-- DYNAMIC_CANAL.lean -- Reference Kernel Spec: SIMD/Fluid Hybrid with Pressure-Adaptive Transport -- Fixed-point only, saturating arithmetic, unified step function -- Integrates DIAT, AVMR, N-DAG, Dynamic Canal, and Throat models import CoreFormalism.FixedPoint import CoreFormalism.Tactics import CoreFormalism.Q16_16Numerics open SilverSight.FixedPoint.Q16_16 set_option linter.dupNamespace false namespace SilverSight.DynamicCanal open SilverSight.FixedPoint.Q16_16 -- ============================================================ -- 1a. Fix16 Type Alias (for NBody compatibility) -- ============================================================ /-- Fix16 is an alias for Q16_16, used in physics contexts. Provides signed fixed-point arithmetic for particle simulations. -/ abbrev Fix16 := Q16_16 namespace Fix16 -- Re-export Q16_16 constants and operations under Fix16 namespace def zero := Q16_16.zero def one := Q16_16.one def epsilon := Q16_16.epsilon def abs := Q16_16.abs def add := Q16_16.add def sub := Q16_16.sub def mul := Q16_16.mul def div := Q16_16.div def sqrt := Q16_16.sqrt def max := Q16_16.max def sat01 := Q16_16.sat01 def ofInt := Q16_16.ofInt def ofNat := Q16_16.ofNat def toInt := Q16_16.toInt def neg := Q16_16.neg def mk (raw : UInt32) : Fix16 := Q16_16.ofBits raw end Fix16 -- ============================================================ -- 2. VECTOR PRIMITIVES -- ============================================================ /-- Small fixed-point vector -/ abbrev VecN (n : Nat) := Fin n → Q16_16 /-- Zero vector -/ def VecN.zero {n : Nat} : VecN n := fun _ => Q16_16.zero /-- Vector addition (component-wise saturating) -/ def vecAdd {n : Nat} (a b : VecN n) : VecN n := fun i => Q16_16.add (a i) (b i) /-- Vector subtraction -/ def vecSub {n : Nat} (a b : VecN n) : VecN n := fun i => Q16_16.sub (a i) (b i) /-- Vector L1 norm (sum of absolute values) -/ noncomputable def vecL1 {n : Nat} (v : VecN n) : Q16_16 := Fin.foldl n (fun acc i => Q16_16.add acc (Q16_16.abs (v i))) Q16_16.zero /-- Vector max absolute component -/ def vecMaxAbs {n : Nat} (v : VecN n) : Q16_16 := Fin.foldl n (fun acc i => Q16_16.max acc (Q16_16.abs (v i))) Q16_16.zero /-- Dot product -/ def vecDot {n : Nat} (a b : VecN n) : Q16_16 := Fin.foldl n (fun acc i => Q16_16.add acc (Q16_16.mul (a i) (b i))) Q16_16.zero -- ============================================================ -- 3. ENUMERATIONS -- ============================================================ /-- Execution regime for lanes -/ inductive Regime | coherent -- Stable transport | stressed -- Distorted transport | throat -- Wormhole transfer deriving Repr, DecidableEq, BEq /-- Execution mode: explicit lanes vs aggregate fluid -/ inductive ExecMode | lane | fluid deriving Repr, DecidableEq, BEq /-- Throat classification -/ inductive ThroatClass | stableBridge | lossyChannel | rupture deriving Repr, DecidableEq, BEq /-- Gradient type for precision metrics -/ inductive GradientType | pressureGradient | thermalGradient | velocityGradient | densityGradient | stressGradient deriving Repr, DecidableEq, BEq -- ============================================================ -- 4. DIAT (Dual-Interval Algebraic Transform) -- ============================================================ /-- DIAT encoding of integer n: shell + distances to adjacent squares -/ structure DIAT where shell : UInt32 -- k = floor(sqrt(n)) a : UInt32 -- n - k² (forward distance) b : UInt32 -- (k+1)² - n (backward distance) prod : UInt32 -- a * b (shell interaction) diff : Int32 -- a - b (signed asymmetry) deriving Repr, DecidableEq, BEq namespace DIAT /-- Integer square root via Mathlib's proven Nat.sqrt. Delegates to Nat.sqrt for O(1)-amortized, proven-correct floor(sqrt). -/ def isqrt (n : UInt32) : UInt32 := UInt32.ofNat (Nat.sqrt n.toNat) /-- isqrt_spec: DIAT.isqrt computes floor(sqrt(n)) for all n < 0x400000. Direct from Mathlib's Nat.sqrt_le and Nat.lt_succ_sqrt. The Nat-level version is already proven in UnifiedCompression.lean. This UInt32 version is a bridge using the same Nat-level lemmas, proven by unfolding UInt32.ofNat/toNat definitions and using Nat.sqrt_le / Nat.lt_succ_sqrt since n is bounded below the overflow threshold (n < 0x400000). -/ theorem isqrt_spec (n : UInt32) (h : n < 0x400000) : let k := DIAT.isqrt n k * k ≤ n ∧ n < (k + 1) * (k + 1) := by intro k have hn_lt : n.toNat < 4194304 := by have h_lt := UInt32.lt_iff_toNat_lt.mp h have h_eq : (0x400000 : UInt32).toNat = 4194304 := rfl rw [h_eq] at h_lt exact h_lt -- Let k_nat = Nat.sqrt n.toNat let k_nat := Nat.sqrt n.toNat have hk_nat_def : k = UInt32.ofNat k_nat := rfl -- Since n.toNat < 4194304, k_nat < 2048 have hk_nat_lt : k_nat < 2048 := by by_contra h_ge have h_ge_nat : k_nat ≥ 2048 := by omega have h_sq_ge : k_nat * k_nat ≥ 2048 * 2048 := by nlinarith have h_sq_le : k_nat * k_nat ≤ n.toNat := Nat.sqrt_le n.toNat omega have hk_toNat : k.toNat = k_nat := by rw [hk_nat_def] unfold UInt32.toNat UInt32.ofNat rw [BitVec.toNat_ofNat] apply Nat.mod_eq_of_lt omega have h_mul_le : k * k ≤ n := by rw [UInt32.le_iff_toNat_le] rw [UInt32.toNat_mul] rw [hk_toNat] have h_mod : k_nat * k_nat % 2 ^ 32 = k_nat * k_nat := by apply Nat.mod_eq_of_lt -- k_nat < 2048, so k_nat * k_nat < 2048 * 2048 = 4194304 < 2^32 nlinarith rw [h_mod] exact Nat.sqrt_le n.toNat have hk1_toNat : (k + 1).toNat = k_nat + 1 := by rw [UInt32.toNat_add] rw [hk_toNat] have h1_toNat : (1 : UInt32).toNat = 1 := rfl rw [h1_toNat] apply Nat.mod_eq_of_lt omega have h_lt_mul : n < (k + 1) * (k + 1) := by rw [UInt32.lt_iff_toNat_lt] rw [UInt32.toNat_mul] rw [hk1_toNat] have h_mod : (k_nat + 1) * (k_nat + 1) % 2 ^ 32 = (k_nat + 1) * (k_nat + 1) := by apply Nat.mod_eq_of_lt -- k_nat < 2048, so (k_nat+1)*(k_nat+1) <= 2049*2049 = 4198401 < 2^32 nlinarith rw [h_mod] exact Nat.lt_succ_sqrt n.toNat exact ⟨h_mul_le, h_lt_mul⟩ /-- Encode integer n as DIAT tuple -/ def encode (n : UInt32) : DIAT := let k := isqrt n let lo := k * k let kp := k + 1 let hi := kp * kp let a := n - lo let b := hi - n { shell := k a := a b := b prod := a * b diff := Int32.ofInt (a.toNat : Int) - Int32.ofInt (b.toNat : Int) } /-- Shell width = 2k + 1 -/ def shellWidth (d : DIAT) : UInt32 := 2 * d.shell + 1 /-- Normalized a: a / (2k+1) -/ def normA (d : DIAT) : Q16_16 := Q16_16.div (Q16_16.ofBits d.a) (Q16_16.ofBits ((2 * d.shell + 1) * 0x10000)) end DIAT -- ============================================================ -- 5. TIMING AND PAYLOAD -- ============================================================ /-- Timing tuple for synchronization -/ structure Timing where slot : UInt16 parity : Bool index : UInt32 deriving Repr, DecidableEq, BEq /-- Lane payload with DIAT and metadata -/ structure LanePayload where diat : DIAT codonWindow : UInt32 -- Packed representation metadata : Q16_16 -- Scalar metadata (generalized from array) deriving Repr, DecidableEq, BEq -- ============================================================ -- 6. CORE DATA STRUCTURES -- ============================================================ /-- SIMD lane state -/ structure Lane where active : Bool node : UInt32 pos : VecN 3 -- 3D position (generalized N-space) vel : VecN 3 -- 3D velocity phase : Q16_16 stress : Q16_16 pressure : Q16_16 lambdaEff : Q16_16 -- Dynamic canal effective resistance energy : Q16_16 mismatch : Q16_16 regime : Regime timing : Timing payload : LanePayload /-- AVMR summary for aggregation -/ structure AVMRSummary where count : UInt32 phaseX : Q16_16 phaseY : Q16_16 coherence : Q16_16 mismatchSum : Q16_16 mismatchMax : Q16_16 massSum : Q16_16 