RCLSeedPostingSemantics
plain-language theorem explainer
A Prop-valued certificate that interprets the first two hierarchy levels as positive posting potentials, takes the canonical seed post as their sum, and requires the posting potential to obey the d'Alembert/RCL identity at those values. Anyone proving the canonical seed-size law from RCL semantics cites it. There is no proof body: it is a hypothesis bundle, not a derived theorem.
Claim. For a nontrivial multilevel composition $M$, the RCL seed-posting semantics assert: (i) level $0$ equals $\mathrm{PostingPotential}(x)$ for some $x>0$; (ii) level $1$ equals $\mathrm{PostingPotential}(y)$ for some $y>0$; (iii) the canonical seed-post level equals level $0$ plus level $1$; (iv) for all $x,y>0$, $\mathrm{PostingPotential}(xy)+\mathrm{PostingPotential}(x/y)=2\,\mathrm{PostingPotential}(x)\,\mathrm{PostingPotential}(y)$.
background
The Unified Forcing Chain module aims to force T0–T8 from the cost foundation, with the Recognition Composition Law (RCL), normalization, and calibration as the axiom bundle. T5 pins the unique cost $J(x)=(x+x^{-1})/2-1$; T6 forces $\varphi$ as the self-similar fixed point of the discrete ledger.
PostingExtensivity already shows that the shifted $J$-cost posting potential satisfies the d'Alembert composition law (the RCL surface written multiplicatively). A multilevel composition $M$ supplies a level sequence; the missing link is an interpretation of the seed levels as posting-work sizes, plus additivity of the seed composite.
This structure is that interpretation certificate: levels 0 and 1 are positive posting potentials, the canonical seed post is their additive posting, and the RCL identity holds on the realizing values. It does not re-prove RCL; it packages the semantic bridge from the posting-potential theorem surface to hierarchy seed data.
proof idea
No proof. The declaration is a structure of type Prop with four fields. Instantiating it means supplying existential witnesses for levels 0 and 1 as positive posting potentials, the equality that the canonical seed-post index is the sum of those levels, and the universal RCL/d'Alembert identity for PostingPotential on positive reals. Downstream theorems pattern-match on these fields rather than deriving them here.
why it matters
This is the theorem-facing bridge from posting-potential RCL semantics to the concrete seed-size law. The immediate consumer is canonical_seed_size_law_of_rcl_posting, which reads off seed_post_additive to obtain CanonicalSeedSizeLaw M.
That seed-size law feeds the T5→T6 self-similarity bridge: once hierarchy levels are realized as additive posting work obeying RCL, closed observable frameworks with a realized hierarchy force the scale ratio to be $\varphi$ (primer landmark T6). The structure also records the formal obstruction noted on the T5→T6 bridge: bare closed-observable data do not force hierarchy fields such as additive posting; those must be supplied explicitly, so nothing is smuggled into the chain.
In the forcing narrative, this sits between T5 (unique $J$ / RCL surface) and T6 ($\varphi$ from self-similarity in the discrete ledger).
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