Every claim, by area

115 cards: 114 claim cards of the knowledge base, plus 1 finding that has reached a status displayed here. The tally below is counted from the cards, not typed:

status	claims	what it means
`refereed`	3	a complete structured proof that an independent referee pass has accepted
`proved`	40	a complete structured proof
`numerical`	25	measured, with an error bar and a convergence order; not a theorem
`conjectural`	11	believed, with partial evidence; not proved
`verified`	5	a check that was carried out and agreed
`open`	5	an unsettled question
`active`	14	work in progress
`done`	5	a finished piece of project work
`superseded`	2	replaced by a later claim
`refuted`	3	shown to be false, kept so the correction stays visible
`heuristic`	2	—
total	115

79 of them are refereed, proved, numerical, conjectural and have a page of their own under results/; the remainder are listed here with their status only.

A referee pass is not implied by a status. Of the cards listed here, 0 have never been refereed and 0 carry a failed referee pass, that is repairs asked for and not yet absorbed. Cards awaiting a pass after a recent change are marked too, but not counted here: that number moves with day-to-day editing of infrastructure cards that have no page. All three are marked in the tables below, as never refereed, referee pass failed and review pending. Nothing labelled numerical is a theorem.

Foundations: CMI as a Markov defect (Notes 1-3)

Longo-Xu conditional mutual information of adjacent intervals, its reading as a quantum-Markov (Petz recovery) defect, and the research programme of Note 3.

status	claim
`proved`	Three faces of the Bures form (type III)
`proved`	CMI as the data-processing defect of tracing out C (Note 2)
`proved`	Longo-Xu CMI of adjacent intervals for the free fermion net A_r, its finite-index subnets and extensions
`proved`	Type-III second-variation lemma (fidelity half), under the vector hypothesis (V)
`conjectural`	Longo-Xu Remark 4.3: adjacent-interval CMI (c/6) log(1+z) for every rational conformal net (conjecture)
`verified`	Cross-ratio conventions used across the notes
`verified`	The Delta^(1/2) fidelity formula was wrong (Wigner-Yanase quantity)

Theorem A: exact recovery error of the free fermion

Zero-collar Petz map = geometric compression; Phi(zeta) = -log F; lim Phi/zeta^2 = c/(12 pi^2); remainder, ultraviolet tail, certified numerics, large-zeta values.

status	claim
`refereed`	Theorem A: lim Phi/zeta^2 = c/(12 pi^2)
`proved`	The C^{1,1} extension of the zero-collar map is implementable on Fock space (uhlmann_upper_bound.tex Cor. 2.7, Prop. 3.1)
`proved`	f2 = c/(12 pi^2)
`proved`	Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r)
`proved`	Zero-collar Petz map of the free fermion is a geometric compression
`proved`	Phi(zeta) and D(zeta) are nondecreasing in zeta (check_note5_theory.tex Prop. 8.1)
`proved`	Exact root fidelity of two quasi-free CAR states from their symbols (Note 5 Thm. 2.1)
`proved`	Remainder of Theorem A: global upper bound Phi <= -(1/2) log(1 - 2 f2 zeta^2); c3 <= 0 as a limsup; no positive log^-2 term
`proved`	Remainder bound in the settled range: Phi(zeta) <= -1/2 log(1 - 2 f2 zeta^2) for 0 <= zeta < (4 f2)^{-1/2} = pi sqrt(3)
`proved`	Ultraviolet tail bound for the windowed fidelity (shape proved; explicit constants 0.173, 0.88 rest on numerical inputs)
`proved`	Two-chirality law for the massless Dirac fermion: -log F^(lambda) = Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})), even in lambda
`proved`	Exact x-space Mellin transform W-hat(k) of the <TT> kernel and its KMS relation
`numerical`	Certified enclosures of the compressed recovery error
`numerical`	Continuum Phi at large zeta and the drifting exponent
`numerical`	Weak link: Phi(zeta) <= 0.0085 zeta^2 on (0, 1] (rigorous on (0, 0.8868], numerical on [0.8868, 1])
`numerical`	Weak link: numerical values of the tail constants A_1 <= 2.0 and A_3 <= 0.0374
`conjectural`	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`	Two-chirality law for a general 2D CFT (rests on an unproved chiral factorisation)
`conjectural`	Two-chirality minimum: for the massless Dirac fermion -log F^(lambda) is minimal at lambda = 0 for every z (unproved)
`verified`	Note 5 numerics independently verified
`open`	Certify the defect-matrix quadrature and the box cutoff
`refuted`	Phi' is NOT nondecreasing on (0, infinity): the step used in prop:chirality <1>8 is false
`heuristic`	The two off-diagonal blocks P_- W~_s P_+ and P_+ W~_s P_- of the push-forward W~_s have the same singular values for every real s (complex-conjugation argument of rate_of_remainder.tex C3 (i), not yet written as a proof)
`heuristic`	Remainder of Theorem A in integer powers, c3 = -0.99(1), c4 = +0.9(1) (computed; scripts not in the repository)

