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

\mathrm{Cl}_t(u)(x) = \sigma\!\big(a_{\mathrm{cl}}\,((1-\eta_{\mathrm{cl}})\,u(x) + \eta_{\mathrm{cl}}\,(P_t u)(x) - \tau_{\mathrm{cl}})\big).

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 $u$ .

Two-panel plot: closure iteration profiles converging to a fixed point, and log-scale convergence rate. — Closure iteration profiles converge to a unique fixed point at geometric rate a_cl/4 in the contraction regime (a_cl below 4). Trajectory carries structural information; destination is unique (commitment notes CN1, CN9).

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

L \;\leq\; (\max \sigma') \cdot a_{\mathrm{cl}} \cdot \max(1 - \eta_{\mathrm{cl}}, \eta_{\mathrm{cl}}) \;\leq\; \frac{a_{\mathrm{cl}}}{4},

since $\max \sigma' = 1/4$ 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 $L \leq a_{\mathrm{cl}}/4$ . 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 ( $a_{\mathrm{cl}} \geq 4$ , 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 $\|J_{\mathrm{Cl}}\|_{\mathrm{op}} < 1$ is strictly PD), T-Bind-Proj/Full (residual bound exploits contraction inversion), every persistence theorem (basin radius depends on contraction rate).

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