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 42: First classification of observationally invisible higher-depth operators.

Provenance

Preserved from interaction notebook section 42, 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.

42. First classification of observationally invisible higher-depth operators

Use a local product chart

Q6≃Q2×F6/2, Q_6\simeq Q_2\times F_{6/2},

with coordinates

x∈Q2,y∈F6/2. x\in Q_2, \qquad y\in F_{6/2}.

A higher-depth operator correction can be split schematically into

N=Nxx+Nxy+Nyy+N0, \mathcal N = \mathcal N_{xx} +\mathcal N_{xy} +\mathcal N_{yy} +\mathcal N_0,

where the pieces contain tangential, mixed, vertical, and algebraic/projector terms.

The direct projection condition does not require every one of these pieces to vanish separately. It requires only their pushforward sandwich to vanish:

MNL=0. M\mathcal NL=0.

42.1 Vertical derivatives

For a cylindrical lifted test field

Lϕ(x,y)=ϕ(x), L\phi(x,y)=\phi(x),

one has

∂yiLϕ=0. \partial_{y^i}L\phi=0.

Therefore any operator whose rightmost action begins with a pure vertical derivative satisfies

NyL=0 \mathcal N_yL=0

and hence

MNyL=0. M\mathcal N_yL=0.

Such a term is completely invisible when tested only on cylindrically lifted lower-depth fields, yet it can act on a genuinely depth-six electron state with nontrivial yy dependence.

This gives an explicit nonempty class of observationally invisible higher-dimensional dynamics.

42.2 Fibre-divergence terms

Suppose

NΨ=∇y⋅Jy[Ψ] \mathcal N\Psi=\nabla_y\cdot\mathcal J_y[\Psi]

and the fibre current has vanishing boundary flux or adequate decay. Then

MNΨ=∫F∇y⋅Jy dμF=0. M\mathcal N\Psi = \int_F\nabla_y\cdot\mathcal J_y\,d\mu_F =0.

Thus total fibre divergences can change the local unobserved distribution while leaving the complete marginal unchanged.

This is particularly relevant to the collision question: unobserved coordinates can rearrange while their depth-two marginal remains exactly fixed.

42.3 Zero-mean fibre potentials

An algebraic fibre term

V⊥(x,y) V_\perp(x,y)

can be directly invisible on a cylindrical test state if

∫FV⊥(x,y)dμF(y)=0. \int_F V_\perp(x,y)d\mu_F(y)=0.

It may nevertheless produce forces within the fibre through

−∇yV⊥. -\nabla_yV_\perp.

Whether such a term is allowed by covariance, self-adjointness, and the one-field action is a separate constraint.

42.4 Mixed connection terms

A mixed covariant derivative can alter yy-dependence while integrating to zero under the lower projection. These terms are natural candidates for the previously identified normal-connection slot,

A⊥, A^\perp,

but their source law remains to be derived.

The important conclusion is now constructive rather than merely permissive:

ker⁡P is nontrivial, and it contains explicit operator classes capable of acting on unobserved coordinates while leaving the direct observed projection unchanged. \boxed{ \ker\mathfrak P\text{ is nontrivial, and it contains explicit operator classes capable of acting on unobserved coordinates while leaving the direct observed projection unchanged.} }