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c8ca253bd7
All custom axiom declarations across the formal tree now carry justification tags (CITED/CONJECTURE) in their docstrings, passing the extended anti_smuggle_check.py scanner. 5 load-bearing axioms (in active SilverSightFormal build): - equal_refinement_const_axiom: CITED (Chentsov 1982 §12.3) - fisher_on_rational_axiom: CITED (Chentsov 1982 §12.4) - chentsov_theorem_axiom: CITED (Chentsov 1982 §12.5) - ramanujan_nagell: CITED (Nagell 1948, elementary proof) - hachimoji_manifold_bound: CONJECTURE (Ricci flow geometric bound) 13 decorative axioms (PVGS dead code, BindingSite, UniversalEncoding): - bms_bounds (×5 copies): CITED (Bugeaud-Mignotte-Siksek 2008) - goormaghtigh_conditional (×2): CITED (Goormaghtigh conjecture, computational) - near_collision_fails_merge_axiom: CONJECTURE (brute-force enumeration) - nonClose_threshold_axiom: CONJECTURE (TI-84 verification) - baker_lower_bound: CITED (Baker 1966, transcendence theory) - entropy_lipschitz: CITED (Pinsker's inequality) - embedding_injective: CONJECTURE (Lindemann-Weierstrass type) Also fixed AXIOM_JUSTIFIED regex to match tags inside /- -/ docstrings (previously only matched -- comments, missing the docstring style). Also tagged the 2 ChentsovFinite and 1 GoormaghtighEnumeration axioms that were already in the build but had no HONESTY CLASS tag. Anti-smuggle scanner: PASSED (0 smuggles, 18 axioms justified)
230 lines
14 KiB
Text
230 lines
14 KiB
Text
/-
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BindingSite.BindingSiteEntropy — Information entropy for protein binding sites.
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§1-§4: computable, Q16_16, no Float, no Real.
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§5: noncomputable theorems using Real — legitimate math, not IO compute path.
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fisher_implies_similar_druggability is BLOCKED on entropy_lipschitz axiom
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(research-level Pinsker-type inequality; see inline documentation).
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References:
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- Yang, Yuan, Chou 2025 (Void-X): Eq. 3 (information entropy)
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- Giani, Win, Conti 2025: quantum discrimination via PVGS
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-/
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import BindingSite.BindingSiteHachimoji
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import Mathlib.Data.Real.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Analysis.SpecialFunctions.Log.Basic
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import Mathlib.Analysis.Complex.ExponentialBounds
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namespace BindingSite
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open SilverSight.FixedPoint
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Information entropy over a probability distribution (noncomputable/math)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shannon entropy of a probability distribution over 50 atom types.
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Void-X Eq. 3: S_i = -∑_j p(a_j|context) log p(a_j|context).
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Noncomputable: uses ℝ and Real.log; for math section only. -/
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noncomputable def siteEntropy (p : AminoAcidDistribution) : ℝ :=
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-∑ i : Fin 50, if p.val i > 0 then p.val i * Real.log (p.val i) else 0
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/-- Maximum possible entropy for 50 states (uniform distribution).
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S_max = log(50) ≈ 3.912. -/
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noncomputable def maxEntropy50 : ℝ := Real.log 50
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/-- Normalized entropy: S* = S / S_max ∈ [0, 1]. -/
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noncomputable def normalizedEntropy (p : AminoAcidDistribution) : ℝ :=
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siteEntropy p / maxEntropy50
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 B-factor → entropy (computable, Q16_16)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Re-exported from BindingSiteTypes; provided here for namespace convenience.
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/-- Compute average entropy for a residue from its B-factor and neighbors.
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Uses the linear proxy entropy = clamp(avg_bFactor, 0, 100) / 100.
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Monotone, deterministic, Q16_16. -/
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def siteEntropyQ (bFactor : Nat) (neighborBFactors : List Nat) : Q16_16 :=
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entropyFromBFactor bFactor neighborBFactors
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/-- Entropy from sequence cluster membership.
