What is open
Every next: field below is quoted verbatim from its knowledge-base card: this page records the concrete next step, not an aspiration. Project-management cards (paper edits, plans, infrastructure) are deliberately left out.
Open questions (5)
The universal corner-correlation kernel Ghat(Delta): closed form, positivity, decay
open · knowledge-base id corner-correlation-kernel
Ghat(Delta) = g_Q(delta_0, delta_Delta)/c, where delta_Delta is the y-translate by Delta of the closed-form first-order corner defect ddot(y1,y2) = -(i/2 pi) sinh(y1/2) sinh(y2/2)/sinh^2((y1-y2)/2) (y1 < 0 < y2, + h.c.) in the modular frame I = (-inf, 1), y = -log(1-x). Ghat(0) = 1/(12 pi^2). Open: closed form (Fourier transform of rho(q) = (1/4) int dp |dhat_0(p,p-q)|^2 / W(p,p-q) with the SLD weight W); sign for all Delta; decay rate (e^{-Delta/2}? e^{-Delta}?), which fixes the order of the cross-term correction in the network law. No closed-form candidate is to be selected by fitting to numerics (lesson of theta-value).
next: G1b + G3 in parallel.
Certify the defect-matrix quadrature and the box cutoff
open · knowledge-base id dhat-quadrature-certification
The certified enclosures refer to the compressed problem; two steps are uncertified: the two-dimensional quadrature producing the defect matrix D-hat and the truncation of the box frame at kappa_max.
next: Interval-arithmetic 2-D quadrature of the defect kernel on the box frame, and a rigorous bound for the modes beyond kappa_max (box asymptotics 0.0641 zeta^2/kappa_max^2).
Multi-interval networks (O10, second half)
open · knowledge-base id multi-interval-networks
Not started. The gap geometry is done; networks of recovery maps and their holonomy remain.
next: Phase 6 (rigor/phase6_brief.md, card phase6-plan): corner calculus (G1a), second-order network law with the universal kernel Ghat (G1b), lattice sequential Petz chains and the VWZ protocol test 1:2:4 (G2), Ghat(Delta) in two frames (G3).
Charged sectors, state dependence, collar Gaussianity (O11)
open · knowledge-base id sectors-o11
Deferred throughout Phases 1-3.
next: Sector-dependent Markov defect for the free fermion (quasi-free or finite rank); write the half-page collar-Gaussianity lemma.
theta as a geometry-free Wiener-Hopf constant in the modular frame of I
open · knowledge-base id theta-modular-wiener-hopf
Frame-free form: ddelta = [Q,d] with d = E_I D_chi E_I, and on the isometric orbit delta^(1) = [Q, d - zeta], so 1 - theta = inf_G g_Q([Q, d + G]) / g_Q([Q,d]), the INFIMUM over BOUNDED anti-self-adjoint G supported in D (not a min; the distinction from theta_HS is open, cf. exact_fidelity_upper_half.tex Rem 7.4(i)). In the modular frame of the half-line I = (-inf,1) — coordinate y = -log(1-x), implemented by the half-density push-forward, A = {y<0}, D = {y>0} — Bisognano-Wichmann makes Q a Fourier multiplier Q = (1 + e^{-2 pi p})^{-1}, p = -i d/dy, E_D = 1_{y>0}, and the compression field becomes w(y) = -4 sinh^2(y/2) 1_{y>0} (C^{1,1} at 0, ~ -e^y as y -> +inf: the moving endpoint x=1 sits at y=+inf). With X^ the kernel in the Fourier variable, g_Q([Q,X]) = (1/4) int int K(p,p') |X^(p,p')|^2, K(p,p') = sinh^2(pi(p-p')) / [cosh(pi(p-p')) (cosh(pi(p+p')) + cosh(pi(p-p')))]. So theta is the value of a parameter-free quadratic Wiener-Hopf problem (minimise over kernels supported in the quadrant y,y' > 0): theta is a pure number, independent of a,b,c,L,z. NUMERICAL VALUE (track F3, numerics/optimality_all/F3_RESULTS.md, f3_t0.py): theta = 0.384 +- 0.003, 1 - theta = 0.616 +- 0.003; the modular Galerkin frame and a direct discretisation of the frame above agree to 3e-4 after extrapolation (order h^0.77, five mesh families), the modular window Lambda is irrelevant, and the residual h-dependence is a BULK effect (the endpoint piece saturates at coarse h). S10's 0.300 +- 0.005 is not a competing estimate but a TAPER-SUPPRESSED LOWER BOUND: relaxing the circle model's smoothstep UV taper moves theta at L=256 from 0.2896 to 0.36 and beyond. Consequently the three circulated candidates are EXCLUDED: 3/pi^2 = 0.30396 and 1/3 far below, 1 - 2/pi = 0.36338 still 6.9 sigma low; that all three fell in the old 0.30-0.40 window is a cautionary datum about unmotivated candidates. Structural reading: theta > 0 iff E_D u E_D != 0 (S10 Thm 6.2 + Cor 11.2, non-vanishing numerical). theta < 1 is NOT proved: only the converse is established (if d were anti-self-adjoint then G = -d would be admissible and theta = 1); d is skew-symmetric but not skew-adjoint, its boundary form sitting at the moving endpoint. K -> 1 as |p-p'| -> inf at fixed p+p', and K -> 0 as |p+p'| -> inf at fixed p-p' — the latter is the direction the circle taper suppresses, consistent with its downward bias. NO closed form is claimed or fitted.
