Purpose: Translate formal definitions from registries (D-xxxx, S-xxxx, A-xxxx) into readable English. This is the "user-facing" documentation for theory terms.
Core SCC Concepts
Cohesion Field u_t
Formal: D-0001 / S-0001
Plain English: A function that assigns a "strength of bonding" value between 0 and 1 to each location in a relational space. Think of it as a heatmap showing where things are "sticking together" at time t. At location x, u_t(x) tells you how much cohesion exists there.
Why it matters: This is the fundamental entity in SCC—we don't start with objects; we start with this field. Objects emerge from the dynamics of this field.
Intuition: If u_t(x) = 0.9, location x is strongly "in the formation." If u_t(x) = 0.1, it's weakly bonded. If u_t(x) = 0, it's outside.
Relational Support Space X_t
Formal: D-0002 / S-0002
Plain English: The graph (network structure) on which the cohesion field lives. It consists of vertices (locations) and edges (connections between them).
Why it matters: The field doesn't exist in a vacuum; it exists on a particular network topology. The topology determines how cohesion spreads via the Laplacian.
Example: A 10×10 grid graph where each cell is connected to its 4 neighbors. Or a social network where people are vertices and friendships are edges.
Energy Functional E(u)
Formal: D-0003 / S-0003:S-0007
Plain English: A function that scores how "expensive" a given cohesion field configuration is. Like a cost function: lower energy = better, more stable configuration.
The four terms:
- E_cl (Closure): penalizes fields that are "scattered"; rewards consolidation
- E_sep (Separation): penalizes fields that deviate from isolated regions; rewards distinctness
- E_bd (Boundary): penalizes sharp boundaries; encourages smooth transitions
- E_tr (Transport): penalizes temporal discontinuity; rewards inheritance from prior state
Why it matters: The cohesion field evolves by flowing downhill on this energy surface (gradient descent). The shape of the energy landscape determines what formations are stable.
Closure Operator Cl_t
Formal: D-0004 / S-0013
Plain English: A transformation that smooths and stabilizes the cohesion field. It "fills in gaps" where cohesion is spotty.
Key property: It's NOT idempotent (applying it twice doesn't give the same result as once). This is deliberate—closure has a "tendency" to stabilize, not a fixed point goal.
Intuition: If your cohesion looks like scattered patches, closure says "tighten up and become more consolidated."
Distinction Operator D_t
Formal: D-0005 / S-0014
Plain English: A measurement of how much "edge" or "boundary richness" exists in the field. It quantifies where the field changes sharply.
Formula: Based on the contrast in (1 - u_t) — the "anti-cohesion."
Intuition: Where u_t = 0.5 (neither cohesive nor dispersed), D_t is large. Where u_t ≈ 0 or ≈ 1 (clearly inside or outside), D_t is small.
Resolvent C_t
Formal: D-0006 / S-0015
Plain English: A global integration measure. It tells you "how well does the cohesion field hang together across the whole relational space?"
Technical: It's the inverse Laplacian applied to the cohesion field.
Intuition: If C_t is high, the field is globally connected and integrated. If C_t is low, it's fragmented.
Transport Kernel M_{t→s}
Formal: D-0007 / S-0016
Plain English: A temporal correspondence: "How much of formation-at-time-s becomes formation-at-time-t?"
Implementation: Computes optimal transport (Wasserstein distance) between the cohesion fields at two time points, using entropy regularization to smooth the solution.
Intuition: If M_{t→s} is high between two fields, the formation has persisted—the old formation "transported" into the new one.
Diagnostic Measurements
Proto-Cohesion Diagnostic d = (Bind, Sep, Inside, Persist)
Formal: D-0008 / S-0021
Plain English: A 4-tuple of numbers (all in [0,1]) that answer: "Is this formation coherent?"
The four measurements:
- Bind: How concentrated is the field? (0 = dispersed everywhere, 1 = sharp point mass)
- Sep: (For K ≥ 2 only) How separated are the formations? (0 = overlapping, 1 = completely disjoint)
- Inside: How persistent is the core across time? (0 = core disappears, 1 = core unchanged)
- Persist: How much of the field carries forward temporally? (0 = completely new, 1 = completely inherited)
Interpretation: A "good" formation has high d values (especially Bind, Inside, Persist). The quadruple d acts like a quality score.
