rcl_seed_posting_surface
plain-language theorem explainer
For positive reals x and y the posting potential obeys the d'Alembert law Π(xy)+Π(x/y)=2 Π(x) Π(y). Anyone working the Recognition Composition Law at potential level, or seed-posting closure in the forcing chain, cites this surface. Proof is a one-line term applying the canonical posting d'Alembert identity. It is the RCL surface demanded by seed posting semantics.
Claim. For all real $x,y>0$, if $\Pi$ is the posting potential, then $\Pi(xy)+\Pi(x/y)=2\,\Pi(x)\,\Pi(y)$.
background
The Unified Forcing Chain module proves T0–T8 as inevitabilities from the cost foundation. The axiom bundle is the Recognition Composition Law (RCL), normalization $F(1)=0$, and calibration $F''(1)=1$. Classically RCL is written for the J-cost $J(x)=(x+x^{-1})/2-1$ (also $\cosh(\log x)-1$); an equivalent surface uses the posting potential $\Pi$, related by the shift $J=\Pi-1$.
Upstream, the posting d'Alembert theorem states exactly $\Pi(xy)+\Pi(x/y)=2\Pi(x)\Pi(y)$ and records that this identity is equivalent to RCL via that shift. Seed posting semantics interpret a level sequence as the posting-potential image of a positive scale $\sigma$ at seed indices 0, 1, and 2; the remaining closure rule is then additive posting at the potential level.
proof idea
One-line term proof. The quantified identity is definitionally the posting d'Alembert theorem (PostingExtensivity.posting_dalembert), so the body is just that theorem used as a term. No extra algebra or case splits.
why it matters
Packages the posting-potential RCL surface for seed posting semantics inside the complete inevitability chain. Downstream it is consumed by the T5-to-T6 self-similarity bridge theorem, which asserts that once J-uniqueness (T5) is available the bridge to $\phi$ as self-similar fixed point (T6) is theorem-backed. T5 itself is uniqueness of $J$ from d'Alembert plus normalization and calibration; having RCL available at the $\Pi$ surface (not only at $J$) lets seed-hierarchy closure speak the same language as the posting-extensivity layer that feeds that bridge.
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