Matched Friction Under Hysteresis: A Programmatic Proposal for a Measurement Schema of Learning Optima

Nine independently-developed traditions in learning theory and belief-revision — Yerkes–Dodson arousal, Vygotsky’s zone of proximal development, Kalyuga’s expertise-reversal, Bjork’s desirable difficulties, the spacing and testing effects among them — have each characterised an inverted-U between challenge and outcome. This paper proposes one substrate-level measurement vocabulary (matched friction under hysteresis) that may organise all nine under a single schema: a research direction with explicit commitments and known gaps, not a mature unification. Across those traditions the recurring shape is an inverted-U: Yerkes–Dodson arousal, Vygotsky's zone of proximal development, Kalyuga's expertise-reversal effect, Bjork's desirable difficulties and spacing effect, Bengio's curriculum learning, Roediger's testing effect, Shannon–Berger rate-distortion, and Brehm-Festinger reactance/cognitive dissonance. Each names its own curve. This paper is a programmatic proposal, not a completed theoretical contribution: it sketches a candidate substrate-level vocabulary — matched friction under hysteresis — under which these traditions might be organised, and specifies the measurement programme such a vocabulary would require to become a falsifiable theory. The schema proposes that the learning optimum is not the matched-filter maximum of reception but a three-way trade-off between reception, learning, and flow, L ∝ 𝟙[|d−m|>c] · coherence(d,m) · r, where d and m are data and model friction structures, c is substrate coercivity, and r is per-event remanence. The expression is read as a measurement vocabulary, not a fully-specified predictive equation, and not a demonstrated three-parameter reduction: the coherence function is not yet specified independently of outcomes (Appendix C proposes a candidate operationalisation for silicon substrates), and the three-parameter orthogonality has not yet been demonstrated on any single substrate (the central live falsifier). Three-grade taxonomy of the nine traditions (§3): Grade (a) re-expression — native curve expressed in the schema's vocabulary, showing parsimony without novel empirical content: Yerkes–Dodson, Vygotsky ZPD, Kalyuga, curriculum learning, testing effect (5 traditions). Grade (b) candidate derivation with surplus-content — schema produces a quantitative prediction the native literature does not make: Bjork spacing as temporal shadow of recovery-time τ(I) (fatigue-bound substrates only); desirable difficulties' descending limb from flow-breakdown χ (2 traditions). Grade (c) speculative extension — schema applied where empirical inverted-U is not established on the relevant substrate: rate-distortion (c → 0 limit is structurally consistent with but does not term-by-term recover R(D)); reactance/dissonance (biological inverted-U over encoding-depth is framework prediction, not result). Empirical status. Empirical work to date is preliminary computational probing rather than empirical validation. The companion empirical paper's parabel-ensemble (Paper 4, Pødenphant Lund 2026g) reports direction-consistent inverted-U patterns on four input-intensity axes of the Qwen2.5 family, with chunking-density additionally replicated on DeepSeek-V3 and Llama-3.1-70B. A fifth tested axis (violation magnitude) produced a null; the within-event / between-event consolidation-layer distinction in §4.10 was developed in response to that null and is treated as a post-hoc framework refinement consistent with the null, not as an independent prediction the null confirmed. A cross-substrate transfer rule (which schema claims generalise silicon↔biology, which do not, and why) is part of the schema and is stated in §4.10. The paper states a candidate schema, grades its empirical readiness, lists the falsifiers it commits to, and operationalises its central term (coherence) on one substrate kind — a research direction worth pursuing, not a confirmed unifying theory. Companion papers in the Friction Theory series: Paper 0 (Behavioural Friction Theory): 10.5281/zenodo.19462499 Paper 1 (Friction Theory substrate): 10.5281/zenodo.20012654 Paper 2 (Capacity scaling): 10.5281/zenodo.20013491 Paper 2B (ICL/FT substrate mechanism): 10.5281/zenodo.20145218 Paper 3 (Friction-guided inference): 10.5281/zenodo.20014121 Paper 4B (Substrates Encode Experience): 10.5281/zenodo.20059861 Paper 5 (Field-theoretic taxonomy of emotions): 10.5281/zenodo.20058825 Paper 10 (Race-architecture across substrates): 10.5281/zenodo.20014567 Paper 13 (Operational Friction Theory): 10.5281/zenodo.20059876 Paper 14 (Logic as Reactance): 10.5281/zenodo.20217712 Target venue: Zenodo preprint as the primary archival location. A compressed journal version exceeds main-text limits of letter-format journals and is a later, separate step; the deposit is the canonical form. v4 (August 2026) — repositioning revision. The paper is aligned front to back as a combined negative report and research prospectus: the introduction no longer describes as an open test a falsifier the body reports as performed and negative, and assertive residues are deflated to what the evidence carries. Three cited-but-unlisted references are added and orphaned references pruned; a per-axis reconciliation and an explicit mechanism-status column are supplied. Editorial pass. Earlier versions remain in the version history.