--- title: "An eight-point obstruction to universality of RBM(4,3)" date: 2026-09-06 version: "0.2.0-candidate" doi: 10.5281/zenodo.22550044 pdf: https://github.com/ipitchford/rbm43-eight-point-obstruction/releases/download/v0.2.0-candidate/rbm43-eight-point-obstruction-0.2.0-candidate.pdf repository: https://github.com/ipitchford/rbm43-eight-point-obstruction archive: https://zenodo.org/records/22550044 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # An eight-point obstruction to universality of RBM(4,3) ## Summary A small probabilistic neural network can have more adjustable parameters than the distribution it is trying to fit and still miss some distributions. This candidate identifies eight of the sixteen patterns on four binary variables that a classical restricted Boltzmann machine with three hidden units cannot approximate arbitrarily well. The obstacle is structural, not a failed training run. If the model puts substantial probability on each of the eight chosen patterns, it must leave some probability outside them. A finite certificate and a written argument make this restriction apply even to limiting distributions reached with unbounded parameters. The work also constructs a strictly positive excluded distribution. Its certified separation is tiny and is not an estimate of practical training loss. The result is an unrefereed candidate, supported by exact producer checks and internal AI review. ## Summary for specialists Let $\mathcal R$ be the visible closure of classical real binary $\operatorname{RBM}(4,3)$ and let $$ S=\{0000,0001,0010,0100,0111,1001,1010,1100\}. $$ The candidate theorem states that every $p\in\mathcal R$ satisfies $$ p(S^c)\geq \left(\frac{\min_{x\in S}p(x)}{8}\right)^4. $$ Consequently every law whose support is exactly $S$ is excluded. For the uniform law $q$ on $S$, the total-variation distance from the model is at least $2^{-25}$. If $u$ is uniform on all sixteen states and $q^+=(1-2^{-25})q+2^{-25}u$, the strictly positive law $q^+$ has distance at least $2^{-26}$. A separate proposition represents arbitrary weights on the eight even-parity vertices together with $0001$. At least four hidden units are therefore necessary for universality on four visible bits, but four-hidden sufficiency is not proved. ## Technical account Each joint visible/hidden state has an augmented sufficient-statistic vector containing a constant, four visible coordinates, three hidden coordinates and twelve interactions. Equality of nonnegative integer sums of these vectors produces an exact identity between products of normalized joint probabilities. The constant coordinate guarantees equal degrees, so normalization cancels. For each selected visible state, choose a hidden state carrying at least one eighth of its visible mass. The certificate covers every such choice: 42 base identities generate 12,096 directly checked transformed identities. After hidden normalization, 4,648 partial assignments cover all 2,097,152 selectors. Two exact coverage implementations agree. In the selected identity, some factors must lie outside $S$. If their total multiplicity is $o$ and the common degree is $D$, the check establishes $D/o\leq4$. The maximum degree is seven, not four. Bounding the products gives the leakage inequality; continuity extends it to the entire visible closure. The parity-plus-one result uses a different mechanism: ten exposed joint states with a full-rank interpolation matrix. This permits arbitrary logarithmic masses, not merely the projected support. ## Evidence, assurance and limitations The package contains the written proof, all 42 printable certificate rows, machine data, standard-library checkers, exact rational controls, normal/optimized replay, adversarial mutations, immutable-receipt tests and a complete current manifest. These checks are producer-coordinated. Five model-based publication roles provide internal editorial review, not external peer review. A supplied AI review reports a full, unnormalized reconstruction, but its linked audit ZIP was not retrieved by this publication workflow; that report does not become an independently retrieved reproduction receipt. The source-to-computation bridge remains a written mathematical argument, not a proof-assistant theorem. The broader support scan contains infeasibility reports and timeouts and is not a certified classification. No smallest excluded support, exact hidden-unit threshold, sharp approximation radius or historical priority is claimed. ## Relationship to earlier work The AIM workshop summary poses the fixed $(4,3)$ question, discusses a parity-plus-one investigation, and records a broader eight-vertex proof outline. This candidate provides a counterexample to the broad assertion suggested by that outline; it does not overturn a completed established theorem. Seigal and Montúfar's smaller-model work already uses real polynomial inequalities to describe representable distributions. The new claimed contribution is this specific four-visible obstruction. Supermodular rank, generic complex Hadamard rank and universality with a growing number of hidden units address different constraints and do not supply fixed-size real universality here. