Reciprocity audit: Hermiticity does not force observed-unobserved exchange symmetry

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 83: Reciprocity audit: Hermiticity does not force observed-unobserved exchange symmetry.

Provenance

Preserved from interaction notebook section 83, 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.

83. Reciprocity audit: Hermiticity does not force observed-unobserved exchange symmetry

Section 82 found that an exact factor-swap symmetry on

$$\mathbf4=(\mathbf2_O,\mathbf2_H)$$

would select

$$q=1.$$

The next question is whether that symmetry already follows from the reciprocal/Hermitian character of the scattering lock or the projected cross-depth interaction.

It does not.

83.1 Hermiticity and factor exchange are different conditions

In the aligned frame, the ideal quadratic lock Hessian is

$$K_{\rm lock} =\kappa \begin{pmatrix} I&-I\\ -I&I \end{pmatrix}.$$

Define the observed-unobserved exchange involution

$$S_{OH} =\begin{pmatrix} 0&I\\ I&0 \end{pmatrix}.$$

For this special equal-diagonal lock,

$$[K_{\rm lock},S_{OH}]=0.$$

So the particular Procrustes Hessian used in sections 56--57 is exchange symmetric after alignment.

However, the most general reciprocal Hermitian block with the same off-diagonal link is

$$K_{\rm gen} =\begin{pmatrix} aI&-\kappa I\\ -\kappa I&cI \end{pmatrix}.$$

It is Hermitian for arbitrary real

$$a,\ c,\ \kappa,$$

but

$$[S_{OH},K_{\rm gen}]=0$$

if and only if

$$\boxed{a=c.}$$

Therefore

$$\boxed{\text{Hermiticity / reciprocal response does not imply }O\leftrightarrow H\text{ exchange symmetry}.}$$

The exchange symmetry of the special lock Hessian comes from the additional choice of equal observed and unobserved self-stiffnesses, not from self-adjointness alone.

83.2 The orbit generator makes the distinction especially clear

The spinor orbit family is

$$\widetilde G_q =\operatorname{diag} \left( q+\frac14, -q+\frac14, -\frac34, \frac14 \right).$$

It is Hermitian for every real $q$.

The spin-factor swap acts by

$$|+-\rangle\leftrightarrow|-+\rangle.$$

Direct calculation gives

$$[S_{OH},\widetilde G_q]=0$$

if and only if

$$\boxed{q=1.}$$

Hence the desired $q=1$ value is selected by exchange invariance, not by Hermiticity.

83.3 The locked complex structure is not itself exchange-even

In the aligned real $3+3$ frame, the locked complex structure is

$$J_{\rm lock} =\begin{pmatrix} 0&I\\ -I&0 \end{pmatrix}.$$

With the same exchange involution

$$S_{OH} =\begin{pmatrix} 0&I\\ I&0 \end{pmatrix},$$

one finds

$$\boxed{\{S_{OH},J_{\rm lock}\}=0,}$$

not

$$[S_{OH},J_{\rm lock}]=0.$$

So exchanging the two locked three-planes reverses the sign of the complex structure:

$$S_{OH}J_{\rm lock}S_{OH}^{-1}=-J_{\rm lock}.$$

This is important conceptually. The frame lock identifies the two three-frames, but it does not automatically make them indistinguishable under an ordinary commuting symmetry. The complex orientation distinguishes the direction of the map

$$H\to O$$

from its reverse.

83.4 The nucleus also distinguishes the two halves

The bound-state module defines

$$O$$

as the nuclear-relative three-plane and

$$H=O^\perp$$

as its orthogonal complement. The construction therefore gives the two subspaces different physical roles before locking.

This means an $O\leftrightarrow H$ symmetry cannot be inferred merely from their equal dimension.

83.5 Status of the exchange selection route

The exact result from section 82 remains true:

$$O\leftrightarrow H\text{ invariance of }G_q \Longrightarrow q=1.$$

But section 83 shows that current V2 does not derive that invariance from:

• self-adjointness of the master Dirac operator;
• reciprocal cross-depth response;
• Hermiticity of the lock Hessian;
• or the existence of the locked complex structure.

The exchange route is therefore conditional and cannot yet be used as a first-principles derivation of the $1{:}1{:}3$ orbit spectrum.

83.6 Revised proof obligation

The integer orbit parameter remains open unless one can derive an additional symmetry or selection law. The remaining possibilities are now sharply separated:

1. derive a genuine $O\leftrightarrow H$ invariance of the stationary Hamiltonian circle from the projected electron interaction;
2. derive $q=1$ from a different representation-theoretic property of the selected circle;
3. derive a dynamical variational rule over the allowed integral circles;
4. or accept $q$ as presently undetermined in the written V2 action.

The next useful test is representation-theoretic rather than variational: classify the integral $U(1)$ subgroups of $SU(4)$ compatible with the locked subgroup and the required $\mathbf1\oplus\mathbf3$ decomposition, and ask whether regularity, primitiveness, or isotropy of the locked orbit reduces the family $(q,-q,-1,0)$ to $q=1$ without invoking factor exchange.

Related

• Research status — current proof boundaries and correction guidance.
• Physical $(3,1)\oplus(1,3)$ audit: beam symmetry leaves $q$ free; frame exchange would select $q=1$ — preceding section in the research sequence.
• Locked-sector preservation theorem: invariance of the $\mathbf1\oplus\mathbf3$ split uniquely selects $q=1$ — following section in the research sequence.
• Leading-source orientation no-go: the written scalar and factorized $3\leftrightarrow6$ structures can select $s$ but cannot select the locked frame orientation — later audit limits or corrects this route.