Provisional book architecture

Proposed 2026-10-01; updated from the maintained notes, 2026-10-02. The chapters below remain a reading and writing plan. The test question is: what survives refinement, and under what conditions can a positive physical scale survive with it? The repository establishes several exact insertion laws, conditional scale bounds and informative closures. It has not established either main necessity/existence-and-gap goal. The Newton record bounds supply positive quantum action. The Kepler completion and classical-limit results leave physical coherence and domain selection as additional premises. The strong-coupling gauge inputs keep Wilson transfer bounds, KS finite-volume bounds and existential volume-uniform theorems separate. The reading paths organize the established inputs and their open extensions.

Choosing a spine

The preferred reading is foundations → Newton records → gauge refinement → physical scales and unfinished exits. An exact identity on one side is never promoted to a proof on the other. Correspondences explains where a common equation exists and where the comparison is structural.

Part I. What a refinement must preserve

1. Two questions and one discipline of limits

Introduce Newton’s positive action necessity and the pure SU(3) continuum existence/gap target, with physical time, cutoff, volume, preparation and observable weight kept separate. The pion illustrates protection of a zero threshold by symmetry. Sources: three continuum limits, official targets, lattice obligations. Purpose: formulate the actual endpoints without identifying a sampling gap, a finite-box gap or a supplied action constant with either endpoint.

2. Insert a variable, then eliminate it

Start with the exact constant-force cut and gauge heat-kernel convolution. Explain compatibility under every admissible partition, the sufficient summable-defect criterion, and the harmonic construction beyond a finite Cauchy estimate. Sources: refinement composition, cut consistency, zero spacing. Purpose: exact local composition and stated limiting hypotheses, without a positive scale being inferred from consistency alone.

3. The state at a cut

Position alone generally loses momentum; eliminating another body can leave memory even after position and momentum are retained. Telegraph bridges give a constructive alternative to scalar convolution rigidity. Sources: classical cut state, reachable composition, three-body memory, telegraph bridge. Purpose: identify the closure data before taking a continuum limit.

Part II. Newton’s cell, and the price of its record

4. A shrinking area with an exact action

Derive the inertial–parabola area, distinguish its matched-endpoint chord lens, and show how the cut measure decomposes the Dirichlet action into orthogonal hats. The polygon’s scalar phase uses the same functional with a supplied phase unit. Sources: insertion action, cut measure, polygon phase. Purpose: additive finite action bookkeeping; the geometry itself reaches zero.

5. From Gaussian marks to general disturbance

Present the sharp invariant Gaussian benchmark first, then arbitrary pointer recoil and arbitrary-instrument disturbance/Bures bounds. Explain which resources belong to the body, apparatus and comparison states, and why a known preparation can escape a preparation-independent cell floor. Sources: mark cost, probabilistic bound, recoil, disturbance, path length. Purpose: strongest current information/disturbance theorem, with its quantum input and preparation dependence visible. The early bounded-support model belongs in a correction box.

6. A unit, a classical limit, and a selection rule

Separate dimensional availability, radiation calibration, composition universality, quantum role and positivity. Newton-compatible deformations and covariant state restrictions expose alternatives to joint determinacy; Leibniz continuity becomes a precise but family-dependent record criterion. Compare complete coherent records with smeared records and the incoherent packet mixture that recovers the same classical swarm for quadratic forces. Then distinguish Kepler’s classical collision from its global quantum completion and fixed-interval centroid limit. Use the Dirac/Klein–Gordon selection table to show how point-source domains decide action thresholds; keep a radial critical operator separate from a full evolution theorem. Sources: fifth postulate, unit and indeterminacy, dimensional selection, rotation composition, Leibniz records, coherence and classical limits, Kepler completion and domains, Newton synthesis, §10a. Purpose: state which preparation and domain choices determine the claim. The October 2 additions are unrefereed written derivations; an independent physical necessity argument and universal calibration remain open.

