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
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:
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.
- Use the detector null-test for experimental design.
- Use the two-macroelement assembly gate for numerical-analysis work.
- Use the source-free torus cut for differential-cohomology generalization.
- Use the loss-port and finite-bath boundary for passive material and bath modeling.