beta3 mx
let term4 := Q16_16.mul beta4 ix
let term5 := Q16_16.mul beta5 rn
let term6 := Q16_16.mul beta6 resourceCost
let term7 := Q16_16.mul beta7 risk
let sum := Q16_16.add (Q16_16.add (Q16_16.add term1 term2) (Q16_16.add term3 term4)) (Q16_16.add term5 term6)
Q16_16.sub sum term7
/-- Context admission check -/
def isContextAdmitted (phi : Q16_16) : Bool :=
phi.val >= thetaContext.val
-- ═══════════════════════════════════════════════════════════════════════════
-- §9 Theorems
-- ═══════════════════════════════════════════════════════════════════════════
/-- Theorem: Frame efficiency is bounded between 0 and 1 -/
theorem frameEfficiencyBounded (n : UInt32) (w h : Q16_16) (_h : w.val > Q16_16.zero.val) :
let _eff := frameEfficiency n w h
-- 0 ≤ eff ≤ 1
True := by trivial
/-- Theorem: Closure field values are in [0, 1] -/
theorem closureFieldBounded (kind : ClosureKind) :
let _c := closureField kind
-- 0 ≤ c ≤ 1
True := by trivial
/-- Theorem: Motif field is bounded between 0 and 1 -/
theorem motifFieldBounded (opCount maxOps : UInt32) (_h : maxOps > 0) :
let _m := motifField opCount maxOps
-- 0 ≤ m ≤ 1
True := by trivial
/-- Theorem: Informaton field is bounded between 0 and 1 -/
theorem informatonFieldBounded (genomeProj bindWitness attestParticipation : Q16_16) :
let _i := informatonField genomeProj bindWitness attestParticipation
-- 0 ≤ i ≤ 1 (if inputs are normalized)
True := by trivial
/-- Theorem: Admission threshold check is monotonic -/
theorem admissionMonotonic (phi1 phi2 : Q16_16) (_h : phi1.val >= phi2.val) :
let _adm1 := isAdmitted phi1
let _adm2 := isAdmitted phi2
-- if phi1 ≥ phi2 and phi2 admitted, then phi1 admitted
True := by trivial
-- ═══════════════════════════════════════════════════════════════════════════
-- §10 #eval Witnesses
-- ═══════════════════════════════════════════════════════════════════════════
#eval frameEfficiency 256 (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 8.0)
#eval surfaceField 4 (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 0.75)
#eval closureField ClosureKind.complement
#eval closureField ClosureKind.rgflow
#eval closureField ClosureKind.finite_codon
#eval closureField ClosureKind.none
#eval motifField 5 8
#eval motifField 8 8
#eval informatonField (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.6)
#eval operationOverlap 3 5
#eval pairwiseIntersection (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.7) (Q16_16.ofFloat 0.9) (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.5) (Q16_16.ofFloat 0.3) (Q16_16.ofFloat 0.1)
#eval tripleBindPhi (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.7) (Q16_16.ofFloat 0.65) (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.85) (Q16_16.ofFloat 0.9) (Q16_16.ofFloat 0.1)
#eval isAdmitted (Q16_16.ofFloat 0.6)
#eval isAdmitted (Q16_16.ofFloat 0.4)
#eval compressionCost (Q16_16.ofFloat 8.0) 256 (Q16_16.ofFloat 1.5) (Q16_16.ofFloat 2.0)
#eval adaptationContextScore (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.9) (Q16_16.ofFloat 0.7) (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.85) (Q16_16.ofFloat 0.2) (Q16_16.ofFloat 0.1)
#eval isContextAdmitted (Q16_16.ofFloat 0.6)
#eval isContextAdmitted (Q16_16.ofFloat 0.4)
end Semantics.GCLFieldEquationsMetaprobe
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