--- title: "The Exact KL Radius of Two Bernoulli Products on Three Bits" date: 2026-09-05 version: "0.1.0-candidate" doi: 10.5281/zenodo.22339153 pdf: https://github.com/ipitchford/rbm31-exact-kl-radius/releases/download/v0.1.0-candidate/rbm31-exact-kl-radius-0.1.0-candidate.pdf repository: https://github.com/ipitchford/rbm31-exact-kl-radius archive: https://zenodo.org/records/22339153 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # The Exact KL Radius of Two Bernoulli Products on Three Bits ## Summary Three yes-or-no variables can exhibit a pattern that a mixture of two independent models cannot capture perfectly. How large can the unavoidable approximation error be? This computer-assisted proof candidate gives an exact answer, approximately **0.831 bits**, and identifies the two worst cases: uniform even and odd parity. The constant was conjectured previously. This release contributes a proposed complete proof, exact certificates and a program that turns a supplied rational target into a certified feasible approximation. It remains an anonymous, AI-assisted, unrefereed candidate. ## Summary for specialists Let $\mathcal M_{3,2}$ be the closed model of two Bernoulli-product components on three bits, equivalently $\overline{\mathrm{RBM}(3,1)}$. With natural logarithms, the candidate theorem is $$ \max_{p\in\Delta_7}\min_{q\in\mathcal M_{3,2}}D(p\Vert q) =c=-\frac34\log(2\sqrt3-3). $$ Only the two uniform parity targets maximize this quantity; each has four translated nearest distributions. For general product mixtures the elementary conditioning consequence is $$ R(n,k)\le c\qquad(n\ge3,\ k\ge2^{n-2}). $$ The higher-dimensional statement is an **upper bound**, not a sharp-radius or maximizer classification. General product mixtures must not be identified with larger RBMs. ## Technical account The proof has three connected mechanisms. A positive two-product mixture satisfies log-supermodular inequalities after coordinate relabelling. Three exact positive multipliers certify the parity lower bound inside all relevant cones, including the boundary by a stated limit argument. For the upper bound, reversing a local binary channel removes a positive target entry while never decreasing the distance to the model. Repetition leaves a zero-distance case or one of six support orbits. The remaining weights are arbitrary real numbers: they are not discretized. A finite cover then assigns one feasible model distribution to each simplex. The checker verifies its divergence at every vertex. Convexity in the target carries the bound to every point of that simplex. Strict convexity and the equality condition for data processing identify the only maximizers. The constructive program follows the same channel and subdivision choices and returns exact mixture parameters. It is a feasible-witness tool, not a general maximum-likelihood solver. Conditioning on the other coordinates transfers the guarantee to the stated all-$n$ component budget. ## Evidence, assurance and limitations The package contains an 11-page paper, aligned Markdown, six exact certificate files, a standalone standard-library checker and an exact query interface. Its 52 roots lead to 16,600 leaves at maximum depth 20. The 93,792 vertex conditions include 552 parity equality incidences; all others are strictly certified. Normal and optimized Python runs, fresh extraction, semantic rejection controls, query examples and PDF inspection are producer-side evidence. The checker was implemented separately from discovery within that workflow. The supplied review and the five-role internal editorial gate are not authenticated external specialist review. No independent-person reproduction, formal verification, journal peer review, exhaustive novelty or priority claim is supplied. All five internal roles recommend Accept with no Critical or Major findings. Read the [editorial decision and review limits](https://github.com/ipitchford/rbm31-exact-kl-radius/blob/v0.1.0-candidate/editorial/DECISION.md) and the [response to their nonblocking notes](https://github.com/ipitchford/rbm31-exact-kl-radius/blob/v0.1.0-candidate/editorial/RESPONSE.md). The exact frozen review ZIP remains a separate release asset. The four-star stretch produced an executable interface and an infinite-family consequence. These demonstrate scoped reuse, not a substantiated four-star rating or measured impact. ## Relationship to earlier work Montúfar's 2018 review, Section 9 item 10, records the proposed constant and asks for maximizers; it does not explicitly identify parity there. Allman and collaborators' nonnegative-rank-two and boundary work supplies important context. The neighboring Seigal–Montúfar result concerns a different model. Alexandr and Hoşten use strict convexity on logarithmic Voronoi polytopes for linear and toric models. Here the simplices carry common feasible witnesses for a nonconvex mixture model; those witnesses need not remain optimal across their cells. The conditioning mechanism is the classical disjoint-support mixture principle of Montúfar, Rauh and Ay. The new numerical consequence depends on the candidate three-bit theorem. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Algebraic statisticians | Inspect a sharp small latent-class approximation problem and its equality cases. | The analytic bridges remain open to unaffiliated review. | | Information geometers | Compare common-witness covers with projection-cell methods. | A cover cell is not claimed to be a logarithmic Voronoi cell. | | Verification and tool builders | Reuse a rational target interface and transparent exact certificate. | Passing a checker is not a proof-assistant theorem. | | Interested readers | See why a finite certificate can cover a continuous problem. | The result is model-specific, not a general claim about AI models. | ## Why the problem matters Small models provide places to understand approximation error exactly. The challenge here is to exclude every more difficult target, not just to fit the parity example well. The channel reduction and continuous-cover argument make that global step inspectable. The larger-model consequence gives a concrete transfer target while leaving its sharpness open. ## How to inspect or reproduce the recorded checks Download the [complete versioned ZIP](https://github.com/ipitchford/rbm31-exact-kl-radius/releases/download/v0.1.0-candidate/rbm31-exact-kl-radius-0.1.0-candidate.zip), verify its [SHA-256 sidecar](https://github.com/ipitchford/rbm31-exact-kl-radius/releases/download/v0.1.0-candidate/SHA256SUMS.txt), extract it and run: ```sh python3 build_release.py --check python3 verify.py python3 test_verify.py python3 test_witness.py ``` Python 3.10 or newer is sufficient; no third-party package is needed. Repeat with `python3 -O` to confirm that proof rejection does not depend on removable assertions. Expected: all six covers pass, 13 verifier tests pass and five constructive test groups pass. The worked target is available through `python3 witness.py 3/16 1/16 0 1/4 0 1/4 3/16 1/16`. Its exact trace ends at parity and leaf `0:00000000`. Output coordinate arrays are explicitly least-significant-bit first. See the environment note for optional discovery and PDF rebuilding. ## The most valuable next projects First reconstruct the analytic-to-code interface without importing the producer implementation. Then seek a smaller certificate or formalize the exact checker and its analytic bridges. Higher-dimensional sharpness at the stated component budget is a substantive mathematical follow-up, not a claim already proved here. ## What is in the evidence package The PDF and Markdown give the proof and attribution; the six JSON trees and verifier carry the finite lemma; the query program and worked example expose constructive use. Tests, replay receipts, references, claim index, review response, internal editorial records, environment notes, licences and a complete manifest explain exactly what can be checked and what remains unassessed. GitHub and Zenodo supply the versioned public artefacts linked alongside this page. ## Open directions for follow-up research - Obtain an authenticated unaffiliated reconstruction of the cone, boundary, channel and equality arguments and a separately written exact checker. - Determine whether the conditioning upper bound is sharp for any n greater than three at the stated component budget. - Compress or formally verify the finite convex-cover certificate and its analytic interfaces. - Study other component budgets without conflating product mixtures with larger RBMs. - Assess broader mathematical significance through specialist review and actual downstream reuse, not a producer-assigned four-star label. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:rbm31-exact-kl-radius - Attempt and metric receipts: ep-attempt:rbm31-exact-kl-radius-revision-stretch-publication: published / positive; scope research-through-publication; target Close mandatory revisions, assess a bounded stretch without inflating rating, and publish the candidate with replay, five-role internal editorial gate, immutable identity, media, hosted CI and exact canonical readback.; active forecast 120 minutes (80-150); Fermi components Review repair, source audit and bounded stretch: 1 x 20/30/40 minutes low/central/high (One minor review; stop stretch after 30 active minutes if no proved result.); PDF, deterministic replay and five-role internal review: 1 x 20/30/40 minutes low/central/high (Existing exact certificate, new interface and manuscript.); GitHub, Zenodo, reader-first page and media: 1 x 20/35/40 minutes low/central/high (One new release using established generators.); Two seal, CI, deploy and readback cycles: 1 x 20/25/30 minutes low/central/high (Mandatory new-slug publication and live-ledger cycles.); positive-signal/closure probabilities 0.9/0.8 within 150 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 36/unknown/unknown/10/3/13; cycles positive/negative/inconclusive 2/0/0; falsification gates 2; architectures tested/rejected 2/0; result target-closed; target reached true; forecast