Current research dependency graph

Navigation map, 2026-10-01. STATE selects work; this map does not change that queue. The catalog covers every note. Solid arrows below indicate recommended reading order; the accompanying text identifies actual mathematical inputs and required corrections. Dashed arrows mark proposed extensions or supporting transfers, not established inputs to a completed continuum construction.

flowchart TD
  R[Exact insertion and arbitrary partitions] --> S[Gauge series / parallel factorization]
  R --> C[Newton cut measure and polygon phase]
  S --> M[Compact-group midpoint identities]
  M --> O[Mid-plane order-t calculation]
  O --> B[Retained-barrier small-field estimates]
  B --> D[All-order weak-strip dressing]
  D --> Q[Quadratic moving-mean source test]
  Q -.-> N[Complex nonlinear saddle and full normalized comparison]
  N -.-> I[Stability, SU3 transfer, iteration]
  S --> G[4D Gaussian composition]
  G -.-> U[Depth-uniform generated-action matching]
  I -.-> V[Continuum construction and physical spectral weight]
  U -.-> V
  T[Strong-coupling target box] -.-> V
  F[Renormalized small-volume Feshbach bridge] -.-> V
  C --> P[Gaussian and general record costs]
  P --> A[Unit separate from indeterminacy]
  A --> Z[Statistical-speed conditional theorem]
  Z --> H[Complete-record and nonlinear recording tests]
  H --> E[Coherence memory and positive-tail stability]
  E -.-> X[Physical history rule with retained memory]
  X -.-> Y[Positive universal action necessity]

Gauge: the next calculation

Read in this order:

1. Refinement composition, exact insertion identities and the summable-error criterion.
2. Series/parallel refinement, Prop. 1: directional halving isolates the interacting mid-plane integral.
3. SU(2) midpoint and, when changing group or cut fraction, centre-sensitive cuts. Exact character traces do not supply all matrix entries or volume control.
4. Order-t mid-plane calculation, including torus holonomy and the normalized remainder obligation.
5. Small-field notebook, §§24–25 for targets (112), (113), (116), then §§33–37 for the actual current inputs.

Durable inputs: uniform convexity and joint bad-set rarity, decaying response kernels with contacts accounted for, an undifferentiated barrier, and a convergent weak-recoupling gas. Section 35 supplies all-order connected bad-component control in its weak strip. Section 36 constructs exact dressed activities satisfying (112). Section 37 supplies a fixed source radius and the subtracted log-barrier bound (177) in a centered quadratic model. It is a test of part of (113), not the complete nonlinear integral.

The next mathematical step is complex centered nonlinear control with decaying dependence of the moving saddle on boundary sources. Dependencies are the fixed-barrier contour/source accounting of §37 and the locality and connected-support reserves of §§33–36. Abandon a proposed estimate if it differentiates the barrier, charges moving-mean amplification to every good connector, assumes contact cancellation, or loses volume-uniform constants. Physical recoupling, target-covering paths, curvature conversion, group integral comparison, perturbed-action stability and iteration remain later obligations. Check group constants before transferring SU(2) results to SU(3).

Keep these constraints within reach: local/global Agmon, blocking criterion, flow versus coarse graining, explicit weak-side thresholds, and failed crossover bridges. The obstruction map gives the scope of each failure.

Gauge: four-dimensional matching and the spectral exit

For atlas cell 4 read Gaussian blocking → parallel logarithm → 4D composition, especially its (14) and §7. Exact Gaussian Schur composition and finite one-step subtractions are inputs; depth-uniform nonlinear contraction and remainder control remain open. A formal one-loop coefficient is not iteration.

For a physical gap exit read lattice obligations, with the scaling-region correction, then strong-coupling target and UV halving/IR confinement. Strong-coupling volume uniformity is durable. Reaching that box with controlled generated interactions from a weak-coupling scaling trajectory is open. T2′ at every bare coupling is stronger than the continuum argument needs. Construction, nontriviality, volume uniformity, physical time and surviving gauge-invariant spectral weight must each be supplied.

