What question did this study set out to answer?

The aim is to analyze mass-like gaps in variational backbone regime closures as summary-interface footprints rather than primitive coefficients.

June 28, 2026Open Access

Variational Backbone and Regime Closures V (Mass): Core-Density Schur Reduction and the Order-Zero Spectral Mass Shift

Puntos clave

The aim is to analyze mass-like gaps in variational backbone regime closures as summary-interface footprints rather than primitive coefficients.
Computed the Schur-reduced retained operator on retained-effective variables, including retained fields.
Utilized a declared representative pullback for unread coordinates.
Applied projector-valued order-zero footprints at an R2-selected effective background.
Determined the lowest positive spectral value of the effective retained operator on the physical retained subspace.
Established that the mass or gap is derived from the summary-mediated spectral gap, not from primitive mass insertion.

Resumen

This paper treats mass-like gaps in VBRC as summary-interface footprints rather than as primitive coefficients. Starting from the fixed core density of Parts I-II, Part V computes the Schur-reduced retained operator on the retained-effective variables: the retained field and its licensed summary. The unread coordinate appears only through a declared representative pullback, not as an independently observable hidden-sector degree of freedom. At an R2-selected effective background, Schur elimination of the summary-side variation produces an effective retained operator with a projector-valued order-zero footprint. The mass or gap is not this footprint itself, but the lowest positive spectral value of the effective retained operator on the physical retained subspace. R3 then provides only the dispersion check. Part V is therefore a calculation of a summary-mediated spectral gap, not a theory of primitive mass insertion or a unique hidden-sector ontology.

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