After Turing: The Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Classical Computational Science from Smithian Fold Theory

After Turing: The Fold Machine is the standalone 396-page paper for the completed Classical Computation branch of the third clean-room reconstruction of Smithian Fold Theory (SFT). From separately admitted Foundation, Mathematics and Information Science receipts it derives, in dependency order, Formal Computation; Computability; Computational Complexity; Algorithms and mathematical data structures; Semantics and mathematical programming theory; Concurrent and Distributed Computation; Cryptography and Computational Security; Learning and Intelligence Theory; and Scientific Computation.

The branch does not import a Turing machine, lambda calculus, conventional complexity class, probability cause, cryptographic hardness assumption, pretrained model, fitted parameter, floating proof value, application answer or earlier SFT derivation as a premise. Every object is an exact generated finite carrier, held label, relation, trace, resource ledger, interface, observation class or proof record. Empty One is structural rather than numerical zero; complementary held labels replace negative proof quantities; and randomized algorithms execute complete registered deterministic schedule support rather than assume an uncaused stochastic transition.

The frozen inventory contains 113 dependency-ordered claims. Their grammars execute 28,928 generated candidates and preserve 28,928 decisions, 113 unique survivors, 113 depth-independent base/successor certificates, 452 adverse controls and 113 implementation-distinct validations. The manuscript documents every claim in full: dependencies; exact theorem; eight structural axes; all first-failure elimination counts; unique survivor; minimality; named-shape uniqueness; operational laws; live witnesses; induction certificate; controls; meaning; correspondence boundary; limitations; and exact source, census, seal, validator and engine-receipt identities.

The central result is a native Fold machine whose states, words, actions and computations preserve complete provenance. The branch derives universal interpretation only after languages, automata, rewriting, recursion, binding, abstract machines, circuits, processes and composition close. Halting and incompleteness follow from exact self-description and held complement operations. Security requires exact adversary and resource grammars. Learning retains complete hypothesis alternatives and sealed evaluation. Scientific simulation proves consequences of its registered model but does not select natural laws.

The accompanying open release contains the PDF, complete Markdown manuscript, frozen inventory, all 113 claim packages and candidate/decision ledgers, executable sources, independent validators, receipts, evidence map, tests and checksum ledger.