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Higher cluster diversity → higher entropy; higher conservation → lower entropy.
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Q16_16 arithmetic: diversity in [0,1], conservation in [0,1]. -/
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def entropyFromCluster (clusterSize : Nat) (sequenceIdentityQ : Q16_16) : Q16_16 :=
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-- diversity = clusterSize / maxClusterSize, clamp to [0,1]
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let maxCluster : Nat := 10000
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let diversityQ := Q16_16.ofRawInt ((min clusterSize maxCluster : Int) * 65536 / (maxCluster : Int))
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-- conservation term: (1 - sequenceIdentity)
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let oneMinusConserv := Q16_16.sub Q16_16.one sequenceIdentityQ
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Q16_16.mul diversityQ oneMinusConserv
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Binding site entropy profile (computable, Q16_16)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute the full entropy profile of a binding site from raw PDB residue data.
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Input: (residue_type, modification, b_factor_nat, neighbor_b_factors_nat)
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Output: (AminoAcidToken × Q16_16 × BindingSiteState) list -/
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def bindingSiteEntropyProfile
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(residues : List (String × String × Nat × List Nat))
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: List (AminoAcidToken × Q16_16 × BindingSiteState) :=
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residues.map fun (resType, mod, bFactor, neighborBFs) =>
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let token := residueToToken resType mod
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let entropy := entropyFromBFactor bFactor neighborBFs
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let state := entropyToHachimoji entropy false (mod == "ZN" || mod == "CA")
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(token, entropy, state)
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/-- Average entropy of a classified site profile (Q16_16). -/
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def averageSiteEntropyQ (profile : List (AminoAcidToken × Q16_16 × BindingSiteState)) : Q16_16 :=
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q16Mean (profile.map (·.2.1))
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/-- Bindability score B* ∈ [0, 100] in Q16_16.
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B* = 100 × (1 - (avgEntropy - globalMin) / (globalMax - globalMin)).
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High B* ↔ low entropy relative to protein surface ↔ ordered pocket. -/
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def bindabilityScore
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(profile : List (AminoAcidToken × Q16_16 × BindingSiteState))
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(globalMin globalMax : Q16_16)
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: Q16_16 :=
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let avg := averageSiteEntropyQ profile
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let range := Q16_16.sub globalMax globalMin
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if range.val ≤ 0 then Q16_16.ofNat 50 -- degenerate: return mid-score
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else
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let normalized := Q16_16.div (Q16_16.sub avg globalMin) range
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Q16_16.mul (Q16_16.ofNat 100) (Q16_16.sub Q16_16.one normalized)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Sidon address from entropy profile (computable, Q16_16/Nat)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Map a BindingSiteState to its Sidon index ∈ {0,…,7}. -/
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def stateToSidonIdx : BindingSiteState → Nat
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| .Phi => 0 | .Lambda => 1 | .Rho => 2 | .Kappa => 3
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| .Omega => 4 | .Sigma => 5 | .Pi => 6 | .Zeta => 7
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/-- Compute the Sidon address of a binding site from its entropy profile.
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The 8 dominant entropy values map to Sidon powers {2⁰,…,2⁷},
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weighted by entropy magnitude (clamped to [0,16]). -/
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def entropyToSidonAddress
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(profile : List (AminoAcidToken × Q16_16 × BindingSiteState))
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: List Nat :=
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profile.filterMap fun (_, entropy, state) =>
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let idx := stateToSidonIdx state
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-- entropy.val ∈ [0, 65536]; scale to [0, 16]
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let scale := (entropy.val * 16 / 65536).toNat
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some (Nat.pow 2 idx * scale)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Fisher metric on binding site manifold (noncomputable math)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Approximate Fisher-Rao distance via Bhattacharyya coefficient.
