42 / C5-F: Compressive-Axis Robustness, Pressure-Signature Locking, and Derivative-Gate Escalation

C5-E has already translated seven-point cancellation entirely into three PDE field-defect routes. C5-F asks a more refined question: does middle-gap degeneration also wipe out C5-D's compressive-axis pressure geometry along with it? Does the direction dispersion that cancellation genuinely requires only need the other two eigenvectors to rotate, leaving the most-compressive axis fixed? And can a shared far-field pressure instead lock the compressive axis in place, forming a second finite-dimensional incompatibility with cancellation? C5-F.1, Uniform Compressive Spectral Gap, first proves that the normalized strain in the entire positive-middle sector (including the gap boundary) has a uniform spectral gap $k_2-k_1\ge1/\sqrt2$ — the most-compressive eigenvalue is everywhere simple, so a standard finite-dimensional spectral-perturbation estimate then gives C5-F.2, the Middle-Gap Limit Preserves the Compressive Axis: gap degeneration only pushes the strain shape toward the degenerate boundary $(-1/\sqrt2,0,1/\sqrt2)$, and does not erase the compressive-axis projection $G(e_1)$ — the middle-gap route therefore remains coupled to the pressure-axis geometry. It then proves the quadratic half-space theorem in both a fixed-axis version (C5-F.3) and an axis-cap version (C5-F.4): as long as the gap is non-degenerate and the compressive axis lies within a projection cap of radius set by the gap, even allowing the other two eigenvectors to rotate arbitrarily, all quadratic directions still fall into the same strict half-space. C5-F.5, Nondegenerate Q Cancellation Forces Axis Anti-Concentration, is therefore stronger than C5-E's result: what seven-point cancellation genuinely requires is dispersion of the compressive axis itself, not merely dispersion of the full strain direction — letting only the other two eigenvectors rotate freely while holding the most-compressive axis fixed can never produce a zero barycenter. On pressure: a harmonic far-field pressure matrix admits only two non-degenerate inertia types — a single negative eigenvalue $(-,+,+)$ or a double negative eigenvalue $(-,-,+)$. The round's headline result, C5-F.6: if the shared far-field pressure signature is $(-,+,+)$ and the negative-direction compensation margin is strong enough, it locks the compressive axis into a projection cap; if this cap is narrower than the scale of dispersion that cancellation requires, the two cannot coexist within the same recurrent limit — this is the C5 series' second finite-dimensional algebraic incompatibility, finer than C5-D's original result, because it does not need to lock the full strain direction, only the compressive axis itself, and it supplies an explicit marginal criterion threshold. Conversely, the double-negative signature $(-,-,+)$ does not force a single axis-cap lock (the negative eigenspace is two-dimensional, so the axis can still disperse significantly within it) — so a pressure survivor that is to coexist with non-degenerate-gap cancellation must escape toward a weak margin, a double-negative signature, a degenerate signature boundary, a split pressure source, or mean rotation. On vorticity-dominant leakage, a Hölder argument forces a genuine $\omega\otimes\omega$ $L^2$-congestion lower bound, which an orthogonal projection then splits into either Miller-operator orthogonal congestion (connecting back to the growth-direction orthogonality) or constrained-complement congestion (explicitly stated to not be the true pressure Hessian, only a provisional placeholder that will need to be jointly analyzed in future work alongside strain-squared, advection, and the genuine pressure). Strain-derivative leakage gives a genuine critical pointwise second-derivative-amplitude lower bound, $R^3\|D^2u\|_\infty/\nu\gtrsim1$, but amplitude is not the same as sparseness of component/sign superlevel sets — this hard rule is retained. The document compares the spatial sparseness exponent of middle-gap cubic intermittency ($2/3$) against Grujić–Xu's fixed-$k=1$ direct regularity scaling exponent ($3/5$); since $2/3>3/5$, this is formally favorable — but it immediately warns that $\nabla u$ contains vorticity, so the raw high-derivative-value set must be covered by the union of the high-strain-value set and a vorticity high-value defect set (C5-F.9, the Field-Conversion Dichotomy), so this is only a scale-favorable conditional interface, not a theorem application. C5-F.10, Fixed-Order Recurrence or Derivative-Order Escape (a pure natural-number-subsequence dichotomy), explicitly states that repeated failure at a fixed order does not by itself imply that the derivative order must escape to infinity — only once every fixed-order recurrent defect has been separately excluded is the research line entitled to formally advance to the asymptotic critical boundary $k\to\infty$, and even that boundary is itself not a contradiction — every generation may still fail on component/sign conversion, late-time analyticity, the derivative chain, or a spatial-support mismatch. It formally hands off to C5-G, Pressure-Signature Defects, Vorticity Constraint Complements, and Fixed-Order Derivative-Gate Closure, with 8 proof targets.