Research Article · Preprint Mathematics
Finite Persistent Memory in Exact Sampling: Countable Rigidity and Algebraic Memory Bounds
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
An exact sampler reads independent fair bits and emits an iid discrete target law, retaining only a fixed finite state between outputs. Temporary computation and the depth of each round's prefix tree are unrestricted. We prove that zero asymptotic expected information loss is possible, on finite or countably infinite support, exactly when every positive mass is a reciprocal integer power of two. The formulation uses nonnegative finite-path loss and therefore also covers infinite entropy. The countable necessity argument reduces infinitely many output kernels to finitely many integer multiplicity types, then produces an exact positive rational normalization. For finite targets we develop a signed-charge cycle bound that remains effective when zero-gap products defeat every marked-symbol bound. For every fixed finite law with algebraic masses outside the reciprocal-power class, it gives a polynomial lower bound on loss in the number of retained states. Together with an existing exact-sampling construction, this determines the order of the minimum persistent memory as Θ_p(log(1/ε)) bits. These bounds transfer uniformly to countable conditional dyadic refinements, including infinite-entropy examples. An established logarithmic-form estimate gives a concrete exponent for a fully resonant example.
Keywords exact sampling · persistent memory · entropy loss · countable distributions · algebraic probabilities · finitary coding
Scope of the work
Finite persistent-state exact iid samplers have zero asymptotic expected path loss precisely for countable reciprocal-power laws. Every fixed finite algebraic nonexceptional target has a polynomial state-count loss obstruction and Θ_p(log(1/ε)) persistent-bit order, with the stated uniform conditional-dyadic refinements.
Reproduction notes
The 12-page manuscript and complete editable source are supplied in this 3-member package. README.md gives LaTeX build instructions. The argument is analytic and requires no external datasets, computational certificates or supplementary numerical programs. The manuscript states the persistent-state model and attributes the matching upper construction.
The manuscript
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The complete manuscript, as published in this edition.
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Author disclosures
AI-assisted tools were used in exploring and internally checking the arguments and in preparing the manuscript. These internal checks are not external refereeing or formal proof-assistant verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Finite Persistent Memory in Exact Sampling: Countable Rigidity and Algebraic Memory Bounds (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/finite-persistent-memory-exact-sampling-countable-rigidity-algebraic-memory-bounds/v/2
BibTeX
@techreport{luminary_finite_persistent_memory_exact_sampling_countable_rigidity_algebraic_memory_bounds_v2,
title = {Finite Persistent Memory in Exact Sampling: Countable Rigidity and Algebraic Memory Bounds},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/finite-persistent-memory-exact-sampling-countable-rigidity-algebraic-memory-bounds/v/2}
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