From changing coordinates to changing meaning.

One mountain can have a photograph, contour map, and coordinates. In quantum mechanics, one physical ray can have many phase-related vector descriptions.

When a representation changes, ask which quantities remain fixed instead of comparing raw coordinates.

A transport law says how to carry a state while local representation conventions change.

No single chart needs to be privileged. Compatible patches and their transition rules can carry global information.

Two atlases can describe the same bundle through an explicit witness without forcing either atlas into a universal master form.

A direct witness and a composed path may establish the same relation. Their evidence can differ while their semantic coordinates agree.

In non-Abelian transport, reversing composition can change the result. Semantic workflows should not assume commutativity without a law saying so.

Legal, operational, audit, scientific, or data-system views can concern one subject while remaining non-identical and sometimes non-equivalent.

Symmetry, inverse, transitivity, entailment, integrity, and composition belong to the relation profile—not to a universal hidden algebra.

A relation may apply only in one jurisdiction, version, product, role, or interval. Historical replay uses the law effective then, not today's law by default.

A useful result says what moved, under which witness and law, which invariants were preserved, what was lost, and what authority the result does—or does not—have.

A retired law can remain necessary for historical replay while becoming unavailable for new use. Still-live dependents require an explicit transition policy; a version timeline alone cannot decide their fate.

A changed rule can reach many nodes in a graph without changing their meaning. Real impact requires a declared causal dependency and a law saying how the consequence propagates; unknown causality should remain unknown.

Several local perspectives can jointly support one coherent account without being rewritten into one master schema. Required overlap must cover the shared roles, and alternate translation paths must agree as laws—not only on the current sample.

Local perspectives can update asynchronously. Authority belongs to semantic roles and effective times, not the largest revision number. A stale projection can remain visible without overriding current meaning; shared-authority conflict and authority gaps fail closed.

A federation can exchange bounded evidence instead of complete local records. Exact disclosure, bounded attestation, deliberate nondisclosure, policy-prohibited disclosure, and genuine ignorance must remain distinct—and a bounded attestation proves only the predicate it actually states.