Book-Balancing Magic Has Nowhere to Hide: Source-Native Accounting and Actual Continuation

Abstract

We construct a whole-ledger continuation chain fulfilled by one actual source: an original residual in the complete future model generates the syntax and responsibility row of the next request; the source program executes and strictly pays the same debt; and its completion event delivers an answer, a receipt, a complete ledger write, and the actual successor. With the source and lawful compilation contract fixed, the named candidates based on event trimming, receipts from another source, hidden responsibilities, equal-budget migration, and zero-budget pseudo-settlement cannot construct the required lawful objects. The construction holds for the original source factory, initial material, configuration, and every natural-number-indexed admission stop in its cofinal runtime sequence, retaining future readings at every allowed action word and site. Original syntax, paid state, the full execution trace, and responsibility ownership enter the same next write. Zero numerical effect does not erase syntax payment, and a new responsibility receives its own source rather than refilling the old debt. Complete event inventories and complete open-responsibility fibres are the accounting objects; finite explicit rows answer the whole ledger through exact remainders, while split and merge preserve every parent, child, occurrence multiplicity, and original typed receipt. For a fixed original debt, finite non-refilling histories enclosing one strict payment make the payment macro-step relation well founded without a total closure law. A separate total law emitting exact settlement or an actual paid macro-step at each current constructs a finite settlement history. Final settlement requires both the original terminal receipt and an empty same-debt fibre in the actual successor. Target six exhibits actual repair, payment, and settlement; an original integer consumer cannot be recovered by any function from the entire family of finite complex observations; and, conditional on the specified right-half-plane zero receipt, four traces of a finite window, inverse residual, two means, and a cochain persist into a correction program, its actual new inventory, and every later reading. The ledger both governs how responsibility ends and generates how it continues to work.