K-Field (Multi-Formation) Concepts
K-Field Configuration {u¹_t, u²_t, ..., u^K_t}
Formal: D-0014
Plain English: K different cohesion fields living on the same relational space, with a constraint that they sum to 1 everywhere (simplex constraint: Σ u^j_t(x) = 1 for all x).
Intuition: You can think of this as K competing formations; at each location, the K fields share the "cohesion budget" (total 1.0).
Why it matters: Extends SCC from single formations (K=1) to multiple interacting formations (K > 1).
Per-Formation Mass m_j
Formal: D-0015 / S-0028
Plain English: The "size" or "volume" of formation j. It's the integral of u^j_t across the entire space.
Formula: m_j = ∫ u^j_t dx
Constraint (resolved by W4): Per-formation mass is a structural assumption of the K-field architecture. The earlier "F-1 / M-1 problems" — that K=1 is energetically preferred when m_j varies — turn out to be the correct theorem T-Merge (b) on pure , not an open problem. Under full SCC, the F=1 single-disk is non-critical (T-PreObj-1), so the dichotomy "K=1 cheaper static minimum vs empirical K>1" is dissolved by CN15 Static/Dynamic Separation.
Repulsion Energy E_rep
Formal: D-0017
Plain English: A penalty term that prevents K formations from overlapping too much. Think of it as a "spring" pushing formations apart.
Formula: E_rep = λ_rep × [measure of overlap between u^j and u^k for j ≠ k]
Role: Without this, all K formations would converge to the same place (lowest energy). With it, they maintain separation.
CV-1.5.1 / CV-1.5.2 Multi-Formation Vocabulary
This section collects the multi-formation framework vocabulary introduced at CV-1.5.1 (Commitment 16 K-status decomposition, σ-tuple, OP-0008 and OP-0009) and at CV-1.5.2 (T-L1-F Hard-Bar / Active-Count Bridge under the L1-J regime). Entries are ordered alphabetically.
Commitment 14 — Orbital character is constitutive
Formal: Canonical §11.1 #14 (W4 04-24, CV-1.5)
Plain English: The σ-tuple — the canonical Hessian signature attached to a formation — is the constitutive identity of that formation, not an analogical decoration. Two single-formation minimizers with different σ-tuples are different formations as a matter of theory, not as a matter of interpretation.
Why it matters: This commitment fixes σ as canonical formation identity, which then propagates into the multi-formation extension (T-Commitment-14-Multi-Static, CV-1.5.1) and into the OP-0008 question of how σ inheritance behaves at K-jump events.
Commitment 15 — Pre-objective commitment is a mathematical theorem
Formal: Canonical §11.1 #15 (W4 04-25, CV-1.3)
Plain English: "Formations exist before discrete objects" is not a stipulation or a modeling preference — it is grounded by T-PreObj-1 and T-PreObj-1G as a Cat A theorem (graph-class independent). Under full SCC parameters, the F=1 single-disk minimizer of pure E_bd is not a critical point, and gradient flow attracts to F ≥ 2 multi-peak configurations.
Why it matters: This converts "pre-objective" from a methodological slogan into a load-bearing theorem and dissolves the F-1 paradox.
Commitment 16 — K-status two-tier decomposition
Formal: Canonical §11.1 #16 (W5 Day 3 EOD, CV-1.5.1, 2026-04-29)
Plain English: The integer K is not a primitive of the theory. K decomposes into two distinct quantities at different layers: K_field (architectural cap, set externally at instantiation) and K_act (dynamic stratum index, derived from u). This resolves the four-month K ontological ambiguity catalogued in OP-0009-K and reconciles the apparent conflict between the I9 architecture statement ("K-field architecture guarantees K > 1 by construction") and CN6 ("K is kinetically determined, not thermodynamically selected").
Why it matters: Without Commitment 16, "K" was used in five conflicting senses across canonical and working files. The two-tier decomposition gives the K-field architecture, K-jump events, σ_multi^A multi-set treatment, and T-L1-F a single shared language.
CN15 — Static / Dynamic Separation
Formal: Canonical §14 CN15 (W4 04-24)
Plain English: The kinetic K (the K_act trajectory under gradient flow) is a separate quantity from the static energy minimum K. T-Merge (b) Cat A always gives K = 1 as the static minimum on pure E_bd, but this does not contradict empirically observed K > 1 because the dynamic protocol-endpoint observables live on a different layer.
