Perception theory

Soft cohesion field on a 2D grid: graded participation values from 0 to 1, with level contours. — The primitive entity of the perception track: a graded soft cohesion field u_t : X_t → [0,1] on a relational support space. Objects emerge as the high-u, stable, temporally persistent limit of this field — not as inputs.

Most accounts of perception start from objects and ask how they relate. This track starts one step earlier and asks: what holds anything together enough to become an object in the first place?

The answer offered here is Soft Cognitive Cohesion (SCC). Its primitive is a graded field

u_t : X_t \to [0, 1]

over a relational support space, where $u_t(x)$ measures the degree to which site $x$ participates in a cohesive formation. It is not a probability, not a segmentation mask, and not a clustering assignment. Objects are a distinguished limit of this formation space, not its starting point.

Formal structure in one paragraph

Definition(Formal universe of SCC)

\mathfrak{C}^{\mathrm{soft}} = \Big( \mathbb{T},\ \{X_t\},\ \{u_t\},\ \{\mathrm{Cl}_t\},\ \{\mathbf{N}_t, \mathbf{D}_t\},\ \{\mathbf{M}_{t \to s}\} \Big)

with operator pair $(\mathrm{Cl}_t, \mathbf{D}_t)$ (self-completion and self-contrast) and a four-term energy $E = \lambda_{\mathrm{cl}} E_{\mathrm{cl}} + \lambda_{\mathrm{sep}} E_{\mathrm{sep}} + \lambda_{\mathrm{bd}} E_{\mathrm{bd}} + \lambda_{\mathrm{tr}} E_{\mathrm{tr}}$ on the volume-constrained simplex $\Sigma_m$ . Formation is read off a diagnostic vector $\mathbf{d} = (\mathrm{Bind}, \mathrm{Sep}, \mathrm{Inside}, \mathrm{Persist}) \in [0,1]^4$ .

Schematic showing the canonical energy E(u) decomposing into four boxed terms: closure, separation, boundary, transport. — The four-term energy of SCC. Each term encodes a structurally distinct requirement; together they define the formation landscape on the constraint manifold Σ_m.

A phase transition separates the two regimes:

\frac{\beta}{\alpha} > \frac{4 \lambda_2}{|W''(c)|}, \qquad c \in \left( \tfrac{3 - \sqrt{3}}{6},\ \tfrac{3 + \sqrt{3}}{6} \right).

Commitments that shape the theory

State of play

Directed acyclic graph of 18 SCC hero theorems across 6 bands: Foundation, Phase+Stability, W4 Capstone, W5 Bridge, W6 Stochastic Foundation, W6 Boundary+Selection. — Hero theorem dependency DAG — 18 nodes, updated to CV-1.11. Foundation (T1, T6, T20) → Phase + Stability (T8-Core, T7-Enhanced, T11, T14) → W4 capstone (T-PreObj-1, T-V5b-T) → W5 bridge (T-L1-F) → W6 stochastic foundation (T-P-F-ε₀, T-PF-A1-AR/SDE/GI/PE) → W6 boundary + selection (T-OP6-B, T-K-Select-PF/OBS).

Sixty-eight theorems are proved fully (Category A) — including existence (T1), closure fixed-points (T6a/T6b), phase transition (T8-Core), axiom consistency (T20), non-idempotent metastability advantage (T7-Enhanced), Γ-convergence (T11), gradient flow convergence (T14), the W4 Pre-Objective Mechanism cluster (T-PreObj-1 + T-PreObj-1G + Lemma 4), the W4-extended Goldstone result (T-V5b-T), the W5 multi-formation count bridge T-L1-F, the W6 additions (CV-1.6–CV-1.11): hard-depth topological locking (T-ST-5a), boundary precision (T-OP6-B, resolving OP-0006), and the stochastic foundation P-F-A1 Package I — well-posed reflected SDE on $\widetilde{\Sigma}_M$ (T-PF-A1-AR + T-PF-A1-SDE), unique Gibbs invariant measure $\pi_{T_*}$ (T-PF-A1-GI), and exponential ergodicity with explicit Poincaré constant (T-PF-A1-PE), plus Gibbs measure continuity at $\varepsilon = 0$ (T-P-F-ε0), and the W7 additions (CV-1.12–CV-1.17): existence of the temporal correspondence T-Temporal-Identity (a) via H-SINK (Bigot–Cazelles–Papadakis Sinkhorn Lipschitz, $L_g \leq L_c$ in the SCC cost class). Nineteen are proved with a structural parameter (Category B) — including the W6 additions smooth-depth barrier raising (T-ST-5b), Kramers exponent stability (T-P-F-ε0-K), equilibrium K-selection partially resolving OP-0005-EQ (T-K-Select-PF), observation-conditioned K-selection partially resolving OP-0005-OBS (T-K-Select-OBS), and the W7 additions kernel-composed compositional consistency (T-CC-StableK-Kernel, closing H-COMP-KERNEL at CV-1.17) plus the conditional lift T-ACT-KERNEL-COMP→REL (CV-1.15 → activated at CV-1.17). Six remain conditional (Category C) — including the temporal-identity interior persistence part (c) (CV-1.12, blocked by OP-0011). Totalling 98 formal claims, ~70% fully proved. Five claims have been retracted.