Theorem B: universality for diffeomorphism-covariant nets

The second-order coefficient f2 = c/(12 pi^2) holds for every diffeomorphism-covariant net on the circle; implementer hypotheses; bosonic check.

status	claim
`refereed`	(H1)-(H3) hold for all diffeomorphism-covariant nets
`refereed`	Theorem B: f2 = c/(12 pi^2) for every diffeomorphism-covariant net
`proved`	g_B = 2c/(3 pi^2) for the compression direction of every diffeomorphism-covariant net (corollary of Theorem B)
`proved`	Stress-tensor normalisation chain
`numerical`	Bosonic check: U(1) current net agrees

Kubo-Mori (relative entropy) coefficient s2

The relative-entropy analogue of f2: exact identity for unitary orbits, the refuted gauge lemma, and the conjecture s2 = c/12 with its bracket, conditional at both ends.

status	claim
`proved`	Exact relative-entropy identity on unitary orbits (bounded generators; affiliated generators with Omega in D(A))
`conjectural`	s2 = c/12 for general nets
`conjectural`	g_KM = c/6 for the compression direction of a general conformal net (conjecture; only the analytically continued value exists)
`refuted`	Kubo-Mori gauge lemma by restriction to A(K) is false

Optimality of recovery channels (O8, Phases 3 and 3b)

Is the Petz map, or the geometric compression, optimal at second order? Quasi-free SDP, reduction of all channels to it, KKT certificate, and what is still open in exact fidelity.

status	claim
`proved`	A-C restriction bound: (Tcond) implies E_AC = 0 exactly; applied on the hopping chain (4,8,2), (4,12,2), (4,16,2)
`proved`	All-channel second-order optimum = quasi-free SDP value
`proved`	Every recovery channel's recovered symbol is quasi-free feasible
`proved`	The first-order compression defect is a commutator with the vacuum symbol: ddelta = [Q, E_I D_chi E_I] (F1 Lemma 6.1)
`proved`	Exterior directions exhaust the invisible space: closure(Ext) = N^perp (question (Q) settled, YES)
`proved`	Exact KKT certificate at the compression; Moebius rigidity
`proved`	KKT sign criterion: when the isometric orbit carries the all-channel second-order optimum
`proved`	Moebius reduction: the fidelity and relative entropy of the geometric recovery depend on (a, L, s) only through zeta = a s/(a+L), including the half-line chart a = infinity (zeta = s) (Note 5 Lemma 1.1; check_note5_theory prop:closed)
`proved`	The Petz map is not second-order optimal (compression 4x better at leading order)
`proved`	Second-order problem over quasi-free channels is a convex SDP
`proved`	Second-order (SLD / quantum Fisher) expansion of the quasi-free root fidelity in finite dimensions (Note 5 Prop. 2.3)
`proved`	Exact single-observable fidelity bound (Alberti-Uhlmann) and its reduction to the recovered symbol
`proved`	Exact (N,Y,V) chart of F and the tangent problem at the compression
`proved`	The theta-problem in the modular frame of the half-line is a parameter-free quadratic Wiener-Hopf problem (F1 Prop. 6.2)
`proved`	(UH) proved: optimised Uhlmann bound = g_Q of the defect, for every isometric quasi-free channel
`numerical`	Hopping chain: T = Q_AB^+ Q_AC satisfies the A-C intertwining conditions (Tcond) for (L_A,L_B,L_C) = (4,8,2), (4,12,2), (4,16,2)
`numerical`	The all-channel second-order optimum equals the isometric-orbit value: kappa_opt = 1 - theta = 0.616 +- 0.003 (numerical; conditional on H1, H2, H3)
`numerical`	S10's circle model: the ultraviolet taper suppresses theta by ~20%, and there is no taper-free limit
`numerical`	E_rec <= (1-theta) f2 z^2 (1+o(1)): the zero-collar compression is NOT optimal in exact fidelity
`numerical`	The compression is not the minimiser of the second-order functional
`numerical`	Exact -1/2 log det(1-V^*V) with the optimised extension confirms (UH) beyond first order
`numerical`	NS-circle model: Ext = N^perp exactly, with the dimension identity (L-\|I\|)^2
`numerical`	mpmath at 60-80 digits: dist((1-Pi)M, Ext) = 0 exactly
`numerical`	Optimised exterior extension: UB_c = (1/2)s^2\|u\|^2 exactly and UB_rot/UB_c = 1-theta
`numerical`	Is the compression beaten in EXACT fidelity?
`numerical`	Finite chains: exterior directions span exactly the invisible space (Ext = N^perp)
`numerical`	Full primal SDP over F in the modular Galerkin frame: min_F g_Q = 8.41 and 7.13 times g_Q(delta_c) at h = 0.4, 0.3 -- the S11 representability obstruction, not a statement about non-isometric channels
`numerical`	KKT criterion numerics: every finite run FAILS (a); the violation extrapolates to 0^- as h -> 0, Ymax -> inf; under H3, (e) follows from the proved E(zeta_inf) >= 0 alone, and the finite-h junction values are positive
`numerical`	Structure of the numerical optimum
`numerical`	Tangent SDP over (G, N1, Y1) in the circle model: orbit-only optimum matches (1-theta) to 1.1e-7, full optimum not resolvable (measured Gram cond 2.4e22)
`numerical`	The first-order gain constant theta = 0.384 +- 0.003 (S10's 0.300 is a taper-suppressed lower bound)
`conjectural`	Hypothesis H1 of the tangent problem: g_Q-differentiability along the scaled family
`conjectural`	Hypothesis H2 of the tangent problem: localisation of the minimisers of the second-order programme
`conjectural`	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
`verified`	Finite spectral models do not resolve exact fidelities
`open`	theta as a geometry-free Wiener-Hopf constant in the modular frame of I
`done`	Phase 4 plan: settle exact-fidelity optimality via the Uhlmann route with arbitrary exterior extensions
`done`	Phase 5 plan: exact value of the all-channel optimum (non-isometric channels) and reconciliation of theta

