proof(merkle): Stage 1.2 — root-fold law (root of buildMerkleTree = foldRoot over leaves)#49
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…oldRoot over leaves) Prove the Merkle root is a deterministic, representation-independent fold over the leaf vector: rootHashWith h (buildMerkleTree hs) = foldRoot h hs, generic over the combiner h (XOR and BLAKE3 alike). - foldRoot: the leaf-only spec of the root (singleton -> element; a 2^(S k) vector splits at 2^k and combines its two sub-folds with h, the same split buildMerkleTree uses). - rootFoldSplit: the inductive step on the already-split halves, so the rootHashWith-of-Node reduction is clean and decoupled from the `with` abstraction (mirrors buildGetLeafSplit; the IH is a higher-order argument). - rootFoldLaw: the theorem; base Refl, step via rootFoldSplit applied to the IH. Pairs with buildGetLeaf to fully characterise buildMerkleTree (leaves + root) and hands the Stage-4 binding argument a leaf-only handle on the root. Machine-checked, axiom-free, total (clean --total build, 19/19). PROOFS.adoc marks Stage 1.2 DONE and adds rootFoldLaw/foldRoot to the proven surface. https://claude.ai/code/session_011z2t8zAxfcCNLJzU7YdpBQ
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June 18, 2026 00:11
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Summary
Stage 1.2 of the proof campaign (see
docs/PROOFS.adoc). Proves the Merkle root is a deterministic, representation-independent fold over the leaf vector:What landed (
ochrance-core/Ochrance/Filesystem/MerkleBuild.idr)foldRoot— the leaf-only spec of the root (singleton → element; a2^(S k)vector splits at2^kand combines its two sub-folds withh, the same splitbuildMerkleTreeuses).rootFoldSplit— the inductive step stated on the already-split halves, so therootHashWith-of-Nodereduction is clean and decoupled from thewithabstraction (mirrors Stage 1.1'sbuildGetLeafSplit; the IH is a higher-order argument). This was the one wrinkle: a naiverewriteunderwithfails becausebuildMerkleTree's internalcaseis left as a stucklet— the split-lemma sidesteps it.rootFoldLaw— the theorem; baseRefl, step viarootFoldSplitapplied to the IH.Why it matters
Pairs with
buildGetLeaf(Stage 1.1) to fully characterisebuildMerkleTree— both its leaves (getLeafHash) and its root (rootHashWith) are pinned to the leaf vector. Hands the Stage-4 binding argument a leaf-only handle on the root ("the root is a function of the leaves").Verification
idris2 --build ochrance.ipkgunder--total, 19/19, exit 0. Nobelieve_me/assert_*/postulate/holes.docs/PROOFS.adocupdated: Stage 1.2 marked DONE;rootFoldLaw/foldRootadded to the proven surface.Next: Stage 1.3 (the IO↔pure bridge — the estate-wide spine).
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https://claude.ai/code/session_011z2t8zAxfcCNLJzU7YdpBQ
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