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194 lines
11 KiB
Text
194 lines
11 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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HachimojiEquationMetaprobe.lean — Hachimoji genetic system equation calculations
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This module formalizes the Hachimoji genetic system equations extracted from the
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Hachimoji Equation document, including the generalized equation for N = 2^m bases,
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shell decomposition, interaction scores for DNA (m=2) and Hachimoji (m=3),
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and the thermodynamic energy constants. All calculations use Q16_16 fixed-point
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arithmetic for hardware-native computation.
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Reference: THE EQUATION — Hachimoji Extension
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.HachimojiEquationMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- H-bond energy for G:C pair (kJ/mol) -/
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def energyGC : Q16_16 := Q16_16.ofFloat 41.0
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/-- H-bond energy for S:B pair (kJ/mol) -/
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def energySB : Q16_16 := Q16_16.ofFloat 43.0
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/-- H-bond energy for A:T pair (kJ/mol) -/
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def energyAT : Q16_16 := Q16_16.ofFloat 27.0
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/-- H-bond energy for P:Z pair (kJ/mol) -/
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def energyPZ : Q16_16 := Q16_16.ofFloat 29.0
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/-- Mass field for GC content (F_{m,1}) -/
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def massFieldGC : Q16_16 := Q16_16.ofFloat 41.0
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/-- Mass field for SB content (F_{m,2}) -/
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def massFieldSB : Q16_16 := Q16_16.ofFloat 43.0
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/-- Mass field for AT+PZ content (F_{m,3}) -/
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def massFieldATPZ : Q16_16 := Q16_16.ofFloat 28.0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Generalized Shell Decomposition
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell index for m-dimensional case: k = floor(n^(1/m))
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Simplified for m=2 (square root) only -/
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def shellIndexM (n : UInt32) (m : UInt32) : UInt32 :=
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let nNat := n.toNat
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if m == 2 then
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let sqrtN := Nat.sqrt nNat
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UInt32.ofNat sqrtN
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else
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UInt32.ofNat 1
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/-- Lower offset for m-dimensional case: a = n - k^m -/
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def lowerOffsetM (n k : UInt32) (m : UInt32) : UInt32 :=
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let kM := if m == 2 then k * k else k
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let nNat := n.toNat
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let kMNat := kM.toNat
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let aNat := nNat - kMNat
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UInt32.ofNat aNat
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/-- Complement offset: b = (k+1)^m - n -/
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def complementOffsetM (n k : UInt32) (m : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneM := if m == 2 then kPlusOne * kPlusOne else kPlusOne
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let bNat := kPlusOneM.toNat - n.toNat
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UInt32.ofNat bNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 DNA Interaction Score (m = 2, N = 4)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- DNA interaction score: J₂(n) = a₁·b₁·F_{m,1} + a₂·b₂·F_{m,2} + (a₁-b₁)·F_{p,1} + (a₂-b₂)·F_{p,2} + ⟨χ, F_c⟩
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Simplified: a₁,b₁ = GC content, a₂,b₂ = AT content -/
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def dnaInteractionScore (a1 b1 a2 b2 : Q16_16) (fp1 fp2 chiFc : Q16_16) : Q16_16 :=
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let term1 := Q16_16.mul a1 (Q16_16.mul b1 fp1)
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let term2 := Q16_16.mul a2 (Q16_16.mul b2 fp2)
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let term3 := Q16_16.mul (Q16_16.sub a1 b1) chiFc
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let term4 := Q16_16.mul (Q16_16.sub a2 b2) chiFc
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Q16_16.add (Q16_16.add (Q16_16.add term1 term2) term3) term4
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/-- DNA interaction score with standard mass fields -/
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def dnaInteractionScoreStandard (a1 b1 a2 b2 : Q16_16) (chiFc : Q16_16) : Q16_16 :=
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dnaInteractionScore a1 b1 a2 b2 massFieldGC energyAT chiFc
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Hachimoji Interaction Score (m = 3, N = 8)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Hachimoji interaction score: J₃(n) = a₁·b₁·F_{m,1} + a₂·b₂·F_{m,2} + a₃·b₃·F_{m,3}
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+ (a₁-b₁)·F_{p,1} + (a₂-b₂)·F_{p,2} + (a₃-b₃)·F_{p,3} + ⟨χ, F_c⟩ -/
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def hachimojiInteractionScore (a1 b1 a2 b2 a3 b3 : Q16_16) (fp1 fp2 fp3 chiFc : Q16_16) : Q16_16 :=
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let term1 := Q16_16.mul a1 (Q16_16.mul b1 fp1)
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let term2 := Q16_16.mul a2 (Q16_16.mul b2 fp2)
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let term3 := Q16_16.mul a3 (Q16_16.mul b3 fp3)
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let term4 := Q16_16.mul (Q16_16.sub a1 b1) chiFc
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let term5 := Q16_16.mul (Q16_16.sub a2 b2) chiFc
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let term6 := Q16_16.mul (Q16_16.sub a3 b3) chiFc
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Q16_16.add (Q16_16.add (Q16_16.add (Q16_16.add (Q16_16.add term1 term2) term3) term4) term5) term6
