Recoverable Memory Under Replacement

Three questions that must not be conflated

A system can preserve its task behavior while losing an earlier internal bit. It can also preserve a recoverable register while changing future outputs. Whether an external audit counts the resulting token as the same continuant is a third question, governed by whatever lineage rule the application supplies.

This manuscript makes those distinctions explicit for finite deterministic systems. Its word memory has a narrow technical meaning: a declared finite register evaluated before and after a state map. It does not mean autobiographical memory, subjective continuity, or consciousness.

The finest memory that every admitted replacement preserves

The system is modeled as a finite Moore machine with named task transitions and an observable output. An admitted replacement is an endomorphism: a state map that commutes with each named transition and preserves the current output.

A register is robust when its old value can be recovered after every admitted replacement, with a decoder allowed to know which replacement occurred. Robust registers are exactly the invariant quotients on which every admitted replacement acts as a permutation.

The manuscript realizes the finest such quotient with an explicit component graph built from any minimum-rank idempotent. The resulting quotient is independent of the chosen idempotent, is compatible with the task dynamics, and refines predictive equivalence. It is the standard universal group-action reflection in an operational recovery form. Priority for the particular single-idempotent graph presentation still requires human specialist review.

When the replacement is hidden

Suppose the decoder observes the final quotient state but is not told which induced permutation occurred. A trusted record may depend on the replacement, but not on the unknown origin. The smallest possible record alphabet is the chromatic number of a functional characteristic graph.

For a regular cyclic action on Z_n and an interval promise containing h possible origins, the optimum is

ceil(n / floor(n / h)).

This is a one-shot zero-error coding result. The graph-coloring reduction and the cycle-power formula are established mathematics; the manuscript specializes them to the replacement-and-origin recovery contract.

Relevance to B-Theory and The MIND

Acquirable Predictive Models separates certainty about a current predictive state from reconstruction of its origin. This manuscript asks the next structural question: after a declared mechanism-preserving change, which old distinctions remain recoverable at all?

That gives a bounded mathematical instance of Being and Becoming. The quotient records the finest declared information that remains reversibly recoverable even when the underlying physical state map is irreversible. The trusted-record result then measures how much authenticated information about a hidden replacement is needed to reconstruct an allowed origin.

The result also supplies a constraint for Reflective Self-Models: before asking how a controller should estimate changing internal machinery, one can ask which distinctions the admitted transformations leave recoverable in principle. Neither the quotient nor a persistent self-report establishes subjective awareness, introspection, numerical personal identity, or moral status.

Read and inspect

Read the full manuscript with rendered mathematics, download the PDF, download the LaTeX and extracted-text sources, or inspect the verification and internal-review material.

The analytic proofs carry the mathematical claims. Five finite implementations reproduce the declared counts and serve as bounded checks, not proofs or external peer review. The public verification archive omits machine-specific path metadata retained in the internal research archive.