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 $T^2_{20}$ 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 $\mathbf u \in \widetilde{\Sigma}^{K_{\mathrm{field}}}_M(G)$ a shared-pool multi-formation state. Let

U(\mathbf u) = \sum_{j=1}^{K_{\mathrm{field}}} u^{(j)},\qquad A^\varepsilon(\mathbf u) = \{j : m_j(\mathbf u) > \varepsilon\},\qquad K_{\mathrm{act}}^\varepsilon = |A^\varepsilon|,

and

K_{\mathrm{bar}}^{\ell_{\min}}(U;G) = \#\{[d,b] \in \mathrm{Bars}_0^{\mathrm{term}}(U;G) : b - d \ge \ell_{\min}\}

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

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

\boxed{\;K_{\mathrm{bar}}^{\ell_{\min}}(U(\mathbf u);G) = K_{\mathrm{act}}^\varepsilon(\mathbf u),\;}

and the map

\mathcal A_{\mathrm{bar}} : A^\varepsilon(\mathbf u) \to \mathrm{Bars}_0^{\mathrm{term}}(U;G), \qquad \mathcal A_{\mathrm{bar}}(j) = \text{the unique dominant bar with birth in } N_j^r,

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

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

Tag	Statement (abbreviated)
P0	Terminal-death convention for $H_0$ superlevel persistence.
P1	Deterministic tie convention $\prec$ on vertex orderings.
P2	Active mass + connected $\delta$ -support per slot.
P3	LG-1 — disjoint active neighborhoods $N_j^r \cap N_k^r = \emptyset$ .
P4	LG-2 — low boundary collar $\max_{\partial N_j^r} U \le b_j - \ell_{\min} - r_{\mathrm{assoc}}$ .
P5	LG-4 — background suppression on $U$ (not just $R_{\mathrm{inact}}$ ).
P6	Birth height $b_j \ge h_{\min} \ge \ell_{\min}$ .
P7	Decay-to-cut (heterogeneous): per-slot tail bound $u^{(\ell)}(x) \le \psi_\ell(d_G(x, S_\ell^\delta))$ + cut bridge bound.
P8	Tightened H6 second-bar bound on $G_j^r$ : $\ell_{j,2}(u^{(j)}; G_j^r) \le \ell_{\min} - 3\rho_{\mathrm{pert}}$ .
P9	NE-2 perturbation control $\\|R_j\\|_{\infty, N_j^r} \le \rho_{\mathrm{pert}}/2$ .
P10	Inactive residual suppression $\\|R_{\mathrm{inact}}\\|_\infty \le \ell_{\min} - \rho_{\mathrm{res}}$ .
P11	Margin ledger $h_{\min} - \max_{k \neq j} B_{jk} \ge \ell_{\min} + r_{\mathrm{assoc}} + r_{\mathrm{birth}}$ .

The L1-J regime is empirically non-vacuous (L1-I 439/1920 = 22.9% feasible on $T^2_{20}$ 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 $j \in A^\varepsilon$ contributes a dominant bar of length $\ge \ell_{\min}$ :

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 $\ge \ell_{\min}$ .
LG-3 inter-neighborhood bridge (a structural consequence of P3 + P4) prevents premature mergers between distinct active neighborhoods at superlevel heights $\ge b_j - \ell_{\min}$ .
$h_{\min} \ge \ell_{\min}$ (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 $|A^\varepsilon| = K_{\mathrm{bar}}^{\ell_{\min}}$ 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 $q_j^U = \arg\max^\prec_{x \in N_j^r} U(x)$ 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 $K_{\mathrm{bar}}^{\ell_{\min}} = K_{\mathrm{act}}^\varepsilon$ holds only on states satisfying every clause of $(P0)$ – $(P11)$ . Outside the L1-J regime, the bridge can fail.
Does NOT establish $K_{\mathrm{soft}}^\phi = K_{\mathrm{act}}$ . The soft-count corollary additionally requires $\phi \in \Phi_{\mathrm{res}}$ 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:

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.
It connects two count regimes. Chart-level active count $K_{\mathrm{act}}^\varepsilon$ (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.
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 $\phi \in \Phi_{\mathrm{res}}$ 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).

Statement#

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

Proof idea#

Status#

Why this is a hero#

Logical dependencies#

See also#