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# Central pressure constants of Lane–Emden polytropes
The single free parameter is the exact rational polytropic index n. The
mathematical domain is 0 <= n < 5; the stored selection is
0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, in ascending order.
W_n = P_c R^4/(G M^2) uses the finite zero-pressure surface and the normalized
positive Lane–Emden solution. Central density and the dimensional length scale
cancel, so neither is sampled. The surface radius is determined by n, not an
independently chosen argument. No additional quantity or convention is indexed.
The selection is mixed, with disjoint exhaustive groups:
* intrinsic: 0, 1. The incompressible and linear equations have exact regular
profiles, giving W_0=3/(8*pi) and W_1=pi/8. See Princeton, p. 2,
equation (poly.9), and Pols, section 4.1, equations (4.8) and (4.9).
* documented_application: 3/2, 3. These are idealized nonrelativistic and
ultrarelativistic adiabatic stellar models (Princeton p. 3). A fully convective
star also motivates 3/2 (Lu p. 3). These are model applications, not
precision descriptions of observed stars.
* reference_sample: 1/2, 2, 5/2, 7/2, 4, 9/2. These are deliberately
exploratory references, not claims of applications for each exponent.
Reference policy:
* purpose: identify central-pressure coefficients of the simplest nonlinear
algebraic density–enthalpy laws beyond the two exact and two stellar cases.
* bounds: integer or half-integer n in [0,9/2], excluding the four members
assigned to the other groups. There are no other combinations of inputs.
* coverage_rationale: rho/rho_c=theta^n becomes an integer monomial after
theta=t^2 for these exponents. The selection covers both integer and
square-root algebraic laws, from stiff models to centrally concentrated
finite models, before adding higher denominators or approaching five more
closely. This gives useful breadth without an interpolation mesh. The six
references complement rather than replace the intrinsic and application
cases. No entry count or output-size target determines the selection.
* noncanonical_acknowledged: true. Denominator two and the upper cutoff 9/2
are practical reference choices, not a theorem selecting stellar equations
of state. Shared indices are justified by invariants of the same background
models, not by copying a mesh between unrelated functions.
* notation_independent: true. Algebraic exponent complexity does not depend
on decimal spelling or numeral base.
* sources_checked: true. Princeton pp. 1–3, Lu equations (1)–(7) and the
convective-envelope paragraph, and Pols section 4.1, Table 4.1 and equation
(4.17) establish the family, normalization, applications and endpoint.
No publication using each exploratory exponent is asserted.
The incompressible case means constant interior density with a surface jump,
not substituting zero into 1/n. Index five has no finite surface and is
excluded, despite its finite mass in the companion table. Models above three
are mathematical equilibria without a claim of radial stability. Numerical
bounds are practical too: 100 stored significant digits, compared using 150
and 210 working decimal digits with different orders and step controls.
The family coverage was read from issue 246 and live T613–T616. Those tables
separate radius, mass, concentration and inertia. Pressure remains one named
quantity; the later mass–radius and tidal-response proposals are not absorbed.
Existing tags physics and dimensionless are used. Astrophysics is still absent;
it would group T613, T614, T615, T616 and T617 (and later family companions).
Policy evidence: ../20261030T035316Z-build-skills/manifest.json, copied here as
skill-manifest.json. Both archived skill-local.md and skill-live.md have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The current source snapshot has the same hash. This is the builder's selection
justification, not independent critique acceptance. The separate critic must
assess every group and bind its sidecar to the live selection and source files.