What survives refinement?

Two proof goals guide this repository: derive a positive universal action scale from independently justified physical premises in Newton’s Galileo comparison, and construct pure SU(3) Yang–Mills in four dimensions with a positive finite physical mass gap.

Exact insertion laws, conditional record bounds and controlled lattice estimates are in hand. Both main goals remain open. The pion is a symmetry benchmark for the comparison.

Strong results to start with

Classical construction and completion

The causal construction and cusp countermodels distinguish existence of motion, reversible state completion and conditional preparation floors. Their exact assumptions remain in the canonical proofs.

Groupoid scaling and inserted points

Exact scalar dilation commutes with full kernel composition and normalized endpoint-bridge marginalization. The local heat expansion retains explicit potential corrections; the finite-scale product retains insertion centres that collapse at the tangent boundary. Quantum-well zooms use different spatial and temporal scales. Physical ℏ remains supplied.

Newton’s action decomposes under every cut

The cut measure gives exact additive refinement costs. Constant-force and harmonic curves can be constructed under the stated laws for arbitrary shrinking partitions: construction and limit. Consistency alone leaves zero action admissible.

A record costs disturbance

General quantum instruments obey a sagitta/impulse disturbance bound with supplied ℏ and explicit comparison-state resources. The sharp Gaussian floor has narrower mark and preparation-invariance hypotheses; prepared packets show why a universal cell-action floor needs more.

Kepler completion and a controlled classical limit

Coulomb quantum evolution exists at every positive phase constant. The October 2 written derivation controls the singular multiplier and both centroids on fixed collision-free intervals with a fixed smooth regularization. Relativistic action thresholds depend on the chosen point-source domain; full critical Klein–Gordon evolution remains open. These additions are unrefereed.

Series closes; parallel carries the generated action

Directional gauge halving factors exactly. SU(2) midpoint traces and general compact-group cuts are exact; interacting mid-plane comparison remains a separate estimate.

Connected bad sets and exact dressing in a weak strip

All-order marked estimates (116) and dressed activities (112) are proved in their retained-barrier weak-recoupling domain. Section 37 gives a quadratic moving-source test for part of (113). These results do not yet give the full nonlinear group integral or stable physical iteration.

Lattice gaps at strong coupling

The Wilson-transfer criteria give explicit uniform regions. One KS treatment separates existential uniform stability, exact finite-volume bounds and the incomplete history counts behind the withdrawn numerical threshold. Controlled weak-trajectory transport and physical continuum reconstruction remain open.

Current open fronts

Newton: independently justify preparation, continuity and complete-record laws, then test whether they select quantum structure and nonzero physical ℏ. Classical histories and the meter and shear tests expose distinct limitations; the incoherent-packet countercase reaches the classical branch for quadratic forces.
Gauge local step: complex nonlinear centered bounds and decaying source dependence of the moving saddle, with the barrier retained. Current quadratic input.
Four dimensions and spectrum: depth-uniform generated-action matching, stable iteration, and the renormalized small-volume bridge. Continuum axioms, nontriviality, volume uniformity and physical spectral weight remain explicit.

Dependencies and exact next obligations · Live STATE. A gap at every bare lattice coupling (T2′) is an optional stronger statement; the continuum argument needs control along the appropriate weak-coupling scaling trajectory: correction.

Decisive corrections to retrieve

Global Agmon tails do not supply uniform local probabilities; the slice transfer has its own limiting assumptions.
Finite crossover certification and monotonicity: mechanisms, source scope and missing transfer.
Naive blocking, invertible flow and bare Feshbach UV control: scoped failures and surviving alternatives.
Small-volume spectra: the periodic centre-sector countertest separates the full gap from local spacing; the adjoint-loop trial targets the neutral sector.
A prior covariance restriction is vulnerable to complete and nonlinear records: exact tests. Thermodynamic costs and phase consistency supply conditional trades or ratios, while positivity remains separate.

The spectral upper-bound treatment keeps spatial Hamiltonian smearing, positive spectral measures and infrared integrability separate. Four-dimensional flow finiteness alone proves no global spectral gap.

A short reading path

The illustrated argument, then the two goals and the spectral benchmark.
Exact insertion and what a limit requires.
Newton: recent consolidated results → record/disturbance synthesis → classical limits and coherence → physical premise and calibration.
Gauge: series/parallel split → order-t input → current normalized-control notebook.

For a specific calculation use the dependency paths and notation/scale map. The provisional book gives a longer reading architecture; the correspondence test states where the analogy is exact.

Complete archive and evidence

Every maintained note, with summary, status and retrieval priority · greppable catalog · syntheses, technical manuscripts and historical PDFs · primary sources and reading coverage · stable older claim IDs · dated established-result map.

The catalog records completed work in short phrases. Maintained arguments and source companions contain the proofs, corrections and reading coverage; git preserves retired process records.

Retrieval priority measures usefulness to the current questions, not novelty or mathematical quality. Full notes govern their hypotheses and later corrections. Comments and corrections are welcome as repository issues.