Discrete Records Alone Do Not Determine Gauge Ontology: Connes Rigidity, Lattice Continuum Limits, and Source--Shadow Reconstruction in TCGS--SEQUENTION

\begin{abstract} This paper tests a composite Rigidity--Necessity Thesis: that continuous gauge groups can be recovered from finite experimental or lattice records only if the group-factor functor \(\Gamma\mapsto L(\Gamma)\) is injective on countable discrete ICC property-(T) groups, and that failure of this injectivity would make quantum field theory physically indeterminate. The thesis is formulated as an explicit inferential target; no claim is made that any one source states every component verbatim. We show that the proposed chain conflates distinct mathematical objects and reverses the direction of the standard construction. Connes' conjecture concerns countable discrete groups whose group von Neumann algebras are type~\(\IIone\) factors. Lattice gauge theory instead begins with a compact Lie structure group \(K\), places continuous variables \(U_e\in K\) on links, and studies a forward scaling limit while retaining the gauge data as part of the theory. A finite lattice with positive-dimensional compact \(K\) is not finite-dimensional at the kinematic level: \(L^2(K)\) is infinite-dimensional by Peter--Weyl theory. Finite-dimensional matrix algebras arise only after a finite-group choice or an explicit representation cutoff, neither of which automatically remembers \(K\) or enters Connes' ICC setting. The continuum limit is not an inverse reconstruction of a gauge group from one operator algebra. Algebraic quantum field theory is specified by a net of local algebras together with states, covariance, locality, inclusions, superselection sectors, and representation data; its local algebras are generically type~III rather than the special type~\(\IIone\) group factors in Connes' problem. Where compact gauge groups are reconstructed, Doplicher--Haag--Roberts and Doplicher--Roberts theory use an enriched symmetric rigid tensor category of localized sectors. The global form of a gauge group additionally depends on matter representations, line operators, generalized symmetries, and bundle sectors. Consequently, noninjectivity of \(\Gamma\mapsto L(\Gamma)\) would establish only that one analytic invariant is incomplete on one category of discrete groups. It would not erase the empirical content of gauge theory. The manuscript has two explicitly separated layers. Part~I is a framework-independent reconstruction analysis and supplies no evidence for TCGS--SEQUENTION. Part~II derives conditional consequences under the four governing TCGS axioms together with a separately declared Analytic Representation Package; the latter supplies functional-analytic realization hypotheses and is not a fifth ontic primitive. The non-experiential Informational Source is typed as the informational organization of an admissible complete source configuration, \(\mathfrak M_{\Xi}:=\Info(\Xi)\), without independent source data or ontic evolution. The Extrinsic Constitutive Law is separated into a constitutive map, its complete profile value, and one selector-evaluated readout, \[ \Read_{S,G}:\mathscr I_{\mathrm{src}}\longrightarrow\mathscr P_{\mathrm{read}}, \qquad \mathscr E_S:=\Read_{S,G}\circ\Info, \qquad \mathcal E_{S,\Xi}:=\mathscr E_S(\Xi). \] The compact framework notation \(\mathfrak M=\mathcal E_S\) is retained only as a conditional ontic co-reference thesis under an exhaustion condition; it is not used as literal equality between objects of different formal type. For a selector \(\sigma\), a declared physical-readout map \(\pi_{\sigma}^{\mathrm{phys}}\) yields \(\mathscr O_S^{\sigma}=\pi_{\sigma}^{\mathrm{phys}}\circ\mathscr E_S^{\sigma}\). Its closable derivative \(\mathcal L_{p_0}^{(\sigma)}\) defines the positive self-adjoint observability operator \(\mathcal O_{p_0}^{(\sigma)}=(\mathcal L_{p_0}^{(\sigma)})^\dagger\mathcal L_{p_0}^{(\sigma)}\); its spectrum measures squared readout sensitivity, not energy or dynamical stability. Physically equivalent selectors are organized by a groupoid and must carry unitarily natural readouts, whereas in