ledger_forcing_principle
plain-language theorem explainer
J-symmetry forces recognition events into reciprocal pairs whose costs match and whose log-ratios cancel, while also guaranteeing a balanced ledger of total cost zero. Researchers tracing double-entry bookkeeping back to the cost functional would cite the result. The proof is a term-mode tuple that directly invokes J_symmetric, reciprocity, paired_log_sum_zero, and the empty-ledger witnesses.
Claim. Let $J(x) := (x + x^{-1})/2 - 1$. Then $J(x) = J(x^{-1})$ for all nonzero real $x$, every recognition event $e$ satisfies event_cost$(e)$ = event_cost$(e^{-1})$ and log(ratio$(e)$) + log(ratio$(e^{-1})$) = 0, and there exists a ledger $L$ that is balanced with ledger_cost$(L)$ = 0.
background
The cost functional is $J(x) = (x + x^{-1})/2 - 1$. RecognitionEvent is a structure carrying source, target, and positive ratio. Ledger is a list of events equipped with the balanced_list predicate enforcing double-entry. The module proves that J-symmetry forces double-entry ledger structure. Upstream J_symmetric states that $J(x) = J(x^{-1})$ for $x > 0$, while reciprocal supplies the automorphism that inverts ratios.
proof idea
The proof is a term-mode construction packaging six components: the functional implication from J_symmetric, the reciprocity property on events, the paired-log-sum-zero identity, and the empty ledger together with its balanced and zero-cost theorems.
why it matters
The declaration supplies the concrete existence witness that closes the ledger-forcing step in the Recognition Science chain. It translates J-symmetry (T5) into the requirement of double-entry conservation. No downstream uses are recorded; the result stands as the direct implementation of the module's stated principle.
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