cft-cmi

Exact (N,Y,V) chart of F and the tangent problem at the compression

Status: proved — a complete structured proof. Conditional. The statement is not unconditional: it rests on H1, H2, quoted below. Read the badge only together with them.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: tangent-problem-exact-parametrisation

Statement

Feasible set: (X,Z) in F iff N := 1 - X X* >= 0 and Y := Z - X Q_BB X* satisfies 0 <= Y <= N. Polar: X = (1-N)^{1/2} V with V : h_B -> h_D the polar partial isometry; V V* = 1 - 1_{{1}}(N), so the POLAR V is a co-isometry iff 1 is not an eigenvalue of N (sharp: for X = the forward shift no co-isometric factor exists); existence of SOME co-isometric factor is the weaker dim ker(1-N) <= dim ker X. Then V* = W U with W the compression unitary h_D -> h_B and U an isometry of h_D (onto iff V unitary). Scaling N = s N_1 (N_1 a bounded positive operator), Y = s Y_1, U_s = 1 + s zeta + o(s) with (U_s) a C_0-semigroup of isometries gives, exactly feasible at every small s, the first-order defect (as sesquilinear forms on C_c^inf) delta^(1) = ddelta + A(zeta,N_1,Y_1) with A_AA = 0, A_AD = -Q_AD zeta + (1/2) Q_AD N_1, A_DD = -(zeta* Q_DD + Q_DD zeta) + (1/2){N_1,Q_DD} - Y_1 (noise enters DD with a MINUS sign); ddelta = [Q,d] with d = E_I D_chi E_I (Lemma 6.1), and ddelta has zero AA block (locality of D_chi, chi = chi' = 0 on A) and zero DD block (Moebius rigidity, S10 Lemma 11.1). Boundary-moving (non-surjective) isometry directions: the absorption identity A(zeta,N_1,Y_1) = A(G, N_1 - 2 sigma, Y_1) holds with G = anti-symmetric part and sigma <= 0 the BOUNDARY FORM <sigma g,g> := Re<zeta* g,g> on dom(zeta*) (sigma <= 0 by Lumer-Phillips for the adjoint contraction semigroup; for BOUNDED zeta, sigma = 0 and zeta is anti-self-adjoint, so bounded isometry families give nothing beyond S10's class). This yields only an INCLUSION of cones (Thm 3.11(3) as amended): its right-hand side consists of the A(G, N_1', Y_1) with G = sigma - zeta* (zeta contained in G + sigma, zeta* = -G + sigma; exact_optimum_tangent_problem.tex:504, :506, :520, :539, :552; until 2026-09-24 the document wrote G := zeta* - sigma at :504, :520 and :552, corrected 2026-09-24, Correction C1) the anti-symmetric part of an ADMISSIBLE zeta, N_1' >= 0 a form and 0 <= Y_1 <= N_1', i.e. G is described by how it arises; that G itself generates an isometry semigroup is not claimed (it would need C_c^inf(D) to be a core for zeta, true for zeta_0 and zeta_inf, not assumed in general), and the right-hand side must not be enlarged to all anti-symmetric G. The inclusion cannot be reversed - +lambda D_chi|_D is anti-symmetric but generates an OUTWARD flow, and admitting it would give delta^(1) = (1-lambda) ddelta and tangent value T = 0. Also the singular boundary N_1 are NOT in the g_Q-closure of the bounded ones (||Q_AD N_1^(n)||_2 -> inf), so they need their own one-sided condition, not a limit. The tangent value T = inf g_Q(delta^(1)) is an upper bound for limsup z^{-2} min_F g_Q under hypothesis H1 (g_Q-differentiability along the scaled family); the matching lower bound needs hypothesis H2 (Def 3.16: a minimiser EXISTS, is of the scaled-family form, with ||N_1||+||Y_1|| <= C and a uniform o(s^2) remainder - i.e. H2 is the lower bound in disguise), which is NOT proved: g_Q >= (1/4)||.||_2^2 gives O(s) defect blocks but does not localise (X,Z) because Q_AB has 0 in its essential spectrum.

Hypotheses

This result is conditional: at least part of the statement holds only where the hypotheses below hold. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H1, H2.

The tangent value T = inf g_Q(delta^(1)) is an upper bound for limsup z^{-2} min_F g_Q under hypothesis H1 (g_Q-differentiability along the scaled family); the matching lower bound needs hypothesis H2 (Def 3.16: a minimiser EXISTS, is of the scaled-family form, with ||N_1||+||Y_1|| <= C and a uniform o(s^2) remainder - i.e. H2 is the lower bound in disguise), which is NOT proved: g_Q >= (1/4)||.||_2^2 gives O(s) defect blocks but does not localise (X,Z) because Q_AB has 0 in its essential spectrum.

How to verify

cd rigor && pdflatex -interaction=nonstopmode -halt-on-error exact_optimum_tangent_problem.tex; check Lemma 2.1 (both directions), Lemma 2.2(1) and (4) with the shift counterexample in Remark 2.3, Prop 3.4 steps <1>1-<1>5 (the o(s) bookkeeping uses E_D ddelta E_D = 0, S10 Lemma 11.1), Lemma 6.1 (ddelta = [Q,d]), Thm 3.11 steps <1>1-<1>4 (Lumer-Phillips for sigma <= 0; the absorption algebra; the INCLUSION, not equality) together with Remark 4.8 (why the cone must not be enlarged: T = 0 and H1 fails for the contraction family), and Thm 3.9 step <1>2 (triangle inequality in g_Q^{1/2} + H1)

Evidence

Related claims

Review

Review record: passed 2026-09-24 by REF-CARD-CFT.

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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