minimalHierarchy_first_closure_law
plain-language theorem explainer
Every minimal discrete hierarchy carries a canonical first-closure law on its geometric scale sequence: there is some natural index that is the first nontrivial closure index and at which the scales close. Hierarchy and φ-ladder arguments cite this to convert the bare isClosed flag into the packaged first-closure data. The proof is a one-line reverse application of the iff linking isClosed to existence of CanonicalFirstClosureLaw.
Claim. If $H$ is a minimal discrete hierarchy (a geometric scale sequence closed under the first nontrivial composition step), then there exists $n \in \mathbb{N}$ such that the scale sequence of $H$ satisfies the canonical first-closure law at $n$: $n$ is the first nontrivial closure index and the scales close at that index.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Mid-chain, discrete geometric scale ladders appear in the φ-forcing layer (T6 and neighbors): self-similarity on a discrete ledger forces a geometric progression of scales.
A minimal hierarchy is exactly such a ladder together with the assertion that it is closed under the first nontrivial composition step (scales.isClosed). The canonical first-closure law packages that fact: the closure index must be the first nontrivial one, and the seed scales close there. Downstream commentary in the module records that this first nontrivial index is uniquely $2$, and that the old isClosed predicate is equivalent to closure at that canonical index.
The cost side of the foundation uses the shifted cost $H(x)=J(x)+1=\frac12(x+x^{-1})$, under which RCL becomes d'Alembert's equation; the hierarchy objects sit on the discrete geometric side forced after unique $J$ and φ.
proof idea
One-line term proof. Apply the reverse direction of the biconditional canonical_first_closure_law_iff_isClosed at H.scales, feeding the field H.minimalClosure (the witness that the scale sequence is closed). That iff converts bare closure into existence of an index $n$ carrying CanonicalFirstClosureLaw.
why it matters
In the Complete Inevitability Chain, discrete self-similar scale structure is the bridge from unique $J$ (T5) and forced φ (T6) toward the eight-tick octave (T7) and $D=3$ (T8). Minimal hierarchies are the lean data for that ladder: geometric scales plus first nontrivial closure.
This theorem upgrades the raw isClosed flag on a minimal hierarchy into the canonical first-closure package (first nontrivial index plus closure there). That package is the interface the module uses when identifying the first nontrivial index with $2$ and aligning old closure language with the canonical index. No downstream consumers are recorded yet; the result is local glue inside Foundation.UnifiedForcingChain rather than a cited leaf of T7/T8.
It does not itself force φ, the eight-tick period, or spatial dimension; it only rephrases minimality of a hierarchy in the language of canonical first closure.
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