t1_analytic_refinement_holds
plain-language theorem explainer
After the canonical scalar defect is available, the analytic T1 existence package still holds: defect diverges at zero, a unique RS-existent exists, and that existent is the unit 1. Anyone citing the old analytic MP surface once cost uniqueness is in play lands here. The proof is a three-field term constructor reusing the Law-of-Existence blow-up and the unique-existence lemmas.
Claim. The analytic T1 refinement holds: (i) for every real bound $C$ there is $\varepsilon>0$ such that the existence defect exceeds $C$ whenever $0<x<\varepsilon$; (ii) there is a unique real $x$ satisfying the RS-existence predicate; (iii) every RS-existent equals $1$.
background
The Unified Forcing Chain module derives T-1 through T8 as forced steps from the cost foundation (Recognition Composition Law, normalization, calibration). In that spine, T1 is the existence layer: nothing has infinite cost, so a unique physical unit must exist.
The analytic refinement is deliberately a downstream surface. Its structure doc states that it preserves the old defect-near-zero / unique-existent payload without placing that payload before cost uniqueness (T5) in the forcing order. The three fields are: defect blow-up near zero, unique RS-existence, and identification of that existent with 1.
Upstream, the Law of Existence supplies the sharp claim that for any bound $C$, defect exceeds $C$ in a right neighborhood of zero ("Nothing costs infinity"). Unique existence and the unit identification come from the ontology predicates on RS-existence.
proof idea
Pure term-mode structure constructor; no tactics. The nothing-infinite field is filled by the Law-of-Existence blow-up theorem (for every $C$, defect exceeds $C$ near zero). The unique-existent field is the RS unique-existence result. The physical-unit field is a one-line lambda that applies the forward direction of the one-point characterization of that unique existent (RS-existence implies equality to 1).
why it matters
Keeps the old analytic MP surface available inside the complete inevitability chain without reordering the spine. Downstream it feeds the T5-to-analytic-refinements bridge, whose doc-comment says that bridge "follows from T5 plus the closed-form identities of the analytic $J$-scaffolding." In primer terms this is the T1 slot (existence from cost), sitting under T5 $J$-uniqueness and well before T6 $\varphi$, T7 eight-tick, and T8 $D=3$. It closes no open scaffold; it is a proved packaging lemma that lets later analytic identities cite T1 cleanly.
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