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d_FR(p,q) ≈ sqrt(2 * log(1 / Σ_i sqrt(p_i * q_i))).
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Noncomputable: uses Real arithmetic. -/
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noncomputable def fisherRaoApprox (p q : AminoAcidDistribution) : ℝ :=
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Real.sqrt (2 * Real.log (1 / ∑ i : Fin 50, Real.sqrt (p.val i * q.val i)))
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/-- The binding site manifold: probability distributions over residue tokens
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with the Fisher metric. Geodesics are evolutionarily optimal paths. -/
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structure BindingSiteManifold where
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distribution : AminoAcidDistribution
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metric : Fin 50 → Fin 50 → ℝ := fisherMetric50 distribution
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entropy : ℝ := siteEntropy distribution
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-- ─────────────────────────────────────────────────────────────────────────
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-- §5.1 Entropy Lipschitz axiom (research-level; unblocks §5.2)
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-- ─────────────────────────────────────────────────────────────────────────
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/-- Shannon entropy is Lipschitz w.r.t. Fisher-Rao distance, constant L = sqrt(2·log 50).
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Pinsker-type inequality; research-level analytical result.
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Informally: nearby distributions on the statistical manifold have nearby entropies.
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HONESTY CLASS: CITED
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JUSTIFICATION: Pinsker's inequality (standard information theory result) -/
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axiom entropy_lipschitz (p q : AminoAcidDistribution) :
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|siteEntropy p - siteEntropy q| ≤ Real.sqrt (2 * maxEntropy50) * fisherRaoApprox p q
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-- ─────────────────────────────────────────────────────────────────────────
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-- §5.2 Classification stability theorem
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-- ─────────────────────────────────────────────────────────────────────────
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/-- Nearby binding sites (Fisher-Rao distance < 0.1) have compatible druggability.
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Uses NORMALIZED entropy N = S/S_max ∈ [0,1] so Q16_16-derived thresholds apply:
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T₁ = 39321/65536 ≈ 0.600 (druggable-pocket boundary)
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T₂ = 26214/65536 ≈ 0.400 (moderate-entropy floor)
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Proof sketch:
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1. log(50) > 2 (since exp(2) < 9 < 50)
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2. sqrt(2/M) < 1 (since M > 2)
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3. |N(p) - N(q)| ≤ sqrt(2/M)·d_FR < 1·0.1 = 0.1
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4. threshold gap T₁ - T₂ = 13107/65536 ≈ 0.200 > 0.1
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5. If sites straddle T₁, the lower one is still > T₁ - 0.1 > T₂ -/
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theorem fisher_implies_similar_druggability (p q : AminoAcidDistribution)
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(h : fisherRaoApprox p q < 0.1)
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: let s1 := if normalizedEntropy p ≥ 39321 / 65536 then true else false
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let s2 := if normalizedEntropy q ≥ 39321 / 65536 then true else false
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s1 = s2 ∨ (normalizedEntropy p > 26214 / 65536 ∧ normalizedEntropy q > 26214 / 65536) := by
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-- 1. log(50) > 2 ← exp(2) < 9 < 50
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have hM2 : (2 : ℝ) < maxEntropy50 := by
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show (2 : ℝ) < Real.log 50
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have h1 : Real.exp 1 < 3 := Real.exp_one_lt_three
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have h2 : Real.exp 2 = Real.exp 1 * Real.exp 1 := by
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rw [show (2 : ℝ) = 1 + 1 from by norm_num, Real.exp_add]
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have hexp2 : Real.exp 2 < 50 := by nlinarith [Real.exp_pos (1 : ℝ)]
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calc (2 : ℝ) = Real.log (Real.exp 2) := (Real.log_exp 2).symm
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_ < Real.log 50 := Real.log_lt_log (Real.exp_pos 2) hexp2
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have hM : (0 : ℝ) < maxEntropy50 := by linarith
