Soft Cognitive Cohesion — Canonical Specification (CV-1.11)

Version note (2026-07-10). This formal text is the CV-1.11 (W6, 2026-05-06) snapshot. The canonical has since advanced to CV-1.17 (sealed 2026-05-15): 68 Cat A / 19 B / 6 C / 5 Retracted = 98 claims, ~70% proved. The CV-1.12–1.17 additions (T-Temporal-Identity (a) Cat A via H-SINK; T-CC-StableK-Kernel Cat B, H-COMP-KERNEL closed; H-MORSE partially closed) are recorded in the current status page; the full CV-1.17 formal text is pending its reseal here.

Overview#

This document is the canonical human-readable formal specification of the theory of Soft Cognitive Cohesion. This text corresponds to the CV-1.11 (2026-05-06) release, incorporating all prior W4–W5 work plus the W6 additions: the Stereo-SCC extension (T-ST-5a hard-depth locking Cat A; T-ST-5b smooth barrier raising Cat B), boundary precision resolution OP-0006 via T-OP6-B (Cat A, phase-separation regime), P-F-A1 Package I fully Cat A (T-P-F-ε0 + T-PF-A1-AR/SDE/GI/PE — well-posed reflected SDE + unique Gibbs measure + exponential ergodicity on $\widetilde{\Sigma}_M$), equilibrium and observation-conditioned K-selection (T-K-Select-PF/OBS Cat B, OP-0005-EQ/OBS partially resolved), and the Observer Moduli Space OMS-2.0 Accepted — Full with Appendix OMS (§A–§M) added to the canonical specification.

Version naming convention. This site and the underlying Perception_theory both use a single release ladder, CV-x.y[.z], incremented per merge:

CV-1.0 (04-01) → CV-1.1 (04-03) → CV-1.2 (04-12) → CV-1.3 (04-25) → CV-1.4 (04-26) → CV-1.5 (04-27) → CV-1.5.1 (04-29) → CV-1.5.2 (05-02) → CV-1.6 (05-06, Stereo-SCC + T-ST-5a Cat A) → CV-1.7 (05-06, T-OP6-B Cat A + T-P-F-ε0 Cat A + T-P-F-ε0-K Cat B) → CV-1.8 (05-06, T-PF-A1-AR/SDE Cat A + GI/PE Cat B) → CV-1.9 (05-06, T-PF-A1-GI/PE Cat B→A, Package I complete) → CV-1.10 (05-06, T-K-Select-PF Cat B) → CV-1.11 (05-06, T-K-Select-OBS Cat B — the release this text seals) → CV-1.12 (05-11/12, T-Temporal-Identity (a) Cat A, H-SINK) → CV-1.13–1.16 (05-12/14, SEAL increments) → CV-1.17 (05-15, T-CC-StableK-Kernel Cat B, H-COMP-KERNEL closed — current; W7 delta in the status page).

Older site-document tags (v2.0/v2.1/v2.2/v2.3) and the Perception_theory internal theory-revision tag (v2.1, marking ontological shifts like T_t demoted, C_t demoted, A1' / σ-framework primary) are retained only as historical change-log markers within the body text below; they are not a separate active version system. Authoritative release index lives at THEORY/canonical/theorem_status.md §Canonical Spec Version History.

The specification declares the primitive ontology, the formal universe, the axiomatic groups (A–E), the derived predicates, the minimal energy principle with volume constraint, and the proved results registry that jointly constitute the theory as it now stands. It separates stable theoretical commitments — those principles that define the identity of the theory and are not subject to routine revision — from open design choices that remain underdetermined but are constrained by the canonical commitments. Provisional concrete operator forms are presented explicitly as currently favored realizations, not as permanently fixed definitions.

Because the full specification is long, it is split across five linked parts. Read them in order for a full pass, or jump directly to the section you need.

