Keep central density as seed data in the n1 sandbox

experiments/CENTRAL_CONSTRAINT_HYPOTHESIS_2026-09-16.md

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+# Optional central-density constraint
+
+The n=1 sandbox now supplies central density only to its Lane–Emden seed. Its
+equilibrium model fixes the polytropic EOS, surface pressure, total mass, and
+angular momentum. The existing public API already supports this combination;
+no operator, normalization, preconditioner, or solver API changes are needed.
+
+`constraint::FixedCentralDensity` remains an explicit, optional model
+specification. For n=3 it can select the scale along the nonrotating homology
+family. Omitting it remains permitted and may expose a singular or poorly
+conditioned problem. Neither the model nor the solver silently inserts a phase
+constraint or regularization based on the polytropic index. Seed-only central
+density sets an initial condition, not an additional equilibrium equation.
+
+The hypothesis is that fixing the spherical central density together with K, M,
+and nonzero J can distort the interior of an n=1 equilibrium. The implemented
+constraint fixes central enthalpy and adds a scalar source to the hydrostatic
+row; a small bordered residual can therefore coexist with a larger physical
+residual after that source is removed. This is separate from the late Krylov
+failure: the initial no-phase J=.01 control retained that failure while reducing
+the unbordered residual, with negligible change in virial balance.
+
+The next matched control is n=1, J=.05 with and without the constraint, using the
+same mesh, seed, normalization, preconditioner, and solver settings. Record formal
+solve status separately from the retained state's physical quality. Compare
+energy and force virial errors against the required 1e-5 milestone, mass and J,
+actual angular velocity, the unbordered residual, and location-resolved central
+enthalpy/EOS/Bernoulli errors. The phased baseline's sampled maximum EOS mismatch
+is about .934% of the density scale despite virial errors near 3e-7; its cause
+requires this control. Higher-spin no-phase cases follow if the matched control
+is satisfactory. J or angular velocity in reference units is not itself a
+measured fraction of critical rotation.
+
+The added user-API regression covers the previously missing explicit-seed,
+no-phase context construction and solver borrowing/release path. Existing tests
+already cover the square fixed-J root and its Jacobian column. This lifecycle
+test performs no Newton solve and makes no physical-convergence claim. Retain
+the existing phased n=3 tests; unphased n=3 convergence is not a requirement.