T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime

Hero · Multi-Formation Flagship group · Cat A conditional. Source: T-L1-F canonical entry (canonical.md §13 line 1482). Verification: L1-I 439/1920 = 22.9% feasible on $T_{20}^2$ raw_gaussian; L1-H2 stress 5/5; L1-J PO-1 6/6; L1-K external audit + L1-K-REPAIR R-1..R-4. Canonical version: introduced in CV-1.5.2 (W5 Day 6, 2026-05-02) — first multi-formation canonical Cat A theorem. Full proof: Canonical Spec — Part 5 · §13 T-L1-F. Working chain: THEORY/working/MF/ (L1-A through L1-L 13-step chain).

Statement#

Let $G=(X,E)$ be a finite graph and $\mathbf{u}\in {\tilde{\Sigma }}_M^{K_{\mathrm{field}}}(G)$ a shared-pool multi-formation state. Let

and

under terminal-death $H_0$ superlevel persistence on the aggregate field $U$.

Theorem (T-L1-F). Under the L1-J regime hypothesis package $(P0)$–$(P11)$,

and the map

equivalently the bar born at $q_j^U=\arg {\max }_{x\in N_j^r}^{\prec }U(x)$, is a bijection from active slots to dominant terminal $H_0$ bars.

Hypothesis package $(P0)$–$(P11)$ — abbreviated#

The L1-J regime is empirically non-vacuous (L1-I 439/1920 = 22.9% feasible on $T_{20}^2$ with raw_gaussian initial states), but production WQ-1 trajectories — which use mass-projection initial states — typically exit the regime, so T-L1-F's reach in production dynamics is narrow and the conditional Cat A status should be read accordingly.

Proof idea#

Lower bound $K_{\mathrm{bar}}\ge |A|$. Each active slot $j\in A^{\epsilon }$ contributes a dominant bar of length $\ge {\ell }_{\min }$:

1. LG-2 boundary collar (P4) ensures the boundary $\partial N_j^r$ of the active neighborhood lies at $U$-height $\le b_j-{\ell }_{\min }-r_{\mathrm{assoc}}$, so each $N_j^r$ supports a sublevel-disconnection of depth $\ge {\ell }_{\min }$.
2. LG-3 inter-neighborhood bridge (a structural consequence of P3 + P4) prevents premature mergers between distinct active neighborhoods at superlevel heights $\ge b_j-{\ell }_{\min }$.
3. $h_{\min }\ge {\ell }_{\min }$ (P6) secures that each birth height clears the bar-length threshold.

This is L1-H §8 step 2 of the working chain.

Upper bound $K_{\mathrm{bar}}\le |A|$. Two arguments:

(α) Coverage from background suppression. LG-7 coverage, derived from LG-4 (P5) + terminal-death convention (P0), guarantees that every dominant bar's birth has $U\ge {\ell }_{\min }$, hence is not in the background set $X_{\mathrm{bg}}$. So all dominant bars are born inside ${\bigcup }_j{N}_j^r$.

(β) Per-neighborhood at-most-one-dominant-bar. Combining:

• L1-H2 Lemma 1 (graph-inclusion). ${\ell }_{\mathrm{glob}}\le {\ell }_{\mathrm{loc}}$ on $G_j^r\subseteq G$ — bar lengths can only shrink under the larger ambient graph.
• L1-H2 Lemma 2 (contradiction-based bottleneck-stability). If two $U$-bars on $G_j^r$ both have length $\ge {\ell }_{\min }$, both must match a $u^{(j)}$-bar of length $\ge {\ell }_{\min }-{\rho }_{\mathrm{pert}}$ (under tightened H6 = P8), but P8 allows only the slot primary — contradiction.

Combining (α) and (β): every dominant bar lives in some $N_j^r$, and each $N_j^r$ admits at most one dominant bar — so $K_{\mathrm{bar}}\le |A|$. The two bounds together give equality.

PO-1 decay-to-cut bridge bound. P7 (decay-to-cut) bounds ${\theta }_{\mathrm{bridge}}^{jk}(U)$ via L1-J §8.1 + L1-B Cat-A cut lemma, ensuring no anomalous topological mergers across the cut $C_{jk}$ between active neighborhoods. This handles the inter-neighborhood interface that LG-3 alone cannot fully discharge.

The bijection. With equality $|A^{\epsilon }|=K_{\mathrm{bar}}^{{\ell }_{\min }}$ established and each $N_j^r$ matched to exactly one dominant bar, the map ${\mathcal{A}}_{\mathrm{bar}}(j)=$ "the unique dominant bar born at $q_j^U$" is well-defined and bijective. P1 (deterministic tie convention) makes $q_j^U=\arg {\max }_{x\in N_j^r}^{\prec }U(x)$ unambiguous.