The Five Parts#

Part 1 · Foundations & Formal Universe →#

§0, §2–§5. The summation convention; the foundational orientation (why the theory does not begin from objects); the formal universe $\mathfrak{C}^{\mathrm{soft}} = (T, \{X_t\}, \{u_t\}, \{\mathrm{Cl}_t\}, \{\mathbf{N}_t, \mathbf{D}_t\}, \{\mathbf{M}_{t\to s}\})$ and primitive structure; the argument for the primacy of the soft form over the crisp system; the derived geometric and morphological notions (core, interior, boundary band, exterior, transition diagnostics).

Part 2 · Axiomatic Groups & Proto-Cohesion →#

§6–§7. The five axiomatic groups: Group A (closure: A1' conditional extensivity, A2 monotonicity, A3 contraction, A4 continuity), Group B (adjacency), Group C (co-belonging — now a derived diagnostic, C1–C4), Group D (distinction, D-Ax1–3), Group E (temporal transport, E1–E4). The proto-cohesion diagnostic vector $\mathbf{d} = (\mathsf{Bind}, \mathsf{Sep}, \mathsf{Inside}, \mathsf{Persist}) \in [0,1]^4$ and its four component predicates, including the $\ell^2$-normed Bind, the $u$-weighted Sep, and the persistence-based morphological quality measure $\mathcal{Q}_{\mathrm{morph}}$.

Part 3 · Energy Principle & Provisional Operators →#

§8–§9. The volume constraint $\Sigma_m$ and its ontological justification; the four-term canonical energy $\mathcal{E} = \lambda_{\mathrm{cl}} \mathcal{E}_{\mathrm{cl}} + \lambda_{\mathrm{sep}} \mathcal{E}_{\mathrm{sep}} + \lambda_{\mathrm{bd}} \mathcal{E}_{\mathrm{bd}} + \lambda_{\mathrm{tr}} \mathcal{E}_{\mathrm{tr}}$; why the four terms are conceptually independent; the information-geometric structure of $\Sigma_m$; currently favored provisional realizations of the local kernel, closure, distinction, co-belonging (resolvent), and temporal transport operators, with parameter-regime constraints.

Part 4 · Structural Interpretation, Commitments & Open Problems →#

§10–§12. The structural interpretation (existence is not first given as objecthood; cohesion precedes individuation; temporal identity is not pointwise). Fixed commitments (13 non-revisable principles) versus open design choices (9 items still underdetermined). Open problems organized by foundational (self-referential transport, kinetic multi-formation, $K$-field architecture, coupling-bound lemma, T-Persist-K-Sep/Weak/Strong), bridging (temporal proto-cohesion, predicate-energy bridge, $\Lambda_{\mathrm{coupling}}$ parametrization), and extension (sharp-interface dynamics, RG analysis, crisp recovery protocol, identifiability) layers.

Part 5 · Proved Results Registry & Closing Notes →#

§13–§15. The 78-item results registry: 54 Category A (fully proved — T1, T6a/b, T8-Core/Full, T14, T3/T6-Stability, T7-Enhanced, T11, T-Merge (a/b), Topological Lock, T-Birth-Parametric, T-Bind-Proj/Full, Propositions 1.1/1.2 and Theorem 3.1, Persistence Threshold Equation, Predicate-Energy Bridge, Deep Core Dominance 2b, T-PreObj-1 / T-PreObj-1G / Lemma 4 (W4 Pre-Objective Mechanism), T-V5b-T (W4-extended Goldstone), T-σ-Lemma-1/2/3 + T-σ-Theorem-3 (W5 σ-framework), T-Commitment-14-Multi-Static / T-σ-multi-A/D-Static (CV-1.5.1 Multi-Static cluster), V5b-T-zero, T-L1-F (CV-1.5.2, first multi-formation Cat A), T-ST-5a (CV-1.6, hard-depth topological locking), T-OP6-B (CV-1.7, boundary precision, OP-0006 resolved), T-P-F-ε0 (CV-1.7, Gibbs measure continuity), T-PF-A1-AR / T-PF-A1-SDE / T-PF-A1-GI / T-PF-A1-PE (CV-1.8–CV-1.9, P-F-A1 Package I complete)), 14 Category B (barrier exponent, general-graph birth, d_min formula, T-Beyond-Weyl, T-σ-Theorem-4, T-ST-5b, T-P-F-ε0-K, T-K-Select-PF (CV-1.10, OP-0005-EQ partial), T-K-Select-OBS (CV-1.11, OP-0005-OBS partial), plus targets T-σ-Multi-1 / V5b-F-empirical), 5 Category C (T-Persist-1(d), T-Persist-Full, T-Persist-K-Sep/Weak/Unified), 5 Retracted (Thm 3.3 general τ, T-Merge (c/d/e), K-Saddle Conjecture). The consolidated Commitment Notes (CN1–CN17) and the closing summary. Appendix OMS (§A–§M): Observer Moduli Space, OMS-2.0 Accepted — Full (static + temporal).

