ONN Canonical Specification

Status. The ONN / ORTSF programme's central question resolved to a boundary (No-Go) result, not a positive framework. onn_ws/ONN is now archived reference material; active work continues under ULR (Unified Latent Representation). This page states honestly what was built, what survived audit, and what did not.

1. What ONN was#

The Ontology Neural Network (ONN) was a learning architecture whose latent state was intended to carry the type structure of a target ontology rather than a flat vector. Each vertex of a weighted ontology graph $\mathcal{O} = (V_{\mathcal{O}}, E_{\mathcal{O}}, w_{\mathcal{O}})$ carries a latent fiber; a composite update operator $T = T_{\mathrm{proj}} \circ T_{\mathrm{consensus}}$ projects onto the constraint manifold and enforces local agreement. The LOGOS solver embedded this in a constraint pipeline (Forman–Ricci curvature conditioning, Deep Delta Learning, CMA-ES). The ORTSF control fabric was meant to turn the reasoning trace into delay-robust commands.

That is the design. The next sections state what the mathematics actually supports.

2. The central question — and its resolution#

The programme's multi-year question was: is measured higher-order relational structure (loops, cohomology classes) a usable discriminative resource beyond the pairwise data? The current paper — "The Two Ceilings: An Identifiability Boundary for Measured Higher-Order Relational Structure" — answers no, under a precisely-scoped contract.

Operational Factorization / No-Go (proven). Under a complete-pairwise observation contract, on consistent, complete, finite-$\pi_1$ scenes, every admissible measured higher-order readout is one of: pairwise-measurable, exact/zero, label-injected, or noise. Hence a Bayes-optimal decoder given the complete pairwise data gains no information by adding it — algebraically higher-order $\neq$ informationally higher-order. The load-bearing witness is the $\operatorname{coker}\delta_1$ area datum: algebraically non-zero yet analytically reconstructible from pairwise positions to the noise floor (analytic residual $0.050 \approx \sigma$).

Scope (important — this is not a blanket refutation). The boundary holds under the stated contract and each escape route violates a named hypothesis: incompleteness / occlusion, inconsistency or dynamics, a genuinely non-pairwise external sensor, infinite / higher-cohomological-dimension topology, or bounded model capacity. It does not claim to refute higher-order GNNs or sheaf networks broadly.

3. What survived#

• A modest, explicitly non-higher-order "direction channel." The direction of the measured curvature field carries discriminative signal beyond magnitude, at matched capacity (E1b: Q-weak $= +0.115 / +0.120$ AUROC across two noise levels, pass-fraction $1.0$, CI excludes 0; the selection-corrected single-arm effect is $\approx +0.08\text{–}0.10$).
• A standard, honestly-scoped ORTSF control certificate. A proved and runtime-enforced delay margin $\Delta t_{\max} = \varphi_{\mathrm{PM}} / \omega_c$ (with $\varphi_{\mathrm{PM}} = \pi/2 - \arctan(\omega_c / p)$ for the dominant plant pole $p = \lambda_2 + \mu$), plus a sufficient, conservative nonlinear small-gain certificate ($\rho_{\mathrm{ONN}} \cdot \gamma_{\mathrm{ortsf}} < 1$ $\Rightarrow$ stable for arbitrary delay). This is systems completion, not novelty — delay-robust control is standard — and the margin is deliberately not coupled to the graph spectrum $\lambda_2$ (see §4).
• A methodology finding. A learner is neither necessary nor sufficient to judge an information / reconstructibility claim (false positive $E_{w_2}$, false negative $P_{\mathrm{coker}}$) — decide such claims analytically.

4. What did not survive audit#

• Higher-order / cohomological structure as a representational advantage — the exact thesis the No-Go disproves on the information axis.
• "Topology preservation" as originally stated (Betti numbers preserved, a "CSR $\to 1.0$" metric). The real result is weaker: harmonic-subspace preservation, and only conditionally (a hard H-aligned anchor); the soft-anchor version is refuted for $\dim H > 0$, and the Harmonic No-Go shows that subspace is specified by the anchor, not learned from data.
• A "cohomological Lyapunov certificate." No such object exists; the stability that holds is the control-theoretic delay margin above, driven by the plant pole, not by cohomology. The old curvature $\leftrightarrow$ persistent-homology bound "is proved by no theorem."
• The original published paper's 7–8 grand theorems were audited and 0 of 8 survive as written (T2 only down-scoped; T4's delay small-gain is physics-wrong and was replaced). The $\lambda_2$-coupling, PROBEFLOOR and XSHEAF claims are refuted or shown to be prior art.

5. Numbers that were removed#

The following figures appeared in an earlier AI-generated draft of this page and its siblings and are not present anywhere in the research source; they have been removed rather than softened: CSR = 1.0, τ_max = 177 μs, "3M-node", 99.75% improvement, topology-loss 11.68 → 1.15, c_J ≤ 0.7. The old published control numbers (dt_max = 52 ms, phase margin 28°) are not regenerable from the code and are likewise dropped.

6. Where this sits now#

ONN implements Layer 3 of the integrated architecture, with ORTSF at the control layer. The programme's live continuation is ULR (Unified Latent Representation). The published ONN + ORTSF paper remains the paper of record for the original framing; this page reflects the current, audited state.