energySum : Q16_16 coherentCnt : UInt32 stressedCnt : UInt32 throatCnt : UInt32 deriving Repr, DecidableEq, BEq /-- Canal section for fluid mode -/ structure CanalSection where density : Q16_16 capacity : Q16_16 flux : Q16_16 siphon : Q16_16 meanEnergy : Q16_16 meanMismatch : Q16_16 meanStress : Q16_16 pressure : Q16_16 lambdaEff : Q16_16 compliance : Q16_16 width : Q16_16 roughness : Q16_16 gradient : Q16_16 throatExposure : Q16_16 unpackScore : Q16_16 unpacked : Bool loopIteration : Nat -- Current loop iteration coarseGrainLevel : Nat -- Current coarse-graining level (0 = full precision) deriving Repr, DecidableEq, BEq /-- Precision metrics based on gradient type -/ structure PrecisionMetrics where pressurePrecision : Q16_16 -- Precision for pressure gradients thermalPrecision : Q16_16 -- Precision for thermal gradients velocityPrecision : Q16_16 -- Precision for velocity gradients densityPrecision : Q16_16 -- Precision for density gradients stressPrecision : Q16_16 -- Precision for stress gradients deriving Repr, DecidableEq, BEq /-- Edge attributes -/ structure EdgeAttr where baseWeight : Q16_16 dPos : VecN 3 dPhase : Q16_16 dEnergy : Q16_16 torsion : Q16_16 loss : Q16_16 mismatchGain : Q16_16 capacity : Q16_16 pressureCoupling : Q16_16 throatBias : Q16_16 prefPhase : Q16_16 isThroat : Bool /-- Graph edge -/ structure Edge where src : UInt32 dst : UInt32 attr : EdgeAttr /-- Node state across universes -/ structure NodeState where diatState : Q16_16 waveState : Q16_16 timeState : Q16_16 torsionState : Q16_16 fluidState : Q16_16 deriving Repr, DecidableEq, BEq /-- N-DAG (N-dimensional directed graph) -/ structure NDAG where nodes : Array NodeState edges : Array Edge /-- Throat state -/ structure ThroatState where edgeId : UInt32 mismatchNorm : Q16_16 dynWeight : Q16_16 healingGain : Q16_16 cls : ThroatClass deriving Repr, DecidableEq, BEq -- ============================================================ -- 7. GLOBAL PARAMETERS -- ============================================================ /-- Kernel configuration parameters -/ structure KernelParams where -- Stress model alphaSurprise : Q16_16 betaRegret : Q16_16 -- Dynamic Canal lambda0 : Q16_16 -- Base resistance canalElasticity : Q16_16 -- ξ: pressure sensitivity canalSaturation : Q16_16 -- σ: minimum fraction pressureDecay : Q16_16 -- γ: memory decay bioWeight : Q16_16 -- External pressure weight -- Regime thresholds coherentThresh : Q16_16 throatThresh : Q16_16 stressThresh : Q16_16 -- Update rates relaxRate : Q16_16 healRate : Q16_16 torsionRate : Q16_16 torsionEnergyExtraction : Q16_16 -- How much energy torsion steals from manifold mismatchRate : Q16_16 energyLossRate : Q16_16 -- Capacity model capacityPressure : Q16_16 capacityRoughness : Q16_16 capacityMismatch : Q16_16 -- Velocity model velGradientGain : Q16_16 velDensityLoss : Q16_16 velRoughnessLoss : Q16_16 velMismatchLoss : Q16_16 velComplianceGain : Q16_16 -- Throat model throatMismatchLoss : Q16_16 throatPressureGain : Q16_16 throatStressLoss : Q16_16 -- Unpack threshold thetaDensity : Q16_16 thetaMismatch : Q16_16 thetaStress : Q16_16 thetaPT : Q16_16 -- Pressure-throat interaction thetaCrit : Q16_16 deriving Repr -- ============================================================ -- 8. DYNAMIC CANAL LAW (Core Constitutive Equation) -- ============================================================ namespace DynamicCanal /-- Dynamic Canal law: λ_eff(P) = λ₀[σ + (1-σ)e^(-ξP)] Uses rigorous Q16_16Numerics.expNeg for the exponential. -/ def dynamicCanalLambda (p : KernelParams) (pressure : Q16_16) : Q16_16 := let ξP := Q16_16.mul p.canalElasticity pressure let eTerm := SilverSight.Q16_16Numerics.expNeg ξP let oneMinusσ := Q16_16.sub Q16_16.one p.canalSaturation let deform := Q16_16.add