Projector-dynamics audit of the remaining unobserved-steering channel

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 35: Projector-dynamics audit of the remaining unobserved-steering channel.

Provenance

Preserved from interaction notebook section 35, 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.

35. Projector-dynamics audit of the remaining unobserved-steering channel

This note examines one conditional projector-derivative response. It is not an exhaustive classification of same-field interactions through extended tails.

V2 assumes a smooth zeroth-order sector projector

$$P_d:\Sigma_\infty|_{Q_d}\to\Sigma_d$$

and defines the tangentially compressed sector operator schematically by

$$\mathcal D_d=P_d\mathcal D_\infty^{\parallel}P_d.$$

Because $P_d$ is order zero, derivatives of $P_d$ enter only in lower-order terms. These are precisely the terms in which one can obtain

$$\nabla P_d, \qquad A^\perp, \qquad \mathrm{II}_d, \qquad H_d,$$

and other normal/extrinsic connection data.

Thus the geometry does provide a legitimate mathematical location for the desired two-electron mixed or vertical curvature.

However, the current V2 action contains no independent variational dynamics for the projector or embedding. In particular, there is no written equation of the form

$$\boxed{ \frac{\delta S}{\delta P_d}=0 }$$

or an equivalent field equation fixing

$$\nabla P_d, \qquad \mathrm{II}_d, \qquad H_d, \qquad A^\perp.$$

The phrase “dynamically defined projector” is therefore presently a construction requirement, not a completed dynamical law.

35.1 Consequence for the two-electron source problem

A candidate operator considered here was

$$P_6\Gamma^A\nabla_A P_6$$

in the background sourced by another electron.

Formally, differentiating the projector allows lower-order pieces of the schematic form

$$P_6\Gamma^A(\nabla_A P_6)$$

and normal-connection terms in the spatial operator,

$$\mathcal D_6 = i\gamma_6^a (\partial_a+\omega_a+A_a^\perp) +\Phi_6.$$

Therefore mixed or vertical response is not forbidden by the projected Dirac geometry.

But V2 presently gives no source equation telling us how an electron configuration changes

$$P_6, \quad A^\perp, \quad \Phi_6,$$

or the associated curvature.

Consequently the theory cannot currently calculate whether electron 1 induces, at electron 2,

$$F_{\parallel\perp}\neq0$$

or

$$F_{\perp\perp}\neq0.$$

Nor can it determine the sign, range, or strength of such a term.

This is a stronger result than merely saying that a coefficient is unknown:

$$\boxed{ \text{the source-to-projector dynamical equation needed for free }e-e\text{ unobserved steering is absent from present V2.} }$$

35.2 The normal connection does not rescue the argument by itself

The normal connection transforms correctly as a gauge potential,

$$A^\perp\mapsto U A^\perp U^{-1}-(dU)U^{-1},$$

with curvature

$$F^\perp=dA^\perp+A^\perp\wedge A^\perp.$$

This proves that higher-dimensional normal geometry can realize gauge-like structure after projection.

It does not prove that a second free electron sources a particular nonzero $A^\perp$ or $F^\perp$.

Transformation law and dynamical field equation are different statements.

Therefore one may not infer unobserved focusing merely from the existence of the normal bundle.

35.3 Exact missing closure

A completed V2 derivation of the proposed beam correlation would require at minimum a master-geometric term whose variation gives a source equation for the projector or normal connection. Schematically, one needs something like

$$\frac{\delta S_{\rm geom}[P,\Psi]}{\delta P_6}=0$$

or

$$\frac{\delta S_{\rm geom}[A^\perp,\Psi]}{\delta A^\perp}=J^\perp[\Psi],$$

with no assumption here that either displayed schematic form is the correct final theory.

That equation would have to determine all of the following without introducing arbitrary fitted functions:

1. whether a free electron sources normal or mixed curvature;
2. its sign under electron--electron versus electron--opposite-charge configurations;
3. its radial/retarded Green function;
4. its dependence on the actual bound/free directions, complete spatial profiles and orientations;
5. its relation to a derived electromagnetic response and its measured observables;
6. whether it conserves or transfers the higher-dimensional transverse Noether charge;
7. the polarization dependence of the resulting full-depth collision rate.

Until such an equation is derived, the mixed-curvature route remains an allowed slot in the geometry, not a prediction.

Related

• Research status — current proof boundaries and correction guidance.
• Updated verdict after testing the current scalar and electromagnetic sectors — preceding section in the research sequence.