phiUniformClosed_seed_size_law
plain-language theorem explainer
The φ-uniform normal form of any nontrivial multilevel hierarchy obeys the canonical seed-size law: level size at the forced seed index equals the sum of the two seed levels. Hierarchy and T6 (φ-forcing) arguments cite this when assembling the full φ-uniform closure certificate. The proof reduces the identity to φ² = φ + 1 and distributes the base scale.
Claim. For any nontrivial multilevel composition $M$ (positive level sizes, at least three levels), the $\varphi$-uniform closed form of $M$ satisfies the canonical seed-size law: if $\ell_k$ denotes its level sizes and $s$ is the canonical seed post index (forced to $2$), then $\ell_s = \ell_0 + \ell_1$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 forces the golden ratio $\varphi$ as the unique positive self-similar scale: self-similarity in a discrete ledger requires $r^2 = r + 1$, and $\varphi$ is the unique positive solution.
A nontrivial multilevel composition is a positive real sequence of level sizes with at least three positive levels. The $\varphi$-uniform normal form rebuilds such a hierarchy so adjacent ratios are constantly $\varphi$. The canonical seed-size law isolates one remaining posting identity: the seed index is already forced to $2$, and posting levels $0$ and $1$ must close with additive size at that index. The structure is kept separate from the posting operation so later steps can derive it from RCL or posting-potential composition.
proof idea
Tactic proof. Rewrite the goal as equality of $\varphi$-uniform level sizes at the canonical seed index versus levels $0$ and $1$. Unfold the uniform level formula and substitute the definition of the canonical seed post index. Invoke $\varphi_\mathrm{satisfies}$ to obtain the golden constraint $\varphi^2 = \varphi + 1$ (commuting the summands), then simplify with distributivity of multiplication over addition. The geometric powers collapse exactly to the additive seed identity.
why it matters
This is one of the three legs of the canonical $\varphi$-uniform closure certificate: that parent packages uniform scale, growth orientation, and this seed-size law into a single PhiUniformClosure witness. The companion characterization theorem uses it on the right-hand side of an iff: the $\varphi$-uniform normal form agrees with the original hierarchy exactly when the original already satisfies uniform scale, growth, and seed-size law.
In the forcing chain this sits under T6 ($\varphi$ forced by self-similarity). It does not itself force $\varphi$; it shows that once levels are rebuilt with ratio $\varphi$, the seed posting identity holds automatically from the golden equation. That clears a named hypothesis interface for later RCL-based derivation of seed closure without free scale parameters.
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