SCC — Research Status Snapshot (April 2026, W5 Day 1 G0 close, CV-1.5) — HISTORICAL

This is a living page. It documents the state of Soft Cognitive Cohesion — the research thread I am currently most active on — at the level of granularity that lets a collaborator (or future me) pick up where things stand without re-reading a year of notes.

Last updated: 2026-04-27 (W5 Day 1 G0 close).   Canonical version: CV-1.5.   Full specification: see the Canonical Spec note.

What SCC is, in one breath#

A mathematical theory of how coherent formations emerge prior to discrete objecthood. The primitive is a soft cohesion field

over a relational support space. Objects are a distinguished limit of the formation space, not its starting point. Four independent energy terms govern closure, separation, boundary, and transport; a diagnostic vector $\mathbf{d}=(\mathrm{Bind},\mathrm{Sep},\mathrm{Inside},\mathrm{Persist})$ reads off formation state.

Theorem ledger (CV-1.5)#

✓ Category A — Fully proved (43)#

Existence (T1), closure fixed point (T6a/T6b), axiom monotonicity (T-A2), phase transition core (T8-Core), axiom consistency (T20), gradient flow (T14), stability duo (T3 / T6-Stability), enhanced metastability (T7, $p=0.037$), $\Gamma$-convergence (T11), four quantum-mechanical results QM1–4, the Predicate–Energy Bridge, Deep Core Dominance 2b, T-Bind-Proj/Full (all τ), Propositions 1.1/1.2 and Theorem 3.1, Persistence Threshold Equation, the W4 Pre-Objective Mechanism cluster (T-PreObj-1, T-PreObj-1G, Lemma 4), the W4-extended Goldstone result (T-V5b-T), and the W5 Day 1 G0 σ-framework supporting structures (T-σ-Lemma-1/2/3

• T-σ-Theorem-3/4 — irrep decomposition, nodal count, Goldstone–ℓ=1 saturation, closed-form spectrum at uniform on $D_4$ free-BC, leading-order σ at first pitchfork).

◇ Category B — Proved with structural parameter (4)#

• Barrier exponent ${\gamma }_{\mathrm{eff}}\approx 0.89$ — empirical fit, branch/path/manifold conditioned.
• T-Birth-Parametric (general non-D₄ graphs) — supercriticality on D₄-symmetric graphs is Cat A; general graphs remain Cat B.
• T-d_min-Formula — branch-conditioned regression fit.
• T-Beyond-Weyl — structured spectral perturbation bound; quantitative 33× factor is config-specific.

◯ Category C — Conditional (5 + 1 new finding)#

• T-Persist-1(d) — exact-threshold preservation under structural condition $\beta >7\alpha$.
• T-Persist-Full — composition through (d).
• T-Persist-K-Sep — well-separated regime conditions (WS, SR).
• T-Persist-K-Weak — weakly-interacting regime conditions (WI, SR, NB-K).
• T-Persist-K-Unified — parametric (PS, ND, BC′-K, TC-K, SR-Λ), 100% geometric-Lambda agreement on exp46–47.
• V5b-F (W4-extended new finding) — Partial Goldstone on boundary-modified graphs (NQ-173 carry).

✗ Retracted (5)#

• Theorem 3.3 (general $\tau$ form of ${\bar{r}}_0$) — falsified.
• T-Merge (c)(d)(e) — merge path manifold error.
• K-Saddle Conjecture — pre-v2.1 retraction.

A 5/5 clean sweep on the five single-formation predictions (historical record):

Open problems — W4 close (2026-04-24): Critical 3 → 0#

The three Critical open problems that gated v2.0 release for nearly a year are all resolved as of W4 close (2026-04-24):

• OP-0001 · F-1 (K=2 vacuity) — SPLIT-RESOLVED. Two layers, both Cat A: pure ${\mathcal{E}}_{\mathrm{bd}}$ portion via T-Merge (b) (canonical, isoperimetric ordering — pre-existing Cat A); full SCC portion via T-PreObj-1 (i) (W4 Cat A, graph-class independent via T-PreObj-1G). Under full SCC parameters the F=1 single-disk minimizer is not a critical point, so the dichotomy "K=1 cheaper vs observed K>1" does not arise.
• OP-0002 · M-1 (K=1 preference) — LAYER-CLARIFIED. M-1 is the correct mathematical statement (T-Merge (b)), not an open problem. Original framing conflated pure ${\mathcal{E}}_{\mathrm{bd}}$ (where M-1 holds) with full SCC (where the comparison is not even framed because F=1 is non-critical). The apparent conflict is resolved by CN15 Static/Dynamic Separation.
• OP-0003 · MO-1 (Morse inapplicability) — SIDESTEPPED. σ-framework operates on the smooth single-formation manifold ${\Sigma }_m$ (no corners). Theorem 2 family does not require Morse on ${\Sigma }_M^K$. Multi-formation extension (stratified Morse, Phase 5) remains genuine open work for W5+.

Net effect: Critical blockers 3 → 0, v2.0 release path unblocked. Resolution path: Option D (premise dissolution) via SCC-intrinsic re-framing — neither Option A (external constraint), B (K-selection), nor C (kinetic reformulation).

Separately, the exp65 validation failure (2026-04-12) prompted the Type A / Type B classification to be retracted (X-0002).

Remaining open problems (post-W4)#

Implementation#

scc/ package — 175 tests passing#

The pipeline is canonical: graph → params → operators → energy → optimizer → diagnostics.

Experiments — exp1 through exp65#

Fifty-seven experiments have run to completion, with exp65 flagged as a validation failure that drove the 2026-04-12 audit. Each experiment is registered under an E-xxxx ID in the experiment registry.

Twelve iterations of method#

The path from brainstorming to the current spec can be read as twelve distinct iterations:

Repository structure (2026-04-20 reorganisation, current)#

The 2026-04-12 "Research OS" reboot — an 8-layer structure with numbered 00_meta/ … 07_archive/ directories and 5-role daily logs — was tried for four days, collapsed under its own scaffolding on 2026-04-16, and was archived on 2026-04-18. The current repository structure (2026-04-20) is a clean CODE/THEORY split:

• CODE/ — executable assets: the scc/ package (12 modules), pytest suite (175 passing), experiments/ (exp1–exp97 plus W4-extended E-0090–E-0097), papers/, and scripts/.
• THEORY/ — theory documents with strict three-way separation:
  • canonical/ (authoritative; canonical.md + theorem_status.md
    • open_problems.md); promoted content only.
  • working/ (in-progress development; one file per topic).
  • logs/ (chronological journal; daily/ + weekly/ + monthly/).
• Promotion pipeline (2026-04-23 weekly rotation): working/<topic>.md → logs/weekly/YYYY-MM-W<n>/weekly_draft_storming.md → weekly_summary.md → canonical.md (one-way only, weekly merge cadence).

The Research OS approach is documented as an honest "tried and abandoned" record in _archive/research_os_2026-04-12/ — no silent reversion to it; current CLAUDE.md policy explicitly forbids re-introducing the numbered-directory / 5-role / per-item registry structure.

Papers on this track#

Dedicated SCC manuscripts are in revision; when posted, they will appear in the papers index. In the meantime, the closest published-side work is the Axiomatic Framework for Understanding Relations via Gauge-Invariant Cohomology, which establishes the relational substrate underpinning SCC.

Related#

• Canonical Specification (CV-1.5) — the full formal text of SCC.
• Part I · RelationWorld Foundations — the discrete-gauge substrate SCC operates over.
• Perception research track — the public-facing overview page.