next: Solve or bound the Wiener-Hopf problem of Prop 6.2 and compare with theta = 0.384 +- 0.003 (F3 is done); a closed form is open, and no candidate is to be fitted to the numerics
Conjectures (11)
Believed, with partial evidence, and not proved.
s2 = c/12 for general nets
conjectural · knowledge-base id conj-s2
next: Renormalised/collar route for a gauge lemma modulo the commutant orbit; then the exact identity applies. Until then paper 2 states s2 = c/12 as a conjecture with the bracket.
g_KM = c/6 for the compression direction of a general conformal net (conjecture; only the analytically continued value exists)
conjectural · knowledge-base id const-gkm-general-nets
next: Decide whether the Kubo-Mori form of the compression direction is gauge-independent and equal to c/6 for general nets: a gauge lemma modulo the commutant orbit (renormalised or collar route, cf. conj-s2) or a counterexample.
Index zero for e^{sK} beyond ||G_s||_2 < 1: for a unitary group e^{sK} with [P_-, K] Hilbert-Schmidt the two off-diagonal blocks have the same singular values also where |s| ||[P_-, K]||_2 >= 1 (open; no document contains an argument)
conjectural · knowledge-base id group-index-zero-large-s
next: Write the Fredholm-index continuity argument, e.g. in rigor/rate_of_remainder.tex as a new correction item for Lemma 6.1 <1>1 and <1>4 (l. 828-832, 842-846) and Prop. 6.2 <1>3 (l. 871-874). A possible route, in no document: by Lemma 6.1 <1>2, s -> U_s P_+ U_s^* - P_+ = -G_{-s} is continuous in S_2, so the index of the pair of projections (P_+, U_s^* P_+ U_s) = (P_+, P_+ - G_s), which equals ind(P_+ U_s P_+|_{P_+ H}) (the pair (P_+, U_s P_+ U_s^*) has index -ind X), is locally constant in s and hence 0, its value at s = 0. The same kind of argument is needed for the product family W~_s e^{ish'} of Cor. 2.9 (C3, l. 1078-1079) and would cover rigor/exact_fidelity_upper_half.tex Lemma 6.1 <1>1. Have it refereed, then promote this card.
Longo-Xu Remark 4.3: adjacent-interval CMI (c/6) log(1+z) for every rational conformal net (conjecture)
conjectural · knowledge-base id longo-xu-cmi-general-c
next: Prove the adjacent-interval formula for a rational net beyond the free-fermion class (Longo-Xu Remark 4.3), e.g. for a finite-index extension of a Virasoro net, or find a counterexample.
Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal
conjectural · knowledge-base id network-second-order-law
next: G1b proof; G3 numerics; then compare with the lattice (G2).
Hypothesis H1 of the tangent problem: g_Q-differentiability along the scaled family
conjectural · knowledge-base id tangent-h1-gq-differentiability
next: Prove H1 for general data (rigor/exact_optimum_tangent_problem.tex:1628-1636): show g_Q(delta(s) - s delta^(1)) = o(s^2) along a minimising sequence of admissible first-order data (zeta, N_1, Y_1) beyond the isometric orbit and beyond finite-rank N_1, Y_1 in a modular window, the form Thm. thm:upper needs (:439-441), i.e. control the remainder in the weighted g_Q-norm (weights w_ik ~ e^{min(|kappa_i|,|kappa_k|)}), not only in Hilbert-Schmidt norm.
