T3_continuity
Conservation in a ledger forces zero net flux on any closed chain. Researchers formalizing loop invariants in recognition structures cite this result when closing cycles on the phi-ladder. The proof is a one-line wrapper that invokes the conserve field of the Conserves class.
claimLet $L$ be a ledger on recognition structure $M$ that satisfies the conservation property. Then for every chain $ch$ in $M$, if the head of $ch$ equals its last element, the flux of $L$ along $ch$ equals zero.
background
Ledger pairs debit and credit maps from units to integers. Conserves is the class whose field states that every chain with equal head and last has zero chainFlux. Chain is a finite sequence of units with consecutive recognition relations; head and last extract the first and final units. chainFlux is defined as phi(last) minus phi(head), where phi maps units to integers.
proof idea
The proof is a one-line wrapper that applies the conserve axiom of the Conserves instance.
why it matters in Recognition Science
T3_continuity supplies the conservation axiom for closed chains, the third step after atomicity in the chain sequence. It guarantees flux cancellation on loops, a prerequisite for the recognition composition law and later forcing-chain steps that reach J-uniqueness and the eight-tick octave. No downstream uses appear yet.
scope and limits
- Does not prove existence of any conserving ledger.
- Does not give the explicit form of phi.
- Does not treat open chains or infinite sequences.
- Does not interact with atomicity or higher T-steps.
formal statement (Lean)
55theorem T3_continuity {M} (L : Ledger M) [Conserves L] :
56 ∀ ch : Chain M, ch.head = ch.last → chainFlux L ch = 0 := Conserves.conserve
proof body
One-line wrapper that applies end.
57
58end IndisputableMonolith