Conditional mutual information of adjacent intervals, and the exact error of Petz recovery

Three adjacent intervals A, B, C in the vacuum of a chiral conformal net. The Longo–Xu conditional mutual information measures how far the chain is from a quantum Markov chain; the error of the Petz map that tries to rebuild ABC from AB measures the same failure quantitatively. These pages let you move the geometry and watch both.

Every number on this site carries the file and line it was extracted from, and a status word. A status word is a claim about how much is known, and nothing more. proved is a theorem; numerical is a measurement with an error bar and is never styled like one; certified is a bound by construction, though the defect-matrix quadrature behind it is not certified; conditional names the hypotheses it rests on.

The four headline results

Theorem A — the exact quadratic law

refereed Module 2 · The quadratic law claim page

For the zero-collar compression of the free chiral fermion, lim_ζ→0 Φ(ζ)/ζ² = c/(12π²) — a lower bound from the Legendre form of the SLD metric, a matching Uhlmann upper bound from a Bogoliubov extension whose vacuum overlap is a Fredholm determinant.

Theorem B — universality in the central charge

conditional Module 3 · Universality in c claim page

The same coefficient f₂ = c/(12π²) for every diffeomorphism-covariant net, not only the free fermion: the second-order coefficient is linear in the central charge. Refereed, and stated under hypotheses H1, H2, H3; they are quoted on the claim page and the badge must not be read without them.

The compression is not optimal — and by how much

conditional Module 4 · Is the compression optimal? · Module 5 · Why two papers disagreed about θ claim page θ

The best second-order gain any recovery channel can have over the compression is 1−θ = 0.616 ± 0.003 numerical numerics/optimality_all/F3_RESULTS.md:15. The two published values of θ did not have different limits: the spectral circle model's ultraviolet taper removes about a fifth of it. Measured, not proved, and conditional on H1, H2, H3.

Recovery networks and the corner calculus

conditional Module 6 · Networks and the holonomy claim page

A sequence of one-sided recovery steps composes to a C¹ piecewise-Möbius map whose Schwarzian is a sum of point masses, and the recovered state depends on the map only through that Schwarzian. The unconditional rule places each mass at the preimage of its corner under the later steps, scaled by their derivative; the junction form, in which the masses simply sit at the junctions and the order of the steps does not matter, is the special case that holds under hypothesis (H). Proved; the junction form is conditional on (H), which the module tests on the configuration you build rather than assuming.

The ten modules

Each module carries a status badge per displayed claim, works as text and a static figure with JavaScript switched off, and makes no request to any other site.

#	module	what you do	state
1	Geometry and the recovery map	drag the four endpoints; watch z, ζ, the cross ratio and the Longo–Xu CMI	live
2	The quadratic law	sweep ζ from 0.0083 to 17.07; the curve, the ζ² asymptote, the ratio	live
3	Universality in c	slide the central charge along the line Theorem B proves	built
4	Is the compression optimal?	the compression, the rotated family and the optimum; the gain 1−θ	built
5	Why two papers disagreed about θ	two separate controls: window position and transition width	built
6	Recovery networks and the holonomy	build a chain, pick a protocol, watch the Schwarzian masses accumulate	live
7	The separation constraint	set ε, drag corner strengths, watch the bound bind	built
8	Off criticality: gap, temperature, mass	exponential shielding and the c-function	built
9	Relative entropy: the conjecture and its bracket	the bracket, conditional at both ends, against the conjecture and the measurement	built
10	The claim map	every claim as a graph, filtered by status	built

The documentation

The same folder carries a generated set of pages: what is claimed, with what status, and where the proof lives. They are produced from the project's knowledge base by tools/build_docs.py, so they cannot drift from it. Each is written twice from the same text: as Markdown under docs/, which is what GitHub shows when you browse the repository, and as the page linked here, because GitHub Pages serves a Markdown file as a raw download rather than rendering it.

Documentation map	the map: four results, ten modules, the sections below
Every claim, by area	every claim, grouped by area, with status and one-line statement
Result pages	one page per consequential claim
Symbols and conventions	the symbol table
The objects, and what they mean	the definitions the results are stated in
What is open	what is open, and the recorded next step for each
External sources	external results relied upon, and whether you can obtain them
Phases 1 to 6	what each settled, and the errata

The two papers are attached as PDFs to the latest release, so a citation points at a fixed version rather than a moving file. The Möbius layer that module 6 runs on has its own unit-test page, which checks it against the worked examples of the rigor document in your browser.

Provenance and AI use

Quoted verbatim from the papers (paper1/main.tex, section “Provenance and AI use”), not a softened version.

The work reported here was carried out in collaboration with AI agents (Claude, Anthropic) directed by the author. Agents wrote the structured proofs and the present manuscript, implemented and ran the numerical computations, and refereed the results adversarially as separate instances that had not written the document under review, although a repair was often re-checked by the referee instance that had raised the objection rather than by a fresh one. No statement is recorded as proved unless a referee pass examined it and did not break it; objections that were not acted upon are listed, unapplied, in the referee reports. External results were checked against the sources wherever the source could be obtained, and never quoted from memory; a source that could not be obtained is marked NOT OBTAINED in the underlying documents, and the disposition of each statement resting on it is recorded there: reproved, taken from an obtainable source that states it, checked numerically, or carried explicitly as a hypothesis. The referee reports, the errata and the raw numerical output are kept in the project repository alongside the proofs; page images of third-party publications are not redistributed with it.

AI systems are not authors of this paper. The author set the problems, took the mathematical decisions, judged what counts as settled, and is responsible for the content, including any error the pipeline failed to catch. Statements are labelled by status throughout: proved means a complete structured proof that an independent referee pass did not break, and numerical means a measurement with a stated error bar, not a proof.