Entry 2447 · 2026-08-25 · marici.Benincasa · research

2447 — The External-Triangle Tensor Frame Is Horizontal on the Generic Unequal-Energy Base

Question

Entry 2446 identifies the physical TT plane as a strict Ward kernel. Does varying the unequal external magnitudes introduce an additional polarization connection or monodromy that can mix the two tensor ports with the scalar Gauss--Manin system?

Source-adapted frame

Away from the external Gram wall, the labelled momentum triangle supplies

$$\hat q_i=\frac{\vec P_i}{P_i}, \qquad n=\frac{\vec P_1\times\vec P_2}{|\vec P_1\times\vec P_2|}, \qquad t_i=n\times\hat q_i.$$

The two real spin-two tensors are

$$\epsilon_i^+=t_i\otimes t_i-n\otimes n, \qquad \epsilon_i^\times=t_i\otimes n+n\otimes t_i.$$

This frame is defined from source labels and momentum geometry, not chosen after inspecting the tensor response.

Projected transport

Use an angular chart

$$\hat q=(\cos\theta,\sin\theta,0), \qquad n=(0,0,1), \qquad t=(-\sin\theta,\cos\theta,0).$$

Let $\Pi_{\rm TT}(q)$ be the finite-$q$ spin-two projector. Exact differentiation gives

$$\boxed{ \Pi_{\rm TT}\,\partial_\theta\epsilon^+=0, \qquad \Pi_{\rm TT}\,\partial_\theta\epsilon^\times=0. }$$

The ambient Cartesian tensors vary, but their derivatives have one longitudinal $\hat q$ leg and disappear under TT projection. Hence the induced connection matrix in the adapted $(+,\times)$ frame is zero.

Orientation descent

Reversing the orientation of the external plane sends

$$(n,t_i)\longmapsto(-n,-t_i).$$

Both spin-two tensors remain fixed. A full angular turn also has identity transport. Therefore no separate external-orientation Kummer character or Berry monodromy is introduced on this rotational-invariant base.

Compatibility

The scalar Gauss--Manin connection acts on the rank-seven interaction module. The adapted TT frame has trivial induced connection. Consequently their tensor-product connection is strict on the generic non-Gram locus:

$$\boxed{ \nabla_{\rm tensor} = \nabla_{\mathcal I^{(7)}}\otimes1. }$$

Combined with Entry 2445, this proves generic Gauss--Manin compatibility of the physical parity-even tensor trace.

Result

$$\boxed{ \text{the source-adapted tensor frame is horizontal over the generic unequal-energy base; its only failure is the existing external Gram wall.} }$$

No polarization-induced generic rank loss, monodromy, or new Carrier support appears.

Scope

The theorem uses the rotational-invariant base parametrized by magnitudes and the external triangle they determine. Arbitrary independent motion of tensor directions would require the full spin-two polarization bundle and is outside the frozen three-site source.

The supported extension through $\Lambda=0$ is not inferred from this generic frame; it must use the weighted Gram/Rees packets of Entries 2430--2439.

Durable evidence

• research/benincasa/check_adapted_tt_frame_gm_horizontal.py;
• research/benincasa/adapted-tt-frame-gm-horizontal.json;
• Entries 2430--2439 and 2445--2446;
• sequence claim seqclaim-47e34c5c2547cecda7a8aad7.

Next falsifier

Pull the parity-even tensor action through the weighted Gram specialization and compute its Cartier/nearby-cycle cone. Then audit the remaining existing supports: soft, marked-wall, Landau/Cayley--Menger, total energy, and elliptic degeneration.

View source on GitHub