Key Changes in CV-1.5.2 (W5 Day 6, 2026-05-02)#

• T-L1-F (Hard-Bar / Active-Count Bridge under L1-J Regime) merged as Cat A conditional under hypothesis package $(P0)$–$(P11)$. This is the first multi-formation canonical Cat A theorem: under the L1-J regime on shared-pool $\widetilde\Sigma_M^{K_{\mathrm{field}}}$, the hard-bar count $K_{\mathrm{bar}}^{\ell_{\min}}(U(\mathbf u);G)$ of the aggregate field equals the active-slot count $K_{\mathrm{act}}^\varepsilon(\mathbf u)$, with a labelled bijection $\mathcal{A}_{\mathrm{bar}}: A^\varepsilon \to \mathrm{Bars}_0^{\mathrm{term}}$ defined by $\mathcal{A}_{\mathrm{bar}}(j)$ = unique dominant bar with birth at $q_j^U$.
• L1-A through L1-L 13-step chain + L1-K external audit + L1-K-REPAIR cycle (R-1..R-4).
• Counts: 45A → 46A, 60 → 61 claims, 75% fully proved.
• Non-claims preserved (explicit): no global $K_{\mathrm{bar}}=K_{\mathrm{act}}$; no global $K_{\mathrm{soft}}^\phi=K_{\mathrm{act}}$; OP-0005 / OP-0008 NOT solved; reservoir theory NOT promoted; P7 NOT generally derived.

Key Changes in CV-1.6 through CV-1.11 (W6, 2026-05-06)#

All six releases occurred on 2026-05-06 (W6 Day 4, Sessions A–Y).

• CV-1.6 — Stereo-SCC extension + T-ST-5a Cat A. Hard-depth topological locking (T-ST-5a): a depth-disconnected stereo graph forces $K = 2$ topologically — $\Delta E = +\infty$ (state-space disconnection; no P-F flag). Stereo field space $\mathcal{F}_M(P)$, stereo adjacency D-ST-1/2, and the backprojection/pullback/prior-likelihood separation (D-ST-5) added as canonical definitions.

• CV-1.7 — OP-0006 RESOLVED; stochastic measure continuity. T-OP6-B (PersRidge Boundary Equivalence, phase-separation regime) promoted Cat B → Cat A conditional under H1–H5: $d_H(\mathrm{PersRidge}, \partial\Omega) \leq 2(\alpha/\beta)^{1/2}$. OP-0006 (Boundary precision) is closed. T-P-F-ε0 merged Cat A: $\mu_\varepsilon \Rightarrow \mu_0$ weakly (Gibbs continuity at $\varepsilon = 0$). T-P-F-ε0-K merged Cat B: Kramers exponent stability under Bernoulli regularization.

• CV-1.8 — P-F-A1 Package I (T-PF-A1-AR + T-PF-A1-SDE Cat A; T-PF-A1-GI + T-PF-A1-PE Cat B, promoted in CV-1.9). Affine reduction proves $\widetilde{\Sigma}_M$ is a bounded convex polytope. Lions-Sznitman reflected SDE (T-PF-A1-SDE) establishes existence and uniqueness of the Langevin diffusion on $\widetilde{\Sigma}_M$. Gibbs invariant measure uniqueness (T-PF-A1-GI) and Poincaré inequality + exponential ergodicity (T-PF-A1-PE) round out the stochastic foundation for $T_*$.

