realizedClosedScale_canonical_seed_size
plain-language theorem explainer
Any realized closed-scale model on a closed observable framework automatically satisfies the canonical seed-size law: the multilevel level at the forced seed index equals the sum of the two seed levels. Hierarchy and φ-ladder arguments cite this when closing seed posting. The proof rewrites the certificate to the model's additive posting identity and discharges it by the realized closed-scale additive-posting lemma plus commutativity of addition.
Claim. Let $F$ be a closed observable framework and $H$ a realized closed-scale model on $F$. The multilevel composition built directly from $H$ obeys the canonical seed-size law: if $s$ denotes the canonical seed post index (already forced to equal $2$), then the composition's level at $s$ equals the sum of its levels at $0$ and at $1$.
background
The Unified Forcing Chain module proves T0–T8 as inevitabilities from the Recognition Composition Law together with normalization and calibration. Hierarchy posting is organized by multilevel compositions whose levels record successive observable values along an orbit of a closed framework.
A closed observable framework supplies a state space, a tick map $T$, and an observable $r$. A realized closed-scale model packages a base state whose orbit under $T$ realizes a closed scale hierarchy. The direct multilevel composition of such a model is the nontrivial multilevel composition with levels $k \mapsto r(T^k(\mathrm{base}))$.
The canonical seed-size law isolates one residual size identity after the seed index has already been forced to $2$: posting the two seed levels is additive at that index. As its doc-comment states, the certificate is "intentionally named separately from the posting operation so that the next closure step can derive this law from RCL/posting-potential composition directly." The concrete identity used here is the additive posting lemma for realized closed-scale models.
proof idea
The structure field asks that multilevel levels at the canonical seed post index equal the sum of levels $0$ and $1$. Unfolding the multilevel composition, that is exactly $r(T^{s}(\mathrm{base})) = r(T^{0}(\mathrm{base})) + r(T^{1}(\mathrm{base}))$ with $s$ the canonical seed post index.
The proof changes the goal to that equality, invokes the upstream lemma additive_posting_of_realized_closed_scale for $F$ and $H$, then closes by simplification using the definition of the canonical seed post index and commutativity of addition. Short rewrite-and-apply; no new calculation.
why it matters
This certificate is the seed-size half of closing realized closed-scale models onto the canonical hierarchy normal form inside the Unified Forcing Chain. Downstream it feeds three siblings: the theorem that the multilevel composition has canonical base ratio $\varphi$ (T6: $\varphi$ forced as the self-similar fixed point); level-equivalence of the realized model to its $\varphi$-uniform normal form; and the bundled normal-form equivalence certificate combining uniformity and growth.
Together these steps connect closed-scale realization to the $\varphi$-ladder hierarchy used throughout RS mass and constant derivations. The local doc-comment frames the point cleanly: a realized closed-scale model directly supplies canonical seed closure, so seed posting need not be postulated separately once closed-scale realization is in hand.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.