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Summary: Research record, section 57: Variational derivation of the scattering-selected frame link.

Provenance

Preserved from interaction notebook section 57, 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.

The surviving scattering-frame map can be promoted from a procedural definition to a covariant variational principle.

Let

O^,H^∈SO(3) \widehat O,\widehat H\in SO(3)

be the orientation-preserving polar factors of the observed and unobserved scattering triads. Define

Qscat=O^H^T. Q_{\rm scat}=\widehat O\widehat H^T.

Consider the frame-matching energy

Vframe(Q)=κ2∥O^−QH^∥F2,Q∈SO(3),κ>0. \boxed{ V_{\rm frame}(Q) =\frac{\kappa}{2}\|\widehat O-Q\widehat H\|_F^2, \qquad Q\in SO(3), \qquad \kappa>0. }

Using orthogonality,

Vframe(Q)=κ[3−tr⁡(QH^O^T)]. V_{\rm frame}(Q) =\kappa\left[3-\operatorname{tr}(Q\widehat H\widehat O^T)\right].

This is the orthogonal Procrustes functional. Its minimum is

Q=Qscat=O^H^T. \boxed{Q=Q_{\rm scat}=\widehat O\widehat H^T.}

Thus the strongest candidate need not be defined by hand after all: it is the unique orientation-preserving minimizer of the simplest quadratic mismatch between the two orthonormalized scattering frames.

57.1 Positive local Hessian

Work covariantly in the frame where the minimum is Qscat=IQ_{\rm scat}=I. Write

Q=exp⁡([a]×),a∈R3. Q=\exp([a]_\times), \qquad a\in\mathbb R^3.

Then

tr⁡Q=3−∣a∣2+O(∣a∣4), \operatorname{tr}Q =3-|a|^2+O(|a|^4),

so

Vframe=κ∣a∣2+O(∣a∣4). \boxed{ V_{\rm frame} =\kappa|a|^2+O(|a|^4). }

The tangent-space Hessian is therefore

Hess⁡Vframe=2κI3. \boxed{\operatorname{Hess}V_{\rm frame}=2\kappa I_3.}

For a full-rank mixed tensor EE with positive determinant and singular values a,b,c>0a,b,c>0, the Procrustes alignment energy

VE(Q)=−κtr⁡(EQT) V_E(Q)=-\kappa\operatorname{tr}(EQ^T)

has quadratic curvatures

κ(b+c),κ(a+c),κ(a+b), \boxed{ \kappa(b+c),\qquad \kappa(a+c),\qquad \kappa(a+b), }

on the three rotational tangent directions. These curvatures are positive. They remain positive for c=0c=0 and a,b>0a,b>0, so rank two also gives a strict orientation minimum.

57.2 The same interaction selects the s=0s=0 electron orbit

The construction can be coupled directly to the electron complex structure rather than only to an abstract SO(3)SO(3) matrix.

In the scattering-selected frame take the mixed block of the canonical electron family to be

Q(s)=diag⁡(1−s,1−s,1). Q(s)=\operatorname{diag}(\sqrt{1-s},\sqrt{1-s},1).

The natural alignment energy is

Vscat(J)=κ[3−tr⁡(QscatQ(J)T)]. \boxed{ V_{\rm scat}(J) =\kappa\left[3-\operatorname{tr}(Q_{\rm scat}Q(J)^T)\right]. }

In the aligned frame Qscat=IQ_{\rm scat}=I this becomes

Vscat(s)=2κ(1−1−s). V_{\rm scat}(s) =2\kappa\left(1-\sqrt{1-s}\right).

Therefore

Vscat(s)=κs+O(s2), \boxed{ V_{\rm scat}(s)=\kappa s+O(s^2), }

so positive κ\kappa selects

s=0. \boxed{s=0.}

This is precisely the orthogonal 3+33+3 frame-lock orbit.

57.3 Equivalent six-dimensional two-form form

From the dynamically selected scattering frame define the antisymmetric mixed tensor

Fscat=(0Qscat−QscatT0). F_{\rm scat} = \begin{pmatrix} 0&Q_{\rm scat}\\ -Q_{\rm scat}^T&0 \end{pmatrix}.

Unlike the earlier hand-chosen FmixF_{\rm mix}, this object is no longer circular: its mixed block is derived from the scattering-frame variational data.

The linear complex-structure contraction satisfies

−12tr⁡(FscatJ)=tr⁡(QscatQ(J)T) -\frac12\operatorname{tr}(F_{\rm scat}J) =\operatorname{tr}(Q_{\rm scat}Q(J)^T)

on the beam-adapted block decomposition. Hence

Vscat(J)=κ[3+12tr⁡(FscatJ)] \boxed{ V_{\rm scat}(J) =\kappa\left[3+\frac12\operatorname{tr}(F_{\rm scat}J)\right] }

up to the sign convention chosen for FscatF_{\rm scat}.

This gives a compact covariant scalar capable of selecting both the orientation of QQ and the s=0s=0 locked orbit.

57.4 What has advanced

The previous candidate had three apparent arbitrary ingredients: a full-rank mixed tensor, its orthogonal frame link, and a locking scalar. They now collapse into one chain:

(pO,bO;pH,bH)⟶(O^,H^)⟶Qscat⟶Fscat⟶Vscat(J). \boxed{ (p_O,b_O;p_H,b_H) \longrightarrow (\widehat O,\widehat H) \longrightarrow Q_{\rm scat} \longrightarrow F_{\rm scat} \longrightarrow V_{\rm scat}(J). }

Every arrow in this chain is geometrically explicit and covariant, and the last interaction has a positive local Hessian when κ>0\kappa>0.

57.5 Remaining dynamical gap

This still does not make the result a theorem of the current V2 action. The unresolved questions are now narrower:

  1. does the projected master current/connection generate the quadratic frame-mismatch functional or equivalent trace coupling;
  2. why is its coefficient κ\kappa positive;
  3. how are the unobserved scattering directions determined dynamically rather than supplied as independent initial data;
  4. what frequency eigenvalue follows from the resulting restoring block;
  5. and can that frequency be fixed relative to the electron anchor without introducing an extra scale.

The orientation-selection problem is therefore essentially solved at the level of covariant geometry and variational mechanics. The remaining issue is embedding this functional into the V2 master operator and deriving its spectral coefficient.