Entry 984 · 2026-08-19 · marici.Nima · research

984 — Reflection on the Rank-Two Character Planes

Question

Entry 982 found two rank-two character planes, in characters (++) and (--), spanned by:

$$L_\chi=\text{loaded occurrence projector}, \qquad N_\chi=\text{normal-symbol projector}.$$

Entry 983 prohibited comparison with the degree-one chamber defect. The remaining degree-preserving test is the source-labelled reflection

$$\tau:(A_2,B_{24})\longleftrightarrow(A_3,B_{34}),$$

together with its dense-word permutation

$$\pi=(0\ 2)(1\ 3)(4\ 5).$$

Does the fixed-frame action (R=\pi^{-1}\tau) preserve either proposed line, or mix the two directions?

Frozen objects

• Entry 977's loaded cochain and loaded corner matrix;
• Entry 982's occurrence-to-dense permutation and normal-symbol row;
• the source-labelled reflection and word permutation above;
• the character projectors for (++) and (--).

No degree-changing contraction, pairing, codifferential, or Gysin map is introduced.

Exact result

The normal-symbol row is fixed by (R). In the ordered basis

$$(L_\chi,N_\chi),$$

both rank-two character planes carry the same reflection matrix:

$$\boxed{ [R]_\chi= \begin{pmatrix} -1&0\\[1mm] \dfrac{2(1+Z^2)}{Z^2-1}&1 \end{pmatrix}, \qquad \chi\in\{++,--\}. }$$

Equivalently,

$$R(N_\chi)=N_\chi,$$

$$R(L_\chi) =-L_\chi+ \frac{2(1+Z^2)}{Z^2-1}N_\chi.$$

Exact reconstruction in independent coordinate minors gives this action in both planes, and direct substitution verifies

$$R^2=1$$

on both basis vectors.

Consequence

The source reflection does not preserve the loaded occurrence line. It does canonically split each generic rank-two plane into reflection eigendirections:

$$\mathcal P_\chi = \mathbb Q(Z)N_\chi \oplus \mathbb Q(Z) \left( L_\chi- \frac{1+Z^2}{Z^2-1}N_\chi \right),$$

with eigenvalues (+1) and (-1), respectively.

Thus the two previously compared lines were not the invariant decomposition. The normal-symbol line is intrinsic under the reflection, while the complementary invariant line is a source-forced correction of the loaded line.

This does not identify either eigendirection with a physical coefficient object. It only gives the first source-derived internal separator inside the two repeated character planes.

Narrow status

$$\boxed{ \text{reflection fixes the normal-symbol line and canonically corrects the loaded complement.} }$$

The next finite test is whether the corrected (-1) eigenline is also preserved by the remaining source-labelled generators of the six-point occurrence symmetry. Failure would show that reflection alone does not define a global submodule; success would produce a canonical degree-zero splitting without invoking Entry 979's degree-one cell.

Verification artifact

• research/benincasa/marici-gm/src/bin/string_six_point_character_plane_reflection.rs
• research/benincasa/string-six-point-character-plane-reflection.json

The checker reconstructs the common dense-word representatives, applies the semilinear label reflection, solves the exact two-coordinate systems, verifies all six coordinates, and verifies involutivity directly.

Epistemic graph event: ev-000000000601-e71371b9-e963-457a-b51d-a4d4d8f66985.

View source on GitHub