cft-cmi

KKT sign criterion: when the isometric orbit carries the all-channel second-order optimum

Status: proved — a complete structured proof. Conditional. The statement is not unconditional: it rests on H1, H2, H3, 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: kkt-noise-sign-criterion

Statement

PROVED (rigor/exact_optimum_tangent_problem.tex, Sec. 4). Let delta_* be the b-orthogonal residual of ddelta on the closure of the LINEAR class L = {[Q,G] : G bounded anti-self-adjoint, supported in D} (no minimiser is posited), t_* := SLD(delta_*) (S_Q(t_*) = delta_*, t_*~_ik = w_ik delta_*~_ik), tau_* := E_D t_* E_D and M_* := (1/2) E_D (t_* Q + Q t_*) E_D = Herm(t_{*,DA} Q_AD) + Herm(t_{*,DD} Q_DD) = Herm((t_* Q)_DD). Since w is unbounded, t_* need NOT be Hilbert-Schmidt: M_* and tau_* are quadratic FORMS, defined intrinsically by <psi,M_* psi> := 4 b(delta_*, A(0,|psi><psi|,0)) and <psi,tau_* psi> := -4 b(delta_*, A(0,0,|psi><psi|)) on the domain D of psi for which those directions lie in the form domain F (Def. def:forms); 'M_* >= 0' is a form inequality and lambda_min may be -inf. Normalisation (Lemma lem:norm): g_Q(delta) = (1/4) Tr(t_delta delta), V(t_delta) = 2 g_Q(delta), d g_Q(delta)[eta] = (1/2) Tr(t_delta eta) (S10 (C4) is internally inconsistent with its own S_Q(t) = delta; the ratio and all sign conditions are unaffected). (i) Lemma lem:deriv: the one-sided derivative along an admissible non-isometric direction A(0,N_1,Y_1) is (1/2)[Tr(N_1 M_*) - Tr(Y_1 tau_*)]; it vanishes along the linear class; along the admissible unbounded generator G = -lambda D_chi|_D it equals 2 lambda (1-theta) g_Q(ddelta) (>= 0; > 0 iff theta < 1). (ii) Thm. thm:kktmain: (a) M_* >= 0 and M_* - tau_* >= 0 <=> (c) Tr(N_1 M_*) >= Tr[(N_1^{1/2} tau_* N_1^{1/2})_+] for all bounded N_1 >= 0; (b) M_* >= (tau_*)_+ is sufficient but strictly stronger (the operator order is not a lattice); AMENDED after the referee report: (d) (zeta_*,0,0) is a GLOBAL minimiser of the tangent programme <=> (a) AND (e), where (e) E(zeta) := 2 b(delta_*, A(zeta,0,0)) >= 0 for every admissible isometry generator zeta with A(zeta,0,0) in F; (a) alone is NECESSARY but NOT SUFFICIENT, because the feasible cone also contains the one-sided boundary-moving isometry directions (unbounded generators). If (a) and (e) hold then T = (1-theta) g_Q(ddelta), i.e. kappa_opt = 1 - theta exactly, and under H1 and H2 (Thm. thm:value) E^(2) = (1-theta) f_2 z^2 (1+o(1)). If (a) fails, a violating psi gives a strictly better feasible datum (Cor. cor:better, eq:gain2); kappa_opt < 1 - theta needs one admissible psi with Delta < 0 AND finite g_Q(d_psi), so a violation that vanishes with the weight cap or the mesh is a regularisation artefact. (iii) Prop. prop:bdryfun: E MUST be evaluated on the blocks of A (A_AD = -Q_AD zeta, A_DD = -(zeta* Q_DD + Q_DD zeta)); the commutator form -2 b(delta_*,[Q,zeta]) of Lemma lem:commQ is valid only for BOUNDED anti-self-adjoint zeta (where E = 0) and deletes exactly the boundary term (2'). E is additive, positively homogeneous and vanishes on the closure of L; when the cyclicity Tr(t_*[Q,zeta]) = Tr([t_*,Q] zeta) is legitimate (zeta bounded, t_* trace class), E(zeta) = (1/2)Tr(E_D[Q,t_*]E_D zeta) = 0 by the stationarity Prop. prop:S: E is exactly the cyclicity anomaly, a pure BOUNDARY functional, equal to (1/2)Tr((-2 sigma_zeta) M_*) whenever -2 sigma_zeta lies in the form domain of M_*. In the modular frame D = {y>0} has two endpoints (y = 0, the junction with A; y = +inf, the moving endpoint x = 1), and modulo the closure of L the VECTOR-FIELD subcone of admissible generators is generated by zeta_0 = -E_D d/dy E_D (the shift semigroup g(y) -> g(y-a), i.e. x -> 1 - e^{-a}(1-x); moves the junction y = 0) and zeta_inf = -lambda D_chi|_D (further compression; moves y = +inf); for the whole admissible cone this is hypothesis H3. E(zeta_inf) = 2 lambda (1-theta) g_Q(ddelta) is