Why it matters: CN15 dissolves the apparent F-1 / M-1 paradox without retracting any theorem. T-Merge (b) is preserved as the correct static statement; the empirical multi-formation regime is the dynamic-layer answer.
F (peak count)
Formal: Canonical §13 inline in T-PreObj-1 + §14 CN17 (W4 04-25)
Plain English: A derived diagnostic counting the local maxima of u — threshold-free and upper-semicontinuous in u. Used in Theorem 2 / T-PreObj-1 to state that the F = 1 single-disk minimizer of pure E_bd is non-critical under full SCC.
Why it matters: F is the single-formation count diagnostic that anchors Pre-Objective theorems. OP-0009-F flags F as a derived diagnostic awaiting full canonical §5 registration via OAT-2 (W6 promotion).
K_act (active formation count)
Formal: Canonical §11.1 Commitment 16 (ii) (CV-1.5.1)
Plain English: The number of formations currently "alive" in a K-field minimizer at time t, defined as the count of slots whose mass exceeds a support threshold:
K_act^ε(u) = #{j : ‖u^(j)‖_1 > ε}
K_act is a derived integer diagnostic, not an externally set parameter. It is right-continuous in t under gradient flow and decreases at K-jump events.
Why it matters: K_act is what CN6 ("K is kinetically determined") actually refers to. It is distinguished from K_field (the architectural cap, external) by Commitment 16. K_act is also one of the two quantities bridged by T-L1-F at CV-1.5.2.
K_bar (hard-bar count)
Formal: Canonical §13 T-L1-F entry (CV-1.5.2, 2026-05-02)
Plain English: The number of "hard" topological bars surviving in the H_0 superlevel persistence diagram of the aggregate field U(u) = Σ_j u^(j) under the terminal-death convention:
K_bar^ℓ_min(U; G) = #{[d, b] : b - d ≥ ℓ_min}
T-L1-F (CV-1.5.2) establishes K_bar^ℓ_min = K_act^ε under the L1-J regime, with a labelled bijection between active slots and dominant terminal H_0 bars.
Why it matters: K_bar is the topological / observable side of the K-count question. T-L1-F links K_bar (what you can read off the aggregate field via persistence) to K_act (the chart-level active-slot count) under explicit hypotheses.
K_field (architectural cap)
Formal: Canonical §11.1 Commitment 16 (i) (CV-1.5.1, 2026-04-29)
Plain English: The maximum number of formation slots the analytical framework will track. K_field is set externally by the modeler at instantiation, comparable to the choice of X_t structure — a modeling-layer commitment, not an ontological primitive. K_field derives the shared-pool space (see Shared-pool space below).
Why it matters: K_field is what the I9 K-field architecture statement ("guarantees K > 1 by construction") actually refers to. It is the upper bound on K_act and is over-provisionable: K_field can be set larger than K_act will ever reach.
K_soft (envelope-weighted smooth count)
Formal: Working draft THEORY/working/MF/ksoft_kact_bridge_L1M_soft_count_corollary.md (CV-1.6 candidate via L-M corollary)
Plain English: A continuous-valued, envelope-weighted count of bars in the H_0 persistence diagram of U:
K_soft^φ(U) = Σ_i φ(ℓ_i)
where ℓ_i are the bar lengths and φ is a soft-step envelope. The L-M working draft bounds |K_soft^φ − K_act^ε| under φ in the Φ_res envelope class.
Why it matters: K_soft is the smooth relaxation of K_bar — it replaces the hard threshold ℓ ≥ ℓ_min with a smooth envelope. T-L1-F gives K_bar = K_act; the L-M corollary extends this to K_soft = K_act under the additional Φ_res hypothesis, but this is currently working-grade Cat-B sketched (not canonical).
L1-J regime (hypothesis package P0–P11)
Formal: Canonical §13 T-L1-F entry (CV-1.5.2)
Plain English: The explicit hypothesis package under which T-L1-F is proved. Eleven conditions on a shared-pool multi-formation state are required:
- P0 terminal-death H_0 superlevel persistence convention.
- P1 deterministic tie convention (fixed total order on X; ties in descending-U broken by ascending order).
- P2 active mass + connected δ-support.
- P3 disjoint active neighborhoods (LG-1): N_j^r ∩ N_k^r = ∅.
- P4 low boundary collar (LG-2): max U on ∂N_j^r ≤ b_j − ℓ_min − r_assoc.
