# Dynatology 02: Corpus Crosswalk

**Date:** 24 October 2026  
**Status:** organizational overlay; source documents retain their own authority

## Programme roles

| Existing lane | Primary dynatological role | What it contributes | What it does not by itself supply |
|---|---|---|---|
| [Admissibility](../admissibility/admissibility_00_foundational_programme.md) | \(A,T\) | allowed regions, transformations, invariants and physical contact gates | a universal objective or unique realized state |
| [Typed two-rail relation carrier](../admissibility/admissibility_126_typed_two_rail_positive_lift_preserves_the_mirror_face.md) | \(X,A,\rho,I\) | a positive pre-quotient carrier retaining source, target and relation type; an exact rail/type collapse and rank scar | path composition, semantic adjoint, dynamics or physical port selection |
| Representation and exceptional algebra | \(X,T\) | ambient carriers, actions, decompositions and compatible maps | physical admission or scale matching |
| Jordan/Spin response | \(A,\rho,L\) | positive cones, causal response carriers and typed symmetry closure | the unique matter embedding |
| [Freudenthal](../freudenthal/freudenthal_parallel_synthesis_decision_memo.md) | \(X,J,\rho\) | exceptional completion of source--dual-response inverse matching | triality orientation, physical \(E_7/E_8\) symmetry or a new scale |
| Orientation | \(T,D,I\) | reversals, conjugations, anti-invariants and detector faces | an automatic identification of distinct physical involutions |
| [Quotient scars](../quotient-scars/README.md) | \(\rho,L,D,I,R\) | loss, memory, recovery, composition defects and complete-erasure controls | a universal rule that every quotient leaves a scar |
| Frontier physics targets | \(D,R\) | apparatus, observables, nuisance models, kill tests and useful nulls | upstream fundamental interpretation |
| Closure | all coordinates | terminal gates, no-go boundaries and reopening requirements | permission to infer across a closed gate |

## The main corpus journey

The current research path can be read as:

\[
\begin{array}{rcl}
X&:&\text{exceptional/Jordan/Lie ambient architecture}\\
\downarrow A&&\text{positive, causal, gauge and representation constraints}\\
\downarrow\rho&&H_2(\mathbb R),\ Sp(4,\mathbb R)\simeq Spin_0(2,3)\text{ response throat}\\
\downarrow L&&\text{source--response, projector and channel dynamics}\\
\downarrow D&&\text{resolved, summed, oriented and boundary detectors}\\
\downarrow I&&\text{spectral, nonlinear, integral, geometric and memory scars}\\
\downarrow R&&\text{composition, scale and frontier experiments}.
\end{array}
\]

This is a navigation path, not a proof that \(E_8\) is the physical background or that every intermediate structure is uniquely selected.

## Representative instantiations

### Triality quotient

- \(X\): \(D_4\) roots, singularity, magic-square sheets or four resolved channels.
- \(A\): declared cyclic action and compatible metric/dynamics.
- \(\rho\): orbit quotient or unresolved detector block.
- \(I\): \(G_2\) long/short ratio, hidden modes, cubic orientation, \(\mathbb Z_3\) lattice scar and monodromy.
- Open gate: a physical operator over the singularity or the four-mode experiment.

### Moving projector and holonomy

- \(X\): projectors and their frame lifts.
- \(A\): gapped control paths and positive state geometry.
- \(T\): the universal connection \([P,dP]\).
- \(J\): independently measured selector datum when contraction is claimed.
- \(I\): quarter-turn holonomy and orientation-sensitive records.
- Open gate: source-native platform implementation with conditioned and unconditional maps.

### Freudenthal inverse response

- \(X\): paired Jordan source and dual-response slots.
- \(A\): positive \(H_2(\mathbb R)\) face and dual congruence.
- \(J\): Gaussian/log-determinant variational functional.
- \(\rho\): restriction from the exceptional parent to the response face.
- \(I\): quartic discriminant and critical-null inverse ray.
- Boundary: the action derives inverse matching but not the exceptional embedding or triality scar.

### Casimir--thermoelectric response

- \(X\): causal material-response tensors across frequency, field and geometry.
- \(A\): passivity, Kramers--Kronig, Onsager, boundary and material constraints.
- \(D\): Hall, Seebeck, Nernst and differential force/torque records.
- \(R\): common carrier and no-refit propagation between experiments.
- Open gate: one broadband tensor and nuisance-complete force prediction.

### Measurement-conditioned selection

- \(X\): system plus apparatus plus record.
- \(\rho\): conditioning on a trajectory versus tracing over outcomes.
- \(L\): Kraus/instrument dynamics.
- \(D\): complete record and ensemble channel tomography.
- \(I\): branch purification may survive in records while unconditional uniqueness does not.
- Boundary: conditioned selection is not unconditional physical selection.

## Current coverage and gaps

| Coordinate | Corpus coverage | Principal gap |
|---|---|---|
| \(X\) ambient carriers | extensive | no theorem selects one universal ambient carrier |
| \(A\) admissibility | extensive and increasingly operational | composition of heterogeneous admissibility predicates |
| \(T\) reachability | partial | viability kernels and resource-bounded reachable sets rarely computed |
| \(J\) objective provenance | uneven | objectives often remain imported, state-dependent or absent |
| \(\rho\) realization/reduction | strong finite examples | physical provenance of several reductions |
| \(L\) dynamics | exact in mini-systems | common source-native action across lanes |
| \(D\) observation | strong detector theory | end-to-end calibrated apparatus models |
| \(I\) scars/recovery | strong quotient-scar programme | theorem-level conditions separating recovery from total erasure |
| \(R\) scale/composition | emerging | renormalization-stable and serial-composition laws |

## Priority signposts

1. **Objective provenance:** derive rather than choose \(J\).
2. **Viability and reachability:** compute the difference among admissible, viable and reachable sets.
3. **Composition:** measure when serial admissibility and direct admissibility disagree.
4. **Scale:** test which scars commute with coarse-graining and renormalization.
5. **Detector-relative actuality:** classify distinctions that exist physically but not in a declared record algebra.
6. **Experimental closure:** execute the four-mode quotient-scar and projector-return protocols.