• CV-1.9 — P-F-A1 Package I fully Cat A. T-PF-A1-GI and T-PF-A1-PE promoted Cat B → Cat A. Uniqueness via Neumann heat kernel positivity + L²(π) semigroup argument; exponential ergodicity rate $\lambda_1 \geq (\pi^2/n)\exp(-\mathrm{osc}(\tilde{\mathcal{E}})/T_*)$ via Payne-Weinberger + Holley-Stroock. P-F-A1 Package II (Eyring-Kramers explicit rates) remains open (OP-0021, W9+).

• CV-1.10 — Equilibrium K-Selection Cat B (OP-0005-EQ partially resolved). T-K-Select-PF: under Package I, the pushforward of $\pi_{T_*}$ under $K_{\mathrm{act}}$ defines sector probabilities $\{p_K\}$; $K^* = \arg\max_K p_K$. Does NOT resolve OP-0005-DYN (Kramers) or OP-0005-OBS.

• CV-1.11 — Observation-Conditioned K-Selection Cat B (OP-0005-OBS partially resolved). T-K-Select-OBS: Bayesian posterior $\pi_{T_*}(\cdot \mid \mathcal{O})$ over observation-compatible states selects $K^*_{\mathrm{obs}}$. exp85 ALL PASSED (numerical anchor). OP-0005 is now 3-way split: EQ (partially resolved), OBS (partially resolved), DYN (open, W9+).

OMS-2.0 Accepted — Full (W6 Day 6, Sessions 7–8): The Observer Moduli Space — the space of symmetry-inequivalent critical formations with stratification by $G_{\mathrm{cw}}$ conjugacy class — is proved finite and carries a codim-1 branch decomposition. Temporal extension ($G_{\mathrm{cw}}^{\mathrm{temp}} = \{e\}$, 19-branch Δ³ map with 7 λ_tr-unique branches) established via VP-11. Appendix OMS (§A–§M) added to canonical.md.

Navigational additions (W6 Day 5): DECLARATION.md (DECL-1.0, 6 epistemological commitments) and hypothesis_tree.md (HT-3.0, dependency structure + critical path for H-SINK / CV-1.12) created.

Counts (CV-1.5.2 → CV-1.11): 46A/5B/5C/5R = 61 → 54A/14B/5C/5R = 78 claims, ~69% fully proved.

Key Changes in CV-1.5.1 (W5 Day 3 EOD, 2026-04-29)#

• D-6a Multi-Static cluster merged: 3 Cat A definitional entries grounding Commitment 14-Multi — T-Commitment-14-Multi-Static (extends σ-signature to multi-formation interior $\widetilde{\Sigma}^{K_{\mathrm{field}},\,\circ}_M$), T-σ-multi-A-Static, T-σ-multi-D-Static.
• T-σ-Multi-1 registered as Cat B target (Goldstone-pair instability).
• V5b-T-zero new sub-statement (Cat A definitional, sub-spinodal $\mu_{\mathrm{Gold}} = 0$ exact) replaces WITHDRAWN V5b-T' (NQ-198f phantom on torus).
• V5b-F-empirical sub-statement added as Cat B target via NQ-198a 1/n scaling.
• T-σ-Theorem-4 retroactive Cat A → Cat B per Critic 7-agent verdict + NQ-187 numerical refutation; Cat A re-promotion deferred to CV-1.7+ post-γ/β/α path audit.
• Commitment 16 (K-status: Two-Tier Decomposition) added: K decomposes into K_field (architectural cap, modeling-layer commitment per I9) and K_act(t) (active stratum index, derived integer diagnostic). Inequality K_act(t) ≤ K_field at all t. CN6 ("K is kinetically determined") refers to K_act; I9 refers to K_field.
• Commitment 14 sub-conventions (O5')(O7) added: multi-irrep eigenspace ordering convention via Mulliken character order; tie-breaking convention.
• OP-0008 (σ^A K-jump non-determinism) + OP-0009 (Multi-Formation Ontological Foundations, 7 sub-items) registered as HIGH OPEN.
• MO-1 re-activation rider added to OP-0003: re-activates on D-6b approval or NQ-248 multi-formation Morse work.
• Counts: 43A → 45A, 4B → 5B, 57 → 60 claims.