- P5 background suppression on U (not just on R_inact).
- P6 birth height: b_j ≥ h_min ≥ ℓ_min.
- P7 decay-to-cut (heterogeneous form): per-slot Combes-Thomas / discrete Agmon decay envelope.
- P8 tightened H6 second-bar bound on G_j^r.
- P9 NE-2 perturbation: ‖R_j‖_∞ on N_j^r ≤ ρ_pert / 2.
- P10 inactive residual suppression: ‖R_inact‖_∞ ≤ ℓ_min − ρ_res.
- P11 margin ledger:
h_min − max_{k≠j} B_jk ≥ ℓ_min + r_assoc + r_birth.
Why it matters: T-L1-F is Cat A conditional on (P0)–(P11). The L1-J regime is empirically non-vacuous (L1-I numerical: 439 / 1920 = 22.9% feasible on T^2_20 with raw_gaussian initial states) but production WQ-1 trajectories typically exit the regime. The conditional Cat A status should be read accordingly.
OP-0008 — σ^A K-jump Inheritance Non-Determinism
Formal: theorem_status.md Open Problems Catalog OP-0008 (HIGH; registered W5 Day 4, CV-1.5.1)
Plain English: Under K-field gradient flow on the shared-pool space, the post-merger σ^A signature is not deterministic in the pre-merger σ^A signature alone — it requires merger-geometry data (cluster centroid, orientation, Wigner-von Neumann data) beyond the eigenvalue tuple. This bifurcates the CV-1.6 release path: D-6b dynamic σ_multi^A(t) Cat A requires σ-rich augmentation.
Status: OPEN. Path B (σ-rich + Φ-rich) is a Cat B target and a CV-1.7 Commitment 18 candidate.
OP-0009 — Multi-Formation Ontological Foundations
Formal: theorem_status.md Open Problems Catalog OP-0009 (HIGH; registered W5 Day 4, CV-1.5.1; 7 sub-items)
Plain English: A parent open problem covering the ontological status of every multi-formation primitive. Seven sub-items:
- OP-0009-K (K-status). RESOLVED via Commitment 16.
- OP-0009-F (F as derived diagnostic). PARTIALLY (OAT-2).
- OP-0009-λ (λ_rep ontology). PARTIALLY (OAT-3).
- OP-0009-A (Architecture: K-field vs Shared-pool). PARTIALLY (OAT-4).
- OP-0009-C (C_t multi-formation status). PARTIALLY (OAT-5).
- OP-0009-Pre (Pre-objective + K-field tension). PARTIALLY (OAT-6).
- OP-0009-Emp (R23 empirical verification). PARTIALLY (OAT-7).
Status: PARTIALLY ADDRESSED at CV-1.5.1 + W5 Day 4 OAT batch — 1 of 7 RESOLVED (OP-0009-K via Commitment 16), 6 of 7 PARTIALLY RESOLVED. Full OP-0009 closure deferred.
Φ_res envelope class — Reservoir-admissible envelopes
Formal: Working draft THEORY/working/MF/ksoft_kact_bridge_L1M_soft_count_corollary.md Definition L-M-D1 (CV-1.6 candidate)
Plain English: The class Φ_res(ℓ_min, τ) of soft-step envelopes φ : [0, 1] → [0, 1] used to define K_soft. Five axioms:
- F1 range bound: 0 ≤ φ(ℓ) ≤ 1 for all ℓ in [0, 1].
- F2 lower normalization: φ(0) = 0.
- F3 monotonicity: φ is non-decreasing on [0, 1].
- F4 sub-threshold suppression: φ(ℓ) ≤ ε_sub^φ on [0, ℓ_min − τ].
- F5 dominant retention: 1 − φ(ℓ) ≤ ε_dom^φ on [ℓ_min + τ, 1].
The pair (ε_sub^φ, ε_dom^φ) is the structural-deviation pair; the class becomes meaningful when ε_sub^φ + ε_dom^φ ≪ 1.
Why it matters: Φ_res isolates the envelope shapes for which the L-M soft-count corollary controls |K_soft^φ − K_act^ε|. The default rational envelope φ_0(ℓ) = ℓ / (ℓ + ℓ_min) is excluded by F4, predicting (and matching) its empirical sub-resolution drift.