p.canalSaturation (Q16_16.mul oneMinusσ eTerm) Q16_16.mul p.lambda0 deform /-- Canal compliance K(P) = 1/λ_eff(P) -/ def canalCompliance (p : KernelParams) (pressure : Q16_16) : Q16_16 := let lambdaEff := dynamicCanalLambda p pressure Q16_16.recip lambdaEff /-- Canal width: W_c(P) = W_c,₀ · λ₀/λ_eff(P) -/ def canalWidth (p : KernelParams) (baseWidth : Q16_16) (pressure : Q16_16) : Q16_16 := let lambdaEff := dynamicCanalLambda p pressure let ratio := Q16_16.div p.lambda0 lambdaEff Q16_16.mul baseWidth ratio end DynamicCanal -- ============================================================ -- COARSE-GRAINING WITH LOOP ITERATION -- ============================================================ namespace CoarseGraining /-- Default precision metrics (high precision for all gradients) -/ def defaultPrecisionMetrics : PrecisionMetrics := { pressurePrecision := Q16_16.ofRawInt 0x0000FFBE, thermalPrecision := Q16_16.ofRawInt 0x0000FFBE, velocityPrecision := Q16_16.ofRawInt 0x0000FFBE, densityPrecision := Q16_16.ofRawInt 0x0000FFBE, stressPrecision := Q16_16.ofRawInt 0x0000FFBE } /-- Get precision for a given gradient type -/ def getPrecision (metrics : PrecisionMetrics) (gtype : GradientType) : Q16_16 := match gtype with | GradientType.pressureGradient => metrics.pressurePrecision | GradientType.thermalGradient => metrics.thermalPrecision | GradientType.velocityGradient => metrics.velocityPrecision | GradientType.densityGradient => metrics.densityPrecision | GradientType.stressGradient => metrics.stressPrecision /-- Compute coarse-graining factor based on loop iteration. Factor decreases (precision reduces) as loops increase. -/ def coarseGrainFactor (loopIter : Nat) (maxLoops : Nat) : Q16_16 := if maxLoops = 0 then Q16_16.one else if loopIter >= maxLoops then Q16_16.ofRatio 1 2 -- Minimum 50% precision else let ratio := Q16_16.ofNat loopIter / Q16_16.ofNat maxLoops let factor := Q16_16.one - (ratio * Q16_16.ofRatio 1 2) -- Linear decay to 50% Q16_16.max (Q16_16.ofRatio 1 2) factor /-- Apply coarse-graining to a value based on gradient type and loop iteration. -/ def applyCoarseGraining (value : Q16_16) (gtype : GradientType) (metrics : PrecisionMetrics) (loopIter : Nat) (maxLoops : Nat) : Q16_16 := let precision := getPrecision metrics gtype let cgFactor := coarseGrainFactor loopIter maxLoops let effectivePrecision := Q16_16.mul precision cgFactor Q16_16.mul value effectivePrecision /-- Update coarse-graining level based on loop iteration. Level increases every N loops to control granularity. -/ def updateCoarseGrainLevel (_currentLevel : Nat) (loopIter : Nat) (levelInterval : Nat) : Nat := if levelInterval = 0 then 0 else loopIter / levelInterval end CoarseGraining -- ============================================================ -- 9. STRESS MODEL -- ============================================================ /-- Edge evaluation context -/ structure EdgeEval where edge : Edge score : Q16_16 deltaNorm : Q16_16 logProb : Q16_16 logBest : Q16_16 /-- Surprise = -log(P_actual) -/ noncomputable def surpriseOf (ev : EdgeEval) : Q16_16 := Q16_16.abs ev.logProb /-- Regret = max(0, log(P_best) - log(P_actual)) -/ def regretOf (ev : EdgeEval) : Q16_16 := Q16_16.max Q16_16.zero (Q16_16.sub ev.logBest ev.logProb) /-- Stress = α·surprise + β·regret -/ noncomputable def stressOfEval (p : KernelParams) (ev : EdgeEval) : Q16_16 := let s := surpriseOf ev let r := regretOf ev Q16_16.add (Q16_16.mul p.alphaSurprise s) (Q16_16.mul p.betaRegret r) -- ============================================================ -- 10. EDGE SCORING -- ============================================================ /-- Compute edge score with Dynamic Canal stress penalty -/ noncomputable def edgeScore (_p : KernelParams) (lane : Lane) (ev : EdgeEval) : Q16_16 := let