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- Prover-Integrated Orchestration Layers (L0-L3): Goedel-Prover-V2 watchdog, BFS-Prover-V2 swarm consensus, bf4prover topology adaptation - FAMM Verilator benchmark: uniform vs preshaped delay comparison (4.4x speedup) - Swarm topological device prober: 11 agents probing traces, caps, delays, errors, vias, PDN - Spec sheet puller: 10 components with key params and topological relevance - Virtual FPGA system tests: 6/6 passed, 134K ops/s throughput - Fixed merge conflicts in AI-Newton test_experiment.ipynb
307 lines
11 KiB
Text
307 lines
11 KiB
Text
import Mathlib.Data.Real.Basic
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import Semantics.CodonOTOM
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import Semantics.PeptideMoE
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noncomputable section
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namespace CodonPeptideConsistency
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open CodonOTOM
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open PeptideMoE
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/-
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Codon -> amino acid -> peptide consistency layer.
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This file connects:
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- codon-level efficiency Φ_codon
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- translation into amino-acid labels
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- peptide-level efficiency Φ_peptide
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through a sequence-level aggregate score.
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-/
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/-- Abstract peptide alphabet label induced by amino acids.
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TODO(lean-port): external biological mapping — replace with concrete genetic code table. -/
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opaque aaToPeptideClass : AminoAcid → Nat
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/-- A coding sequence is a list of codons. -/
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abbrev CDS := List Codon
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/-- Codon-dependent translation speed (strongest biological defensibility).
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TODO(lean-port): external simulator measurement — replace with empirical data. -/
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opaque translationSpeed : Codon → ℝ
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/-- Local folding delay (clearest simulator signal).
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TODO(lean-port): external simulator measurement — replace with empirical data. -/
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opaque foldingDelay : Codon → ℝ
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/-- Synonymous-codon-specific structural bias (most ambitious structural claim).
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TODO(lean-port): external structural model — replace with empirical data. -/
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opaque structuralBias : Codon → ℝ
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/-- Expert bias for codon-specific structural effects. -/
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structure CodonBias where
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b_k : ℝ -- codon-specific bias for expert k
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/-- Translate a coding sequence into amino acids. -/
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noncomputable def translateCDS (s : CDS) : List AminoAcid :=
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s.map translate
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/-- Average codon-level score over a coding sequence. -/
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noncomputable def phiCDSCodon
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(s : CDS) : ℝ :=
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match s.length with
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| 0 => 0
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| n => (s.map (fun c => phiCodon w (fs c) c)).sum / n
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-- Forward-declare empty values for opaque types
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-- (needed because `noncomputable def` in this section requires Nonempty instances)
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private noncomputable def emptyPeptideState : PeptideState :=
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{ phi := (0 : ℝ), psi := (0 : ℝ), internalEnergy := (0 : ℝ),
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conformationalEntropy := (0 : ℝ), structuralCoherence := (0 : ℝ),
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stericEnergy := (0 : ℝ), bondEnergy := (0 : ℝ) }
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noncomputable instance : Nonempty PeptideState := ⟨emptyPeptideState⟩
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/-- Abstract peptide state induced by a translated coding sequence with codon dynamics.
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TODO(lean-port): external biological model — replace with concrete folding simulator. -/
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opaque buildPeptideStateWithDynamics :
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List AminoAcid → (Codon → ℝ) → (Codon → ℝ) → (Codon → ℝ) → PeptideState
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/-- Abstract peptide state induced by a translated coding sequence (legacy, no dynamics).