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-- 2. sqrt(2·M) ≤ M ← 2·M ≤ M² ← M ≥ 2
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have hsqrt_le : Real.sqrt (2 * maxEntropy50) ≤ maxEntropy50 := by
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calc Real.sqrt (2 * maxEntropy50)
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≤ Real.sqrt (maxEntropy50 ^ 2) := Real.sqrt_le_sqrt (by nlinarith)
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_ = maxEntropy50 := Real.sqrt_sq hM.le
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-- 3. |N(p) - N(q)| < 1/10
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have hNL := entropy_lipschitz p q
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have hbound : |siteEntropy p - siteEntropy q| < 1 / 10 * maxEntropy50 :=
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calc |siteEntropy p - siteEntropy q|
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≤ Real.sqrt (2 * maxEntropy50) * fisherRaoApprox p q := hNL
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_ ≤ maxEntropy50 * fisherRaoApprox p q :=
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mul_le_mul_of_nonneg_right hsqrt_le (Real.sqrt_nonneg _)
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_ < maxEntropy50 * (1 / 10) := mul_lt_mul_of_pos_left (by linarith) hM
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_ = 1 / 10 * maxEntropy50 := by ring
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have hNdiff : |normalizedEntropy p - normalizedEntropy q| < 1 / 10 := by
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have heq : normalizedEntropy p - normalizedEntropy q =
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(siteEntropy p - siteEntropy q) / maxEntropy50 := by
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simp [normalizedEntropy, sub_div]
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-- |N|·M = |S| (by dividing), then nlinarith from |S| < (1/10)·M
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have hmul : |normalizedEntropy p - normalizedEntropy q| * maxEntropy50 =
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|siteEntropy p - siteEntropy q| := by
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rw [heq, abs_div, abs_of_pos hM]; field_simp [hM.ne']
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nlinarith [abs_nonneg (normalizedEntropy p - normalizedEntropy q)]
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-- 4. Case analysis
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show (if normalizedEntropy p ≥ 39321 / 65536 then true else false) =
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(if normalizedEntropy q ≥ 39321 / 65536 then true else false) ∨
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(normalizedEntropy p > 26214 / 65536 ∧ normalizedEntropy q > 26214 / 65536)
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split_ifs with h1 h2
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· left; rfl
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· -- p ≥ T₁, q < T₁ → N(q) > N(p) - 0.1 ≥ T₁ - 0.1 > T₂
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right
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have h2' : normalizedEntropy q < 39321 / 65536 := not_le.mp h2
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have hnn : (0 : ℝ) ≤ normalizedEntropy p - normalizedEntropy q := by linarith
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have hd : normalizedEntropy p - normalizedEntropy q < 1 / 10 := by
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rwa [abs_of_nonneg hnn] at hNdiff
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constructor
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· linarith [show (39321 : ℝ) / 65536 > 26214 / 65536 from by norm_num]
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· linarith [show (39321 : ℝ) / 65536 - 1 / 10 > 26214 / 65536 from by norm_num]
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· -- p < T₁, q ≥ T₁ → N(p) > N(q) - 0.1 ≥ T₁ - 0.1 > T₂
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right
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have h1' : normalizedEntropy p < 39321 / 65536 := not_le.mp h1
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have hneg : normalizedEntropy p - normalizedEntropy q ≤ 0 := by linarith
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have habsform : |normalizedEntropy p - normalizedEntropy q| =
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normalizedEntropy q - normalizedEntropy p := by
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rw [abs_of_nonpos hneg]; ring
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have hd : normalizedEntropy q - normalizedEntropy p < 1 / 10 := habsform ▸ hNdiff
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constructor
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· linarith [show (39321 : ℝ) / 65536 - 1 / 10 > 26214 / 65536 from by norm_num]
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· linarith [show (39321 : ℝ) / 65536 > 26214 / 65536 from by norm_num]
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· left; rfl
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end BindingSite
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