seedClosureEquiv_self_iff_seed_size_law
plain-language theorem explainer
A nontrivial multilevel hierarchy is seed-closure-equivalent to itself if and only if its seed post already has additive size (level at the canonical seed index equals level 0 plus level 1). Hierarchy and T5–T6 bridge arguments cite this as the fixed-point criterion for seed closure. The proof is a two-line constructor: forward projects the level-2 field; reverse applies the reflexivity lemma under the size law.
Claim. Let $M$ be a nontrivial multilevel composition (positive level sizes, at least three levels). Then $M$ is seed-closure-equivalent to itself if and only if $M$ satisfies the canonical seed-size law: the level at the canonical seed post index equals the sum of levels $0$ and $1$.
background
The module UnifiedForcingChain develops the complete inevitability chain T-1 through T8 from the Recognition Composition Law plus normalization and calibration. Mid-chain, hierarchies of positive real level sizes encode discrete self-similar structure that later forces $\varphi$ (T6) after unique $J$ (T5).
A nontrivial multilevel composition is a map $\mathbb{N}\to\mathbb{R}$ with strictly positive values and at least three occupied levels. The canonical seed-size law isolates the remaining size constraint once the seed index is fixed at $2$: posting levels $0$ and $1$ closes with additive size at that index.
Seed-closure equivalence is the forcing-relevant quotient between a hierarchy and a seed-closed replacement: non-seed levels agree, and the seed level is the canonical additive closure of levels $0$ and $1$. Self-equivalence is therefore a fixed-point statement for that quotient, not a trivial reflexivity of an ordinary equality.
proof idea
Term-mode constructor on the biconditional.
Forward: from a self seed-closure equivalence $h$, project the level2 field. By definition that field is exactly $N.\mathrm{levels}(\mathrm{canonical_seed_post_index})=M.\mathrm{levels},0+M.\mathrm{levels},1$ with $N=M$, which is the seed-size law certificate.
Reverse: apply the existing reflexivity lemma seedClosureEquiv_refl_of_seed_size_law at $M$, which builds a self seed-closure equivalence once the size law is given.
No further algebraic work; the two structure fields are definitionally the same additive identity.
why it matters
This lemma is the self-equivalence leg of the canonical seed-closure preservation certificate, which packages closure equivalence, exact level preservation iff the size law, and self-equivalence iff the size law. That certificate is how the forcing spine treats seed closure as a quotient rather than an ad hoc rewrite.
Downstream it is consumed by t5_to_t6_bridge_holds, the theorem-backed T5-to-T6 self-similarity bridge: unique $J$ (T5) plus internal hierarchy dynamics force $\varphi$ as the self-similar fixed point (T6 in the primer chain). Without a clean criterion for when a hierarchy is already seed-closed relative to itself, the bridge cannot separate genuine self-similarity from residual seed-size slack.
In framework terms this is bookkeeping on the T5→T6 step of the complete inevitability chain, not a new physical constant. It closes the seed-size interface so later hierarchy dynamics can derive the size law from RCL/posting-potential composition rather than assume it.
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