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TODO(lean-port): external biological model — replace with concrete folding simulator. -/
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opaque buildPeptideState :
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List AminoAcid → PeptideState
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noncomputable instance : Nonempty (List AminoAcid → (Codon → ℝ) → (Codon → ℝ) → (Codon → ℝ) → PeptideState) :=
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⟨buildPeptideStateWithDynamics⟩
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noncomputable instance : Nonempty (List AminoAcid → PeptideState) :=
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⟨buildPeptideState⟩
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/-- Peptide-level score induced by the translated coding sequence with dynamics. -/
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noncomputable def phiCDSPeptideWithDynamics
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(s : CDS)
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(v : Codon → ℝ) -- translation speed
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(τ : Codon → ℝ) -- folding delay
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(b : Codon → ℝ) -- structural bias
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: ℝ :=
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phiPeptide tp ap (buildPeptideStateWithDynamics (translateCDS s) v τ b)
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/-- Peptide-level score induced by the translated coding sequence (legacy, no dynamics). -/
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noncomputable def phiCDSPeptide
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(s : CDS) : ℝ :=
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phiPeptide tp ap (buildPeptideState (translateCDS s))
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/-- Combined sequence-level score with codon dynamics. -/
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noncomputable def phiCDSWithDynamics
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(α β : ℝ)
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(v : Codon → ℝ) -- translation speed
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(τ : Codon → ℝ) -- folding delay
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(b : Codon → ℝ) -- structural bias
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(s : CDS) : ℝ :=
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α * phiCDSCodon w fs s + β * phiCDSPeptideWithDynamics tp ap s v τ b
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/-- Gate weight for expert k at codon c_i with folding delay. -/
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noncomputable def gateWeightWithFolding
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(z_k : PeptideState → ℝ) -- base gate weight
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(b_k : CodonBias) -- codon-specific bias
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(η : ℝ) -- folding sensitivity
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(P_t : PeptideState)
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(c_i : Codon) : ℝ :=
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let base := z_k P_t + b_k.b_k
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let folded := η * foldingDelay c_i
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-- softmax (simplified as exponential for single value)
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Real.exp (base - folded)
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/-- Peptide dynamics: ∂Θ_t/∂t = Σ_k g_k(P_t; c_i) Advice_k(P_t; c_i) + ξ_t -/
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noncomputable def peptideDynamics
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(P_t : PeptideState)
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(c_i : CDS)
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(z_k : PeptideState → ℝ)
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(b_k : CodonBias)
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(η : ℝ)
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(Advice_k : PeptideState → ℝ)
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(ξ_t : ℝ) : ℝ :=
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let g_sum := (c_i.map (fun c => gateWeightWithFolding z_k b_k η P_t c)).sum
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let advice_sum := Advice_k P_t * c_i.length
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g_sum * advice_sum + ξ_t
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/-- Theorem: zero folding delay reduces to standard gate weight. -/
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theorem gateWeight_zero_folding
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(z_k : PeptideState → ℝ)
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(b_k : CodonBias)
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(P_t : PeptideState)
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(c_i : Codon)
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(h : foldingDelay c_i = 0) :
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gateWeightWithFolding z_k b_k 0 P_t c_i = Real.exp (z_k P_t + b_k.b_k) := by
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unfold gateWeightWithFolding
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rw [h]
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ring_nf
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/-- Theorem: zero codon bias reduces to base gate weight. -/
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theorem gateWeight_zero_bias
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(z_k : PeptideState → ℝ)
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(η : ℝ)
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(P_t : PeptideState)
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(c_i : Codon) :
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gateWeightWithFolding z_k (CodonBias.mk 0) η P_t c_i = Real.exp (z_k P_t - η * foldingDelay c_i) := by
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unfold gateWeightWithFolding
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ring_nf
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/-- Combined sequence-level score. -/
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noncomputable def phiCDS
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(α β : ℝ)
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(s : CDS) : ℝ :=
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α * phiCDSCodon w fs s + β * phiCDSPeptide tp ap s
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/-- A synonymous mutation preserves the translated amino acid. -/
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def synonymous (c₁ c₂ : Codon) : Prop :=
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translate c₁ = translate c₂
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/-- Mutation at a single site in a coding sequence. -/
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def pointMutate (s : CDS) (i : Nat) (c' : Codon) : CDS :=
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s.take i ++ c' :: s.drop (i + 1)
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/-- Codon-local beneficial mutation. -/
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def beneficialAtCodon
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(c₁ c₂ : Codon) : Prop :=
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0 < phiCodon w (fs c₂) c₂ - phiCodon w (fs c₁) c₁
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/-- Sequence-level beneficial mutation. -/
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def beneficialAtCDS
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(α β : ℝ)
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(s s' : CDS) : Prop :=
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0 < phiCDS tp ap w fs α β s' - phiCDS tp ap w fs α β s
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/-
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Consistency property:
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a synonymous mutation that improves local codon score and leaves the peptide
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builder invariant should improve the combined CDS score when α > 0 and β ≥ 0.
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This is an external biological invariant that depends on the concrete
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buildPeptideState implementation.