Key Changes in CV-1.5 (W5 Day 1 G0, 2026-04-27)#

• σ-framework supporting structures canonical-merged. Five new Cat A entries grounding Commitment 14: T-σ-Lemma-1 (irrep decomposition well-defined via Maschke + Schur orthogonality on finite stabilizer; finite-graph hypothesis essential), T-σ-Lemma-2 (nodal count properties — sub-statements (i)–(iv) Cat A; (v) Courant + (vi) $G_u$-orbit divisibility Cat C riders within parent entry), T-σ-Lemma-3 (Goldstone–ℓ=1 angular saturation, Cat A in continuum, anchors T-V5b-T-(e) universal nodal count = 2; dimension-general 1D/2D/3D), T-σ-Theorem-3 (closed-form spectrum at uniform on $D_4$ free-BC grid: $\mu_k = 4\alpha\lambda_k^{\mathrm{Lap}} + \beta W''(c)$), T-σ-Theorem-4 ($D_4 \to \mathbb{Z}_2$ symmetry breaking at first pitchfork; trivial vs sign irrep split; $K_0 = K_1$ leading-order degeneracy on $D_4$ with $A_2/A_1 = 4$).
• Decision: Option α — five separate §13 entries (mathematically independent statements deserve individual canonical visibility for paper §4 σ-framework reference). T1 explosion: 3 → 8.
• Two erratum/refinement rounds (2026-04-27 evening + night). Caught 3 substantive math errors (IBP factor in Lemma 3, $K_1$-vs-$K_0$ degeneracy in Theorem 4, irrep-table off-diagonal entries in Theorem 3) plus 7 structural refinements (dimension-general extension, anchoring footer, spinodal hypothesis discussion, orbit-representative remark, well-definedness flag for $K_0 = K_1$). All five σ structures remain Cat A; theorem status unchanged.
• Pre-brainstorm corrections folded. T-σ-Lemma-1 finite-graph hypothesis explicit; T-σ-Lemma-2 sub-statement (iii) reframed as lower-bound $\mathcal{N} \geq 2$ from $\mathbf{1}^\perp$ constraint (was incorrectly stated as "$n_k = 1$ iff constant" in plan templates); T-σ-Lemma-3 IBP interpretation B adopted.
• Proved results registry (at CV-1.5): 43 Category A, 4 Category B, 5 Category C (+ 2 sub-statement riders bundled), 5 retracted; total 57 claims (75% fully proved). Subsequently upgraded to 45A/5B/5C/5R = 60 claims at CV-1.5.1 (2026-04-29) and 46A/5B/5C/5R = 61 claims at CV-1.5.2 (2026-05-02) — see "Key Changes in CV-1.5.1" and "Key Changes in CV-1.5.2" above.
• Commitment-level changes deferred to user decision (not yet merged): Commitment 14 (O5') multi-irrep eigenspace convention, Commitment 14 (O7) tie-breaking convention by canonical irrep order. NQ-187..NQ-190 spawned as carry.