Hypothesis H2 of the tangent problem: localisation of the minimisers of the second-order programme
conjectural · knowledge-base id tangent-h2-minimiser-localisation
next: Either prove H2 (localisation of the minimisers of min_F g_Q in the scaled-family form with ||N_1|| + ||Y_1|| <= C and a uniform o(s^2) remainder, overcoming Rem. rem:noloc: Q_AB has 0 in its essential spectrum), or bypass it by a finite-s dual certificate: one self-adjoint trace-class t (t_* at finite s) with Delta_z(t) >= 0 whose multipliers satisfy the sign conditions of S10 Thm. 3.2 at the optimal point, which gives the lower bound min_F g_Q >= Delta_z(t)^2/(8V(t)) through the weak-duality bound eq:weakdual (Rem. rem:dual, rigor/exact_optimum_tangent_problem.tex:657-671; To be supplied :1628-1636).
Hypothesis H3 of the tangent problem (cone hypothesis): every admissible isometry generator with a feasible first-order datum lies in the model cone modulo the linear class
conjectural · knowledge-base id tangent-h3-cone-hypothesis
next: Evaluate E(zeta) = 2 b(delta_*, A(zeta,0,0)) directly on shift semigroups of multiplicity > 1 (Cooper-Wold; boundary forms of infinite rank) and on unbounded skew-adjoint zeta = i m(y), or prove the cone reduction for all admissible generators (rigor/exact_optimum_tangent_problem.tex:1633-1635 and Rem. rem:H3 :1258-1260, 'the sharpest remaining analytic task').
Two-chirality law for a general 2D CFT (rests on an unproved chiral factorisation)
conjectural · knowledge-base id two-chirality-law-general-cft
next: Prove the chiral factorisation of the rotated Petz map for 2D nets of the form A_chiral x A-bar_chiral (and decide the non-tensor-product case), and establish the zero-collar collapse (normal extension of the glued map) for a general chiral net, so that Phi_c is defined.
Two-chirality minimum: for the massless Dirac fermion -log F^(lambda) is minimal at lambda = 0 for every z (unproved)
conjectural · knowledge-base id two-chirality-minimum-at-lambda-zero
next: Prove that zeta Phi'(zeta) is nondecreasing on (0, infinity) (e.g. from the determinant representation, Note 5 Thm 3.1), or bound g(lambda) - g(0) from below with certified values of Phi on a grid plus monotonicity (card phi-monotone-in-zeta); until then the minimum at lambda = 0 is conjectural.
VWZ four-interval protocols: Phi(1):Phi(2):Phi(3) -> 1:2:4; middle-out is the worst order
conjectural · knowledge-base id vwz-protocol-ordering
next: G2 run; G1a proof of the exact reductions.
Settled claims that still record a next step (4)
These carry a status of their own; the step recorded is what would sharpen or extend them.
E_rec <= (1-theta) f2 z^2 (1+o(1)): the zero-collar compression is NOT optimal in exact fidelity
numerical · knowledge-base id compression-not-optimal-exact-fidelity
next: kappa_opt <= 0.616 +- 0.003; whether non-isometric channels lower it further is the subject of Phase 5 -- the KKT criterion's endpoint condition is still being evaluated (F1/F2), numerically no (card kkt-noise-criterion-numerics). Do not record kappa_opt = 1 - theta as settled.
Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions
proved · knowledge-base id corner-calculus
next: G2 to test the general rule (Thm 5.2) on the lattice, including union-conditioning and single-interval-start protocols where (H) fails.
Is the compression beaten in EXACT fidelity?
numerical · knowledge-base id exact-fidelity-optimality-open
next: Remaining: the exact value of the all-channel optimum (non-isometric channels; bracket [D_z, (0.616 +- 0.003) f2 z^2]) and an analytic proof that E_D u E_D is not self-adjoint (theta > 0). kappa_opt = 1 - theta is NOT settled: the KKT criterion's endpoint condition is still being evaluated (Phase 5, F1/F2); numerically non-isometric channels do not lower it further (card kkt-noise-criterion-numerics).
KKT sign criterion: when the isometric orbit carries the all-channel second-order optimum
proved · knowledge-base id kkt-noise-sign-criterion
next: Analytic proofs of (a) and of theta < 1 (both numerical now), and H3, H1 for general data, H2 or the finite-s certificate (rigor/exact_optimum_tangent_problem.tex:1628-1636)