Shared-pool space \widetilde{\Sigma}_M^{K_field}(G)
Formal: Architecture I9' (working proposal THEORY/working/MF/shared_pool_canonical_proposal.md; T-L1-F operates on this space at CV-1.5.2)
Plain English: The shared-pool multi-formation state space, an alternative architecture to I9. A state is u = (u^(1), …, u^(K_field)) with total mass Σ_j m_j = M shared across the K_field slots; per-slot masses m_j are not fixed externally, and K_act is derived from the state via the support threshold.
Why it matters: Distinguished from K-field architecture I9 (the canonical Σ^K_M product manifold with fixed individual masses). Under I9' the K_act / K_field decomposition is intrinsic: K_act varies dynamically inside a fixed K_field budget. T-L1-F is stated and proved on the shared-pool space.
σ-rich augmentation
Formal: Working draft THEORY/working/MF/sigma_rich_augmentation.md (Path B candidate; CV-1.7 Commitment 18 candidate)
Plain English: An expansion of the canonical σ-tuple to include cluster centroid, orientation, and Wigner-von Neumann data — beyond the (frequency, irrep, eigenvalue) eigenvalue tuple of Commitment 14. Required by OP-0008 to make σ^A K-jump inheritance deterministic.
Status: Path B Cat B target. Not canonical.
σ-tuple (canonical Hessian signature)
Formal: Canonical §11.1 Commitment 14 (W4 04-24, CV-1.5)
Plain English: The constitutive identity of a single-formation minimizer u*, defined as
σ(u*) = (F(u*); {(n_k, [ρ_k], λ_k)}_{k=1}^K)
a frequency × irrep × eigenvalue triple per relevant Hessian eigenmode. F(u*) is the peak count; n_k is the nodal count; [ρ_k] is the irrep label under the formation's stabilizer; λ_k is the eigenvalue.
Why it matters: σ is the canonical formation identity per Commitment 14. The multi-formation extension T-Commitment-14-Multi-Static (CV-1.5.1) lifts σ to (σ^A, σ^D) on the shared-pool / K-field interior. OP-0008 asks whether σ^A is a complete dynamic invariant under K-jump.
Status & Category Concepts
Category A (Unconditional)
Formal: D-0019
Plain English: A theorem proved without needing any special assumptions beyond the basic SCC axioms. It's "true as stated" with no caveats.
Example: T-1 (existence of minimizers) — given any parameters, a minimizer always exists.
Category B (Conditional)
Formal: D-0020
Plain English: A theorem proved, but only under stated conditions (e.g., "for generic parameters"). The conditions are assumed to hold, but they need to be checked in practice.
Example: T-σ-Theorem-4 — Cat B in the ε-small regime per the Critic 7-agent verdict at CV-1.5.1 (continuum-vs-discrete grid caveat added 2026-05-04 per NQ-187 audit). Note: T-Persist-K-Sep was previously listed here but was reclassified Cat C per Erratum 2026-04-07; T-Bind-Proj and T-Bind-Full are now Cat A per the Phase 13 erratum (W6 G2 audit reconfirmed 2026-05-04), not Cat B.
Category C (Very Conditional)
Formal: D-0021
Plain English: A theorem that holds in a very restricted regime. These are specialty results, not broadly applicable.
Examples (Cat C per Erratum 2026-04-07): T-Persist-K-Sep, T-Persist-K-Weak, T-Persist-K-Unified — the K-field persistence family. T-Persist-K-Weak only applies in the weakly-interacting regime; T-Persist-K-Sep requires the well-separated regime AND per-formation persistence; T-Persist-K-Unified is parametric over the coupling measure but holds only in a restricted regime.
Critical Open Problems — All Resolved in W4 (2026-04-24)
F-1: "K=2 is Vacuous" — ✅ SPLIT-RESOLVED
What it was: K=2 global stability requires per-formation mass m_j held fixed externally; if m_j varies freely, energy prefers m_j → 0 (K collapses to 1).
W4 resolution (2026-04-24): F-1 decomposes into two layers, each Cat A: (i) the pure portion is the correct theorem T-Merge (b) (isoperimetric ordering, pre-existing Cat A) — not an open problem; (ii) the full SCC portion is resolved by T-PreObj-1 (i): under full SCC parameters, the F=1 single-disk minimizer of pure is not a critical point, so the dichotomy "K=1 cheaper vs observed K>1" does not arise. The premise of F-1 collapses.
Current status: ✅ SPLIT-RESOLVED. Resolution path: Option D (premise dissolution).
