Argument dependencies
STATE owns priorities; the catalog owns coverage. Arrows below indicate mathematical input or reading order as specified, not an implication between physical premises. Former names resolve to canonical notes.
Multiscale control connects the repository’s three settings: gauge construction and spectral transfer, tangent-groupoid and refinement limits and fluid scaling and concentration. Each asks which estimates survive refinement; their states, clocks, regulators and target operators require separate identifications.
| Question | Argument path | Dependency that remains explicit |
|---|---|---|
| Classical initial-value construction | Polygons and regularity | Stability/residual estimates; the logarithmic arrival test replaces a sufficient Lipschitz hypothesis only in its conservative family |
| Endpoint elimination | Time blocking → nonlinear cell | Dirichlet contraction and normalized full conditional law; endpoint caustics differ from IVP failure |
| Quantum lift | C1,1 certificate and smooth construction → groupoid states | Operator toolkit and supplied action; weaker and stronger regularity give different rates |
| Groupoid scaling | Interacting kernels and normalized bridges → well-resolving dilation | Specified kinetic energy and potential; local strong/kernel limits differ from global spectral control and physical action calibration |
| Cusp preparation floors | Merger, history and continuity → Gaussian-sector construction → meter/shear test | Different state spaces, weak/W2 closure, actual Gaussian transport, independent copying and selective instruments |
| Record geometry and bounds | Cut energy → mark model; general quantum resources | Gaussian availability and nuisance invariance differ from hybrid-state aperture and conjugate-disturbance hypotheses |
| Classical noise universality | Composition → exact return bridge | Mass/preparation equivalence, microscopic clocks and additive force-noise law; zero remains allowed |
| Detector events | Mechanical works → threshold/shared/local receivers; CHSH | Conditional independence and fixed routing differ from independent settings with shared readiness |
| Complete classical apparatus | Receiver mechanics → autonomous readout → calibration fibres | Actual hidden/controller state and calibrated access; no conversion of scalar reconstruction risks into covariance by definition |
| Force-limited information | Reachable lenses and minimax | Specified force, delay, record precision and independent/shared/block budget geometry |
| Complete retained records | Recording → score-constrained action | Conditional terminal noise, conjugate recoil, path limits, nonlinear curvature and branch-dependent policies |
| Quantum/classical records | Coherence countercase, Coulomb completion | Coherence, topology, singular-domain selection and fixed-interval localization remain separate |
| Gauge local step | Series/parallel → compact-group characters and SU2 matrix sectors → order-t action → retained-barrier estimates | Exact traces are not matrix entries; weak-strip (112)/(116) and quadratic (177) do not prove nonlinear (113), full recoupling or iteration |
| Four-dimensional running | Gaussian coefficient → one-step parallel matching → generated-action composition | Uniform subtracted cubic/quartic contractions, endpoint matching and remainder sum are open |
| Physical gauge spectrum | Lattice and limit obligations with Wilson, Dobrushin and KS inputs | Operators, physical units, volume/cutoff order, nontriviality and observable weight must match |
| Solved gauge models | Matrix, classical area and singular comparisons → 3D field limit | A classical sublevel bound is not a quantum gap; invariant sectors, nonzero modes and physical limits remain separate |
| Small-volume transfer | Feshbach → torus valley | Renormalized H1/H2 complement/Schur control, centre sector and physical observable algebra; H3 is not ground energy or one-patch spacing |
| Gauge vacuum probabilities | Magnetic identities, Agmon and slice transfer → action cost and operator comparison | An extensive global tail differs from a normalized local event and a signed Hamiltonian inequality |
The gauge atlas gives the dimension/group map; three continuum limits retains the distinct pion Ward identity and physical spectral criteria. The obstruction map identifies failed routes and countermodels that constrain every proposed extension. Source reading and mathematical proof status are recorded in each canonical treatment.