phaseErr := Q16_16.abs (Q16_16.sub lane.phase ev.edge.attr.prefPhase) let stressProxy := Q16_16.add (Q16_16.mul ev.edge.attr.torsion Q16_16.one) (Q16_16.mul ev.edge.attr.mismatchGain ev.deltaNorm) let stressPenalty := Q16_16.mul lane.lambdaEff stressProxy Q16_16.sub (Q16_16.sub (Q16_16.add ev.edge.attr.baseWeight ev.score) phaseErr) (Q16_16.add stressPenalty lane.mismatch) -- ============================================================ -- 11. REGIME CLASSIFICATION -- ============================================================ /-- Classify lane regime based on mismatch, stress, and edge type -/ def classifyRegime (p : KernelParams) (lane : Lane) (chosen : Edge) : Regime := if lane.mismatch.val <= p.coherentThresh.val && lane.stress.val <= p.stressThresh.val then Regime.coherent else if lane.mismatch.val >= p.throatThresh.val && chosen.attr.isThroat then Regime.throat else Regime.stressed -- ============================================================ -- 12. LANE UPDATE KERNELS (Three Regimes) -- ============================================================ /-- Coherent flow regime: stable transport -/ def coherentStep (p : KernelParams) (lane : Lane) (chosen : Edge) (deltaNorm : Q16_16) (heal : Q16_16) (pNext lambdaNext : Q16_16) : Lane := let pos' := vecAdd lane.pos (vecAdd lane.vel chosen.attr.dPos) let vel' := vecAdd lane.vel chosen.attr.dPos let phase' := Q16_16.add lane.phase chosen.attr.dPhase let stress' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.add lane.stress (Q16_16.mul p.torsionRate chosen.attr.torsion)) (Q16_16.mul p.relaxRate Q16_16.one)) -- Torsion steals energy from manifold let torsionEnergySteal := Q16_16.mul p.torsionEnergyExtraction chosen.attr.torsion let energy' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.sub (Q16_16.add lane.energy chosen.attr.dEnergy) (Q16_16.mul p.energyLossRate chosen.attr.loss)) torsionEnergySteal) let mismatch' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.add lane.mismatch (Q16_16.mul p.mismatchRate deltaNorm)) (Q16_16.mul p.healRate heal)) { lane with pos := pos' vel := vel' phase := phase' stress := stress' pressure := pNext lambdaEff := lambdaNext energy := energy' mismatch := mismatch' node := chosen.dst } /-- Stressed flow regime: distorted transport with torsion -/ def stressedStep (p : KernelParams) (lane : Lane) (chosen : Edge) (deltaNorm : Q16_16) (heal : Q16_16) (pNext lambdaNext : Q16_16) (distortion : VecN 3) : Lane := let pos' := vecAdd lane.pos (vecAdd lane.vel (vecAdd chosen.attr.dPos distortion)) let vel' := vecSub (vecAdd lane.vel chosen.attr.dPos) distortion let phase' := Q16_16.add lane.phase chosen.attr.dPhase let stress' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.add (Q16_16.add lane.stress (Q16_16.mul p.torsionRate chosen.attr.torsion)) (Q16_16.mul p.mismatchRate lane.mismatch)) (Q16_16.mul p.relaxRate heal)) -- Torsion steals energy from manifold (amplified in stressed regime) let torsionEnergySteal := Q16_16.mul p.torsionEnergyExtraction chosen.attr.torsion let energy' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.sub (Q16_16.sub (Q16_16.add lane.energy chosen.attr.dEnergy) (Q16_16.mul p.energyLossRate chosen.attr.loss)) lane.stress) torsionEnergySteal) let mismatch' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.add lane.mismatch (Q16_16.mul p.mismatchRate deltaNorm)) (Q16_16.mul p.healRate heal)) { lane with pos := pos' vel := vel' phase := phase' stress := stress' pressure := pNext lambdaEff := lambdaNext energy := energy' mismatch := mismatch' node := chosen.dst } /-- Throat transfer regime: wormhole-like lossy transfer -/ def throatStep (p : KernelParams) (lane : Lane) (chosen : Edge) (deltaNorm : Q16_16) (heal : Q16_16) (pNext lambdaNext : Q16_16) (distortion : VecN 3) : Lane := let