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204 lines
7.6 KiB
Text
204 lines
7.6 KiB
Text
/-
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VoxelEncoding.lean - Voxel, Seed, Sieve, and Topological Encoding
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Ports rows 124-133 from MATH_MODEL_MAP.tsv (Python → Lean).
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-/
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import Semantics.Bind
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import Semantics.FixedPoint
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namespace Semantics.VoxelEncoding
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open Q16_16
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-- Row 124: Voxel Key Encoding (30-bit packed)
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-- key = ((x+512) &&& 0x3FF) <<< 20 ||| ((y+512) &&& 0x3FF) <<< 10 ||| ((z+512) &&& 0x3FF)
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-- x,y,z ∈ [-512, 511]
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structure VoxelKey where
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val : UInt32
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deriving Repr, DecidableEq, Inhabited, BEq
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def encodeVoxel (x y z : Int) : VoxelKey :=
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let cx := (x + 512).toNat &&& 0x3FF
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let cy := (y + 512).toNat &&& 0x3FF
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let cz := (z + 512).toNat &&& 0x3FF
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⟨UInt32.ofNat (cx <<< 20 ||| cy <<< 10 ||| cz)⟩
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def decodeVoxel (k : VoxelKey) : (Int × Int × Int) :=
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let cx := (k.val.toNat >>> 20) % 0x400
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let cy := (k.val.toNat >>> 10) % 0x400
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let cz := k.val.toNat % 0x400
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(Int.ofNat cx - 512, Int.ofNat cy - 512, Int.ofNat cz - 512)
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-- Row 125: Microvoxel Seed 4-Byte Encoding
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-- 32-bit: delta_p[9:0]|region[13:10]|gamma[18:14]|activation[22:19]|polarity[26:23]|confidence[30:27]|flag[31]
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structure MicrovoxelSeed where
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deltaP : UInt32 -- 10 bits [9:0]
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region : UInt32 -- 4 bits [13:10]
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gamma : UInt32 -- 5 bits [18:14]
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activation : UInt32 -- 4 bits [22:19]
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polarity : UInt32 -- 4 bits [26:23]
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confidence : UInt32 -- 4 bits [30:27]
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flag : Bool
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deriving Repr, Inhabited, DecidableEq
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def encodeSeed (s : MicrovoxelSeed) : UInt32 :=
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(s.deltaP &&& (0x3FF : UInt32)) |||
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((s.region &&& (0xF : UInt32)) <<< 10) |||
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((s.gamma &&& (0x1F : UInt32)) <<< 14) |||
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((s.activation &&& (0xF : UInt32)) <<< 19) |||
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((s.polarity &&& (0xF : UInt32)) <<< 23) |||
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((s.confidence &&& (0xF : UInt32)) <<< 27) |||
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(if s.flag then (0x80000000 : UInt32) else 0)
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inductive SeedClass | Exclude | Explore | Promote deriving Repr, DecidableEq, Inhabited
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def classifySeedByEfficiency (eff : Q16_16) : SeedClass :=
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-- eff < 0.8 → 52429; eff < 1.2 → 78643
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if eff.val < 52429 then .Exclude
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else if eff.val < 78643 then .Explore
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else .Promote
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-- Row 126: DCVN Verification Invariant Survival
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-- 4 invariants: completeness(c), consistency(s), freshness(f), provenance(p)
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structure DCVNState where
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completeness : Q16_16
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consistency : Q16_16
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freshness : Q16_16
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provenance : Q16_16
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deriving Repr, Inhabited, DecidableEq
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inductive DCVNParticipation | Full | Partial | Observer | Absent
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deriving Repr, DecidableEq, Inhabited
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def dcvnThreshold : Q16_16 := ⟨52429⟩ -- 0.8 * 65536
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def dcvnSurvivalMask (s : DCVNState) : UInt8 :=
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(if s.completeness.val >= dcvnThreshold.val then 0b1000 else 0) |||
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(if s.consistency.val >= dcvnThreshold.val then 0b0100 else 0) |||
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(if s.freshness.val >= dcvnThreshold.val then 0b0010 else 0) |||
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(if s.provenance.val >= dcvnThreshold.val then 0b0001 else 0)
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def dcvnParticipation (s : DCVNState) : DCVNParticipation :=
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let bits := (dcvnSurvivalMask s).toNat
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let count := (if bits &&& 8 != 0 then 1 else 0) + (if bits &&& 4 != 0 then 1 else 0) +
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(if bits &&& 2 != 0 then 1 else 0) + (if bits &&& 1 != 0 then 1 else 0)
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if count == 4 then .Full
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else if count >= 2 then .Partial
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else if count >= 1 then .Observer
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else .Absent
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-- Row 127: Watanabe Total Correlation + Kolmogorov complexity approximation
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-- TC ≈ (0.4 · kolmogorov + 0.4 · entropy/8 + 0.2 · CV) in Q16.16
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def totalCorrelationEstimate (kolmogorov entropy cv : Q16_16) : Q16_16 :=
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-- 0.4 = 26214; 0.2 = 13107
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let w1 : Q16_16 := ⟨26214⟩
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let w2 : Q16_16 := ⟨26214⟩
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let w3 : Q16_16 := ⟨13107⟩
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let entropyNorm := div entropy ⟨8 * 65536⟩
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add (add (mul w1 kolmogorov) (mul w2 entropyNorm)) (mul w3 cv)
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-- Row 128: Relation Sieve 5-Symbol packing
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-- Pack 5×2-bit symbols into 10-bit: sig = (T<<<8)|(D<<<6)|(C<<<4)|(A<<<2)|R
