T6 — Closure Fixed Point (Banach Contraction)

Hero · Foundation group · Cat A. Source: C-0006a / P-0006a (existence) + C-0006b / P-0006b (uniqueness, contraction). Verification: E-0005, E-0006. Canonical version: CV-1.0. Full statement and proof: Canonical Spec — Part 5 · §13.

Statement#

Let ${\mathrm{Cl}}_t:[0,1{]}^n\to [0,1{]}^n$ be the sigmoid closure operator

T6a (existence). ${\mathrm{Cl}}_t$ has at least one fixed point on $[0,1{]}^n$.

T6b (uniqueness + contraction). When $a_{\mathrm{cl}}<4$, ${\mathrm{Cl}}_t$ has a unique fixed point, and iterates ${\mathrm{Cl}}_t^{(k)}(u)$ converge geometrically at rate $a_{\mathrm{cl}}/4$ for every initial $u$.

Proof idea#

T6a. Brouwer fixed-point theorem. ${\mathrm{Cl}}_t$ maps $[0,1{]}^n$ to itself (sigmoid output is in $(0,1)$) and is continuous. Brouwer gives at least one fixed point.

T6b. Banach contraction mapping theorem. The Lipschitz constant of ${\mathrm{Cl}}_t$ is

since $\max {\sigma }^{\prime }=1/4$ for the logistic sigmoid. When $a_{\mathrm{cl}}<4$, $L<1$, so ${\mathrm{Cl}}_t$ is a strict contraction and Banach's theorem applies: unique fixed point, geometric convergence at rate $L\le a_{\mathrm{cl}}/4$. The bound $a_{\mathrm{cl}}<4$ is tight: at $a_{\mathrm{cl}}=4$ the Lipschitz bound saturates and contraction fails. $\square$

Why this is a hero#

Closure is the central self-referential operator of SCC: it is the mechanism by which the cohesion field becomes self-supporting. T6 is the convergence guarantee for that operator. Without T6, "self-completion" is just a description; with T6, it is a well-defined mathematical operation with a unique destination and a known convergence rate.

The parameter constraint $a_{\mathrm{cl}}<4$ is a signature commitment of the theory. SCC operates in the contraction regime by design — the alternative ($a_{\mathrm{cl}}\ge 4$, multiple fixed points) is reserved for explicit multi-formation work via the K-field architecture, not for the single-formation core. T6 makes that design choice rigorous: contraction is provable below threshold, and the threshold is sharp.

A subtle but important point: the closure operator is a contraction, not a projection (CN1 commitment). The trajectory carries structural information even though the destination is unique. Path-dependence in SCC arises at the energy landscape level (T8-Core multi-well, T7-Enhanced metastable basins), not at the closure-operator level. This two-landscape distinction (CN9) is essential and easily confused.

Logical dependencies#

• Builds on: T20 (axiom consistency — A3 contraction must be compatible with A1', the conditional extensivity, at $a_{\mathrm{cl}}<4$).
• Builds into: T7-Enhanced (the closure Hessian at a non-idempotent fixed point with $\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}<1$ is strictly PD), T-Bind-Proj/Full (residual bound exploits contraction inversion), every persistence theorem (basin radius depends on contraction rate).

See also#

• Full proof in canonical: Canonical Spec §13 T6a/T6b/T6-Stability
• Why non-idempotence matters: T7-Enhanced hero page
• The two-landscape structure (closure FP vs energy minima): Canonical Spec Part 2 §6 Group A · CN1, CN9