Hosting versus Deriving the Born Rule in Registration Theory: A Conditional Theorem, Native Quadratic Invariants, and Obstructions
Status report on whether Registration Theory derives the qubit Born rule or only hosts a deterministic construction that reproduces it. The paper proves a conditional Born theorem under three explicit bridge conditions—additive seed, the required spherical-shadow fine-state law, and angle locking—then identifies native quadratic projector-overlap and square-root ledger structures while documenting the obstructions that prevent those ingredients from yet constituting a derivation.
Scope
Exact and conditional results are stated separately. The paper concludes that canonical RT does not yet derive Born frequencies; the missing object is the preparation–measurement map M_PM. Claims marked [R] are project-audit results rather than independent theorem proofs in this status paper.
Relation to other papers
- Builds on Deterministic Registration-Cycle Realization of Quadratic Projective Weights in Registration Theory: An Explicit Born-Form Existence Construction and the Remaining Linear-Coupling Problem
- Builds on Unoriented Phase Connection in Registration Theory: Exact Compact-Phase Descent, a Bilateral Circle Invariant, and Holonomy Constraints in Canonical ACRF-4
- Builds on Bilateral Participatory Registration in ACRF-4
- Builds on Retained Relational Geometry as a History Functional
Keywords
- Registration Theory
- Registration Theory (Physical Foundations)
- Eugene Catrambone
- Born rule
- quantum probability
- deterministic measurement
- ACRF-4
- preparation-measurement map
- M_PM
- receiver seed
- bilateral phase
- projector overlap
- quadratic invariant
- square-root ledger
- fine-state measure
- spherical shadow
- angle locking
- measurement foundations
- U10
- U12
- T-B17
- T-B18
- T-B21