M-1: "K=1 Always Wins" — ✅ LAYER-CLARIFIED
What it was: The K=2 energy landscape is monotonically decreasing as one formation mass shrinks; K=1 always energetically cheaper than balanced K=2.
W4 clarification (2026-04-24): M-1 is not an open problem — it is the correct mathematical statement (T-Merge (b)) about isoperimetric ordering on the constraint manifold. Original framing conflated pure (where M-1 holds) with full SCC (where the comparison is not even framed because F=1 is non-critical). The apparent paradox between this proved theorem and empirically observed K>1 is resolved by CN15 Static/Dynamic Separation: static global minimum on pure ≠ dynamic protocol-endpoint observables under full SCC.
Current status: ✅ LAYER-CLARIFIED. Proved theorem misframed as problem.
MO-1: "Morse Theory Fails" — ⚪ SIDESTEPPED
What it was: The K=2 constrained manifold has corners (smooth Morse theory inapplicable).
W4 sidestep (2026-04-24): The W4 work introduced the σ-framework — Hessian eigenvalue/irrep/nodal-count signature — which operates on the smooth single-formation manifold (no corners). The Theorem 2 family (Cat A, graph-class independent) does not require multi-formation Morse analysis. MO-1 is therefore not a blocker for current scope.
Current status: ⚪ SIDESTEPPED for single-formation σ scope. Multi-formation extension to (stratified Morse, Phase 5) remains genuine open work for W5+.
Operational Assumptions
"Fixed K" Assumption
Formal: A-0012
What it means: K (number of formations) is decided beforehand and held constant. Not derived from energy minimization.
Why stated: Because all K-field theorems assume this.
Status (post-W4): The "How does K get decided beforehand?" question is now reframed via CN15 Static/Dynamic Separation: the static global minimum need not equal the dynamic protocol-endpoint observables. Full K-selection mechanism (OP-0005) remains partially open, but no longer a Critical blocker.
"Fixed m_j" Assumption
Formal: A-0013
What it means: Each formation's mass m_j is held constant (or Σ m_j = M is constant). Not allowed to vary.
Why stated: Because allowing m_j → 0 causes K to collapse.
Problem: Again, no mechanism for why masses are "fixed." Is it imposed externally? By what?
Deprecated Concepts (Rejected)
"Type A" and "Type B" (Formations)
Status: REJECTED (exp65 invalidated)
What it was: Proposed categorization of K=2 configurations as "centered, stable" (Type A) vs "off-center, swap-prone" (Type B).
Why rejected: exp65 showed no Type B configurations ever emerge; all experiments clustered at Type A. The divergence between exp62 and exp63 was due to optimizer strategy, not formation type.
Lesson: Don't propose unvalidated classifications without empirical backing.
Quick-Reference by Question
"What is SCC trying to explain?"
→ How coherent formations emerge from relational structure, prior to discrete objects.
"What's the primitive entity?"
→ The soft cohesion field u_t : X_t → [0,1].
"How does it work?"
→ The field evolves by gradient descent on an energy functional E(u). The energy has four independent terms (closure, separation, boundary, transport).
"What about multiple formations?"
→ Use K-field architecture: K fields on the same space, summing to 1 everywhere (simplex constraint).
"What's broken?"
→ Critical-3 (F-1, M-1, MO-1) all resolved in W4 (2026-04-24): SPLIT-RESOLVED / LAYER-CLARIFIED / SIDESTEPPED via T-Merge(b) + T-PreObj-1 + σ-framework. Remaining: K-Selection mechanism (OP-0005, partially addressed by σ-framework + CN15 Static/Dynamic Separation), boundary precision (OP-0004), branch selection theory (OP-0006). v2.0 release path now unblocked.
"Are the theorems right?"
→ 46 Cat A / 5 Cat B / 5 Cat C / 5 Retracted = 61 claims / 75% fully proved (CV-1.5.2, 2026-05-02). Single-formation results are solid. Multi-formation persistence regimes are conditional, the foundational F-1 / M-1 / MO-1 issues are resolved (W4), and the first multi-formation Cat A theorem (T-L1-F Hard-Bar / Active-Count Bridge under the L1-J regime) was promoted at CV-1.5.2.
Last updated: 2026-05-04 (CV-1.5.2 multi-formation vocabulary sync)
Audience: Non-specialists, students, reviewers