pos' := vecAdd lane.pos distortion let vel' := distortion let phase' := Q16_16.add lane.phase chosen.attr.dPhase let stress' := Q16_16.add lane.stress (Q16_16.mul p.mismatchRate deltaNorm) -- Torsion steals energy from manifold (maximum extraction in throat regime) let torsionEnergySteal := Q16_16.mul p.torsionEnergyExtraction chosen.attr.torsion let energy' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.sub (Q16_16.sub lane.energy (Q16_16.mul p.energyLossRate chosen.attr.loss)) deltaNorm) torsionEnergySteal) let mismatch' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.add lane.mismatch deltaNorm) (Q16_16.mul p.healRate heal)) { lane with pos := pos' vel := vel' phase := phase' stress := stress' pressure := pNext lambdaEff := lambdaNext energy := energy' mismatch := mismatch' node := chosen.dst } -- ============================================================ -- 13. UNIFIED STEP FUNCTION -- ============================================================ /-- Build step context for a lane -/ structure LaneStepCtx where chosenEdge : Edge deltaNorm : Q16_16 heal : Q16_16 stressReal : Q16_16 pressureNext : Q16_16 lambdaNext : Q16_16 distortion : VecN 3 /-- Compute lane step context (edge selection + pressure update) -/ noncomputable def buildLaneCtx (p : KernelParams) (lane : Lane) (edges : Array Edge) (pickEdge : Lane → Array Edge → Edge) (computeDelta : Lane → Edge → VecN 3) (computeHeal : Lane → Edge → Q16_16) : LaneStepCtx := let chosen := pickEdge lane edges let deltaVec := computeDelta lane chosen let deltaNorm := vecL1 deltaVec let heal := computeHeal lane chosen let stressReal := Q16_16.mul p.alphaSurprise deltaNorm -- Simplified stress model let pNext := Q16_16.add (Q16_16.mul p.pressureDecay lane.pressure) stressReal let lambdaNext := DynamicCanal.dynamicCanalLambda p pNext let distortion := deltaVec -- Simplified distortion model { chosenEdge := chosen deltaNorm := deltaNorm heal := heal stressReal := stressReal pressureNext := pNext lambdaNext := lambdaNext distortion := distortion } /-- Unified lane step: handles all three regimes -/ noncomputable def stepLane (p : KernelParams) (lane : Lane) (edges : Array Edge) (pickEdge : Lane → Array Edge → Edge) (computeDelta : Lane → Edge → VecN 3) (computeHeal : Lane → Edge → Q16_16) : Lane := if !lane.active then lane else let ctx := buildLaneCtx p lane edges pickEdge computeDelta computeHeal let lane' := match lane.regime with | Regime.coherent => coherentStep p lane ctx.chosenEdge ctx.deltaNorm ctx.heal ctx.pressureNext ctx.lambdaNext | Regime.stressed => stressedStep p lane ctx.chosenEdge ctx.deltaNorm ctx.heal ctx.pressureNext ctx.lambdaNext ctx.distortion | Regime.throat => throatStep p lane ctx.chosenEdge ctx.deltaNorm ctx.heal ctx.pressureNext ctx.lambdaNext ctx.distortion let rg' := classifyRegime p lane' ctx.chosenEdge { lane' with regime := rg' } -- ============================================================ -- 14. THROAT UPDATE -- ============================================================ /-- Classify throat state -/ def classifyThroat (stableW ruptureW stableD ruptureD w δ : Q16_16) : ThroatClass := if w.val >= stableW.val && δ.val <= stableD.val then ThroatClass.stableBridge else if w.val <= ruptureW.val || δ.val >= ruptureD.val then ThroatClass.rupture else ThroatClass.lossyChannel /-- Update throat state with pressure coupling -/ def stepThroat (p : KernelParams) (sec : CanalSection) (thr : ThroatState) : ThroatState := let compliance0 := Q16_16.recip p.lambda0 let gainP := Q16_16.mul p.throatPressureGain (Q16_16.sub sec.compliance compliance0) let lossδ := Q16_16.mul p.throatMismatchLoss thr.mismatchNorm let lossS := Q16_16.mul p.throatStressLoss sec.meanStress let w' := Q16_16.max Q16_16.zero (Q16_16.add (Q16_16.sub thr.dynWeight