proved, hence E(zeta_inf) >= 0; E(zeta_inf) > 0 is equivalent to theta < 1, which is NOT proved (rem:whytheta (i), :1461-1464; verdict gap :1504-1507). So, ON THE MODEL CONE {alpha zeta_0 + beta zeta_inf + closure of L : alpha, beta >= 0}, (e) is equivalent to the single inequality E(zeta_0) = 2 b(delta_*, A(zeta_0,0,0)) >= 0; the identity E(zeta_0) = (1/2) Tr((-2 sigma_{zeta_0}) M_*) would need -2 sigma_{zeta_0} = delta_0 (x) delta_0 in the form domain of M_*, which fails (delta_0 is not in h_D). HYPOTHESIS H3 (cone hypothesis, Def. hyp:H3, read in F): for every admissible isometry generator zeta with A(zeta,0,0) in F there are alpha, beta >= 0 and eta in the b-closure of the linear class with A(zeta,0,0) = alpha A(zeta_0,0,0) + beta A(zeta_inf,0,0) + eta. (No boundary-form reformulation: that would be strictly weaker, trivially true for zeta = i m(y), and would not imply (e).) The reduction to the two model directions is PROVED ONLY for generators of vector-field type; shift semigroups of multiplicity > 1 (Cooper-Wold) and unbounded skew-adjoint zeta = i m(y) with [Q, i m] outside the closure of L escape both arguments, and excluding them is exactly the content of H3. Under H3, if A(zeta_0,0,0) is not in F then alpha = 0 is forced and (e) holds OUTRIGHT, E(zeta) = beta E(zeta_inf) >= 0 (rem:H3space), the junction inequality never being invoked; that A(zeta_0,0,0) is not in F (E(zeta_0) = +inf, so (e) is vacuous at zeta_0 rather than satisfied with a margin) rests on F2's divergent corner term (rem:T5 (ii)), i.e. it is numerical. E(zeta_0) >= 0 remains the operative test for REGULARISED junction directions (finite mesh), which is what F2 T5 measures. (iv) Prop. prop:S: stationarity on the LINEAR class <=> E_D[Q,t_*]E_D = 0, i.e. (t_* Q)_DD = M_* self-adjoint: S10's condition (S) holds automatically at the new point. (v) Prop. prop:s10 (trace-class t; S10 Thm 3.2 and Prop 5.2), dictionary to S10 Thm 3.2: Sigma_2 -> (tQ)_DD is the loss-of-isometry (N_1) condition M >= 0; Sigma_1 = Sigma_2 - t_DD >= 0 is the NOISE (Y_1) condition M >= tau; Sigma_1 = Sigma_1* is the rotation condition. NUMERICAL, NOT PROVED HERE (recorded on kkt-noise-criterion-numerics, all-channel-optimum-value, theta-value and compression-not-second-order-minimiser): (a) holds in the limit, although every finite run fails it (F2 T4, rigid frame of Prop 6.2: lambda_min(M_*)/||M_*|| = -4.1e-3, -1.9e-3, -8.2e-4, -3.7e-4 at h = 0.24, 0.12, 0.06, 0.03; violation ~ h^1.15 and ~ 1/Ymax, rank-one gain ~ h^1.85); E(zeta_0) > 0 on the model cone (F2 T5: E(zeta_0)/g_Q(ddelta) = +34.1, +66.9, +131.3, +258.0 at h = 0.24, 0.12, 0.06, 0.03 for the admissible backward/dissipative scheme, a scheme-dependent total ~ h^-0.97 dominated by an upwind-viscosity term; centred = 0, the linear class; forward = inadmissible, negative; the continuum piece is the corner term (1/2) M_*(0,0)/h = +10.3, +16.1, +24.7, +37.2, ~ h^-0.60, so M_*(y,y) -> +inf as y -> 0+ and E(zeta_0) = +inf: the junction direction is infinitely penalised at first order); E(zeta_inf) reproduces 2(1-theta_h) g_Q(ddelta) to 1e-15, and the A-form and boundary-form expressions agree to 2e-10 ... 7.5e-8 (f2_t5_endpoint.out:7, :37; the tex's :1056 and :1181 read the same since 2026-09-24, Correction C7); the commutator form returns, even for zeta_inf, -9.1e-6, +1.9e-4, -9.9e-3, +7.4e-2 at h = 0.24, 0.12, 0.06, 0.03 and -63, -1.6e6 on other rungs of the scan (O(1), no information on the true 2 lambda (1-theta) g_Q(ddelta) = 1.26-1.37 g_Q(ddelta); prop:bdryfun (2')); theta = 0.384 +- 0.003, hence kappa_opt = 1 - theta = 0.616 +- 0.003 conditional on H1, H2 AND H3 and on (a) and (e), where (a) is extrapolated from meshes, on each of which it fails, and (e) is verified on meshes (tex :1624, corrected 2026-09-24, Correction C9); and (S) FAILS at the compression, i.e. E_D u E_D != 0 (||E_D u E_D||_2^2/||u||_2^2 = 0.0143, S10 cor:maximal eq:thetalb).

