T7-Enhanced — Non-Idempotent Metastability Advantage

Hero · Phase + Stability group · Cat A. Source: C-0007 / P-0007. Verification: E-0008, E-0009 (residence time, $p=0.037$). Canonical version: CV-1.0. Full proof: Canonical Spec — Part 5 · §13.

Statement (canonical T7-Enhanced)#

T7-Enhanced (canonical line 1042). Non-trivial constrained minimizers of the SCC energy have strictly larger minimum Hessian eigenvalue than corresponding Allen-Cahn minimizers, due to the self-referential closure correction. Concretely, the closure term ${\lambda }_{\mathrm{cl}}{\mathcal{E}}_{\mathrm{cl}}$ adds a positive-definite Hessian contribution at closure fixed points (Gram matrix $2(I-J_{\mathrm{Cl}}{)}^T(I-J_{\mathrm{Cl}})$), which raises the minimum eigenvalue of the constrained Hessian beyond the Allen-Cahn baseline.

This is a local curvature result; the gap between minimum Hessian eigenvalue and actual energy barrier height (saddle energy minus minimizer energy) requires Morse-theoretic analysis that has not been carried out at the multi-formation level (open).

Distinction from T3/T6-Stability — eigenvalue COUNT vs eigenvalue MAGNITUDE#

T7-Enhanced is often confused with T3/T6-Stability (a separate Cat A theorem). The two address different aspects of the same closure-Hessian:

• T3/T6-Stability — eigenvalue count. At a non-idempotent closure fixed point with $\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}<1$, the closure Hessian $H_{\mathrm{cl}}=2(I-J_{\mathrm{Cl}}{)}^T(I-J_{\mathrm{Cl}})$ is strictly positive definite with $n$ of $n$ positive eigenvalues. For idempotent closure, $J_{\mathrm{Cl}}$ is a projection of rank $k<n$, so $H_{\mathrm{cl}}$ is semi-definite with at most $(n-k)/n$ positive eigenvalues. This is a count statement (no flat directions in the non-idempotent case).

• T7-Enhanced — eigenvalue magnitude. Among non-trivial constrained minimizers, the minimum Hessian eigenvalue is strictly larger under SCC than under Allen-Cahn — the closure correction not only removes flat directions (T3/T6) but also raises the smallest curvature among all directions. This is a magnitude statement (deeper basin).

The two are complementary: T3/T6 gives the count, T7-Enhanced gives the magnitude floor.

Proof idea#

Gram matrix structure. Write $A=I-J_{\mathrm{Cl}}$. Then $H_{\mathrm{cl}}=2A^{T}A$ is the doubled Gram matrix of $A$. By standard linear algebra, $A^{T}A$ is positive definite iff $A$ has trivial kernel (full rank).

Non-idempotent case. $\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}<1$ implies $J_{\mathrm{Cl}}$ has all eigenvalues strictly inside the unit disk, so $A=I-J_{\mathrm{Cl}}$ has all eigenvalues bounded away from 0 (real parts $\ge 1-\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}>0$). Hence $A$ is invertible, $A^{T}A$ is positive definite, and $H_{\mathrm{cl}}$ has $n$ positive eigenvalues.

Idempotent case. If $J_{\mathrm{Cl}}^2=J_{\mathrm{Cl}}$, then $J_{\mathrm{Cl}}$ is a projection of some rank $k$. $A=I-J_{\mathrm{Cl}}$ is also a projection (the complementary projection), with rank $n-k$. So $A^{T}A$ has rank $n-k$ and at most $n-k$ positive eigenvalues; the remaining $k$ eigenvalues are zero. $\square$

Why this is a hero#

T7-Enhanced is the mathematical payoff of SCC's signature commitment to non-idempotent closure. The theory deliberately avoids the classical topological closure axiom ${\mathrm{Cl}}^2=\mathrm{Cl}$, framing closure as a stabilization tendency (A3) rather than a completed operation. This commitment is philosophical (closure is dynamic, not static; trajectory carries information) — but T7-Enhanced shows it is also strictly stronger mathematically.

Concretely: at a closure fixed point, the energy landscape's Hessian receives a strictly positive-definite contribution from the closure term, which deepens the basin (T7-Enhanced raises the minimum eigenvalue) and removes flat directions (T3/T6 secures the count). Together they give SCC formations a curvature floor strictly above the Allen-Cahn baseline.

Corroborating empirical evidence (not part of the theorem statement). Residence-time measurements in E-0008 / E-0009 ($p=0.037$) exceed the prediction from Hessian curvature alone, consistent with the picture that non-idempotent fixed points sit at locally curved minima with strictly fewer flat directions than idempotent counterparts. The residence-time framing is empirical corroboration of the variational statement, not a re-statement of T7-Enhanced — the theorem itself is purely about the local Hessian eigenvalue floor.

Caveat (Hessian vs barrier). T7-Enhanced is a local curvature result. The gap between minimum Hessian eigenvalue and actual energy barrier height (saddle energy minus minimizer energy) requires Morse-theoretic analysis that has not been carried out at the multi-formation level — see open problems on multi-formation σ-framework Phase 5. The single-formation case is uncontroversial.

Logical dependencies#

• Builds on: T6 (closure has unique fixed point at $a_{\mathrm{cl}}<4$ with $\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}<1$).
• Builds into: T-Persist-1(b) (basin radius via energy sublevel set uses Hessian eigenvalues), T-Persist-K-Sep (per-formation persistence inherits T7-Enhanced curvature), CN14 (closure expands multi-formation stability).

See also#

• Full proof in canonical: Canonical Spec §13 T7-Enhanced
• A3 stabilization tendency commitment: Canonical Spec Part 2 §6 Group A · Interpretive Remark
• T6 closure fixed point (the prerequisite): T6 hero page