Ledgeral Mathematics: A Finite Algebra of Recursion, Admissibility, Projection, and Survivor Structure This repository contains the complete public edition of Ledgeral Mathematics, a foundational mathematical monograph that develops a finite algebra of recursion, admissibility, projection, survivor formation, residue retention, transport, composition, optimization, falsification, and audit. The theory begins from the retained finite record, an explicitly formed object whose carrier, addresses, entries, active support, inactive structure, status, formation history, comparison discipline, readout route, and audit relation remain part of its mathematical identity. Ledgeral Mathematics begins at a more primitive level than mathematical systems that take numbers, points, sets, spaces, functions, graphs, trajectories, or continua as already available objects. Those structures may be constructed and used within the theory, though they do not receive automatic foundational standing. Every object must first declare what carries it, what occupies each retained address, how it was formed, what operations may act upon it, what transformations are permitted, and what information must remain available after those transformations have occurred. The central admission principle is straightforward. Nothing enters the mathematics by implication. Every lawful object must have a finite retained form. Every operation must declare its input region, carrier rule, entry rule, legality conditions, invalidity conditions, and output status. Every comparison must identify the equality relation being used. Every readout must preserve a trace to the record from which it was produced. Every projection must identify what survives, what is rejected or displaced, and how the full event can be audited. This discipline allows Ledgeral Mathematics to preserve distinctions that conventional notation may compress or erase. A lawful null record is different from an invalid expression. A missing object is different from a retained object with inactive support. Candidate status is different from survivor status. Residue is different from error, absence, or nonexistence. Carrier equality, support equality, entry equality, readout equality, provenance equality, and full record equality are separate mathematical claims. The relevant comparison must therefore be declared rather than assumed. One of the central structures of the theory is the survivor-residue-audit form of projection. A candidate record is submitted to a declared admissibility rule and projection procedure. The projection produces a survivor, a residue, and an audit packet. The survivor contains the structure admitted by the projection. The residue retains rejected, displaced, suppressed, obstructed, unresolved, or otherwise excluded structure. The audit records the candidate, the governing admissibility conditions, the projection route, the resulting survivor, the resulting residue, and the verification status of the event. Projection therefore does more than select an accepted output. It retains the mathematical consequences of exclusion. Loss becomes inspectable. Rejection becomes information. Suppression remains traceable. A lawful null survivor may coexist with nonempty residue. An active survivor may retain displaced structure outside its support. A mixed event may preserve admitted components, rejected components, and formation failures under different statuses. These distinctions allow later analysis of irreversibility, obstruction, instability, hidden coupling, model disagreement, implementation failure, measurement conflict, and operation-order dependence. Recursion is developed through the same finite retained discipline. A process does not receive an unbounded history in advance. It is represented through finite depth carriers, finite update words, finite survivor chains, finite branch records, finite residue histories, and finite continuation audits. Persistence is established through repeated admitted continuation across retained recursion depth. Branching, merging, recurrence, stabilization, obstruction, termination, return, cyclic behavior, and irreversible loss remain available as explicit finite structures. The monograph extends this foundation into operator-word algebra, holonomy calculus, finite transport and boundary accounting, constitutive algebra, branching and capacity calculus, co-admissibility, convergence, directed persistence, signal and readout calculus, finite recursion-spectral analysis, regime classification, construction and optimization, audit and falsification, and representation-layer quarantine. The full work is organized across twenty-three major sections, a global closure, and five technical appendices devoted to notation, dependency tracking, result indexing, verification, reproduction, serialization, archiving, implementation boundaries, and execution audit. Representation remains available throughout the theory, though its role is controlled. Equations, arrays, tables, coordinates, diagrams, graphs, curves, spectra, statistical models, analytic expressions, and continuous systems may be generated as readouts from ledgeral records. A representation does not become a native object merely through familiarity or usefulness. It may enter native calculation only after it has been reconstructed as a finite retained record with a declared carrier, entries, role, formation rule, and audit trace. This separation preserves the distinction between a mathematical object and the representation used to inspect, communicate, or calculate with it. Ledgeral Mathematics was developed partly in response to the foundational requirements of Post-Temporal Physics, though it is presented here as an independent mathematical system. Its potential applications extend across foundational mathematics, algebra, logic, proof theory, discrete systems, physics, computation, artificial intelligence, formal verification, data provenance, system assurance, engineering, sensing, control, optimization, scientific measurement, model comparison, reproducibility, and falsification. The theory does not claim that established mathematical systems are unnecessary. It presents a distinct foundational program organized around finite formation, retained accountability, explicit admissibility, preserved residue, and auditable transformation. This repository contains the foundational public volume. Implementation-oriented methods, domain-specific extensions, and the separate companion program known as Applied Ledgeral Mathematics are outside the scope of this release and are not presently being distributed openly. Portions of that work may carry significant dual-use implications. Any future distribution of unpublished applied material may therefore be considered individually following appropriate legal, export-control, security, intellectual-property, and end-use review. This publication-scope notice does not designate the public monograph or any unpublished companion material as classified, ITAR-controlled, EAR-controlled, export-controlled, or otherwise restricted by the United States Government. Any legal determination of that kind must be made by qualified authorities or professional counsel. The published monograph is released under the Creative Commons Attribution 4.0 International License. That license applies only to the material contained in the publicly released volume. It does not apply to unpublished manuscripts, software, datasets, implementation packages, technical materials, or companion works unless those materials are separately released under the same license.
Adib Enayati (2026) studied this question.