error -84 minutes; ratio 0.3; inside interval false; positive-signal/target-closure Brier scores 0.01/0.04; missing telemetry activeHumanMinutes: No task-local human-work timer.; computeMinutes: Substantive compute was not measured separately from ordinary build and publication processes; no retrospective estimate.; deduplicatedModelTokens: No supported fork-aware task-local token counter supplied; rollout token events were not summed.; uncachedInputTokens: No supported task-local uncached-input counter supplied.; appended measurement corrections measurement.reworkMinutes -> metrics.outcome.reworkMinutes: The terminal metrics.outcome records 13 minutes as an upper attribution bound for recorded generation in that repair window; the earlier measurement.reworkMinutes remains zero. Neither number is a complete labor total. (reason: The published pre-deployment measurement recorded zero rework. Later recorded generation intervals include the CI repair and reconciliation window. Preserve the earlier field rather than overwrite a public historical observation.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: certificate — A finite exact convex-cover certificate and constructive target-to-witness interface for a continuous probability-model radius. Scope: Exact three-bit two-product-mixture radius; specified non-sharp all-n conditioning upper bound. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication, translation - Semantic bridge: explicit — Necessary log-supermodularity plus three multipliers gives the lower bound; reverse local channels give six support orbits; exact midpoint trees cover continuous simplices; first-argument convexity and strict data processing complete the value and equality proofs; disjoint-context KL cancellation proves the conditioning lift. Remaining risks: Analytic and code interfaces remain unaffiliated-unverified.; The checker is not proof-assistant certified.; Bounded novelty searches may miss differently phrased or unpublished work.; No higher-dimensional sharpness or general-RBM transfer is proved.. - Human judgement gates: Check necessary cone inclusion and boundary limits, not just the candidate star fit.; Check that simplex coverage and common-witness convexity justify the continuous quantifiers.; Keep diagnostic floating-point comparisons out of exact proof decisions.; Distinguish feasible approximation from optimal projection and product mixtures from larger RBMs.; Preserve anonymous authorship, rights, prior attribution and separate assurance dimensions. - Next assurance action: Obtain unaffiliated specialist reconstruction and a separate checker before stronger assurance claims; investigate higher-dimensional sharpness as new research. - Claim ceiling: Anonymous AI-assisted unrefereed computer-assisted theorem candidate for the exact three-bit two-product-mixture KL radius, with a constructive rational interface and a non-sharp conditioning corollary. Not unaffiliated verification, formal proof, authenticated specialist review, journal peer review, exhaustive novelty, priority, a four-star rating or demonstrated impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — More reusable exact approximation-error certificates in Producer-coordinated mathematical research and publication; design none; comparator No matched conventional workflow comparator was assigned.; estimand No causal discovery-time, effort, reliability, uptake or field-impact effect was estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://arxiv.org/abs/1806.07066; inherited claim: Section 9 item 10 records the proposed constant and asks for maximizers.; inherited ceiling: The recorded conjecture is not a proof or an explicit parity-maximizer identification.; extends-result https://arxiv.org/abs/1303.0268; inherited claim: Lemma 6 supplies the classical disjoint-support mixture decomposition used in the conditioning transfer.; inherited ceiling: This mechanism does not establish the candidate sharp three-bit constant or higher-dimensional sharpness. ## Verification status Anonymous AI-assisted unrefereed computer-assisted theorem candidate after actioning the supplied minor-revision review. The sharp result is three-bit and two-component; the all-n consequence is an upper bound for the specified product-mixture budget. Four-star potential was considered but is not claimed achieved. ## References 1. Montúfar, G. (2018). Restricted Boltzmann Machines: Introduction and Review, Section 9 item 10. 2. Allman, E. S., Rhodes, J. A., Sturmfels, B., and Zwiernik, P. (2015). Tensors of Nonnegative Rank Two. 3. Allman et al. (2019). Maximum Likelihood Estimation of the Latent Class Model through Model Boundary Decomposition. 4. Seigal, A., and Montúfar, G. (2018). Mixtures and products in two graphical models. A different model. 5. Alexandr, Y., and Hoşten, S. (2025). Maximum information divergence from linear and toric models, Proposition 2. 6. Montúfar, G., Rauh, J., and Ay, N. (2013). Maximal Information Divergence from Statistical Models Defined by Neural Networks, Lemma 6.