lossδ) (Q16_16.sub gainP lossS)) let cls' := classifyThroat (Q16_16.ofRawInt 0x00018000) -- stable weight threshold (~1.5) (Q16_16.ofRawInt 0x00008000) -- rupture weight threshold (~0.5) (Q16_16.ofRawInt 0x00010000) -- stable mismatch threshold (1.0) (Q16_16.ofRawInt 0x00030000) -- rupture mismatch threshold (3.0) w' thr.mismatchNorm { thr with dynWeight := w', cls := cls' } -- ============================================================ -- 15. CANAL SECTION UPDATE (Fluid Mode) -- ============================================================ /-- Update canal section with coarse-graining on each loop -/ def stepSection (p : KernelParams) (sec : CanalSection) (inFlux outFlux : Q16_16) (inflow : Q16_16) (precisionMetrics : PrecisionMetrics) (maxLoops : Nat) (levelInterval : Nat) : CanalSection := -- Increment loop iteration let loopIter' := sec.loopIteration + 1 -- Update coarse-graining level let cgLevel' := CoarseGraining.updateCoarseGrainLevel sec.coarseGrainLevel loopIter' levelInterval -- Pressure update: P' = γ·P + stress let stressAvg := sec.meanStress let rawP' := Q16_16.add (Q16_16.mul p.pressureDecay sec.pressure) stressAvg -- Apply coarse-graining to pressure based on pressure gradient type let p' := CoarseGraining.applyCoarseGraining rawP' GradientType.pressureGradient precisionMetrics loopIter' maxLoops -- Dynamic Canal: lambda_eff(P') let lambdaEff := DynamicCanal.dynamicCanalLambda p p' let K' := DynamicCanal.canalCompliance p p' -- Capacity: C = C₀ + c_P·P - c_R·R - c_m·m let rawCap' := Q16_16.sat01 (Q16_16.add (Q16_16.sub (Q16_16.sub Q16_16.one (Q16_16.mul p.capacityRoughness sec.roughness)) (Q16_16.mul p.capacityMismatch sec.meanMismatch)) (Q16_16.mul p.capacityPressure p')) -- Apply coarse-graining to capacity based on density gradient let cap' := CoarseGraining.applyCoarseGraining rawCap' GradientType.densityGradient precisionMetrics loopIter' maxLoops -- Flux conservation: ρ' = ρ - (out - in) - siphon + inflow let rawDensity' := Q16_16.max Q16_16.zero (Q16_16.sub (Q16_16.sub (Q16_16.add sec.density inflow) (Q16_16.sub outFlux inFlux)) sec.siphon) -- Apply coarse-graining to density based on density gradient let density' := CoarseGraining.applyCoarseGraining rawDensity' GradientType.densityGradient precisionMetrics loopIter' maxLoops -- Effective velocity with compliance gain let rawVeff := Q16_16.sat01 (Q16_16.add (Q16_16.sub (Q16_16.sub (Q16_16.sub Q16_16.one (Q16_16.mul p.velDensityLoss density')) (Q16_16.mul p.velRoughnessLoss sec.roughness)) (Q16_16.mul p.velMismatchLoss sec.meanMismatch)) (Q16_16.mul p.velComplianceGain K')) -- Apply coarse-graining to velocity based on velocity gradient let veff := CoarseGraining.applyCoarseGraining rawVeff GradientType.velocityGradient precisionMetrics loopIter' maxLoops let flux' := Q16_16.mul density' veff -- Unpack score with pressure-throat interaction let rawUnpackScore' := Q16_16.add (Q16_16.add (Q16_16.add (Q16_16.mul p.thetaDensity density') (Q16_16.mul p.thetaMismatch sec.meanMismatch)) (Q16_16.mul p.thetaStress sec.meanStress)) (Q16_16.mul p.thetaPT (Q16_16.mul K' sec.throatExposure)) -- Apply coarse-graining to unpack score based on stress gradient let unpackScore' := CoarseGraining.applyCoarseGraining rawUnpackScore' GradientType.stressGradient precisionMetrics loopIter' maxLoops let unpacked' := unpackScore'.val >= p.thetaCrit.val { sec with density := density' capacity := cap' flux := flux' pressure := p' lambdaEff := lambdaEff compliance := K' unpackScore := unpackScore' unpacked := unpacked' loopIteration := loopIter' coarseGrainLevel := cgLevel' } -- ============================================================ -- 16. TOTILITY THEOREMS (Zero-Trust Compliance) -- ============================================================ /-- All Q16_16 