T20 — Axiom Consistency (A1' resolves A1↔A3 incompatibility)

Hero · Foundation group · Cat A. Source: C-0020 / P-0020. Verification: E-0025. Canonical version: CV-1.0. Full statement and proof: Canonical Spec — Part 5 · §13.

Statement

The sigmoid closure operator $\mathrm{Cl}_t(u)(x) = \sigma\!\big(a_{\mathrm{cl}}\,((1-\eta_{\mathrm{cl}})u(x) + \eta_{\mathrm{cl}}(P_t u)(x) - \tau_{\mathrm{cl}})\big)$ satisfies:

A1' (conditional extensivity): $\mathrm{Cl}_t(u)(x) \geq u(x)$ whenever $u(x) \leq c^*$ and $(P_t u)(x) \geq u(x)$ , where $c^*$ is the unique scalar fixed point of $\sigma(a_{\mathrm{cl}}(c - \tau_{\mathrm{cl}}))$ in $(0,1)$ .
A2 (monotonicity): $u \leq v$ pointwise $\Rightarrow$ $\mathrm{Cl}_t(u) \leq \mathrm{Cl}_t(v)$ pointwise — unconditional.
A3 (contraction): iterates form a Cauchy sequence; geometric rate $a_{\mathrm{cl}}/4$ when $a_{\mathrm{cl}} < 4$ .
A4 (continuity): $\mathrm{Cl}_t$ is continuous in any $\ell^p$ topology — unconditional.

The original axiom A1 (weak extensivity, $\mathrm{Cl}_t(u) \geq u$ unconditionally) and A3 (contraction, $a_{\mathrm{cl}} < 4$ ) are incompatible: A1 at $u(x) = 0.9$ requires $a_{\mathrm{cl}} \geq 5.49$ , contradicting $a_{\mathrm{cl}} < 4$ .

A1' resolves the tension by restricting extensivity to sites below the self-support threshold $c^*$ .

Proof idea

Direct computation. Define $g(u) = \sigma(a_{\mathrm{cl}}(u - \tau_{\mathrm{cl}})) - u$ . Then $g(0) = \sigma(-a_{\mathrm{cl}}\tau_{\mathrm{cl}}) > 0$ and $g(c^*) = 0$ . Since $g'(c^*) = a_{\mathrm{cl}} \sigma'(a_{\mathrm{cl}}(c^* - \tau_{\mathrm{cl}})) - 1 < 0$ (because $a_{\mathrm{cl}}/4 < 1$ ), $g$ is positive on $[0, c^*)$ . When $(P_t u)(x) \geq u(x)$ and $u(x) \leq c^*$ , the pre-activation $z(x) \geq a_{\mathrm{cl}}(u(x) - \tau_{\mathrm{cl}})$ , so $\mathrm{Cl}(u)(x) \geq u(x) + g(u(x)) \geq u(x)$ . This proves A1'.

A2 from monotonicity of $\sigma$ and $P_t$ . A3 from $\max \sigma' = 1/4$ giving Lipschitz constant $\leq a_{\mathrm{cl}}/4 < 1$ . A4 from composition of continuous functions.

A1 fails: $\sigma^{-1}(0.9)/\!(0.9 - \tau_{\mathrm{cl}}) \approx 5.49 > 4$ . $\square$

Why this is a hero

T20 is the theory's foundational legitimacy. Without it, the axiom system is just a wishlist; with it, the wishlist is shown to be coherent.

The substantive content of T20 is the identification of A1↔A3 incompatibility and the resolution via A1'. This is not a routine consistency check — it is a discovery that the original axiom system was inconsistent for the sigmoid realization, and a non-trivial refinement that captures the correct structural intent (closure builds up cohesion below the self-support threshold $c^*$ , corrects above it) while preserving compatibility with contraction.

The conditional form of A1' is also load-bearing for the theory's interpretation. SCC's closure is self-regulating, not merely extensive: it is bounded above by the scalar fixed point. This bounded-above property is what allows closure to coexist with the volume constraint $\Sigma_m$ (saturating $u \equiv 1$ everywhere is forbidden) and with the multi-formation regime (closure does not collapse formations to total dominance).

Logical dependencies

Builds on: nothing — this is the consistency theorem at the bottom of the stack.
Builds into: T6 (which uses A3 contraction at $a_{\mathrm{cl}} < 4$ ), every existence/stability result that invokes the closure operator.

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