Theorem 1: Ledger Necessity & Uniqueness
A unique, positive, double-entry ledger follows from MP + Composability + Finiteness
Statement
Any recognition structure satisfying the Meta‑Principle (Nothing cannot recognize itself), Composability, and Finiteness admits a unique (up to order‑isomorphism) positive, double‑entry ledger with an immutable generator.
In plain English
Reality keeps receipts. If something changes somewhere, there’s a matching counter‑change somewhere else. The only honest way to keep score is a two‑column ledger with a smallest coin (δ). You can’t mint extra coins out of thin air, you can’t hide them in a third column, and you can’t secretly change what a coin is worth. Under the bare‑minimum assumptions (no free miracles, you can chain recognitions, and no infinite rabbit‑holes), that ledger isn’t optional—it’s forced.
- Why inevitable: two columns because every give has a take; closed loops must sum to zero.
- No magic credits: “modulo” or k‑column tricks let you erase real cost; that violates the meta‑principle.
- One denomination: rescaling δ would let you fake progress by changing units mid‑story.
Proof Sketch (informal)
- Construction: Build the ordered abelian “book” via Grothendieck completion on the set of oriented recognitions. Choose a positive generator δ to induce the positive cone.
- Double‑entry / Conservation: Postings telescope on closed recognition chains, enforcing dual‑balance and yielding a discrete conservation law.
- Uniqueness: Follows by the universal property of the completion; any other admissible ledger is order‑isomorphic.
Formal objects
- Recognition structure (R): Objects U, relation recog ⊆ U×U, with composition and MP; finiteness excludes infinite strictly advancing chains.
- Ledger L(R): A linearly ordered abelian group with distinguished generator δ > 0, Page maps ϕ, ψ: U → L, and double‑entry: for any a→b, ϕ(b) − ψ(a) = δ.
- Positive cone: The submonoid generated by δ induces the order; Archimedean property follows from finiteness.
Key lemmas (outline)
- Free abelian construction: The Grothendieck completion of oriented recognitions modulo [a→b] + [b→a] = 0 yields an abelian group with natural posting maps.
- Telescoping on closed chains: For any finite closed chain, the sum of postings is zero; hence conservation on cycles.
- Order and positivity: Selecting δ > 0 defines a positive cone; MP forbids rescaling automorphisms δ ↦ s·δ with s ≠ 1.
- Uniqueness: Any other ledger with the same properties factors uniquely through the completion (universal property), giving an order‑isomorphism.
Implications
- Ground of conservation: Local conservation (Theorem 3) is a corollary of double‑entry on chains.
- Quantized postings: Combined with Finiteness, forces countable, unit postings (Theorem 2).
- Formal backbone: A single ordered “book” underlies all measurement and dynamics.
Checks & boundaries
- No k‑ary ledgers (k ≥ 3): Would produce orphan costs or modular loops, violating MP.
- No modular costs: Posting modulo m admits nontrivial closed loops of apparent zero cost, contradicting ledger order.
- Archimedeanity: Finiteness excludes pathological orders; ensures δ sets the atomic scale.
Where this is used next
- T2 Atomicity: one posting per tick forces quantization.
- T3 Conservation: continuity law from double‑entry.
- T4 Cost Uniqueness: convexity and dual‑symmetry over the positive cone.