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Algebraic statisticians | A concrete inequality and excluded support for a small latent-variable model. | The full real-model geometry is not classified. | | Machine-learning theorists | A counterexample to reasoning from parameter count alone. | No typical-data or large practical error claim. | | Verification researchers | A compact integer object with an exhaustive selector cover. | Check the written semantic bridge as well as the code. | | Interested readers | A precise example of structural limits in a small neural model. | Candidate publication is not independent acceptance. | ## Why the problem matters Universal approximation asks whether an architecture can approach every target distribution, not whether optimization usually succeeds. An explicit obstruction answers that question for a fixed architecture and identifies the mechanism forcing failure. The contrasting nine-point representability result shows why counting supported patterns is also insufficient. The result is a bounded theoretical advance. Its eventual influence on other architectures or practical learning requires separate work and evidence. ## How to inspect or reproduce the recorded checks Download the versioned archive or check out the immutable candidate tag. Python 3.10 or newer is enough for the exact checks; no solver is needed. From the package root, run the main verifier on certificate.json, the auxiliary verifier, then test_release.py and the package inventory check as documented in README.md. The expected main receipt records 42 identities, 12,096 transforms and complete coverage of 2,097,152 normalized selectors. The regression suite must reject corrupted inputs in normal and optimized modes, leave package bytes unchanged, and refuse existing output paths. Fresh receipts go to stdout unless an explicitly new output path is requested. ## The most valuable next projects First, reproduce the certificate and analytic bridge in an unaffiliated implementation with a public receipt. Specialist assessment should also examine the AIM source correspondence and bounded priority claim. Further mathematical questions include the exact hidden-unit threshold, smaller excluded supports, sharper distances and extensions of the leakage method. ## What is in the evidence package The linked GitHub release and Zenodo deposit carry the current PDF, Markdown proof, printable appendix, JSON certificate, verifiers, regression tests, manifest, claim index, source audit, licences and internal editorial records. Earlier discovery outputs and research metrics remain explicitly historical. The release page separately exposes the prospectively scoped assurance/publication forecast and measured outcome. Audio and artwork explain the result; they are not additional mathematical evidence. ## Open directions for follow-up research - Obtain an unaffiliated reconstruction with a publicly retrievable receipt and specialist review. - Determine the exact minimum universal hidden-unit count for four visible bits. - Find sharp approximation distances or smaller excluded supports. - Generalize the certificate-to-leakage method while checking the new selector domain separately. ## 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:rbm43-eight-point-obstruction - Attempt and metric receipts: ep-attempt:rbm43-eight-point-obstruction-assurance-publication: published / positive; scope assurance-through-publication; target Review repairs, exact replay, internal five-role approval, public GitHub/Zenodo and first canonical Evidence Press release. Second ledger-sealing deployment and final administrative closeout measured separately.; active forecast 150 minutes (95-210); Fermi components Review repairs and deterministic assurance: 1 x 25/40/60 minutes low/central/high (Existing finite proof and two bounded checker defects.); Five-role frozen editorial review: 1 x 20/30/40 minutes low/central/high (One substantive round; three concurrent reviewer slots.); Archive identity, page and media: 1 x 30/50/70 minutes low/central/high (Established repository generators.); First guarded deployment and canonical readback: 1 x 20/30/40 minutes low/central/high (Composite seal and hosted CI.); positive-signal/closure probabilities 0.95/0.85 within 210 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 44/unknown/unknown/3/0/1; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -106 minutes; ratio 0.29333333333333333; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human direction and review time not instrumented.; computeMinutes: Individual exact-check wall/CPU times are retained in