7. Can a physical apparatus keep the restriction?

Use the two-pointer escape and exact Gaussian posteriors as reference cases, then physical feedback, adaptive records, nonlinear generators and weak copies. Finish at coherence memory in the supplied quantum free-evolution reference: positive tails permit phase reconstruction but fail uniform stability in the stated local-data metric. Sources: indeterminacy routes, recording closure, SED calibration. Purpose: a concrete physical history law remains missing; a covariance ceiling or a stationary bath alone does not discharge it.

Part III. Gauge refinement and the generated action

8. Solved lower-dimensional tests

Explain exact two-dimensional subdivision, compact-group Lévy alternatives, and why three-dimensional U(1) monopole suppression disappears under its fixed-coupling continuum trajectory while the same total-variation budget fails in four dimensions. Sources: zero spacing, Villain refinement, halving atlas. Purpose: dimension and group change what survives; a solved regulator limit does not automatically produce a nontrivial physical spectrum.

9. Series closes; parallel creates interactions

Factor directional halving into exact series moves and an interacting mid-plane integral. Develop the exact SU(2) midpoint character and general group/cut identities, including centre torsion at thirds for SU(3). Sources: series/parallel, SU(2) midpoint, centre cuts, order-t calculation. Purpose: isolate the nonlinear RG obligation; traces and one-loop coefficients are not the full normalized matrix or group-integral comparison.

10. Retain the barrier and normalize the clusters

Organize the large small-field notebook by logical dependency: convexity and rarity, response contacts, weak recoupling, marked connected sets, exact dressing, and moving Gaussian sources. Sources: small-field notebook, especially §§24–37; large-field action cost as a scoped supporting input. Purpose: (116) and (112) in their weak strip, then the quadratic source test for part of (113). Full nonlinear comparison, physical recoupling and stable iteration remain the chapter’s explicit open exit.

11. The four-dimensional logarithm

Compose Gaussian block maps before contracting generated vertices. Explain tree-level scale invariance, the conditional one-loop logarithm and the continuum-subtracted matching problem at arbitrary block depth. Sources: Gaussian blocking, parallel logarithm, 4D composition, UV/IR ladder. Purpose: depth-uniform nonlinear matching and remainder control, rather than an assumption that a formal running coupling is already a constructive RG.

Part IV. Which scales can survive?

12. A rigorous strong side and a difficult arrival

State the Wilson transfer bound, the continuous-time finite-volume Hamiltonian bound and Yarotsky’s existential volume-uniform theorem with their own normalizations, then the existential target box. The October 2 KS numerical uniform threshold is withdrawn. Explain why blocking without coupling flow, global Agmon suppression, conjugate flow and physical crossover certification do not bridge the weak trajectory. Sources: Wilson gap, KS gap, target box, reasons to stop, mass-gap position. Purpose: a physical positive gap needs controlled arrival and construction; all-bare-coupling T2′ is an optional stronger statement.

13. Spectral weight, valleys and the ultraviolet bridge

Contrast trial-state/moment upper bounds with lower-gap criteria from a complete observable frame. Develop the matrix-model valley mechanism and the torus/Feshbach test, retaining the UV dressing obstruction. The displayed cutoff calculation uses SU(2); corresponding SU(3) operator estimates remain an additional transfer obligation. Sources: moments, susceptibility, low-dimensional gap, Feshbach, torus valley, UV Schur error. Purpose: distinguish finite positive spectral weight and an upper energy scale from a volume/cutoff-uniform positive lower edge.

14. What the comparison has earned

Return to the two independent proof goals. Tabulate exact correspondences, structural analogies and the still-open physical premises, using three continuum limits, Newton record/parallel move, action floor and matrix gap, tangent groupoid. Purpose: explain why RG appears through elimination without claiming that its appearance selects positive action or proves a continuum mass gap. The comparison must distinguish a coherent-record contrast, an admissible interval in a selected point-source model, and a spectral gap of one Hamiltonian. None supplies the missing premise of another chapter.

Material outside the main spine

Maintained proof notes remain the sources for chapter claims. Synthesis opportunities identifies candidate chapters that would connect those proofs and their assumptions.