Remembering information and reaching it

FPRD Lab, 5 September 2026. Research note, revision 4.

Read residual-counts-and-access.pdf or its Markdown source first. The note contains the grammar and all local proofs. mathematical-audit.md states the audit findings, dependency boundaries, and prior failed conversions.

The terminal-marker construction studied here has the following property: easy and hard canonical families match at every update cut, every query prefix, and after conditioning on any initial update segment. Uniform completions induce the same multiset of residual probabilities. The access exclusion depends on Ko’s theorem. The non-PEG corollary also uses the Loff–Moreira–Reis reversal characterization. The note extends Kim–Park’s explicit separation and adapts their recursive parity rules. Its tree-addressed selection and residual-profile comparison have not been established as novel in the literature. The short simulation is proved directly; the overall lower-bound strategy follows Kim–Park.

Run the focused checker with Python 3 and no additional packages:

python3 check_conditioned_profiles.py

The checker imports the original independent traversal/grammar implementation in check_tree_witness.py. Its saved output is conditioned-profile-audit.json. To reproduce the original grammar checks:

python3 check_tree_witness.py

To regenerate the editions with Pandoc and XeLaTeX installed:

pandoc residual-counts-and-access.md -s -o residual-counts-and-access.tex
xelatex -interaction=nonstopmode -halt-on-error residual-counts-and-access.tex
xelatex -interaction=nonstopmode -halt-on-error residual-counts-and-access.tex

The TeX source is generated from the Markdown source. The site HTML edition uses the same Markdown and its existing KaTeX rendering path. The note contains eight numbered statements; it does not claim formal verification, external expert review, or confirmed novelty.

MANIFEST.sha256 covers every file in the packet except itself and the ZIP. No third-party paper or code is included. Primary sources are linked in the paper’s bibliography. All numerical checks are corroboration, not proofs of universal claims or lower bounds.

Expanded reading edition

Revision 2, 5 September 2026, adds worked examples and background on Ford, Ko and the Kim–Park strategy. It corrects the opening terminology about the number of distinct residuals; the numbered mathematical statements and their dependencies are unchanged. Markdown, HTML, TeX and PDF are regenerated together. The site build script is scripts/build-residual-reading.py.

Revision 3 prints the complete grammar for the actual non-PEG witness, L = U^R, at the front. The original grammar for U remains in Proposition 1. The new display reverses every production right-hand side simultaneously; it does not change the language or theorem.

Revision 4 retains the full work as a research note within FPRD Lab. It identifies Kim–Park’s explicit witness and the adapted parity rules at the opening and beside the grammar. The eight mathematical statements and their dependencies are unchanged. The current reading URL is /research-notes/residual-counts-and-access; the earlier reading and download URLs remain available.