Grassmannian sigma-model and Jacobi audit: where the eigenvalue four actually enters

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 63: Grassmannian sigma-model and Jacobi audit: where the eigenvalue four actually enters.

Provenance

Preserved from interaction notebook section 63, 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.

63. Grassmannian sigma-model and Jacobi audit: where the eigenvalue four actually enters

Section 62 found the canonical Einstein relation

$$\operatorname{Ric}=4g$$

on

$$\mathrm{Gr}(3,6)=SO(6)/(SO(3)\times SO(3))$$

with the projector-tangent normalization

$$\langle X,Y\rangle=-\frac12\operatorname{tr}(XY).$$

The next question is whether this geometric eigenvalue four is automatically the Hessian of the simplest projector-locking action.

It is not.

63.1 Jacobi/Ricci operator

For the compact symmetric-space curvature convention

$$R(X,Y)Z=-[[X,Y],Z]_{\mathfrak m},$$

the Ricci/Jacobi trace through an orthonormal tangent basis is isotropic. Direct finite-matrix evaluation on all nine elementary mixed tangent generators reproduces the same magnitude-four operator found in section 62, up to the overall sign convention chosen for $R$:

$$\boxed{\operatorname{Ric}^{\sharp}=4I_9.}$$

Thus the number four genuinely belongs to the curvature/Jacobi structure of the metric of the full projector manifold.

63.2 Hessian of the simplest projector mismatch

For $V_P(P)=\frac{\eta}{2}\|P-P_0\|_F^2$ and a unit elementary observed--unobserved plane generator $X$, set $P(t)=e^{tX}P_0e^{-tX}$. Direct calculation gives

$$\|P(t)-P_0\|_F^2=2\sin^2t,\qquad V_P(t)=\eta\sin^2t,\qquad V_P''(0)=2\eta.$$

An additional factor $1/2$ in the potential would give Hessian $\eta$, not $2\eta$. The mismatch Hessian depends on the stated action coefficient and is not the Ricci operator $4I_9$.

63.3 Ricci curvature is not automatically the locking Hessian

The geometric identity

$$\operatorname{Ric}^{\sharp}=4I_9$$

and the projector mismatch Hessian are different mathematical objects.

The former is fixed by the homogeneous geometry. The latter is fixed only after an action functional and its normalization have been specified.

Therefore the tempting chain

$$\operatorname{Ric}=4g \quad\Rightarrow\quad \widehat\kappa=4$$

is not valid without an action-level theorem.

Likewise,

$$g_{66}=\frac14$$

being the reciprocal of the Ricci eigenvalue remains an exact and interesting normalization relation, but not yet a derived dynamical identity.

63.4 Where the eigenvalue four could legitimately enter

The Ricci eigenvalue can enter a physical fluctuation operator if the effective projector dynamics contains a standard geometric Jacobi term. For example, second variation of a harmonic-map/sigma-model action about a nontrivial background map contains schematically

$$\mathcal J\xi =\nabla^*\nabla\xi -\operatorname{tr}R(\xi,dP)dP.$$

On a sufficiently isotropic background whose normalized tangent directions fill the projector manifold symmetrically, the curvature trace can reduce to a multiple of the Ricci operator,

$$\operatorname{tr}R(\xi,dP)dP \propto4\xi.$$

But the proportionality depends on the background derivative norm and on how spacetime/source directions populate the nine projector tangents.

So the number four becomes dynamically relevant only after the master theory determines the background projector map $P(x)$ and its kinetic normalization.

63.5 Result

The Grassmannian sigma-model test therefore gives a mixed outcome:

$$\boxed{ \begin{array}{ll} \operatorname{Ric}=4g & \text{exact and universal},\\ \text{projector-lock Hessian}\propto\eta & \text{action-normalization dependent},\\ \text{Jacobi curvature term}\propto4 & \text{possible on an isotropic dynamical background},\\ \widehat\kappa=4 & \text{not yet derived}. \end{array} }$$

This prevents overinterpreting the $4\leftrightarrow1/4$ coincidence while preserving the genuinely useful geometric result.

The next proof obligation is now sharper than before: derive or infer the effective projector kinetic/action term generated by the existing V2 projected Dirac/contact system, then compute its second variation. Only that calculation can decide whether $4I_9$ actually enters the physical lock-frequency generator.

Related

• Research status — current proof boundaries and correction guidance.
• Canonical projector geometry: the rank-three Grassmannian has isotropic Ricci eigenvalue four — preceding section in the research sequence.
• Dirac-projector commutator: a conditional sigma-model construction — following section in the research sequence.