Abstract

Multiple sectors in the Axis Model suite (electroweak normalization, strong-sector normalization, emergent gravity, and scalar-projected neutrinos) are written using a shared scalar-coherent kernel language. This raises a legitimate structural concern: are distinct sectoral constraints being silently identified as one object, leading to hidden parameter sharing or over-constraint? We show that, at the level where the suite derives boundary conditions and effective couplings, the kernel is a background operator rather than an observable. Each sector accesses it only through projector-restricted contractions (e.g. PsK−1 Ps and scalar functionals thereof), not through the full operator content or eigenstructure of K on the full internal space. Consequently, no sector fixes the kernel uniquely. Instead, sectoral results constrain only a finite collection of projected invariants. We formalize this as a Principle of Projector-Restricted Observability: sector observables depend only on PsK−1 Ps, not on K−1 as a full operator. We formalize an independence principle for non-identical projectors and state a compatibility criterion as an existence problem: whether the intersection of the sectoral feasible sets in SPD is nonempty. The existence status is illustrated by the Schur-complement kernel constructions presented in Appendix J of the companion electroweak-normalization paper, and by a minimal non-commuting-projector toy model given here. This paper introduces no new parameters, fits, RG running, or phenomenological predictions. It is a logical-status and consistency note aimed at clarifying what the suite currently does and does not claim about kernel universality.

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Keywords: Axis Model, scalar-coherent kernel, projector-restricted observability, formalism, effective field theory, gauge normalization, emergent gravity, neutrino mass generation, linear algebra consistency, sectoral compatibility