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-/
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structure SynonymousCodonImprovesCDSHypothesis where
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property (tp : ThermoParams) (ap : AdmissibilityParams) (w : CodonWeights)
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(fs : Codon → CodonFeatures) (α β : ℝ) (hα : 0 < α) (hβ : 0 ≤ β)
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(s : CDS) (i : Nat) (c₁ c₂ : Codon) (hi : i < s.length)
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(hget : s.get ⟨i, hi⟩ = c₁) (hsyn : synonymous c₁ c₂)
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(hlocal : beneficialAtCodon w fs c₁ c₂)
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(hpep : buildPeptideState (translateCDS (pointMutate s i c₂)) =
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buildPeptideState (translateCDS s)) :
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beneficialAtCDS tp ap w fs α β s (pointMutate s i c₂)
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/-- A zero peptide weight reduces the CDS score to codon-average selection. -/
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theorem phiCDS_zero_peptide_weight
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(α : ℝ)
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(s : CDS) :
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phiCDS tp ap w fs α 0 s = α * phiCDSCodon w fs s := by
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unfold phiCDS
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ring
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/-- A zero codon weight reduces the CDS score to peptide-level selection. -/
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theorem phiCDS_zero_codon_weight
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(tp : ThermoParams)
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(ap : AdmissibilityParams)
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(w : CodonWeights)
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(fs : Codon → CodonFeatures)
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(β : ℝ)
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(s : CDS) :
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phiCDS tp ap w fs 0 β s = β * phiCDSPeptide tp ap s := by
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unfold phiCDS
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ring
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/-- Kinetic cost term: Σ_i (ln 64 + λ ln d(c_i) + γ τ(c_i)) + C_0 -/
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noncomputable def kineticCost
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(lam γ C_0 : ℝ)
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(d : Codon → ℝ) -- degeneracy function
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(τ : Codon → ℝ) -- folding delay
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(s : CDS) : ℝ :=
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match s.length with
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| 0 => C_0
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| _n => (s.map (fun c => Real.log 64 + lam * Real.log (d c) + γ * τ c)).sum + C_0
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/-- Cotranslational folding window: at step t, only first t codons exist. -/
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noncomputable def cotranslationalWindow
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(t : Nat)
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(s : CDS) : CDS :=
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s.take t
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/-- Cotranslational peptide state at step t. -/
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noncomputable def cotranslationalPeptideState
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(t : Nat)
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(s : CDS)
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(v : Codon → ℝ)
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(τ : Codon → ℝ)
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(b : Codon → ℝ) : PeptideState :=
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buildPeptideStateWithDynamics (translateCDS (cotranslationalWindow t s)) v τ b
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/-- Theorem: cotranslational window is prefix of original sequence. -/
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theorem cotranslationalWindow_is_prefix
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(t : Nat)
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(s : CDS) :
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(cotranslationalWindow t s).length = min t s.length := by
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unfold cotranslationalWindow
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simp [List.length_take]
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/-- Theorem: empty cotranslational window has empty translation. -/
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theorem cotranslationalWindow_empty
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(s : CDS) :
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translateCDS (cotranslationalWindow 0 s) = [] := by
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unfold cotranslationalWindow
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simp [List.take, translateCDS]
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/-- Theorem: full cotranslational window equals original sequence. -/
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theorem cotranslationalWindow_full
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(s : CDS) :
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cotranslationalWindow s.length s = s := by
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unfold cotranslationalWindow
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simp
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/-- Theorem: Φ_CDS is bounded when codon and peptide components bounded.
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This follows from the triangle inequality; the proof is straightforward. -/
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theorem phiCDS_bounded
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(tp : ThermoParams) (ap : AdmissibilityParams) (w : CodonWeights)
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(fs : Codon → CodonFeatures) (α β : ℝ)
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(M_codon M_peptide : ℝ)
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(h_codon : ∀ s, |phiCDSCodon w fs s| ≤ M_codon)
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(h_peptide : ∀ s, |phiCDSPeptide tp ap s| ≤ M_peptide) :
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∃ M, ∀ s, |phiCDS tp ap w fs α β s| ≤ M := by
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refine ⟨|α| * M_codon + |β| * M_peptide, fun s => ?_⟩
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unfold phiCDS
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have h_c : |phiCDSCodon w fs s| ≤ M_codon := h_codon s
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have h_p : |phiCDSPeptide tp ap s| ≤ M_peptide := h_peptide s
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calc
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|α * phiCDSCodon w fs s + β * phiCDSPeptide tp ap s|
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≤ |α * phiCDSCodon w fs s| + |β * phiCDSPeptide tp ap s| := abs_add_le _ _
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_ = |α| * |phiCDSCodon w fs s| + |β| * |phiCDSPeptide tp ap s| := by
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rw [abs_mul, abs_mul]
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_ ≤ |α| * M_codon + |β| * M_peptide := by
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have h_nonneg_alpha : 0 ≤ |α| := abs_nonneg _
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have h_nonneg_beta : 0 ≤ |β| := abs_nonneg _
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nlinarith
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end CodonPeptideConsistency
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