The Observer Moduli Space (OMS-2.0) reached Accepted — Full at W6 close: the space of symmetry-inequivalent critical formations is proved finite and stratified, with both the static branch decomposition ( $G_{\mathrm{cw}} = \{e\}$ generic symmetry, codim-1 strata) and the temporal extension ( $G_{\mathrm{cw}}^{\mathrm{temp}} = \{e\}$ , 19-branch Δ³ map) established. Appendix OMS (§A–§M) is now part of the canonical specification.

Per Commitment 16 (CV-1.5.1, 2026-04-29), the formation count $K$ decomposes into $K_{\mathrm{field}}$ (architectural cap, modeling-layer commitment) and $K_{\mathrm{act}}$ (dynamic stratum index, derived diagnostic), resolving the long-standing OP-0009-K ontological ambiguity.

Open problems: Critical-3 cleared, OP-0006 resolved, OP-0005 split, H-COMP-KERNEL closed

The three Critical open problems — F-1 (K=2 vacuity), M-1 (K=1 preference), MO-1 (Morse inapplicability) — are all resolved, clarified, or sidestepped at W4 close (2026-04-24). OP-0006 (Boundary precision) is RESOLVED at W6 via T-OP6-B (PersRidge Boundary Equivalence, Cat A, phase-separation regime). OP-0005 (K-Selection) is 3-way split: EQ partially resolved via T-K-Select-PF (Cat B), OBS partially resolved via T-K-Select-OBS (Cat B), DYN open (Eyring-Kramers, W9+). H-COMP-KERNEL is CLOSED at W7 (CV-1.17) via T-CC-StableK-Kernel (Cat B); H-MORSE is partially closed at W7 (CV-1.13) via OP-HMORSE-BROADNESS RESOLVED, with six new OP-HMORSE-* sub-items registered. The currently active High set is 3 (OP-0005 K-Selection — EQ/OBS partial, DYN open; OP-0008 σ^A K-jump; OP-0009 Multi-Formation Foundations) + 9 Medium + 4 Low (the Medium count grew because the H-MORSE umbrella was deliberately decomposed into seven separable sub-items). Next consolidation target: **CV-1.18 via T-Temporal-Identity (b)(d)

σ-inheritance partial + Package II first sub-result.**

SCC in detail

The most active thread right now. Curated entry point: SCC Hero Theorems — the major results (foundation / phase transition + stability / W4 capstone / W5 multi-formation bridge / W6 stochastic foundation + OMS / W7 compositional closure). The full status page — theorem ledger (68A/19B/6C/5R = 98 claims), implementation state (215 passed + 1 xfailed out of 216 collected), OMS-2.0 Full, H-COMP-KERNEL CLOSED, H-MORSE partially closed, and the remaining open problems — is kept at Part 0 · SCC status (May 2026). The authoritative formal text lives at the Canonical Specification, mirroring THEORY/canonical/canonical.md (sealed CV-1.17, with Appendix OMS §A–§M from CV-1.11) in the Perception_theory repository. The most recent narrative is the Week 7 weekly post (2026-05-15); the previous one is the Week 6 weekly (2026-05-08).

Two parallel pipelines: clustering operates on pre-given objects, SCC operates on a graded soft cohesion field. — SCC vs clustering / segmentation: a categorical inversion at the input level. Clustering takes individuated objects and outputs partitions; SCC takes a graded soft field and outputs emergent multi-peak formations.

The larger goal

SCC is the pre-objective layer of a larger design target: a single cognitive-reasoning architecture that unifies SCC with Ontology Neural Networks. The north-star document for that unification is Integrated Architecture — SCC × ONN as one cognitive-reasoning system. Everything on this page is in service of the integration claim there.

Papers on this track

An Axiomatic Framework for Understanding Relations via Gauge-Invariant Cohomology — the formal bridge from SCC to RelationWorld.
In preparation: the two principal SCC manuscripts (mathematical and cognitive-science treatments) are being revised for submission.

Ontology Neural Networks — how the outputs of SCC become inputs to learned semantic reasoning.
Mathematical foundations — the RelationWorld framework that gives SCC a concrete relational substrate.

Formal structure in one paragraph#

Commitments that shape the theory#

State of play#

The larger goal#

Papers on this track#

Related#

Formal structure in one paragraph

Commitments that shape the theory

State of play

The larger goal

Papers on this track

Related