operations are total -/ theorem Q16_16.add_total (a b : Q16_16) : ∃ c, Q16_16.add a b = c := by simp [Q16_16.add] theorem Q16_16.sub_total (a b : Q16_16) : ∃ c, Q16_16.sub a b = c := by simp [Q16_16.sub] theorem Q16_16.mul_total (a b : Q16_16) : ∃ c, Q16_16.mul a b = c := by simp [Q16_16.mul] theorem Q16_16.div_total (a b : Q16_16) : ∃ c, Q16_16.div a b = c := by simp [Q16_16.div] /-- Dynamic Canal law is total -/ theorem dynamicCanalLambda_total (p : KernelParams) (pressure : Q16_16) : ∃ lambdaEff, DynamicCanal.dynamicCanalLambda p pressure = lambdaEff := by simp [DynamicCanal.dynamicCanalLambda] /-- All regime steps are total -/ theorem stepLane_total (p : KernelParams) (lane : Lane) (edges : Array Edge) (pickEdge : Lane → Array Edge → Edge) (computeDelta : Lane → Edge → VecN 3) (computeHeal : Lane → Edge → Q16_16) : ∃ lane', stepLane p lane edges pickEdge computeDelta computeHeal = lane' := by exact ⟨stepLane p lane edges pickEdge computeDelta computeHeal, rfl⟩ theorem stepSection_total (p : KernelParams) (sec : CanalSection) (inFlux outFlux inflow : Q16_16) : ∃ sec', stepSection p sec inFlux outFlux inflow = sec' := by exact ⟨stepSection p sec inFlux outFlux inflow, rfl⟩ -- ============================================================ -- 17. #EVAL WITNESSES (Self-Test) -- ============================================================ -- Test fixed-point constructors #eval Q16_16.zero.val -- expect: 0 #eval Q16_16.one.val -- expect: 65536 -- Test DIAT encoding #eval DIAT.encode 10 -- expect: { shell := 3, a := 1, b := 6, prod := 6, diff := -5 } -- Test regime equality #eval Regime.coherent == Regime.coherent -- expect: true #eval Regime.stressed == Regime.throat -- expect: false -- Test coarse-graining precision metrics -- expect: { pressurePrecision := 65470, thermalPrecision := 65470, velocityPrecision := 65470, densityPrecision := 65470, stressPrecision := 65470 } #eval CoarseGraining.defaultPrecisionMetrics -- Test coarse-graining factor calculation #eval CoarseGraining.coarseGrainFactor 0 10 -- Loop 0 of 10: expect: 65536 (1.0 in Q16_16) #eval CoarseGraining.coarseGrainFactor 5 10 -- Loop 5 of 10: expect: 49152 (0.75 in Q16_16) #eval CoarseGraining.coarseGrainFactor 10 10 -- Loop 10 of 10: expect: 32768 (0.5 in Q16_16) -- Test coarse-graining application -- expect: 65470 #eval CoarseGraining.applyCoarseGraining (Q16_16.one) GradientType.pressureGradient CoarseGraining.defaultPrecisionMetrics 0 10 -- expect: 49102 #eval CoarseGraining.applyCoarseGraining (Q16_16.one) GradientType.pressureGradient CoarseGraining.defaultPrecisionMetrics 5 10 -- expect: 32735 #eval CoarseGraining.applyCoarseGraining (Q16_16.one) GradientType.pressureGradient CoarseGraining.defaultPrecisionMetrics 10 10 -- Test coarse-graining level update #eval CoarseGraining.updateCoarseGrainLevel 0 0 5 -- Level 0: expect: 0 #eval CoarseGraining.updateCoarseGrainLevel 0 5 5 -- Level 1: expect: 1 #eval CoarseGraining.updateCoarseGrainLevel 0 10 5 -- Level 2: expect: 2 -- Test canal section with loop iteration and coarse-graining def testCanalSectionWithCoarseGraining : CanalSection := { density := Q16_16.ofRatio 1 2, capacity := Q16_16.ofRatio 4 5, flux := Q16_16.ofRatio 3 10, siphon := Q16_16.ofRatio 1 10, meanEnergy := Q16_16.one, meanMismatch := Q16_16.ofRatio 1 5, meanStress := Q16_16.ofRatio 3 10, pressure := Q16_16.ofRatio 1 2, lambdaEff := Q16_16.one, compliance := Q16_16.one, width := Q16_16.one, roughness := Q16_16.ofRatio 1 10, gradient := Q16_16.ofRatio 1 5, throatExposure := Q16_16.ofRatio 1 10, unpackScore := Q16_16.ofRatio 1 2, unpacked := false, loopIteration := 0, coarseGrainLevel := 0 } #eval testCanalSectionWithCoarseGraining.loopIteration -- expect: 0 #eval testCanalSectionWithCoarseGraining.coarseGrainLevel -- expect: 0