canonical_growth_closure_preservation
plain-language theorem explainer
Packages four growth-closure facts into one certificate: after geometric closure, any nontrivial multilevel hierarchy is growth-oriented, has base ratio φ, equals the original levelwise iff the first step was already the φ-step, and is idempotent under re-closure. The T5→T6 self-similarity bridge cites it. Term-mode structure assembly from four prior lemmas.
Claim. For every nontrivial multilevel composition $M$ (a sequence of positive real levels with at least three levels), the canonically growth-closed form of $M$ satisfies the growth-closure preservation certificate: it is growth-oriented, its base ratio equals $\varphi$, its levels equal those of $M$ for all indices if and only if $M$'s first adjacent step is multiplication by $\varphi$, and re-applying growth closure leaves the levels unchanged.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. 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.
A nontrivial multilevel composition is a positive real sequence of levels with at least three entries. Growth closure keeps the original base level and base ratio, then generates every higher level geometrically. The certificate structure GrowthClosurePreservation packages four properties of that normal form: canonical growth orientation, base ratio equal to $\varphi$, exact levelwise preservation equivalent to an original $\varphi$-step, and idempotence of the closure.
Upstream lemmas already prove each field separately: the closed hierarchy grows at the base step, its base ratio is $\varphi$ by direct simplification, levels match the original iff the first step is $M_0\cdot\varphi$, and closing twice does nothing.
proof idea
Pure term-mode structure construction. Each field of the certificate is filled by an existing lemma applied to $M$:
- growth orientation from
growthClosedMultilevelComposition_growth; - base ratio $\varphi$ from
growthClosedMultilevelComposition_base_ratio; - exact-preservation equivalence from
growthClosedLevels_eq_original_iff_phi_step; - idempotence from
growthClosedMultilevelComposition_idempotent_levels.
No new algebra; the proof is a one-shot witness assembly.
why it matters
Sits on the T5→T6 bridge. Downstream, t5_to_t6_bridge_holds needs a theorem-backed self-similarity certificate once J-uniqueness is in hand; this declaration supplies the canonical growth-closure half of that bridge (realized closed scale forces $\varphi$, and exact preservation is equivalent to already having the $\varphi$-step).
In the forcing chain, T5 uniqueness of $J$ plus discrete self-similarity force $\varphi$ (T6). Growth closure is the normal-form operation that makes “self-similar hierarchy” precise: keep the base, expand geometrically, and demand the base ratio be $\varphi$. Without this certificate, the T5→T6 step would remain a named hypothesis rather than a proved bridge.
No open scaffold: claim status is proved, zero sorry.
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