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structure SieveSymbols where
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torsion : UInt8 -- 2-bit [0..3]
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drift : UInt8 -- 2-bit
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coherence : UInt8 -- 2-bit
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angmom : UInt8 -- 2-bit
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radius : UInt8 -- 2-bit
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deriving Repr, Inhabited, DecidableEq
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def packSieveSymbols (s : SieveSymbols) : UInt16 :=
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(s.torsion.toUInt16 <<< 8) |||
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(s.drift.toUInt16 <<< 6) |||
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(s.coherence.toUInt16 <<< 4) |||
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(s.angmom.toUInt16 <<< 2) |||
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s.radius.toUInt16
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inductive SieveDecision | Pass | Hold | Reject deriving Repr, DecidableEq, Inhabited
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def classifySieve (s : SieveSymbols) : SieveDecision :=
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if s.torsion == 3 || s.angmom == 3 || s.coherence == 3 ||
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(s.torsion >= 2 && s.coherence >= 2) ||
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(s.drift == 3 && s.angmom >= 2) ||
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(s.radius == 3 && s.coherence >= 2)
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then .Reject
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else if s.torsion == 2 || s.drift == 2 || s.coherence >= 1
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then .Hold
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else .Pass
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-- Row 129: Proxy Extraction
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def proxyExtractTorsion (torsionSamples : Array Q16_16) : UInt8 :=
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let sum := Array.foldl (fun acc s => acc + s.val) 0 (torsionSamples.take 32)
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let scaled := (sum / 65536) * 100
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UInt8.ofNat (Nat.min 255 scaled.toNat)
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def proxyExtractCoherence (torsion : UInt8) : UInt8 :=
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255 - torsion
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-- Row 130: SEISMIC Shell Detection bounds
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-- 0.35 ≤ φ_corr < 0.47 in Q16.16: [22938, 30801]
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def seismicLow : UInt32 := 22938 -- 0.35 * 65536
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def seismicHigh : UInt32 := 30801 -- 0.47 * 65536
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def isSeismicShell (phiCorr : Q16_16) : Bool :=
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phiCorr.val >= seismicLow && phiCorr.val < seismicHigh
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-- Row 131: Half Möbius Closure Integral ∮τ·ds = π
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-- Accumulate until torsion integral reaches π (≈205887 in Q16.16)
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def piQ : Q16_16 := ⟨205887⟩ -- π * 65536
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def halfMobiusClosure (torsionSamples : Array Q16_16) (stepSize : Q16_16) : Option Nat :=
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let rec go (i : Nat) (acc : Q16_16) : Option Nat :=
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if i >= torsionSamples.size then none
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else
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let newAcc := add acc (mul torsionSamples[i]! stepSize)
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if newAcc.val >= piQ.val then some i
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else go (i + 1) newAcc
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go 0 zero
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-- Row 132: Regret Field Blink Cycle
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def baselineMs : Q16_16 := ⟨500 * 65536⟩ -- 500ms
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def regretMs : Q16_16 := ⟨700 * 65536⟩ -- 700ms
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def decayLambda : Q16_16 := ⟨2 * 65536⟩ -- λ = 2.0
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def blinkDuration (regretMagnitude : Q16_16) : Q16_16 :=
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let range := sub regretMs baselineMs
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let offset := mul range regretMagnitude
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add baselineMs offset
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def regretDecay (regret dt : Q16_16) : Q16_16 :=
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let ldt := mul decayLambda dt
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if ldt.val >= one.val then zero
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else mul regret (sub one ldt)
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-- Row 133: Hugoniot Shock — kinetic energy harvesting
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-- E_kinetic = ½ · I · ω²; E_harvested = E_stored · efficiency (Q16.16)
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def kineticEnergy (momentOfInertia omega : Q16_16) : Q16_16 :=
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mul ⟨32768⟩ (mul momentOfInertia (mul omega omega)) -- ½ * I * ω²
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def harvestedEnergy (stored efficiency : Q16_16) : Q16_16 :=
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mul stored efficiency
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-- Bind wrappers
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def voxelInvariant (k : VoxelKey) : String := s!"voxel:{k.val}"
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def voxelCost (a b : VoxelKey) (_m : Metric) : UInt32 :=
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if a.val > b.val then a.val - b.val else b.val - a.val
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def voxelBind (a b : VoxelKey) (m : Metric) : Bind VoxelKey VoxelKey :=
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geometricBind a b m voxelCost voxelInvariant voxelInvariant
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def sieveInvariant (s : SieveSymbols) : String :=
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s!"sieve:{s.torsion}{s.drift}{s.coherence}{s.angmom}{s.radius}"
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def sieveCostFn (a b : SieveSymbols) (_m : Metric) : UInt32 :=
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(packSieveSymbols a).toUInt32 + (packSieveSymbols b).toUInt32
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def sieveControlBind (a b : SieveSymbols) (m : Metric) : Bind SieveSymbols SieveSymbols :=
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controlBind a b m sieveCostFn sieveInvariant sieveInvariant
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-- Verify
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#eval encodeVoxel 0 0 0
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#eval decodeVoxel (encodeVoxel 100 (-50) 200) -- expect (100, -50, 200)
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#eval classifySieve { torsion := 3, drift := 0, coherence := 0, angmom := 0, radius := 0 }
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#eval isSeismicShell ⟨26214⟩ -- 0.4 * 65536 → should be true
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end Semantics.VoxelEncoding
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