Claim status map — Geometric QCD

Claim status after the revision

This map distinguishes a course calculation from a source assertion. Exact calculation means the stated model was calculated, not that it proves the full QCD proposal. Formal means the displayed argument needs the stated functional domain and limiting assumptions. Source reconstruction identifies provenance, not independent proof.

Claim or step	Course location	Status in this edition	Boundary or missing evidence
Ribbon counting and leading normalized-trace factorization	I.1, I.5	Standard topological expansion; explicit counting example	Fixed ‘t Hooft coupling and usual expansion assumptions
Proper-time inverse and free worldline kernel	I.2	Exact calculation	Interacting Brownian transport needs ordering and regulator
Ordered insertion and plaquette curvature	I.3–I.4	Exact finite expansion; formal operator limit	Transport orientation and operator domain
Loop-operator product rule	I.4	Formal under stated continuity hypotheses	Singular continuum functionals need a common regulated domain
Finite-color splitting	I.5	Exact color algebra	Loop equation is distributional; source coupling normalization must be translated
All-orders bootstrap including ghosts	I.6	Source claim	Gauge-fixed comparison, ghost signs, symmetry factors, regulator
Failure of naive crossing subtraction	I.7	Explicit local calculation	Does not exclude operator mixing or every coordinate-space formulation
Functional momentum transform and measure splitting	II.1–II.3	Formal source construction; exact finite analogues	Continuum measure, gauge volume, limits
Closedness removes every second-order integral	II.4	Incorrect in general; rectangle counterexample	Symmetric contractions can vanish while coefficients remain nonzero
Low-order MLE tensors	II.5	Source-reported; version discrepancy retained	Complete equations, convention map, sixth-order coefficients and residuals
Eighth-order obstruction	II.6	Reported rank calculation	Exact matrices and independent certificate not reproduced
Roughness or finite-matrix limitations imply nonanalyticity	II.7	Inference not established	Gaussian and Cayley–Hamilton counterexamples separate the claims
Free-word creation relations and vacuum-state distinction	II.7	Exact representation calculation	Not by itself a QCD master-field construction
Duality plus Bianchi implies a homogeneous zero mode	III.2	Conditional derivation	Existence of the intended area derivative and regulated identities
Additive dressing preserves splitting structure	III.3	Exact conditional algebra	Does not factorize a single traced holonomy
Vacuum equals an instanton-like surface ensemble	III.4	Interpretation/source proposal	State, normalization, finite action and boundary data
Harmonic extension of a fixed boundary	III.5	Exact Fourier calculation	Conformality is separate; boundary correspondence must be varied
Literal extended tensor is automatically self-dual	III.5	Fails the explicit circle test	Missing or changed definition would require a new area variation
Null-vector spinor factorization	III.6	Exact local matrix calculation	Global patches, reality, gauge measure, and surface interpretation
OPE contraction $(d-1)(d-4)$	III.7	Exact tensor calculation	Only the displayed leading structure is canceled
Condensate fixes the geometric tension	III.7	Source matching, conditional	Area normalization and the unresolved geometric construction
Majorana theory reproduces QCD loop dynamics	IV.1–IV.7	Source proposal	Operator domain, normalization, geometry, cancellation, contact coefficient
Weyl transformation of the massless Dirac operator	IV.2	Exact local calculation	Massive source equivalence and full anomaly cancellation require more input
Pfaffian, determinant, and connected-loop relation	IV.3	Exact finite calculations	Continuum regulator, spin structure, boundary terms
All nonplanar configurations cancel	IV.4	Source claim; local Grassmann example verified	Full weight-preserving cancellation with all labels and fixed configurations
Large-mass A-term cancellation	IV.5	Conditional source reduction	Uniform limits and the relevant Bianchi/tensor identities
Radial kernel becomes a unit four-dimensional delta	IV.6	Proved for the corrected normalized model	Source kernel without the factor has total mass $3{\pi }^2$
Actual bridge kernel yields the MM coefficient	IV.6	Unresolved source matching	Geodesic bound does not determine normalization or tensor alignment
Two-loop RG scale	IV.7	Derived asymptotic relation	Identifying the induced coupling with that beta function requires matching
Finite $Q$ matrix spectrum and pseudodeterminant	V.2	Exact matrix calculation	Full real quotient measure and zero-spinor patches remain additional
Local radial and propagator weights cancel	V.3	Exact homogeneity calculation	Does not establish full functional anomaly cancellation
Liouville model variation	V.4	Exact bulk, boundary, and constant-variation calculations	Positive bulk and boundary weights have no unconstrained real stationary point with free boundary data; the source saddle needs its own domain
Real rotating surface	V.5	Exact classical example	Does not prove spectral positivity or absence of quantum instabilities
Monodromy and stationary Regge equations	V.6–V.7	Exact within the displayed phase	Phase’s QCD origin and branch selection remain assumptions
Exact WKB and only the lowest level	V.6	Conjectured in the source construction	Full fluctuation determinant, integration cycle, and competing saddles
36-state fit and quoted RMS	V.7	Source-reported	Original state table, objective, optimizer and residuals not reproduced
Mixed-flavor families are held-out predictions	V.7	Not established by a global fit	Reserve data before fitting and evaluate without retuning
Large $N_c$ excludes a massless pion	V.1, V.7	Incorrect as a general statement	Chiral symmetry breaking and Goldstone modes are compatible with large $N_c$

Dependency of the open claims

The geometric problem in III.5 affects the proposed zero-mode realization and subsequent surface-fermion QCD identification. It does not invalidate the independent finite calculations in the course. The eighth-order algebra and the meson fit require their own reproducibility work; neither can establish the missing geometric identity.

The source PDF and published/preprint records are preserved as evidence. A future repair must state which definition or assumption changes and rerun the affected arguments; it must not erase the counterexample by silently changing conventions.