canonical_seed_posting_of_level_two
plain-language theorem explainer
Given a nontrivial multilevel composition whose level-2 size equals the sum of the sizes at levels 0 and 1, the canonical seed-posting certificate holds at index 2. Anyone assembling the T5→T6 hierarchy bridge cites this to pin the seed post without an arbitrary index parameter. The proof is a three-field structure constructor: a prior index lemma, reflexivity for equality with 2, and the given additive size law.
Claim. Let $M$ be a nontrivial multilevel composition (a positive level-size map $\mathbb{N}\to\mathbb{R}$ with at least three levels). If the level size of $M$ at the canonical seed-post index equals the sum of its sizes at levels $0$ and $1$, then $M$ admits a canonical seed-posting operation at that index: the index equals $2$, and the posted seed level has additive size $M_0+M_1$.
background
The Unified Forcing Chain module proves that T0–T8 are forced from the cost foundation (Recognition Composition Law with normalization and calibration). In that chain, T5 pins the unique cost $J(x)=(x+x^{-1})/2-1$, and T6 forces $\varphi$ as the self-similar fixed point of the discrete ledger hierarchy.
A nontrivial multilevel composition is a positive real-valued level-size function with at least three levels. The full all-pairs posting operation is stronger than the T5→T6 bridge needs. As the seed-posting structure records: the hierarchy recurrence only uses the primitive closure of the seed pair—level $0$ posted with level $1$ closes at level $2$, and the posted size is the sum of the two seed sizes.
The certificate isolates exactly that local datum: a seed index certified as canonical, the equality of that index with $2$, and the additive size law at the posted level.
proof idea
Term-mode structure construction with three fields. The seed-index field is discharged by the prior fact that the canonical seed-post index satisfies the canonical-seed-post-index predicate. The field asserting the post index equals $2$ is reflexivity, since that canonical index is definitionally $2$. The additive size law is the input hypothesis, passed through unchanged. No further algebraic work is required.
why it matters
This removes an arbitrary post-index parameter from the hierarchy bridge between unique $J$ (T5) and $\varphi$-forcing (T6). Downstream consumers of the seed-posting certificate can work at the fixed index $2$ rather than quantifying over an unspecified post index. The module's complete inevitability chain treats T6 as forced self-similarity in the discrete ledger; pinning the seed post supplies the local size-law input to that recurrence. No recorded used-by edges yet, so the declaration is presently a leaf certificate inside the forcing-chain assembly rather than a named parent of a later theorem.
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