Conditional mixed-bilinear singlet selector

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Research record, section 93: Conditional mixed-bilinear singlet selector.

Provenance

Preserved from interaction notebook section 93, recorded 2026-09-05 and corrected 2026-09-06. Original sequence contents. This is an exploratory research record, not an independently established V2 result. Read the related later audits before using an intermediate conclusion.

93. Conditional mixed-bilinear singlet selector

On the carrier $\mathbf4=(\mathbf2,\mathbf2)$, define, for $Q\in SO(3)$,

$$T_Q=\sum_{a,i}Q_{ai}\sigma_a\otimes\sigma_i.$$

Choose the spin lift convention $U_Q\sigma_iU_Q^\dagger=\sum_aQ_{ai}\sigma_a$ and let $|S_Q\rangle=(U_Q\otimes I)|S\rangle$, where $|S\rangle$ is the normalized singlet. Then

$$T_Q=(U_Q\otimes I)T_I(U_Q^\dagger\otimes I) =I-4P_Q,\qquad P_Q=|S_Q\rangle\langle S_Q|.$$

Therefore $\operatorname{spec}(T_Q)=\{-3,1,1,1\}$. For a supplied coefficient $\kappa>0$,

$$\kappa T_Q+3\kappa I=4\kappa(I-P_Q)\succeq0$$

has the unique ground ray $[S_Q]$ and gap $4\kappa$.

The bilinear pairing of two $(\mathbf3,\mathbf3)$ tensors is unique up to scale, giving $\operatorname{tr}(MQ^T)$. These are exact representation statements for a supplied tensor and coefficient.

They do not establish that the projected Dirac operator contains $\kappa T_Q$. In particular, the torsion-free spatial Levi--Civita mixed connection contracts to a mean-curvature vector (Section 94), not a free nine-component lock coefficient. No independent time-component tensor avoiding that contraction has been derived here.

A field-fibre ground vector must also not be silently identified with the base coordinate $q\in Q_6$: a base lock requires a derived $q$-dependent spectral energy or another explicit base/field map (Section 96). The second-order constructions in Sections 95--97 are conditional tests of a possible source, not a completed derivation of this operator.

Related

• Research status — current proof boundaries and correction guidance.
• Correction to Section 93: the Levi--Civita mixed connection contracts to mean curvature and does not by itself supply the nine-component lock operator — following section in the research sequence.