Receipt knot algebra

Read a receipt chain as a knotted khipu cord and apply Reidemeister moves R1/R2/R3 — local rewrites that must leave the chain's knot-invariant tag unchanged.

Headline number: 3 moves, 1 invariant tag — R1/R2/R3 preserve the 16-hex knot signature.

The Inka khipu is a knotted-cord record whose primary cord value equals the sum of its pendant values. SZL reads a receipt DAG as exactly that structure, and the knot-calculus-v1 recipe in szl-cookbook ships a runnable TypeScript implementation. This recipe explains the algebra and how to run it.

Honest scope. The Reidemeister invariance for receipt chains is Conjecture R1/R2/R3 in Lean (lutar-lean/Lutar/Knot/ReidemeisterConjecture.lean, all sorry-tagged, target v16). The code checks invariance on concrete chains; it does not prove it for all chains.

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Prerequisites

cd recipes/knot-calculus-v1/code && npm install

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Quickstart (runnable, in this repo)

cd recipes/knot-calculus-v1/code
npx tsx tests/demo.ts
# Prints:
#   1. a 3-organ × 5-decision khipu root receipt with the sum-of-sums invariant verified (TH11 pass)
#   2. the 16-hex knot-invariant tag of that root
#   3. a McAllester-1999 PAC-Bayes upper bound (see recipe 09)
#   4. a TH11 failure demo: a tampered pendant value is rejected by verifySumInvariant

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Full walkthrough

Step 1 — The khipu summation invariant (TH11)

A receipt DAG is a three-tier pendant-cord tree: decisions → organ pendants → root. The invariant is sum-of-sums: each pendant value equals the sum of its decision values, and the root equals the sum of the pendants. The check is verifySumInvariant in code/src/khipu-receipt.ts; the Lean obligation is TH11 khipuReceipt_checksum_invariant (two routine sorrys).

Step 2 — The knot-invariant tag

The pendant–subsidiary tree is a chord diagram of Vassiliev–Bar-Natan type (Bar-Natan 1995; [Vassiliev 1990]); the summation cord supplies the 4T closure relation. The 16-hex tag is a knot invariant of the chain: two chains with the same knot type get the same tag.

Step 3 — Reidemeister moves on receipt chains

A move is a local rewrite of the chain that does not change what it records:

Move	Receipt-chain meaning	Invariant requirement
R1 (twist)	insert/remove a no-op self-referential receipt	tag unchanged
R2 (poke)	add then immediately revoke a paired receipt	tag unchanged
R3 (slide)	reorder two independent, commuting receipts	tag unchanged

// Conceptual check (pattern mirrors the demo): a valid move preserves the tag.
import { buildRoot, knotTag } from "./src/khipu-receipt";
const before = knotTag(buildRoot(chain));
const after  = knotTag(buildRoot(applyR3(chain)));   // R3: swap two commuting receipts
console.assert(before === after, "R3 must preserve the knot-invariant tag");

Step 4 — When the invariant should break

Tampering is not a Reidemeister move. Editing a pendant value changes the recorded total, so verifySumInvariant fails (TH11 failure) and the tag changes — exactly the demo's step 4. This is the difference between a legitimate rewrite and a forgery.

Step 5 — Tie it to live receipts

Pull a live receipt chain from a11oy's memory cortex (the a11oy Memory role) (recipe 05), build its khipu tree, and compute the tag. Two ticks that differ only by commuting steps share a tag; a tampered tick does not.

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Lean obligations referenced

Theorem	File	Status
TH11 khipuReceipt_checksum_invariant	lutar-lean/Lutar/Khipu/SummationInvariant.lean	two routine sorrys
Conjecture R1/R2/R3	lutar-lean/Lutar/Knot/ReidemeisterConjecture.lean	all sorry-tagged, target v16

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See also

• 09 — PAC-Bayes confidence margin — the demo's step 3.
• 01 — Verify a receipt end-to-end
• Code: knot-calculus-v1 (runnable TypeScript in szl-cookbook)

Cite this recipe

@misc{szl_cookbook_knot_algebra_2026,
  title        = {Receipt knot algebra (SZL Cookbook recipe 08)},
  author       = {{SZL Holdings}},
  year         = {2026},
  howpublished = {\url{https://github.com/szl-holdings/szl-cookbook/blob/main/recipes/08-receipt-knot-algebra.md}},
  note         = {Reidemeister R1/R2/R3 are Conjecture in Lean (target v16). Doctrine v11 c7c0ba17.}
}

References: Reidemeister 1927, Abh. Math. Sem. Univ. Hamburg 5:24–32; Bar-Natan 1995, Topology 34:423–472; Vassiliev 1990, Adv. Sov. Math. 1:23–69; Ascher & Ascher 1981, Code of the Quipu; Urton 2003, Signs of the Inka Khipu.

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Doctrine v11 LOCKED — 749/14/163 — kernel c7c0ba17 · Λ = Conjecture 1 · SLSA L1 (honest)