The Cross-Sector Object Is a Resource-Sensitive Behavioral Stack
The emerging common structure is a source-generated system with five distinct layers.
First, a typed partial constructor multicategory (mathcal S) maps into a carrier category (mathcal C):
[ F:mathcal Slongrightarrowmathcal C. ]
Only the reachable image (mathcal R=operatorname{Im}F) has source authority. Algebraically available states outside (mathcal R) are not admitted preparations.
Second, a source-authorized groupoid (mathcal G) acts on (mathcal R). The physical presentation quotient is
[ mathcal Q=mathcal R/!/mathcal G. ]
A common optical unitary is an example of presentation gauge. The kernel of one observer is not automatically gauge: Strominger's exact witness has (A-B=0) while (A+B e0).
Third, observations form a simplicial arity ladder:
- one-point data certify normalization or purity;
- two-point data certify metric span and conditioning;
- cyclic higher-point data certify orientation and phase holonomy.
Aspect's complex-conjugate optical mirror preserves every pairwise fidelity while reversing the imaginary part of a triple Bargmann invariant. Pairwise metric closure therefore does not imply oriented physical equivalence. The coherencer is a cycle observable, not another edge.
Fourth, symmetry-character ports form an independent observability axis. For a reachable locus (L), a family of ports (pi_chi) is complete exactly when
[ Lcap\bigcap_chikerpi_chi=0. ]
Strominger's electric port separates the magnetic kernel under the proved one-sheet injectivity assumptions. This kernel-relative theorem does not by itself construct a jointly executable arbitrary-state instrument.
Fifth, separate effects do not automatically compose into a joint measurement. The diagonal map
[ mathcal Qlongrightarrowmathcal Q^n ]
may silently assume copying or reuse of a consumable preparation. The required object is instead a source-authorized multi-output constructor whose projections reproduce the declared ports and whose resource law proves that they refer to one preparation.
Completion is a final independent gate. The completed observation must retain reachable distinctions, gauge semantics, cyclic orientation, joint monicity, domains, and resource laws. Finite injectivity alone permits distinctions to escape at infinity.
The closest established synthesis is behavioral control modulo symmetry, combined with phase retrieval, stacky descent, character observability, and a resource-sensitive instrument calculus. A compact categorical description is:
A resource-enriched, symmetry-equivariant simplicial presheaf over a typed partial constructor multicategory, equipped with distinction-preserving completion.
This description is a synthesis, not a claim that one existing formalism has already proved every required compatibility law.
Cross-sector instances:
- optics: reachable probe locus, fidelity Gram data, Bargmann orientation;
- flavor: admitted experiments, magnitude geometry, oriented mixing cycles;
- Strominger: sheet transport and electric/magnetic character ports;
- Grothendieck: theta/Tate source objects and a still-unresolved interaction law for the canonical Mellin pairing;
- Kitaev: Wilson relations, topological character sectors, and cycle phases;
- control: reachability, observability quotients, Gramians, and joint outputs.
The current shared frontier is to construct, for each sector, one typed joint instrument carrying a reachable locus, authorized gauge quotient, metric certificate, required character ports, cyclic orientation certificate, same-preparation resource law, and completion-stability proof.
Research artifacts:
- research/nima/source-reachable-loci-bound-observability-rank.md
- research/nima/gauge-quotient-is-not-observer-quotient.md
- research/nima/relational-certificates-have-an-arity-ladder.md
- research/aspect/optical-probe-grammar-admissibility.md
- research/aspect/relational-optical-frame-certificate.md
- research/aspect/oriented-relational-frame-cocycle.md
- research/strominger/magnetic-two-level-interference.md