background_object_audit
plain-language theorem explainer
Seventeen background mathematical objects (continuum, point, space, sets, equality, infinity, functions, probability, amplitude, comparison, completion, complexes, Hilbert spaces, manifolds, measures, physics displays) receive explicit commitment tags in the objecthood registry. Anyone citing registry completeness or the strong-closure certificate needs this audit. The proof is pure definitional reflexivity on the classification table.
Claim. The objecthood map sends continuum, infinity mode, conservative completion, infinite Hilbert space, and measure display to completion; point, space, complex numbers, finite Hilbert space, and manifold display to display; set-object and finite amplitude to permitted; equality regime to quotient; function transport, valid comparison, and physics display object to observable; and finite probability to forced.
background
The objecthood registry classifies every mathematical object the program builds by a seven-valued commitment: forced (uniquely fixed by the law), permitted (admissible free choice), quotient (identification under equivalence), completion (limit-point closure, an independent axiom), display (rendering or instrument, not a native ingredient), observable (defined by what can be measured), and convention (gauge or labeling choice).
The carrier of that classification is an inductive type of program objects, ranging from finite-distinction rationals and protocol reals through physical quotients, generable carriers, and the continuum interface, extended here to Hilbert displays, manifolds, measures, and physics display objects. The classification assignment itself is a total function from those objects to commitments; the present theorem simply records the values on the background layer.
Module context is the Primitive Recognition Calculus foundation: objects must not enter the theory untyped. Upstream display constructions (finite Hilbert display of $F_{RS}$ amplitudes) and completion interfaces supply the concrete carriers being tagged.
proof idea
Term-mode proof: a 17-fold pair of rfl. Each conjunct is definitional equality against the clauses of the classification function, so no lemmas are invoked. The audit is a compiled checklist that the table already says what the doc-comment claims.
why it matters
Without explicit tags, continuum, infinity, Hilbert spaces, and measures could slip into native status. The audit freezes them as completion or display commitments, matching the completion-plan request that complex numbers, finite and infinite Hilbert spaces, manifolds, measures, and physics display objects be typed.
Downstream, the strong-closure certificate assembles the closed Delta-native theorem surface and consumes this registry discipline: every background object appearing in that certificate already carries a commitment. In the broader Recognition chain this is bookkeeping rather than a T0–T8 forcing step, but it is the gate that keeps completion axioms and display instruments from being misread as forced native law.
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