The full chain L1-A through L1-L is recorded in THEORY/working/MF/, including the L1-K external audit and the L1-K-REPAIR cycle (R-1..R-4) that hardened the proof against the original audit findings.

$\square$

Status#

Cat A conditional under the L1-J regime hypothesis package $(P0)$–$(P11)$.

Read the conditionality strictly:

• NOT a global identity. The equality $K_{\mathrm{bar}}^{{\ell }_{\min }}=K_{\mathrm{act}}^{\epsilon }$ holds only on states satisfying every clause of $(P0)$–$(P11)$. Outside the L1-J regime, the bridge can fail.
• Does NOT establish $K_{\mathrm{soft}}^{\varphi }=K_{\mathrm{act}}$. The soft-count corollary additionally requires $\varphi \in {\Phi }_{\mathrm{res}}$ per WQ-LAT-1.B. The L-M soft-count corollary working draft is sketched at Cat B and is the CV-1.6 promotion target; it is not part of T-L1-F.
• Does NOT solve OP-0005 (K-Selection). T-L1-F is a count-bridge under a fixed state, not a mechanism that selects $K$.
• Does NOT solve OP-0008 (${\sigma }^A$ K-jump non-determinism). Independent open problem.
• P7 (decay-to-cut). Adopted as a safe technical regime hypothesis. L1-L Combes-Thomas / discrete Agmon analysis provides backing under strong stationarity, but P7 is not asserted for all SCC states — it is a regime-defining assumption, not a derived fact.

The first multi-formation canonical Cat A theorem in SCC theory therefore lives under explicit regime conditions, not as a universal identity. The conditional Cat A status is the canonical reading.

Why this is a hero#

T-L1-F is the first multi-formation canonical Cat A theorem in SCC theory. Three reasons it earns hero status:

1. It closes the W5 working chain. L1-A through L1-L is a 13-step working chain that was the substantive content of W5. T-L1-F is the canonical promotion of the chain's terminal claim — passing through L1-K external audit and the L1-K-REPAIR R-1..R-4 cycle along the way.

2. It connects two count regimes. Chart-level active count $K_{\mathrm{act}}^{\epsilon }$ (per-slot mass above threshold) and aggregate-field topological count $K_{\mathrm{bar}}^{{\ell }_{\min }}$ (terminal-death $H_0$ bar count on $U={\sum }_j{u}^{(j)}$) live at very different layers: the former is a slot-resolved indicator on the K-field architecture, the latter is a topological invariant of the aggregate scalar field. T-L1-F is the bridge between SCC's per-slot dynamics and field-level morphology — under explicit regime conditions.

3. It is the first canonical multi-formation theorem promoted via the working pipeline. All earlier canonical multi-formation results (T-Persist-K-Sep / Weak / Unified) are Cat C conditional; the σ-multi-static entries (D-6a, CV-1.5.1) are Cat A definitional. T-L1-F is the first entry in the multi-formation block that is theorem-grade Cat A (conditional on a stated regime), promoted through the standard canonical path including external audit and repair.

T-L1-F is deliberately not a K-selection mechanism. It is a count-bridge that presupposes a multi-formation state and tells you that, under explicit regime conditions, the chart-level and field-level counts agree. The K-selection problem (OP-0005) and the ${\sigma }^A$ K-jump non-determinism (OP-0008) remain open.

Logical dependencies#

• Builds on: L1-A through L1-L working chain (THEORY/working/MF/), L1-B Cat-A cut lemma, L1-H2 Lemmas 1 and 2 (graph-inclusion + bottleneck-stability), L1-J PO-1 (decay-to-cut), L1-K external audit + L1-K-REPAIR R-1..R-4, L1-L Combes-Thomas / discrete Agmon analysis (P7 backing under strong stationarity), terminal-death $H_0$ superlevel persistence machinery.
• Builds into: L-M soft-count corollary working draft (Cat B sketch, CV-1.6 promotion target — additionally requires $\varphi \in {\Phi }_{\mathrm{res}}$ per WQ-LAT-1.B); future K-Selection and ${\sigma }^A$ K-jump work (OP-0005, OP-0008 — T-L1-F is a count-bridge, not a selection mechanism).

See also#

• Full proof in canonical: Canonical Spec §13 T-L1-F (canonical.md line 1482)
• Multi-formation prerequisites: T-PreObj-1 hero page, T-V5b-T hero page
• Hero index: SCC Hero Theorems