C03.2 · Derivation

S1: What a Lie-triple certificate proves

Put \(\mathfrak h=[\mathfrak m,\mathfrak m]\). Cartan parity puts \(\mathfrak h\) in \(\mathfrak k\), disjoint from \(\mathfrak m\). Lie-triple closure gives \([\mathfrak h,\mathfrak m]\subset\mathfrak m\). Jacobi then gives \([\mathfrak h,\mathfrak h]\subset\mathfrak h\). Consequently \(\mathfrak h\oplus\mathfrak m\) is a symmetric Lie subalgebra.

The standard integration result produces the corresponding totally geodesic geometry with the appropriate local assumptions; [TL] Theorem 3.3 states this in the positive-matrix setting. Applying it to the programme requires the ambient Cartan realization. It does not select the basepoint or identify matter fields.

Trilinearity means that a basis check is sufficient. Antisymmetry in the first pair reduces the closure computation to unordered distinct pairs followed by a third basis vector. For the source's complementary plane, the resulting count is the one explicitly printed beside its certificate on p.181. A negative result needs only one exact nonzero leakage witness; a positive result needs the complete tensor or an analytic proof.

S2–S4: What the source actually reports

LIB2-007 identifies the complement within the full scalar arena. The subtraction in its title is a count of identity-anchored tangent sectors; it is not a proof that discarding one sector is dynamically consistent.

The source then reports the visible-plane failure:

> The Jordan-54 is NOT a Lie-triple system: the projection of [[54, 54], 54] onto the missing-16 has order-one components

— Rev32.7, PDF p.181 [Q010].

This negative is the main geometric obstruction, not a minor numerical imperfection. Were an exact witness supplied, it would be enough to rule out the named plane as a Lie-triple system.

For the complementary plane it reports:

> The missing-16 IS a Lie-triple system, to machine precision (4.9 × 10−15 over all 1,920 triples).

— Rev32.7, PDF p.181 [Q011].

And for the generated algebra:

> It generates g′ = [16, 16] ⊕ 16 of dimension 28, closing at 7.4 × 10−15 , with compact part of dimension 12, trivial center, rank 4, and Cartan character +4: g′ ∼ = so(4, 4), [16, 16] ∼ = so(4) ⊕ so(4).

— Rev32.7, PDF p.181 [Q012].

Those quotations preserve their original numerical wording. This round **does not replace small residuals with exact zeros**. A conversion of the old numerical result into an exact certificate needs exact source arrays or an independently justified exact reconstruction.

The local isometry in Paper 4 does not by itself prove that the corresponding subgroup is closed in the ambient group, identify a global quotient, or establish the absence of global identifications. Nor does the Lie-triple test include vector couplings, a scalar potential or constraints that were not part of the supplied computation.

S5: A negative control for over-reading identification fingerprints

The quoted generated-algebra report lists a dimension, rank, centre and Cartan character. Those descriptors are useful but not, by themselves, a complete classification certificate.

An explicit standard control compares split \(D_4\) with \(\mathfrak g_{2(2)}\oplus\mathfrak g_{2(2)}\). Using the Cartan matrices displayed in the executable code, both have the same dimension, split rank, zero centre, compact/noncompact dimensions and Cartan character. The code reports the shared tuple under `identical_split_fingerprints`.

They are nevertheless different: one Dynkin diagram is connected, the other has two components. Their Cartan determinants also differ. The displayed matrices and the elementary split orthogonal model are **not proposed replacements for the programme's plane**. They test only the sufficiency of a fingerprint-based identification rule.

Thus this control does not refute LIB2-010. It specifies what a stronger certificate must include: actual root/ideal data, a simplicity proof, or an explicit invertible bracket-preserving map to the stated algebra. The source may have such additional evidence in its unshipped payload; this article cannot infer it.

S6: Exact certificate needed for the registered object

A reproducible registered-plane test requires the tangent generators and their embedding into the ambient matrix representation, exact coefficients for both projectors, and the metric used in projection. It must establish projector idempotence, complementarity and the source's ranks, then compute the leakage tensor without a tolerance.

For the generated algebra, collect the independent commutators together with the complementary generators, solve exact closure coefficients, test Jacobi, and compute the centre and **intrinsic adjoint Killing form**. Report root/ideal data sufficient to distinguish the control above. An ambient trace form can supplement this certificate but must not silently replace the intrinsic Killing form.

None of those missing matrices is manufactured here. Consequently the registered visible-plane failure, complement closure and generated-algebra identification remain **[NOT VERIFIED] by this return**, while the cited source reports remain intact.