Maurer--Cartan test: the scattering frame alone cannot generate its own restoring curvature

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 60: Maurer--Cartan test: the scattering frame alone cannot generate its own restoring curvature.

Provenance

Preserved from interaction notebook section 60, 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.

60. Maurer--Cartan test: the scattering frame alone cannot generate its own restoring curvature

The strongest remaining route was to derive the missing dimensionless lock coefficient from a mixed connection constructed directly from the scattering-selected frame

$$Q_{\rm scat}(x)\in SO(3).$$

There is an immediate canonical connection candidate,

$$\boxed{ A=dQ_{\rm scat}\,Q_{\rm scat}^T }$$

or, equivalently in the left convention,

$$a=Q_{\rm scat}^T dQ_{\rm scat}.$$

Both are Lie-algebra-valued one-forms in $\mathfrak{so}(3)$.

60.1 Exact curvature test

For the right Maurer--Cartan form,

$$A=dQ\,Q^{-1},$$

the Maurer--Cartan identity is

$$\boxed{ dA-A\wedge A=0. }$$

For the left form,

$$a=Q^{-1}dQ,$$

one has

$$\boxed{ da+a\wedge a=0. }$$

This was checked explicitly with a noncommuting two-parameter frame field

$$Q(x,y)=e^{xJ_x}e^{yJ_y},$$

where $[J_x,J_y]\ne0$. Both curvature tensors vanish identically.

Therefore a connection built only from the local frame map $Q_{\rm scat}$ is pure gauge.

60.2 Consequence for the lock mechanism

The scattering frame can still define parallel transport and angular velocity. For a one-parameter rotation

$$Q(t)=R_z(\omega t),$$

one obtains

$$A_t=\dot Q Q^T$$

with

$$-\frac12\operatorname{tr}(A_t^2)=\omega^2.$$

So the frame map supplies a perfectly good kinematic rate variable.

But it does not supply a preferred nonzero rate or a restoring potential. In one dimension there is no curvature two-form, and in multiple dimensions the Maurer--Cartan curvature still vanishes when the connection is constructed purely as $dQ\,Q^{-1}$.

Hence

$$\boxed{ Q_{\rm scat}\text{ determines orientation and transport, but not a dynamical stiffness by itself.} }$$

60.3 What would make the connection physical rather than pure gauge

A nontrivial mixed/extrinsic field must contain information beyond the frame map itself. At least one of the following is required:

1. an independently dynamical connection whose horizontal distribution is not simply the Maurer--Cartan form of $Q_{\rm scat}$;
2. a nonzero second fundamental form or projector derivative $\nabla P$ fixed by an embedding equation;
3. coupling to another sector field whose curvature survives after choosing the scattering frame;
4. a topologically nontrivial connection not globally representable as one smooth $Q^{-1}dQ$;
5. or an induced effective action in which integrating out other fields generates a potential for $Q$.

The current V2 corpus has kinematic locations for items 1--3, but no written variational dynamics fixing their magnitude.

60.4 Combined implication of sections 57--60

The investigation now separates four logically distinct layers:

$$\boxed{ \begin{array}{ll} \text{scattering data} &\Rightarrow Q_{\rm scat}\text{ uniquely},\\ Q_{\rm scat} &\Rightarrow \text{covariant frame-mismatch potential shape},\\ Q_{\rm scat}\text{ alone} &\not\Rightarrow \text{nonzero connection curvature},\\ \text{projected master geometry} &\text{must still derive the physical coefficient.} \end{array} }$$

This rules out one tempting shortcut: the missing $\widehat\kappa$ cannot be obtained merely by differentiating the already constructed frame map and calling the resulting Maurer--Cartan norm a curvature eigenvalue.

The remaining viable source is a genuinely dynamical mixed/extrinsic field, projector geometry, or induced interaction whose curvature is not pure gauge.

Related

• Research status — current proof boundaries and correction guidance.
• Audit of existing V2 candidates for the dimensionless lock coefficient — preceding section in the research sequence.
• Projector-curvature candidate: nonzero geometry beyond the pure-gauge frame map — following section in the research sequence.