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136 lines
4.9 KiB
Text
136 lines
4.9 KiB
Text
/-
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StructuralAttestation.lean - Mechanical Merkle Trees & Structural Cryptography
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Formalizes the bridge between physical structural integrity and computational validity.
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Based on Tech Note: Mechanical Merkle Tree (Proof-of-State).
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-/
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import Semantics.FixedPoint
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import Semantics.Bind
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namespace Semantics.StructuralAttestation
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open Q16_16
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/--
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A 6-axis stress vector representing strain gauge data.
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(σx, σy, σz, τxy, τyz, τzx)
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-/
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structure StressVector where
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sigmaX : Q16_16
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sigmaY : Q16_16
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sigmaZ : Q16_16
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tauXY : Q16_16
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tauYZ : Q16_16
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tauZX : Q16_16
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deriving Repr, DecidableEq
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/--
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A Mechanical Hash (structural signature).
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In hardware, this is derived via Blake3(vector).
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In the formal core, we use a sum-reduction for reachability proofs.
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-/
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def mechanicalHash (v : StressVector) : UInt32 :=
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v.sigmaX.val ^^^ v.sigmaY.val ^^^ v.sigmaZ.val ^^^
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v.tauXY.val ^^^ v.tauYZ.val ^^^ v.tauZX.val
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/--
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A node in the Mechanical Merkle Tree.
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Each node has a local stress state and a combined hash of its children.
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-/
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inductive MechanicalMerkleTree
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| leaf (id : Nat) (stress : StressVector)
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| node (hash : UInt32) (left right : MechanicalMerkleTree)
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deriving Repr
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/-- Compute the root hash of a Mechanical Merkle Tree. -/
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def rootHash : MechanicalMerkleTree → UInt32
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| .leaf _ stress => mechanicalHash stress
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| .node h _ _ => h
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/--
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Build a node from two subtrees.
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Root hash is the XOR-sum of children's hashes (simplified hardware-native hash).
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-/
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def mkNode (l r : MechanicalMerkleTree) : MechanicalMerkleTree :=
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.node (rootHash l ^^^ rootHash r) l r
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/--
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The Ideal Manifold: The target structural state (zero stress baseline).
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-/
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def idealManifoldHash : UInt32 := 0
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/--
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Admissibility: A structural state is admissible if its root hash
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is within the allowed stability epsilon of the ideal manifold.
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-/
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def isStructurallyAdmissible (tree : MechanicalMerkleTree) (epsilon : UInt32) : Bool :=
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let h := rootHash tree
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h <= epsilon -- Simplified stability check
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/--
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The Security Veto: Computational results are only valid
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if the physical structure is intact.
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-/
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def securityVeto (tree : MechanicalMerkleTree) (epsilon : UInt32) : Bool :=
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not (isStructurallyAdmissible tree epsilon)
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/--
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Mechanical Bind: Chains structural integrity to semantic validity.
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-/
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def structuralBind (tree : MechanicalMerkleTree) (epsilon : UInt32) (g : Metric) : Bind MechanicalMerkleTree String :=
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controlBind tree "structural_attestation" g
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(fun t _ _ => if isStructurallyAdmissible t epsilon then zero.val else one.val)
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(fun t => if isStructurallyAdmissible t epsilon then "structural_attestation" else "VETO:PHYSICAL_INTEGRITY_COMPROMISED")
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(fun t => t)
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-- #eval Witness:
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-- Healthy state (all zeros) vs Damaged state (high stress)
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def healthyLeaf : MechanicalMerkleTree := .leaf 0 { sigmaX := zero, sigmaY := zero, sigmaZ := zero, tauXY := zero, tauYZ := zero, tauZX := zero }
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def healthyTree : MechanicalMerkleTree := mkNode healthyLeaf healthyLeaf
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def damagedLeaf : MechanicalMerkleTree := .leaf 1 { sigmaX := ⟨0xFFFFFFFF⟩, sigmaY := zero, sigmaZ := zero, tauXY := zero, tauYZ := zero, tauZX := zero }
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def damagedTree : MechanicalMerkleTree := mkNode healthyLeaf damagedLeaf
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#eval rootHash healthyTree
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#eval rootHash damagedTree
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#eval isStructurallyAdmissible damagedTree 1000
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/--
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Theorem: Any change in structural state (leaf stress)
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is reflected in the root hash.
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-/
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theorem structural_integrity_reflected (id : Nat) (s1 s2 : StressVector) (h : s1 ≠ s2) :
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mechanicalHash s1 ≠ mechanicalHash s2 := by
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-- This depends on the hash function properties.
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-- For XOR-sum it might have collisions, but for Blake3/formal proof we assume
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-- injectivity for the semantic model.
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-- TODO(lean-port): UNPROVABLE AS STATED. XOR-sum is NOT injective (collisions exist).
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-- Weakened theorem: single-component change guarantees hash change.
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sorry
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/--
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Weakened version: a single-component change in stress is reflected in the hash.
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XOR is injective in each argument when the other is fixed.
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-/
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theorem structural_integrity_reflected_single_component
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(s1 s2 : StressVector)
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(hX : s1.sigmaX ≠ s2.sigmaX)
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(hY : s1.sigmaY = s2.sigmaY)
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(hZ : s1.sigmaZ = s2.sigmaZ)
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(hXY : s1.tauXY = s2.tauXY)
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(hYZ : s1.tauYZ = s2.tauYZ)
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(hZX : s1.tauZX = s2.tauZX) :
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mechanicalHash s1 ≠ mechanicalHash s2 := by
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simp [mechanicalHash] at *
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intro h_eq
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rw [hY, hZ, hXY, hYZ, hZX] at h_eq
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have h_cancel : s1.sigmaX.val = s2.sigmaX.val := by
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apply (UInt32.xor_right_inj (s2.sigmaY.val ^^^ s2.sigmaZ.val ^^^ s2.tauXY.val ^^^ s2.tauYZ.val ^^^ s2.tauZX.val)).mp
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simp [UInt32.xor_comm] at h_eq ⊢
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exact h_eq
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have h_eq_stress : s1.sigmaX = s2.sigmaX := by
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have h1 : s1.sigmaX = ⟨s1.sigmaX.val⟩ := by cases s1.sigmaX; rfl
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have h2 : s2.sigmaX = ⟨s2.sigmaX.val⟩ := by cases s2.sigmaX; rfl
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rw [h1, h2, h_cancel]
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contradiction
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end Semantics.StructuralAttestation
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