Lattice, thermal, massive and gapped geometries (O7, O9, O10)

Gaussian lattice tests of the two-branch law, KMS states and shielding, the Markov c-function along the massive flow, and separated intervals.

status	claim
`proved`	Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions
`proved`	Separation constraint: all-weak L->R Petz chains have e^{Delta_k} >= 2/eps (sharp)
`proved`	Amplification: the corner strength in the chain frame is zeta_step (1 + z')
`proved`	Corner data of all one-sided protocols for n = 4 and n = 5: the holonomy is the starting block
`proved`	Angle form of the sequential recovery bound, conditional on Hypothesis (U) (vacuous: (U) is refuted)
`proved`	Type-III sequential recovery bound: Bures distances add along a chain of von Neumann inclusions (chordal and second-order forms, no Hypothesis (U))
`proved`	Vacuum rigidity: the pulled-back vacuum is the vacuum iff the map is Moebius; no protocol recovers exactly
`numerical`	Separated intervals: CMI with a gap and linearity off criticality
`numerical`	Markov c-function along the massive flow
`numerical`	Thermal states: two-branch law at the cylinder cross ratio, exponential shielding
`numerical`	Two-branch law confirmed on the lattice
`conjectural`	Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal
`conjectural`	VWZ four-interval protocols: Phi(1):Phi(2):Phi(3) -> 1:2:4; middle-out is the worst order
`open`	The universal corner-correlation kernel Ghat(Delta): closed form, positivity, decay
`open`	Multi-interval networks (O10, second half)
`open`	Charged sectors, state dependence, collar Gaussianity (O11)
`active`	Phase 6 plan: multi-interval networks as a corner calculus
`refuted`	Hypothesis (U) is false: the natural-cone inner product is not the fidelity

Papers and notes: status and pending edits

Paper 1 (free fermion), paper 2 (universality), the eight notes and their errata; what must be reworded before submission.

status	claim
`active`	Note 8: open problems O1-O11 and phases
`active`	Paper 1 (free fermion): refereed draft, edits pending
`active`	Paper 2 (universality): refereed draft, KM as conjecture
`done`	Errata to Notes 1-7 applied and logged
`superseded`	Note 3 programme: remove the collar (superseded by the plan of Note 8)
`superseded`	Strategy note (Note 6): the four-step programme for universality

Project infrastructure: rigor pipeline, numerics stack, agents

How proofs, referee passes, findings and numerics are produced and checked in this project; tool limits.

status	claim
`verified`	Headless screenshots and shell pitfalls on this Mac
`active`	Generated documentation subfolder: docs/*.md from the knowledge base, with a drift check
`active`	Interactive site skeleton: shared lib/, the data generator, and modules 1, 2, 6
`active`	Interactive site: modules 3 (universality), 4 (optimality) and 5 (the theta reconciliation)
`active`	Interactive site: modules 7 to 10 and the extra data extractor
`active`	Numerics stack and its limits
`active`	Plan page artifact and its source
`active`	Plan: public presentation (README with figures, interactive Pages site, generated docs/)
`active`	Public README and the five generated SVG figures
`active`	Two git repositories over one work tree: public project, private third-party material
`active`	Rigor pipeline: Lamport proofs, referee passes, findings ledger
`done`	Continuous integration passes, for the first time since the workflow was added
`done`	The generated documentation is rendered to HTML for Pages, with the Markdown still the source

Documentation map · What is open · Sources · History