# RDG--Maxwell Bridge 15: exhaustion ledger and handoff

**Date:** 26 October 2026

**Status:** current finite bridge pass complete; external and analytical
frontiers isolated

**Programme:** [RDG--Maxwell bridge](README.md)

## Outcome

The arm has reached a useful stopping boundary. Every distinct branch opened
by the RDG--Maxwell comparison has either:

- an exact passing certificate;
- an exact counterexample or no-go;
- an identification with established mathematics;
- or a named empirical/analytical burden that cannot be closed by another
  small finite matrix.

Continuing to manufacture toy complexes would add examples without changing
the frontier.

## Exhaustion ledger

| Branch | Closed result | Remaining burden | Why another toy is insufficient |
|---|---|---|---|
| source recovery | RDG/MFE/Q vocabulary recovered; no Maxwell theorem found | derive a physically selected bridge | requires theory, not relabelling |
| carrier complex | periodic (T^3) complex, Betti data and Wilson sector exact | continuum/mesh-family estimates | one more finite mesh does not prove convergence |
| selector | symmetry-derived projector commutes; component projector fails | physical selection of (P) | requires a device or microscopic model |
| constitutive dynamics | uniform medium closes; inhomogeneous medium mixes | calibrated dispersive material model | parameters must come from measurement |
| sources | continuity-preserving and violating cases explicit | real source preparation and uncertainty | experimental burden |
| memory/resonance | extended Lorentz state closes; varying resonance leaks | nonlinear, thermal and noisy material state | model identification burden |
| remeshing | chain retract, reconstruction and diagonal no-go exact | approximation and stability theory | needs mesh-family analysis |
| sparse metric | minimal macroelement escape and overlap pass exact | conditioning under refinement | needs bounds, not another instance |
| topology | (H^1), spherical (H^2), and source-free torus (H^2) receipts exact | arbitrary cut/glue histories | relative/differential cohomology already supplies framework |
| loss | energy-output port and positive-real transfer exact | microscopic bath/noise law | nonunique realization; requires constitutive evidence |
| irreversibility | finite conservative recurrence explicit | continuum/thermodynamic limit | finite matrices cannot prove permanent decay |
| detector | eight conventional leakage channels and held-out rule compiled | same-device execution | requires hardware and statistics |
| literature | nearest mature fields and novelty distance mapped | systematic priority review if publishing novelty | current claim is integration, not priority |

## The joined picture

The useful chain is now:

\[
 \begin{aligned}
 &\text{cellular carrier and incidence}\\
 &\xrightarrow{\text{commuting selector}}
   \text{admitted and defect subcomplexes}\\
 &\xrightarrow{\text{metric/constitutive state}}
   \text{energy and Maxwell evolution}\\
 &\xrightarrow{\text{typed transport}}
   \text{remesh and topology receipts}\\
 &\xrightarrow{\text{ports and detector}}
   \text{prospective residual}.
 \end{aligned}
\]

At every arrow the programme now knows both a passing condition and at least
one ordinary failure mode. That is the strongest result of the arm.

## What would move the corpus next

Only three next tranches would materially change the conclusion.

### A. Numerical-analysis tranche

Define a genuine family of changing meshes and prove or measure:

- uniform boundedness of transport and restriction;
- conditioning versus refinement depth and vertex degree;
- approximation order for Maxwell fields;
- stability with nonconforming interfaces and moving boundaries.

Success would turn the finite certificates into a numerical method. Failure
would close the remeshing branch without affecting the detector discipline.

### B. Topological-state tranche

Implement fields as differential characters and cut/glue operations as maps
of relative differential-cohomology data. The retained record must include
curvature, integral class and holonomy. Success would generalize the finite
topology receipts; failure would show that the current record schema is too
small.

### C. Experimental tranche

Choose one coherent multi-mode device, freeze a calibration and predict a
held-out complex transfer matrix under loop-phase reversal. Carry all eight
nuisance channels with uncertainties. This is the only route that tests
whether the admissibility/defect grammar improves physical discovery rather
than merely organizing known mathematics.

## Stop rule

Do not reopen this finite arm merely to:

- add a larger complex with the same commuting identities;
- rediscover another standard compatible discretization;
- rename a port, cohomology class or passive transfer as an RDG effect;
- treat nonzero (Q)-power as anomalous before closing the nuisance ladder;
- infer fundamental discreteness, aether, gravity or new particles from the
  existence of a useful discrete carrier.

Reopen it when there is a new theorem, calibrated device dataset, materially
different topology-changing operation, or a failure that survives the current
controls.

## Final implication

The arm supports a restrained claim:

> Admissibility can be implemented as a sequence of commuting, energetic,
> topological and observational compatibility tests. In Maxwell-like systems,
> these tests do not replace known physics; they make the loss of a relation
> explicit and experimentally auditable.

That is enough to justify the programme as an interface between corpus
grammar and current physics. It is not evidence that the grammar is
fundamental.

## Handoff

- Start with the [primary literature and frontier-distance map](14_primary_literature_map_and_frontier_distance.md).
- Use the [detector null-test](13_detector_null_test_and_residual_budget.md)
  for experimental design.
- Use the [two-macroelement assembly gate](10_two_macroelement_assembly_gate.md)
  for numerical-analysis work.
- Use the [source-free torus cut](11_source_free_torus_flux_cut_gate.md) for
  differential-cohomology generalization.
- Use the [loss-port and finite-bath boundary](12_loss_port_and_finite_bath_boundary.md)
  for passive material and bath modeling.