The PG(3,2) Plant: isotropics, null polarity, Peirce decomposition and idempotent frames of the two-qubit symplectic geometry, with a pre-registered null on an entropy-collapse screen

A complete, exhaustively verified structural map of PG(3,2) — the projective 3-space over GF(2), 15 points / 35 lines / 15 planes — under its symplectic (Pauli-commutation) polarity, together with the stabilizer, Peirce and idempotent structure the geometry carries, and a pre-registered null result on an entropy-collapse screen applied to it. Four families were built: the totally isotropic subspaces (ISOS), the perps and the null polarity (ORTHOS), the Peirce decomposition (PEIRCE) and the idempotents (ID). Principal results. The symplectic polarity is a null polarity: all 15 points are absolute, every perp is a Fano plane, and perp is a polarity of the whole 67-element subspace lattice; the lines fixed by perp are exactly the 15 totally isotropic lines. The 35 lines split 15 isotropic / 20 anisotropic with no totally isotropic planes, so the isotropic lines are the Lagrangians; with the 15 points they form the generalized quadrangle GQ(2,2) = W(2), verified on all 180 non-incident pairs with zero violations, carrying exactly 6 spreads and 6 ovoids. Each spread is a complete set of 5 mutually unbiased bases with every cross-basis overlap measured at exactly 1/4, and each has the identical profile of 3 local and 2 entangled bases. Two counts in the prior record are corrected. The complete orthogonal families (frames) of the geometric idempotents number 105 — 15 line-aligned plus 90 mixed — not 15, established exhaustively over all C(60,4) = 487,635 four-subsets. And PG(5,2) has 315 totally isotropic lines but 135 Lagrangians; the two are commonly conflated. An impossibility result with a positive control. There is no Sp(4,2)-equivariant map between the 15 line-aligned frames and the 15 planes in either direction: they are the two inequivalent 15-element actions of Sp(4,2) isomorphic to S_6. Orbits on the product set give [45, 180] with no size-15 orbit, while the positive control (planes x points) gives [15, 90, 120] and finds the size-15 orbit that the polarity guarantees. Mixed frames do map to planes, 6 per plane, equivariantly under the full 11,520-element Clifford group. An exact algebraic obstruction. On the octonion and sedenion plus-algebras (spin factors over a negative-definite form) idempotency forces the squared norm of the imaginary part to equal -1/4, impossible over the reals; hence the idempotent set is {0, 1}, capacity is 1, and the Peirce decomposition is trivial. This was pre-declared as the expected outcome and is reported as an obstruction rather than an unfinished computation. On H_4(C) Peirce closes and lands on the geometry: for e_n = (I + g_n)/2 the half-eigenspace is the span of the 8 Paulis anticommuting with n, i.e. the complement of the plane n-perp, on all 15; the Peirce dimensions from the eigenvalues of L_e are (1,6,9), (4,8,4) and (9,6,1), constant within each rank because U(4) acts transitively on rank-r projectors. On the Albert algebra J_3(O) Peirce also closes, with off-diagonal part of real dimension 24, reported as an exact dimension match and nothing more. A pre-registered null, four times over. Four pre-registered 24-number measurement vectors were built and read through a 24-dimensional entropy-gated collapse routine; all four returned a null, an exclusion, or no interpretable reading. The cause is mathematical rather than instrumental: Sp(4,2) is transitive on the planes, the Clifford group on each frame class, U(4) on rank-r projectors, and at two qubits bipartite entanglement takes only the values 0 and 1. An invariant that cannot vary carries no information. Three control-design defects are documented alongside: the pre-registered positive control fails to separate on three of four bases; the built-in control pool is a single fixed set of dense Gaussians and is mis-specified for non-negative, sparse or two-valued inputs; and the primary channel is bounded below at -1.983 sigma against that pool, so a downward two-sigma criterion cannot fire. Honest scope. Everything here is Clifford/stabilizer structure and is classically simulable under the Gottesman-Knill theorem. No quantum advantage is claimed and none is available from this geometry. The [[2,1,1]] codes present have distance 1 — no error detection and no error correction — and are named as structure only. Non-associativity of octonions has no quantum counterpart, and the Jordan product provably cannot see it. All entanglement statements are relative to a fixed qubit factorization, not to the geometry. Results are for the two-qubit rung only and must not be extrapolated. Algebraic identities are verified numerically to stated residuals; the combinatorial counts are exact integer enumerations. Every load-bearing correction was independently re-derived along a second code path, three adversarial reviews were run before assembly, and every accepted refutation is listed with the corrected statement in a WITHDRAWN section. Package: full source, complete stdout logs, all emitted JSON and CSV, the pre-registration written before any number was computed, and a timestamped amendment log flagging for each entry whether it was written before or after the relevant number was known.