cft-cmi

Hypothesis H1 of the tangent problem: g_Q-differentiability along the scaled family

Status: conjectural — believed, with partial evidence; not proved. Stated under hypotheses H1, quoted below.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: conjecture · Knowledge-base id: tangent-h1-gq-differentiability

Statement

For the first-order datum (zeta, N_1, Y_1) under consideration, the exact defect delta(s) of the scaled feasible family satisfies g_Q(delta(s) - s delta^(1)) = o(s^2) as s -> 0; this is known for the isometric orbit (S10 Lemma 6.1) and for finite-rank N_1, Y_1 with range in a modular window, and is a hypothesis in general because the g_Q-norm is strictly stronger than the Hilbert-Schmidt norm.

Hypotheses

The statement is made under named hypotheses. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H1.

Hypothesis H1 of the tangent problem: g_Q-differentiability along the scaled family.

How to verify

rigor/exact_optimum_tangent_problem.tex Def. 3.7 (hyp:H1) and Remark 3.8 (rem:H1), :417-431: known for the isometric orbit (S10 Lemma 6.1 <1>3-<1>4, numerically S10 Sec. 7(d) to 1e-3) and for finite-rank N_1, Y_1 with range in a modular window, a hypothesis in general; used by Thm. 3.9 (thm:upper, :433-456, minimising-sequence clause :439-441) and Thm. 3.17 (thm:value); status in Sec. 7, Conditional item 1 (:1601-1606: it fails for the inadmissible contraction family of Rem. 4.8) and To be supplied (:1628-1636).

Evidence

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A4.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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