FRC Paper 21a - Symbolic Fossils of Finite Reality: Collatz, Kaprekar, Ramanujan and the Three-Memory Grammar, with Supporting Information and FRC Programme Map Rev07

FRC Paper 21a - Symbolic Fossils of Finite Reality: Collatz, Kaprekar, Ramanujan and the Three-Memory Grammar, with Supporting Information and FRC Programme Map Rev07 By Joe Bloggs Abstract Paper 21 introduced the Finite Observation Audit and a provisional three-part event-memory structure consisting of inner memory, boundary memory and outer memory. Paper 21a applies that method to three independent bodies of finite-number mathematics and asks whether they preserve related finite grammars without falsely promoting symbolic resemblance into physical identity. Collatz supplies exact local alternation between enlargement and contraction, valuation-based repayment, path merger, reverse-source ambiguity and a recurrent terminal basin. Kaprekar supplies finite collection into fixed points, quotient compression, enclosure-dependent attractors, exact expansion-cut-return identities, self numbers and junction numbers. Ramanujan supplies bounded internal multiplicity, boundary recovery of hidden coefficients, residue memory, alternate admissibility grammars producing equal outer counts, record-survivor structures and modular completion. The paper formalises a bounded-freedom gradient: larger finite enclosures may admit rapidly increasing numbers of internal arrangements, decompositions and histories while every realised state remains finite and boundary-conditioned. It also audits the phrase “multiple inner histories merging into one outer reality”. That statement is exact for declared deterministic many-to-one maps, quotient projections, predecessor multiplicity, junction structures and finite coarse-graining maps. It is not established as a literal physical description by path-integral notation, consistent-histories formalisms, decoherence models or Everettian many-worlds ontology. Those remain calculational or interpretive languages placed around finite receipts unless a discriminating observation requires their ontology. The accompanying Supporting Information supplies the mathematical definitions, finite proofs, history-merger tests, identifiability and rank conditions, bounded-history counts, worked Collatz, Kaprekar and Ramanujan derivations, quantum-language receipt corrections, stochastic and attractor distinctions, negative controls, falsifiers and reproducibility forms. The included FRC Programme Map Rev07 places Paper 21a and its Supporting Information within Phase V - Reality Audit, Correspondence and Empirical Accountability - and records their programme dependencies and qualified claim boundaries. No operator identity between Collatz, Kaprekar, Ramanujan mathematics and physical reality is claimed. The proposed single grammar - retained relation, active boundary and surviving outer receipt - remains a provisional research hypothesis until a discriminating physical test requires it. Introduction Paper 21 began Phase V of the Finite Reversible Closure Programme by separating what was recorded from what was calculated, interpreted or added as ontology. It established that exact cross-language correspondence requires a declared map preserving the relevant states and transitions; preservation of selected relations permits close correspondence; shared morphology alone remains rhyme; and a proposed universal explanation beyond present discrimination remains speculation. This record contains the first substantial symbolic application of that audit. Paper 21a examines Collatz, Kaprekar and selected Ramanujan structures as independent finite mathematical domains. It does not ask whether the same attractive numbers happen to appear in several subjects. It asks whether each domain can be decomposed reproducibly into what is retained within a state, what is decided or resisted at an active boundary, and what survives outward as a finite receipt. The comparison reveals several exact symbolic forms: multiple predecessors producing one receipt, distinct paths sharing a later suffix, large visible state populations compressing into small quotient grammars, finite boundaries determining terminal basins, and expanded structures retaining exact recoverability of their source through a specific cut or relation. These exact results remain exact only within their own mathematical systems. Their interpretation as evidence for a wider FRC memory grammar is separately classified. The accompanying mathematical Supporting Information makes the comparison independently attackable. It defines the finite event-memory object, tests whether inner, boundary and outer components are identifiable from the available receipt, distinguishes deterministic merger from statistical convergence, and provides finite downgrade conditions where the proposed grammar is redundant, non-unique or insufficient. The FRC Programme Map Rev07 places this work directly after Paper 21. Paper 21 supplies the audit instrument; Paper 21a applies it to finite symbolic structures; later programme work must determine whether measured physical systems provide discriminating evidence for the same grammar.