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, H3.

If (a) and (e) hold then T = (1-theta) g_Q(ddelta), i.e. kappa_opt = 1 - theta exactly, and under H1 and H2 (Thm. thm:value) E^(2) = (1-theta) f_2 z^2 (1+o(1)).

In the modular frame D = {y>0} has two endpoints (y = 0, the junction with A; y = +inf, the moving endpoint x = 1), and modulo the closure of L the VECTOR-FIELD subcone of admissible generators is generated by zeta_0 = -E_D d/dy E_D (the shift semigroup g(y) -> g(y-a), i.e. x -> 1 - e^{-a}(1-x); moves the junction y = 0) and zeta_inf = -lambda D_chi|_D (further compression; moves y = +inf); for the whole admissible cone this is hypothesis H3.

HYPOTHESIS H3 (cone hypothesis, Def. hyp:H3, read in F): for every admissible isometry generator zeta with A(zeta,0,0) in F there are alpha, beta >= 0 and eta in the b-closure of the linear class with A(zeta,0,0) = alpha A(zeta_0,0,0) + beta A(zeta_inf,0,0) + eta.

(No boundary-form reformulation: that would be strictly weaker, trivially true for zeta = i m(y), and would not imply (e).) The reduction to the two model directions is PROVED ONLY for generators of vector-field type; shift semigroups of multiplicity > 1 (Cooper-Wold) and unbounded skew-adjoint zeta = i m(y) with [Q, i m] outside the closure of L escape both arguments, and excluding them is exactly the content of H3.

Under H3, if A(zeta_0,0,0) is not in F then alpha = 0 is forced and (e) holds OUTRIGHT, E(zeta) = beta E(zeta_inf) >= 0 (rem:H3space), the junction inequality never being invoked; that A(zeta_0,0,0) is not in F (E(zeta_0) = +inf, so (e) is vacuous at zeta_0 rather than satisfied with a margin) rests on F2's divergent corner term (rem:T5 (ii)), i.e. it is numerical.

E(zeta_inf) reproduces 2(1-theta_h) g_Q(ddelta) to 1e-15, and the A-form and boundary-form expressions agree to 2e-10 ... 7.5e-8 (f2_t5_endpoint.out:7, :37; the tex's :1056 and :1181 read the same since 2026-09-24, Correction C7); the commutator form returns, even for zeta_inf, -9.1e-6, +1.9e-4, -9.9e-3, +7.4e-2 at h = 0.24, 0.12, 0.06, 0.03 and -63, -1.6e6 on other rungs of the scan (O(1), no information on the true 2 lambda (1-theta) g_Q(ddelta) = 1.26-1.37 g_Q(ddelta); prop:bdryfun (2')); theta = 0.384 +- 0.003, hence kappa_opt = 1 - theta = 0.616 +- 0.003 conditional on H1, H2 AND H3 and on (a) and (e), where (a) is extrapolated from meshes, on each of which it fails, and (e) is verified on meshes (tex :1624, corrected 2026-09-24, Correction C9); and (S) FAILS at the compression, i.e. E_D u E_D != 0 (||E_D u E_D||_2^2/||u||_2^2 = 0.0143, S10 cor:maximal eq:thetalb).

How to verify

cd rigor && pdflatex -interaction=nonstopmode -halt-on-error exact_optimum_tangent_problem.tex; check Lemma 1.1 (SLD normalisation, factors 1/4 and 1/2), Lemma 4.2 steps <1>3-<1>5 (block trace giving M_* = Herm((t_*Q)_DD)) and its amended last clause (the derivative vanishes on the LINEAR class, and its vanishing is not automatic along all anti-symmetric G: along G = -lambda D_chi it is 2 lambda (1-theta) g_Q(ddelta), >= 0, > 0 iff theta < 1, which is not proved; :716-721, corrected 2026-09-24, Correction C2), Def 4.9 (M_*, tau_* as forms), Thm 4.4 steps <1>1-<1>5 (Douglas, rank-one, (a)<=>(c), (d) <=> (a) and (e), the conditional value), Cor 4.5, Prop 4.3, Prop 4.10 (the endpoint functional, the vector-field subcone and its two model directions, E(zeta_inf) = 2 lambda (1-theta) g_Q(ddelta) in <1>4; (2') uses Lemma 6.1), Rem 6.3(i) and the verdict gap after Rem 6.4 (theta < 1 not proved), Def 4.12 + Rem 4.13-4.14 (H3; alpha = 0 needs A(zeta_0,0,0) not in F; the two families H3 excludes), Rem 4.11 (F2 T5: corner vs viscosity term; A(zeta_0,0,0) not in F is numerical), Prop 4.6. Numerically (NOT part of this card's proof): numerics/optimality_all/F2_RESULTS.md T4/T5 and the .out files cited by kkt-noise-criterion-numerics; theta from F3 (theta-value).

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.

Documentation map · All claims by area · What is open · Symbols · Definitions