receipts; complete nonoverlapping substantive-compute and inference-hardware accounting is unavailable, and ordinary builds/CI/deployment are not substantive research compute.; appended measurement corrections reworkMinutes interpretation: The numeric value 1 records only the rounded dedicated 19:11:39–19:12:39 UTC CI diagnosis/repair interval. Treat it as a minimum, not total rework. Subsequent validation and publication repair overlapped other scoped work. PR 223 is a recorded exception to the normal two-PR route; research assets and claims did not change. (reason: Earlier PDF/test concurrency repair, art-palette correction, README repair and metadata validation repairs were not separately timed. Deployment also required completing shallow Git history and a second pre-deployment PR for an omitted method-registry changeLog entry. Zero would be false; total active rework is unavailable.); measurement.agentRuns -> metrics.outcome.agentRuns: The terminal metrics outcome records agentRuns=6. Agent runs include the coordinator and five bounded reviewers; reworkMinutes remains the explicitly disclosed measured minimum, not total rework. (reason: The already public intake measurement agentRuns=1 is a historical opening snapshot, not the completed-attempt total. It must remain unchanged.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: The terminal metrics outcome records reworkMinutes=1. Agent runs include the coordinator and five bounded reviewers; reworkMinutes remains the explicitly disclosed measured minimum, not total rework. (reason: The already public intake measurement reworkMinutes=0 is a historical opening snapshot, not the completed-attempt total. It must remain unchanged.). 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, communication - Decision object: obstruction — Finite integer certificate giving a visible leakage inequality and explicit excluded distributions. Scope: Classical real binary RBM(4,3) including the entire visible closure. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Counterexample- and proxy-first analysis (counterexample-proxy-first); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Augmented sufficient-statistic identities imply normalized probability monomials; exhaustive hidden selectors supply an identity at every parameter value; leakage and continuity give the closure-wide obstruction. Remaining risks: Written analytic bridge is not formally verified.; Coordinated checkers/reviewers may share errors.; Bounded novelty search does not establish priority.; No practical modelling penalty follows from tiny certified lower bounds.. - Human judgement gates: Check the full-model-to-selector and continuity arguments.; Separate real probability laws from generic complex decompositions.; Assess source correspondence and historical priority.; Preserve rights, Anonymous authorship and assurance boundaries. - Next assurance action: Unaffiliated exact reconstruction and specialist assessment; larger architectures remain new research. - Claim ceiling: Unrefereed candidate for classical real binary RBM(4,3), including the full visible closure. No four-hidden sufficiency, minimal excluded support, complete classification, sharp error radius, practical training penalty, formal proof, unaffiliated reproduction or historical priority is claimed. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Reusable fixed-architecture obstruction in Producer-coordinated mathematical candidate publication; design none; comparator No matched comparator.; estimand No research acceleration, reuse or impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://aimath.org/pastworkshops/boltzmannrep.pdf; inherited claim: Fixed (4,3) universality question and workshop proof outline supply historical problem framing.; inherited ceiling: Source framing, not a theorem dependency or priority clearance. ## Verification status Unrefereed candidate for classical real binary RBM(4,3), including the full visible closure. No four-hidden sufficiency, minimal excluded support, complete classification, sharp error radius, practical training penalty, formal proof, unaffiliated reproduction or historical priority is claimed. ## References 1. American Institute of Mathematics. Boltzmann machines workshop summary, page 2: fixed-size question, parity-plus-one investigation and eight-vertex proof outline. 2. Montúfar (2018). Restricted Boltzmann Machines: Introduction and Review, Sections 6 and 9. 3. Seigal and Montúfar (2018). Mixtures and products in two graphical models. Real inequalities for RBM(3,2). 4. Sonthalia, Seigal and Montúfar (2023). Supermodular Rank: Set Function Decomposition and Optimization. 5. Antolini, Montúfar and Oneto (2025). Hadamard ranks of algebraic varieties. Generic complex rank is a different boundary. 6. Reinhart and De les Coves (2026). Universal spin models are universal approximators in machine learning. Hidden size may vary.