Key Changes in CV-1.3 + CV-1.4 (W4 + W4-extended close, 2026-04-19 ~ 04-26)#

• Critical 3 open problems all resolved (2026-04-24). F-1 (K=2 vacuity) SPLIT-RESOLVED via T-Merge (b) (pure $\mathcal{E}_{\mathrm{bd}}$ portion, pre-existing Cat A) plus the new T-PreObj-1 (full SCC portion). M-1 (K=1 preference) LAYER-CLARIFIED as a proved theorem (T-Merge (b)) misframed as a problem. MO-1 (Morse inapplicability) SIDESTEPPED — the σ-framework operates on the smooth single-formation manifold $\Sigma_m$ (no corners). Critical blocker count 3 → 0, v2.0 release path unblocked.
• New Cat A (W4 close, 2026-04-24): T-PreObj-1 (Pre-Objective Multi-Peak Formation Mechanism — F=1 single-disk minimizer non-critical under full SCC; gradient flow attracts to multi-peak F≥2; IC-protocol dichotomy adaptive bounded vs random $\sim L^{2.8}$), T-PreObj-1G (graph-class independent — conclusions hold on any finite connected graph under (G1)–(G4)), Lemma 4 ($M \in \mathbb{R}^{2 \times 2}$ positive definite under linear independence; destabilization magnitude $\Lambda^T M \Lambda > 0$), F-1 Resolution Corollary (split-resolution).
• New Cat A (W4-extended close, 2026-04-26): T-V5b-T (Pre-Objective Goldstone on Translation-Invariant Graphs) — sub/super-lattice spectral dichotomy on torus T^d / cycle C_n; 2D commensurability split; 1D Goldstone; nodal count = 2 universal. After 8 V5b iterations (V1 → V5b'').
• New Cat C finding (W4-extended): V5b-F (Partial Goldstone on Boundary-Modified Graphs) — boundary lifting mechanism qualitative observation. NQ-173 carry.
• New commitment note: CN15 Static/Dynamic Separation — static global minimum on pure $\mathcal{E}_{\mathrm{bd}}$ need not equal dynamic protocol-endpoint observables ($\widehat{K}$, $\mathcal{F}$).
• σ-framework introduced (canonical-ready, Cat A definitional). Hessian eigenvalue/irrep/nodal-count signature $\sigma(u^*) = (\mathcal{F}; \{(n_k, [\rho_k], \lambda_k)\})$ on $\Sigma_m$.
• Reproducibility crisis identified+resolved (NQ-172, 2026-04-26). Mode-indexing artifact in NQ-170 analysis script; mode-agnostic detection adopted.
• Proved results registry (CV-1.4 final): 38 Category A, 4 Category B, 5 Category C, 5 retracted; total 52 claims (73% fully proved). Subsequently upgraded to 43A/57/75% in CV-1.5 — see "Key Changes in CV-1.5" above.

Key Changes in CV-1.0 to CV-1.2 era (preserved for history)#

• Category upgrades (3 theorems B→A): Deep Core Dominance 2b (isoperimetric inequality on $\mathbb{Z}^d$), T8-Full ($\mu_0(H_{\mathrm{bd}})$ at $\mathcal{E}_{\mathrm{bd}}$ minimizer $> 0$ in all tested configs), Predicate-Energy Bridge (Sep bidirectional exact; Bind reverse at minimizers).
• Retraction: Theorem 3.3 ($\bar{r}_0 = O(n^{-1/d})$ for general $\tau$) — experimentally falsified; $\bar{r}_0$ is genuinely $O(1)$ for $\tau \neq 1/2$.
• T-Persist-K-Unified: Parametric theorem unifying Sep/Weak/Strong persistence regimes via the coupling parameter $\Lambda_{\mathrm{coupling}} = \lambda_{\mathrm{rep}} \cdot \omega_{jk} / \min(\mu_j, \mu_k)$.
• Unified regime parametrization: $\Lambda_{\mathrm{coupling}}$ adopted as canonical coupling measure with $\mu$-floor regularization $\mu_{\mathrm{floor}} = w_{\mathrm{cl}} \cdot 2(1 - a_{\mathrm{cl}}/4)^2$. Experimental validation: 100% geometric-Lambda agreement across 69 configurations (exp46-47).