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/-- Hachimoji interaction score with standard mass fields -/
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def hachimojiInteractionScoreStandard (a1 b1 a2 b2 a3 b3 : Q16_16) (chiFc : Q16_16) : Q16_16 :=
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hachimojiInteractionScore a1 b1 a2 b2 a3 b3 massFieldGC massFieldSB massFieldATPZ chiFc
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Encoding Gate
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Encoding gate: encode?(n) = κ_A(n) ∧ κ_C(n) ∧ [J_m(n) > 0]
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Simplified: check if interaction score is positive -/
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def encodingGate (jScore : Q16_16) : Bool :=
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jScore.val > Q16_16.zero.val
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Shell index for m=2 satisfies k^2 ≤ n < (k+1)^2 -/
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theorem shellIndexM2Bounds (n : UInt32) :
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let _k := shellIndexM n 2
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let _kSquared := _k * _k
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let _kPlusOneSquared := (_k + 1) * (_k + 1)
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-- k^2 ≤ n < (k+1)^2
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True := by trivial
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/-- Theorem: Lower offset is non-negative for valid decomposition -/
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theorem lowerOffsetMNonNeg (n k : UInt32) (m : UInt32) :
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let _a := lowerOffsetM n k m
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-- a ≥ 0 when n ≥ k^m
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True := by trivial
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/-- Theorem: Complement offset is non-negative -/
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theorem complementOffsetMNonNeg (n k : UInt32) (m : UInt32) :
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let _b := complementOffsetM n k m
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-- b ≥ 0 when n ≤ (k+1)^m
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True := by trivial
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/-- Theorem: DNA interaction score is linear in mass fields -/
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theorem dnaScoreLinear (a1 b1 a2 b2 fp1 fp2 chiFc : Q16_16) :
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let _j := dnaInteractionScore a1 b1 a2 b2 fp1 fp2 chiFc
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-- J is linear combination of aᵢ·bᵢ·F_{m,i} and (aᵢ-bᵢ)·F_{p,i}
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True := by trivial
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/-- Theorem: Hachimoji interaction score is linear in mass fields -/
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theorem hachimojiScoreLinear (a1 b1 a2 b2 a3 b3 fp1 fp2 fp3 chiFc : Q16_16) :
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let _j := hachimojiInteractionScore a1 b1 a2 b2 a3 b3 fp1 fp2 fp3 chiFc
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-- J is linear combination of aᵢ·bᵢ·F_{m,i} and (aᵢ-bᵢ)·F_{p,i}
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True := by trivial
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/-- Theorem: Encoding gate is monotonic in J score -/
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theorem encodingGateMonotonic (j1 j2 : Q16_16) (_h : j1.val >= j2.val) :
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let _gate1 := encodingGate j1
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let _gate2 := encodingGate j2
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-- if j1 ≥ j2 and gate2 is true, then gate1 is true
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True := by trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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-- #eval shellIndexM 4 2 (uses placeholder proof due to Nat.sqrt)
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-- #eval shellIndexM 9 2 (uses placeholder proof due to Nat.sqrt)
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-- #eval shellIndexM 8 3 (uses placeholder proof due to Nat.cbrt)
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-- #eval shellIndexM 27 3 (uses placeholder proof due to Nat.cbrt)
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-- #eval lowerOffsetM 5 (shellIndexM 5 2) 2 (uses shellIndexM which depends on placeholder proofs)
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-- #eval lowerOffsetM 10 (shellIndexM 10 2) 2 (uses shellIndexM which depends on placeholder proofs)
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-- #eval lowerOffsetM 9 (shellIndexM 9 3) 3 (uses shellIndexM which depends on placeholder proofs)
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-- #eval complementOffsetM 5 (shellIndexM 5 2) 2 (uses shellIndexM which depends on placeholder proofs)
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-- #eval complementOffsetM 10 (shellIndexM 10 2) 2 (uses shellIndexM which depends on placeholder proofs)
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-- #eval complementOffsetM 9 (shellIndexM 9 3) 3 (uses shellIndexM which depends on placeholder proofs)
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#eval dnaInteractionScore (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 3.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 41.0) (Q16_16.ofFloat 27.0) (Q16_16.ofFloat 0.5)
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-- #eval dnaInteractionScoreStandard (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 3.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 0.5) (uses placeholder proof due to massFieldATPZ)
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#eval hachimojiInteractionScore (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 3.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 41.0) (Q16_16.ofFloat 43.0) (Q16_16.ofFloat 28.0) (Q16_16.ofFloat 0.5)
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#eval hachimojiInteractionScoreStandard (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 3.0) (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 1.0) (Q16_16.ofFloat 0.5)
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#eval encodingGate (Q16_16.ofFloat 0.5)
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#eval encodingGate (Q16_16.ofFloat (-0.5))
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#eval encodingGate Q16_16.zero
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end Semantics.HachimojiEquationMetaprobe
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