C05.2 · Derivation

T2: the square root is an implication of the quadratic relation

On the active block, \[ (A+P)^2=A^2+2A+P=5A, \qquad (A+P)(4P-A)=5P. \] Consequently \[ H^2=W_h,\qquad H^{-1}=Q+\frac{4P-A}{\sqrt5}, \qquad W_h^{-1}=Q+3P-A. \] These identities are checked by polynomial remainders in `universal_square_root_residual` and `universal_inverse_residual`; they apply to any matrices satisfying the stated projector and polynomial hypotheses.

Suppose, additionally, that the registered \(W_h\) is symplectic. The polynomial relation gives \[ P=W_h+W_h^{-1}-2I. \] It follows that \(P^\dagger=P\), \(Q^\dagger=Q\), and \(A^\dagger=3P-A\). Hence \(H^\dagger=H^{-1}\), so the same polynomial root is symplectic. This proof uses the symplecticity of the **actual** word as a premise. The local script verifies it for the explicitly displayed example only; the frozen-word premise remains [NOT VERIFIED] in this return.

The active polynomial has the two distinct positive roots \[ \frac{3+\sqrt5}{2}=\phi^2,\qquad \frac{3-\sqrt5}{2}=\phi^{-2}. \] The displayed branch therefore has active eigenvalues \(\phi\) and \(\phi^{-1}\), and eigenvalue one on the complement. Symplectic reciprocal pairing gives balanced active multiplicities once the registered active dimension is assumed. No multiplicity is inferred from the polynomial alone. Positive eigenvalues and \((\det H)^2=\det W_h=1\) then give \(\det H=1\).

This is a derivation conditional on the trace-three relation. It does not derive why that word, normalization or active carrier was selected.

T3: the Gram law includes a metric convention

Under the transpose hypotheses in Definitions, \[ (G_T)_{ii}=\operatorname{tr}(T^TP T)/24 =\|PT\|_F^2/24. \] The matrix \(W_0W_1\) is skew-symmetric, so \(\operatorname{tr}(T^TW_0W_1T)=0\). Thus \[ G_T=\frac{\|PT\|_F^2}{24}I_2. \] This proof is for every real carrier map \(T\) of any column count. It explains the source's one-column and ten-column coefficients when the active columns are orthonormal. It does not establish that arbitrary columns have unit norms.

The script tests the complete bilinear polynomial for an arbitrary two-by-three matrix in a symmetric Clifford block. It then conjugates one Clifford generator by \(\operatorname{diag}(2,1)\). Clifford squares and anticommutators survive, but for the first coordinate column the Euclidean Gram is \[ \operatorname{diag}(1/24,1/96), \] not a scalar matrix (`nonorthogonal_Clifford_control_Gram`). This is not a refutation of LIB2-194: the frozen metric and matrices are unavailable. It proves why abstract Clifford relations alone are insufficient to verify a **Frobenius** identity in an unspecified frame.

T4: an explicit braid model, and precisely what it tests

The executable model is \[ U=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad V=\begin{pmatrix}1&0\\-1&1\end{pmatrix}, \qquad UV^{-1}=\begin{pmatrix}2&1\\1&1\end{pmatrix}. \] The actual checks establish \(UVU=VUV\), preservation of the displayed symplectic form, the trace-three polynomial, and \((UV)^3=-I_2\). The entries are not taken from the programme's absent matrix archive.

Padding this model with the declared active copies and an identity complement gives \[ (U_{\rm pad}V_{\rm pad})^3=I-2P. \] Since the two subspaces are nonzero, the result is nonscalar. It therefore cannot kill the central full twist by a single projectivization of the whole vector space. On the active component it is scalar and disappears projectively. The identification of this centre quotient with the modular-group reading is the programme's statement; the script proves the nonscalar obstruction for this explicit representation, not a faithful identification of every possible quotient.

The source prints the corresponding registered result:

> The centre is computed and is not scalar — (B1 B2 )3 = I − 2Pact — so there is no global PSL(2, Z); only the active sector projectivizes.

— Rev32.7, PDF p.313 [Q023].

An explicit intertwiner matching the model, symplectic form, active projector and generator normalizations to the frozen carrier would make the model a check of that realization. It is absent. The model is therefore a standard algebraic control, never a replacement certificate for LIB2-014 or LIB2-195.

T5: finite tick and parity results are not geometric bundles

The source's generator-level tick claim is:

> The Z4 clock weight of the banked grading behaves as a momentum-like quantum number modulo 4: the grading law, additivity under composition, and reversal under the mirror are all exact on the full generator set.

— Rev32.7, PDF p.269 [Q024].

The arithmetic check in the script tests all residue pairs for reversal of addition. It does **not** inspect the banked products or verify the source's full-generator census. That certificate is [NOT VERIFIED] here.

The cylinder-condition report is also carried only as LIB2-055. A dimension tuple alone does not reproduce its comparison-class test. Likewise the scale-class result requires the selected winding symmetry model and its actual cochain complex. The area cocycle is not a census of all possible cylinder models, and no replacement toy cohomology computation is used to manufacture that census.

The parity result has a particularly important boundary:

> what is exhibited is the parity pattern a spin structure would produce, not a spin structure. No Spin-bundle map is constructed

— Rev32.7, PDF p.269 [Q025].

No Spin-bundle or compact-coordinate construction is supplied in this article. This restriction is not an argument that such structures are impossible; it is the distinction between the finite result that exists and a geometric identification that has not been constructed.

The repeated polynomial does not establish a common carrier

> These are the same trace-3 shape and they are not connected here.

— Rev32.7, PDF p.313 [Q026].

The programme's braid word and its separately registered winding/knot object can share a polynomial without being the same operator, representation or functor. The article neither identifies their domains nor transports a physical meaning across that gap.