Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear

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 73: Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear.

Provenance

Preserved from interaction notebook section 73, 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.

73. Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear

Section 72 found the corrected beam-axis decomposition

$$\widetilde G =\frac14\sigma_z\otimes I +\frac14 I\otimes\sigma_z +\frac34\sigma_z\otimes\sigma_z.$$

Because the adopted depth-six self-coupling is

$$g_{66}=\frac14,$$

the bilinear coefficient obeys the numerical relation

$$\frac34=3g_{66}.$$

The simplest possible explanation would be that the factor three comes from summing the same depth-six coupling over the three locked spatial directions. That hypothesis can be tested directly.

73.1 Isotropic three-axis contraction

For two spin-$1/2$ frame factors, the canonical rotationally invariant bilinear obtained by summing three equal directional channels is

$$\boxed{ H_{\rm iso} =g_{66}\sum_{a=1}^3\sigma_a\otimes\sigma_a. }$$

This is the ordinary isotropic exchange operator. With

$$g_{66}=\frac14,$$

its eigenvalues are the familiar singlet/triplet values

$$\boxed{-\frac34,\quad\frac14,\quad\frac14,\quad\frac14}$$

up to ordering.

This operator is not equal to

$$3g_{66}\,\sigma_z\otimes\sigma_z.$$

The latter has eigenvalues

$$\frac34,-\frac34,-\frac34,\frac34,$$

and explicitly singles out the beam axis.

Therefore merely saying “there are three spatial directions, so sum three copies of $g_{66}$” does not derive the required orbit-generator bilinear.

73.2 What the factor three would actually require

A scalar isotropic contraction gives

$$\sum_a g_{66}\sigma_a\otimes\sigma_a,$$

whereas section 72 requires, after the beam axis is selected,

$$\boxed{3g_{66}\sigma_z\otimes\sigma_z.}$$

To obtain the latter from three directional channels, the dynamics would have to project or coherently transfer all three channel weights onto the selected axial component rather than retain the rotationally invariant sum.

That is an additional dynamical statement. It is not implied by dimension counting, by $n_s=4$, by the existence of three spatial axes, or by the value $g_{66}=1/4$.

73.3 Relation to the seed identity $n_s=4$

The current V2 implementation defines

$$n_s=n_d+n_u=4$$

and adopts

$$g_{66}=\frac1{n_s}=\frac14.$$

Elsewhere the representation algebra notes the exact numerical identity

$$\dim S_6^+=4=n_s.$$

This makes

$$g_{66}=\frac1{\dim S_6^+}$$

an appealing possible normalization. But the current files do not derive the coupling from a normalized trace over the four-dimensional chiral spinor representation. In the code, candidate_self_couplings() explicitly labels these couplings as hypotheses supplied to the conditional scale test.

Consequently the chain

$$\dim S_6^+=4 \Longrightarrow g_{66}=\frac14$$

is not yet a theorem of V2.

73.4 Important distinction from the isotropic exchange spectrum

There is nevertheless an interesting representation coincidence. The isotropic operator

$$\frac14\sum_a\sigma_a\otimes\sigma_a$$

has one eigenvalue $-3/4$ and a triply degenerate eigenvalue $+1/4$, precisely the characteristic singlet/triplet splitting of

$$(\mathbf2,\mathbf2)\downarrow_{Spin(3)_{\rm diag}}=\mathbf1\oplus\mathbf3.$$

This is geometrically natural for the locked frame decomposition, but it is not the same spectrum as the orbit generator

$$\widetilde G:\qquad \left(\frac54,-\frac34,-\frac34,\frac14\right).$$

Thus the depth-six value $1/4$ fits naturally as an isotropic two-frame exchange normalization, while the orbit generator additionally requires one-body axial terms and an axial enhancement.

73.5 Verdict

The proposed shortcut

$$\boxed{3g_{66}\text{ comes automatically from three spatial directions}}$$

fails.

The surviving stronger possibility is instead:

1. an isotropic locked-frame interaction supplies a natural $1\oplus3$ exchange operator;
2. the scattering/beam state breaks the diagonal rotational symmetry to an axial subgroup;
3. the stationary projected dynamics then reshuffles or splits that exchange operator together with one-body axial terms into the exact orbit generator.

That third step is still missing and is now the precise place where the $3/4$ coefficient must be derived.

Related

• Research status — current proof boundaries and correction guidance.
• Beam-axis spinor decomposition: the orbit generator requires a bilinear lock term — preceding section in the research sequence.
• Orbit generator as a polynomial of the total locked axial spin — 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.