The Unwritten Page: The First-Order Variation of the Henneaux–Teitelboim Einstein–Cartan Action, and the Solution Space of Experiment Two in Unimodular Gauge

Paper #2 of this corpus, Two Bounds and an Integration Constant, states its unimodular Einstein–Cartan construction in four lines — vary the Henneaux–Teitelboim (HT) action and the traceless equations, the unmodified Cartan equation, and an integration constant fall out — and flags the general derivation in its own text: "the general first-order derivation for arbitrary geometries is companion work in preparation, and no claim below depends on it." The design memorandum deposited with paper #6 names the same page twice in its one-week build plan: "(1) full first-order (tetrad + connection) variation of the HT-EC action, displaying the traceless Cartan equation alongside the metric one (the spin sector's unimodular behaviour is the one genuinely unwritten page)" and "(2) the solution-space statement for Experiment Two's families in unimodular gauge," the second flagged in paper #2 as "a to-verify, not asserted as proven." This paper writes that page. It carries the first-order variation of the action exactly as I print it — tetrad and spin connection independent, every variation shown, every boundary term displayed rather than dropped — and it proves the solution-space statement at the precision the memos need. Three findings are not the ones the memos' shorthand suggests, and they are printed in full rather than harmonized. First, in the HT form I actually use, the direct tetrad variation does not yield the traceless equation alone: it yields the full Einstein–Cartan equation carrying the multiplier field, and paper #2's Eq. (9) is its trace-free projection — correct as a projection, incomplete as a statement of the whole variation. (In the strict, constrained-determinant form of unimodular gravity the traceless equation is the whole tetrad equation, because the trace variation is not free; the two forms have the same solution space under the conditions proved here.) Second, there is no traceless Cartan equation in this action: the connection variation yields the Cartan equation of paper #2, Eq. (10), whole and unprojected — the HT term contains no spin connection, so nothing in the unimodular sector touches, traces, or projects the spin sector — and this paper solves Eq. (10) in closed form (the operator is invertible on the 24-dimensional torsion space; the trace coefficient its own contraction forces is 1/4; for both source classes I use — Dirac spin and the Weyssenhoff spin fluid — the trace term vanishes identically, so torsion is the spin tensor times κ² with no correction at all). Third, the solution-space statement for Experiment Two's families is a theorem with an obstruction attached, not a blanket verification: every family member carries explicit HT data on its stationary chart (τ^t = t·e solves ∂_μτ^μ = e there), its printed completion numbers (horizons, surface gravities, a_max) are functionals of the geometry alone and are exactly unchanged, and the lifts are fibered by one constant Λ_int per connected component of spacetime — but a member is a full HT-EC solution with a physical spin source only if a spin tensor exists on that geometry reproducing its pressures through the Cartan equation, and the record owns no such spin tensor for the rotating members (paper #1: no rotating Einstein–Cartan solution is known; the family is an ansatz). That step is an obstruction, named with its missing input, and the proof stops there, as it must. Nothing here derives the value of Λ_int, matches the conjugate pair (Λ_int, ∫e d⁴x) across the bounce, or touches the value or coincidence tiers of the vacuum problem; quantum gravity and the vacuum catastrophe remain not solved, and the memorandum's quantum caveats — the path-integral measure and the superselection of Λ_int — are stated in §8, not solved, exactly as the build plan ordered.