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 62: Canonical projector geometry: the rank-three Grassmannian has isotropic Ricci eigenvalue four.

Provenance

Preserved from interaction notebook section 62, 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.

62. Canonical projector geometry: the rank-three Grassmannian has isotropic Ricci eigenvalue four

Section 61 left an apparent ambiguity between adding the three unobserved curvature channels coherently or adding their quadratic norms. A more invariant diagnostic is available: compute the curvature of the entire projector manifold rather than selecting an unobserved phase prescription.

The rank-three orthogonal projectors in six dimensions form the real Grassmannian

Gr(3,6)≅SO(6)/(SO(3)×SO(3)). \mathrm{Gr}(3,6) \cong SO(6)/(SO(3)\times SO(3)).

At

P0=diag⁡(I3,0), P_0=\operatorname{diag}(I_3,0),

the nine tangent generators can be written

Xai=(0Eai−EaiT0),a,i=1,2,3. X_{ai}= \begin{pmatrix} 0&E_{ai}\\ -E_{ai}^T&0 \end{pmatrix}, \qquad a,i=1,2,3.

With the invariant normalization

⟨X,Y⟩=−12tr⁡(XY), \langle X,Y\rangle=-\frac12\operatorname{tr}(XY),

the nine XaiX_{ai} are orthonormal.

62.1 Exact sectional-curvature pattern

For the compact symmetric-space metric, the sectional curvature of two orthonormal tangent generators is represented by the squared isotropy commutator

K(X,Y)=∥[X,Y]∥2. K(X,Y)=\|[X,Y]\|^2.

Direct evaluation of all 3636 unordered tangent pairs gives a simple rule. For a fixed XaiX_{ai}, the commutator is nonzero precisely for:

  • the two generators XbiX_{bi} with b≠ab\ne a, sharing the same unobserved direction;
  • the two generators XajX_{aj} with j≠ij\ne i, sharing the same observed direction.

Each nonzero sectional contribution has unit norm.

Thus every unit tangent direction has exactly four unit-curvature partners.

62.2 Ricci curvature

Summing sectional curvatures through an orthonormal tangent basis gives

Ric⁡(Xai,Xai)=4 \operatorname{Ric}(X_{ai},X_{ai})=4

for all nine tangent directions. Hence

Ric⁡=4g \boxed{\operatorname{Ric}=4g}

in this unit projector normalization, and the scalar curvature is

R=9×4=36. \boxed{\mathcal R=9\times4=36.}

This removes the unobserved-channel phase ambiguity at the level of canonical projector geometry. The full Grassmannian curvature is isotropic even though individual sectional curvatures are either zero or one.

62.3 A new dimensionless geometric number

The projector manifold therefore supplies a distinguished dimensionless eigenvalue

4 \boxed{4}

without fitting and without choosing a coherent unobserved-frame phase.

This number is more defensible than either the ad hoc coherent norm 99 or the channelwise norm sum 33, because it is the Ricci eigenvalue of the complete homogeneous projector geometry.

There is a numerically striking relation to the adopted charged-lepton coupling,

g66=14, g_{66}=\frac14,

namely

Ric⁡Gr(3,6)=g66−1=4. \boxed{\operatorname{Ric}_{\mathrm{Gr}(3,6)}=g_{66}^{-1}=4.}

At present this is only an exact numerical/normalization coincidence. No written V2 operator theorem identifies the depth-six coupling with the inverse Ricci curvature of the rank-three projector manifold, so it must not yet be used as a derived coupling relation.

62.4 Does Ricci curvature close the lock coefficient?

Not by itself. A geometric action such as

SP∝∫tr⁡(∇P∇P) S_P\propto\int\operatorname{tr}(\nabla P\nabla P)

or an induced curvature functional would still require its normalization and its precise coupling to the electron complex structure to be derived from the master action.

What has improved is that the candidate coefficient is no longer an arbitrary number extracted from a chosen pair of projector directions. The canonical projector target is now sharply fixed:

Gr(3,6):Ric⁡=4g. \boxed{\mathrm{Gr}(3,6):\quad \operatorname{Ric}=4g.}

A future V2 projector equation can therefore be tested against a concrete invariant eigenvalue four.

62.5 Current status

The projector route now supplies all of the following without free orientation choices:

  1. a nine-dimensional mixed tangent space;
  2. nonzero induced curvature from P(dP∧dP)PP(dP\wedge dP)P;
  3. unit normalized nonzero sectional curvature;
  4. isotropic Ricci eigenvalue four;
  5. scalar curvature thirty-six.

It still lacks the master-action theorem saying which of these curvature invariants enters the stationary lock-frequency generator and with what normalization relative to ωe\omega_e.

So the strongest next target is no longer to search for another integer. It is to derive the effective projector action or linearized projector equation and see whether its spectral Hessian contains the canonical Ricci operator 4I94I_9.