C03.1 · Derivation

F1–F3: The alternating carrier and its Lagrangian half

The trace form identifies the second Jordan copy with the dual of the first. It is nondegenerate, so \(\Omega\) is nondegenerate. Set \[ L=\{(a,X;0,0)\}. \] Both terms in its alternating pairing vanish identically. If \((c,U;d,V)\) is orthogonal to every \((a,X;0,0)\), independent variation of \(a\) forces \(d=0\), and nondegeneracy of \(T\) forces \(V=0\). Thus \(L^\perp=L\): maximal isotropy is proved by the pairing, not merely by counting half the dimension.

The executable test repeats this argument on the complete coordinate matrix. Its keys `FTS_dimension` and `boundary_rank_and_symplectic_perp_dimension` report the full dimension and boundary rank. This verifies the standard coordinate construction underlying LIB2-011. It does **not** verify an unknown change of basis to the programme's frozen matrices.

F4: What a frozen exceptional representation must supply

The standard exceptional-group construction in [K04] identifies the invariant group of the Freudenthal module. LIB2-006 makes the more specific statement that the programme's registered generators realize it. A standard existence theorem is not an equality test on those generator bytes.

Similarly, to verify LIB2-012 one must have the actual matrix \(B\), the source cubic and the shell cubic in matched coordinates. The required assertion is coefficientwise equality of their polarized tensors, plus the stated rank. Defining a shell cubic by pulling back the Albert norm would make that equality tautological and would not test the registered solder. No such substitute is used.

LIB2-013 is a local assertion at a transported point, with a shell-exchange ambiguity. The positive paired-shell space is not implied merely by the indefiniteness or dimension of the full carrier. Its defining polarization is absent; the original assertion remains **[NOT VERIFIED]** in this generation round.

F5–F6: Positive congruence is a standard construction with a conditional group consequence

For positive definite symmetric matrices \(A,M\), define \[ B=A^{-1/2}\bigl(A^{1/2}MA^{1/2}\bigr)^{1/2}A^{-1/2}. \] Then \(BAB=M\). Conversely, a positive symmetric solution makes \(A^{1/2}BA^{1/2}\) the positive square root of \(A^{1/2}MA^{1/2}\), proving uniqueness. In geometric-mean notation it is \(A^{-1}\#M\), the Riccati characterization in [LL].

Now assume the supplied matrix group \(G\) has the stated Cartan property \(G\cap\mathrm{SPD}=\exp\mathfrak p\), and \(A,M\in\exp\mathfrak p\). Their positive powers belong to \(G\). The sandwich \(C=A^{1/2}MA^{1/2}\) is both positive and in \(G\); the same Cartan property puts \(C^{1/2}\) in \(G\). The formula then puts \(B\) in \(G\cap\mathrm{SPD}\), hence \(\log B\in\mathfrak p\).

This proves the conditional mechanism of LIB2-197. It does not recompute the premise that the particular supplied-but-unattached matrices form that Cartan pair. Generic positive matrices cannot replace the two group-Gram hypotheses.

F7: A zero mass-coordinate velocity is not a zero full Hamiltonian vector

Simultaneously scale the conjugate variables as \((b,Y)\mapsto(t b,tY)\). Cubic and quadratic homogeneity give the exact polynomial \[ I_4(a,X;tb,tY) =-4aN(X)+t^2\bigl[-(ab-T(X,Y))^2+4T(X^\#,Y^\#)\bigr] -4t^4bN(Y). \] There is no linear transverse term. Therefore every first conjugate derivative vanishes on \(L\), so the **initial mass-coordinate velocity** is zero under the convention above.

But \(-\partial_q I_4\) need not vanish. The full-coordinate witness \(a=1,X=I_J,b=0,Y=0\) gives \(\dot q=0\) and nonzero conjugate velocity, printed under `witness_pdot_nonzero_entries`. The flow need not remain in \(L\), and the calculation does not assert that later mass-coordinate velocities vanish.

This distinguishes two readings of the p.348 phrase “zero initial velocity.” The source explicitly differentiates in conjugate directions; the supported interpretation is zero **mass-coordinate** velocity. Reading it as a stationary point of the full quartic Hamiltonian would be false for the exhibited witness. This is a precision note, not a claimed house-adopted erratum.

The weights \((-6,+2,+6,-2)\) on \((a,X,b,Y)\) preserve the alternating pairing and the quartic: every displayed monomial has total weight zero. This formal scaling symmetry is kinematics, not a calibration of a physical time or mass unit.