An Exact Tangential Gradient Phase Diagram and Pointwise Finite-Frequency Spectral Obstruction in a Gravity Model with Two Tensorial Degrees of Freedom

We analyze the tangential sector of the exact finite-radius background of the P-059 parent action of a gravity model with two tensorial degrees of freedom, inside its deposited Hamiltonian–Dirac reduction, and we certify every claim by exact number-field computation. The paper delivers one exact classification and one exact obstruction, joined by a diagnostic, and it states the logical type of each result explicitly. (1) Exact tangential gradient phase diagram (region level). The tangential gradient coefficient factorizes as G_tan = (k²/4)·Π²·(W² − Y²) with exact normalization, and its shape splits into two explicit affine factors of a; the failing a-set at each (s, α) is therefore a single interval or ray with closed-form endpoints, organizing the (s, α)-plane into exactly three occupied regions plus an empty region. The negative-gradient region is nonempty (explicit rational witness), with a complete exact enumeration of the failure islands and rays. (2) Pointwise finite-frequency spectral obstruction (constrained parent and relaxed quotient; certified dictionary). From the constrained parent's frozen 10×10 assembly, two exact spectral objects are analyzed side by side: the parent pencil P10 itself, and the relaxed quotient pencil P9 obtained by an invertible multiplier reparametrization followed by the specialization δ = 0 and the drop of one constraint row. The organizing correction of this part: the multiplier difference δ = (λ_A − λ_B)/2 is not a gauge direction — P10(e)·g_gauge = (0, 2e, −2e, 0⁷)ᵀ, and the constraint dynamics forces δ = (c₀·G_r/2)·y on any parent solution with e ≠ 0 — so the gauge-fixed systems cannot stand in for the parent, and the parent pencil is analyzed directly. At all fourteen frozen-dictionary points, in either sign convention and at two wavenumber freezes (the deposited two-slot point and the consistent single-mode slice k_X² = κ = 1), the classification is obtained exactly over the number field K_s = Q(u)/(u⁴ − s): the parent is a regular pencil, det P10 = e³·q̃ with q̃ of degree 2, square-free, both roots real and negative — exactly two nonzero modes, both nonoscillatory (ω² < 0), both decaying; the relaxed quotient has det P9 = e³·q₃ with q₃ of degree 3, all three roots real, signature (N₊, N₋) = (1, 2) at ten points and (0, 3) at four; and gcd(q₃, q̃) is a constant — the quotient's nonzero spectrum is disjoint from the parent's, so the quotient is neither the parent's spectrum nor a superset of it. The obstruction verdict is common to both objects: no oscillatory (real-frequency) tangential mode exists at any certified point; the only real frequency is the zero-frequency constraint sector (in the parent's e = 0 kernel: a two-dimensional gauge/multiplier/auxiliary plane plus one algebraically slaved static deformation). The quotient pencil is a regular quadratic eigenvalue problem with a singular leading coefficient: its twelve infinite eigenvalues (grade-2 count 18 − 6 = 12) are the algebraic constraint/descriptor structure, not missed propagating modes, and the classification delivered is finite-eigenvalue. A new exact lemma certifies that the frozen assembly is mixed-order and carries no weighted-homogeneity (principal-symbol scaling) identity, so this second result is a pointwise finite-wavenumber, finite-frequency spectral obstruction — not a principal-characteristic theorem, and no PDE-level hyperbolicity verdict is claimed or implied. The diagnostic: the one-pair T2 block is positive at every scan point, and the contrast is structural — the quotient pencil carries a nonconservative odd-in-frequency velocity coupling (not a conservative gyroscopic one), so no ω²-type reconstruction equals it. Exact divisibility certificates over the frozen assembly show that the constant-ratio exceptional locus is exactly the zero-frequency sector for both signs of the ratio (a reduction/reconstruction obstruction carried by the parent-derived relaxed quotient assembly, explaining why no quotient-level ω²-type reconstruction can succeed); the corresponding spectral obstruction of the fully unreduced parent P10 itself is established independently, by the direct exact parent analysis (det P10 = e³·q̃ with both nonzero modes nonoscillatory), not by lifting the quotient's structure. Every verdict is pointwise-exact (exact polynomial determinants, Euclid square-freeness, Sturm counts over K_s); the region-level statements of result (1) are exact semialgebraic consequences of the certified factorizations. No open-region full-system spectral verdict, no principal-characteristic claim, no claim resting on real roots alone, and no sign-convention-structural claim is made anywhere.