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 ${\mathrm{Cl}}_t(u)(x)=\sigma (a_{\mathrm{cl}}((1-{\eta }_{\mathrm{cl}})u(x)+{\eta }_{\mathrm{cl}}(P_{t}u)(x)-{\tau }_{\mathrm{cl}}))$ satisfies:

• A1' (conditional extensivity): ${\mathrm{Cl}}_t(u)(x)\ge u(x)$ whenever $u(x)\le c^{\ast }$ and $(P_{t}u)(x)\ge u(x)$, where $c^{\ast }$ is the unique scalar fixed point of $\sigma (a_{\mathrm{cl}}(c-{\tau }_{\mathrm{cl}}))$ in $(0,1)$.
• A2 (monotonicity): $u\le v$ pointwise $\Rightarrow$ ${\mathrm{Cl}}_t(u)\le {\mathrm{Cl}}_t(v)$ 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, ${\mathrm{Cl}}_t(u)\ge u$ unconditionally) and A3 (contraction, $a_{\mathrm{cl}}<4$) are incompatible: A1 at $u(x)=0.9$ requires $a_{\mathrm{cl}}\ge 5.49$, contradicting $a_{\mathrm{cl}}<4$.

A1' resolves the tension by restricting extensivity to sites below the self-support threshold $c^{\ast }$.

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 $g(c^{\ast })=0$. Since $g^{\prime }(c^{\ast })=a_{\mathrm{cl}}{\sigma }^{\prime }(a_{\mathrm{cl}}(c^{\ast }-{\tau }_{\mathrm{cl}}))-1<0$ (because $a_{\mathrm{cl}}/4<1$), $g$ is positive on $[0,c^{\ast })$. When $(P_{t}u)(x)\ge u(x)$ and $u(x)\le c^{\ast }$, the pre-activation $z(x)\ge a_{\mathrm{cl}}(u(x)-{\tau }_{\mathrm{cl}})$, so $\mathrm{Cl}(u)(x)\ge u(x)+g(u(x))\ge u(x)$. This proves A1'.

A2 from monotonicity of $\sigma$ and $P_t$. A3 from $\max {\sigma }^{\prime }=1/4$ giving Lipschitz constant $\le a_{\mathrm{cl}}/4<1$. 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 $c^{\ast }$, 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.

See also#

• Full statement of Group A axioms with proofs: Canonical Spec Part 2 §6 Group A
• The scalar fixed point $c^{\ast }$ and self-regulation interpretation: same page, A1' commentary
• T6 closure fixed point (uses A3): T6 hero page