[Depreciated and replaced by V3] After Turing: The Fold Machine

[Depreciated and replaced by V3] This pre-V3 paper is replaced by the corresponding V3 clean-room reconstruction: After Turing: The Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Classical Computational Science from Smithian Fold Theory; The Quantum Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Reversible and Quantum Computation from Smithian Fold Theory. The V3 source platform is https://github.com/MettaMazza/ernos-labs-sft-platform. The original DOI, concept DOI, version number and files are preserved for transparent historical provenance; this record must not be presented or cited as current V3 work.

Classical and quantum computation are derived here from one exact finite relation rather than installed as separate machine and physical formalisms. Smithian Fold Theory begins with one machine-checked self-proven theorem - there is no nothing - which forces the One, the exact positive-rational domain and the Fold. From the Fold's two-position fibre, the paper generates state, transition, observation, symbols, information, languages, automata, recursion, universality, computability boundaries, complexity, algorithms, semantics, distributed computation, security, learning, scientific computation, reversibility and quantum computation.

The declared fundamental-computation census contains 164 obligations: 163 are internally closed and the foundational Fold-uniqueness statement is correctly conditional on its mechanically generated 84-form composition grammar through size three. Steps 325-407 execute 83 suites and 691 focused checks. The complete synchronized corpus executes 409 suites and 2,693 checks with zero failures; all 409 readable sources regenerate their C certificates byte-identically.

Four depth-independent native results close the formerly named internal frontier: BB_F(k)=k for every positive finite Fold description depth; P_F=NP_F inside the admitted Fold evaluator/proof grammar; exact depth, width, and size lower bounds for every circuit assembled from lawful Fold edges; and the unique minimum fault width 2t+1 for every positive finite fault order. The paper proves each result constructively, supplies adverse controls, and states the exact boundary separating it from conventional Turing-machine, external complexity-class, arbitrary gate-basis, and stochastic hardware claims.

A standalone Fold Computational Laboratory consumes the corpus as immutable authority. Its native tape uses the empty One as blank and the two forced fibre labels as symbols. A proof kernel constrains every transition and a bounded autonomous constructor cannot alter its law or escape its declared resources. The same machine supplies a reversible/quantum mode through complete word support, period-two phase, predecessor interference, joint-word entanglement, record-based measurement, branchwise gates and the general repetition law. Twelve closed-law demonstrations and eight exact finite investigations pass 25 end-to-end tests, 20 unfavorable controls and an independently compiled 34-check C certificate.

The title honors Turing, Church, Goedel, Shannon, von Neumann, Landauer, Bennett, Feynman and Deutsch. Their work supplies the historical correspondence boundary; it does not select the SFT derivations. Scientific author and publication authority: Maria Smith, Ernos Labs. Open source: Smithian Fold Theory of Everything.