For STATE’s supporting small-volume bridge read Feshbach reduction → torus valley, together with the UV Schur correction. H3 is the effective zero-mode valley-gap estimate. The full transfer also needs H1’s nonzero-mode separation and a renormalized/fibered H2 Schur bound; the bare-vacuum error is not uniform as the cutoff is removed. Spectral moment or trial-state upper bounds do not replace a positive lower bound. The displayed cutoff torus calculation uses SU(2); applying this supporting bridge to the main SU(3) goal also requires the corresponding group-dependent operator estimates.

Newton: the next mechanism

Read in this order:

1. Fifth postulate: joint determinacy is the statement under examination; the deformation and state-restriction alternatives both leave positivity as an additional obligation.
2. Planck paper, for the geometric observable and the resource-scoped Gaussian, recoil and general disturbance theorems.
3. Unit and indeterminacy, separating a radiation action unit from a complete-record restriction.
4. Indeterminacy routes, for the conditional statistical-speed bound and its affine Gaussian instance.
5. Recording notebook, §§8.11–8.23: physical feedback, complete records and nonlinear countertests. In the supplied quantum free-evolution reference, the packet tests retain phase memory and distinguish positive-tail reconstruction from uniform stability in the stated local-data metric.
6. Score-constrained ensembles: the canonical field action is a restricted classical action, not an invariant Newtonian reduction. Full population-conjugate action includes reservoir transport; fixed compact transfer shapes do not close even after adding Gaussian width. Sections 4–5 give an exact positive non-Gaussian forward orbit, then the quartic tail and recoil energy generated by a nonlinear coordinate record.

Section 8.22 realizes (117)’s score correction by a shared-sign classical preparation and canonical copies. Section 8.23 repairs its free-motion fourth-moment failure by a stationary switching process with a prescribed force. The preparation-dependent force obstruction (151) identifies the apparatus obligation: a closed implementation retaining that state and energy exchange, tested on two unread widths with complete controller records. The score-constrained note’s positive completion survives free field evolution but nonlinear recording leaves its finite mode family. Its next mechanism is a dynamical realization retaining the generated fields, actual sheet velocities, force/switch energy and unread labels. Terminal-conjugate recovery (140) remains a test before claiming a universal floor. The earlier phase-memory and unread-energy tests (119), (129), (132)–(135) still apply at separation and reunion. Assuming Fisher ensemble dynamics or an exact low-density phase readout does not supply this rule; an unexcluded zero branch does not establish necessity. The Gaussian repair supplies neither the apparatus nor phase reunion; read its assumptions before reuse.

Durable support: cut measure, recoil, disturbance, and path length. The Gaussian action floor uses non-adaptive independent marks and preparation-invariant tests. General instrument bounds allow wider protocols but account for body/apparatus resources on the specified comparison states.

Retrieve Leibniz continuity only with its admissible Gaussian family and optimized invariant-test verdict. Every fixed finite protocol can be continuous even at zero mark cost; the positive-cost equivalence is a statement about that optimized family. Uniformity over arbitrary adaptive complete-record apparatus is an additional physical premise. The stochastic route supplies composition of a common coefficient, not its nonzero value. The SED calibration retains its disputed Planck-spectrum and resonant-variable hypotheses.

Avoid restarting prior-only covariance restrictions, thermodynamic record costs, or phase/prequantization consistency as positivity proofs. Their precise surviving statements are in obstructions.

Orientation and supporting branches

The principal proof inputs can be read separately from the diagram’s routing:

Three continuum limits keeps the pion comparison as a symmetry/spectral benchmark rather than a third construction programme. Halving atlas locates open cells; refinement results is a compact theorem entry point, with later notebook sections needed for the current frontier. Historical proof architecture in ../newtonlean is source/formalization context, not a Lean-build dependency of these modern calculations.