Key Changes from 2026-04-02 Audit (Paradigm Shift: Kinetic Multi-Formation)#

• Paradigm shift: Multi-formation is kinetic (barrier-based), not thermodynamic (energy minimization). K*=1 universally on all connected graphs (exp51, 10 configurations: grids, SBM, barbell, random geometric). K>1 coexists as metastable local minima with barrier height $\propto \beta^{0.89}$ (exp38, exp55). Persistence is maintained by kinetic barriers, not energy preference. Well-separated bumps remain stable under noise $\sigma \leq 0.5$ indefinitely (exp55: zero merges in 5000 iterations, barrier height ≈20 >> noise scale ≈0.5). Three kinetic pillars: Nucleation (spectral eigenvectors), Metastability (inter-formation barriers), Coarsening (noise-driven merge dynamics).
• P-Unified-1 FALSIFIED (exp49-50): Persist degradation is NOT monotone in $\Lambda_{\mathrm{coupling}}$. Λ_coupling reclassified as structural classifier, not dynamical predictor. Replaced by kinetic predictions MK-1–MK-4: nucleation via spectral modes (exp51), coarsening \alpha < 1/2 (exp57), barrier scaling $\propto \beta^{0.89}$ (exp38), enhanced metastability from closure (exp57 corrected).
• CN6 resolved: K is kinetically determined. K emerges from initial conditions, spatial structure, and barrier dynamics, not energy optimization. On single field: K>1 well-separated formations metastable when $d > d_{\min}^*$ (~5 nodes, $\beta=30$). K-field architecture ensures K>1 by construction.
• CN14 added: Closure expands multi-formation stability. Closure reduces $d_{\min}^*$ by ~30% (from ≈7 to ≈5 nodes, $\beta=30$; exp57), enabling formation coexistence at closer distances. Barrier height increases ($\beta^{0.89}$ vs $\beta^{0.85}$ for Allen-Cahn). Multi-formation realization of T7-Enhanced metastability.
• §12 Multi-formation restructured: Kinetic 3-pillar framework adopted as primary conceptual narrative. Replaced Erratum-bracketed patches with clean kinetic framing. T-Persist-K-Sep/Weak/Unified cover regimes below merge bifurcation; strongly-interacting merge dynamics remain open (Kramers rates, barrier crossing).

Foundational Changes — Pre-CV-1.0 Heritage (retained from theory-revision v1.0 → v2.0 transition)#

• A1 replaced by A1' (conditional extensivity) — original A1 proved incompatible with A3 for sigmoid closure.
• A3 strengthened to Cauchy convergence of iterates with proved contraction rate.
• Volume constraint $\sum_x u_t(x) = m$ added as a structural axiom with ontological justification.
• The transition operator $\mathbf{T}_t$ is demoted from the formal universe to a derived diagnostic — it had zero realizations, zero theorems, and zero energy/predicate roles across five iterations.
• Co-belonging $\mathbf{C}_t$ receives a provisional resolvent realization, revised axiom C3'' (local monotonicity), and new axiom C5 (symmetry).
• The morphological quality measure $\mathcal{Q}_{\mathrm{morph}} = \ell_{\max} \cdot \mathrm{Artic}$ receives a provisional definition.
• Separation predicate $\mathsf{Sep}$ revised to $u$-weighted form (corrected from intermediate $\mathbf{C}_t$-weighted version, which was diagnostically degenerate).
• Binding predicate $\mathsf{Bind}$ norm specified as $\ell^2$.
• E3 (core inheritance) reclassified from operator axiom to solution constraint.
• Proto-cohesion reformulated as a diagnostic vector $\mathbf{d} \in [0,1]^4$, replacing the Boolean conjunction.
• Distinction parameter $b_D = 0$ (or $\varepsilon$-smoothed) for energy analyticity.
• Ordered-pair summation convention declared (see Part 1, §0), resolving the critical ratio ambiguity in T8-Core.
• Proved results registry added (see Part 5, §13): 12 theorems from Iterations 1–2.
• Commitment Notes consolidated (see Part 5, §14).

This specification is intended to serve as the authoritative reference for future formalization, implementation, and theoretical extension. It is written to be readable by mathematicians, theoretical cognitive scientists, and system designers alike.

Start reading → Part 1 · Foundations & Formal Universe