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# Argument selection for T618
The sole free parameter is the rational polytropic index n, with mathematical
domain 0<n<5. The quantity N_n uses the equation-of-state normalization
K=N_n G M^((n-1)/n) R^((3-n)/n), with P=K rho^(1+1/n), and the central
Lane–Emden normalization theta_n(0)=1, theta_n'(0)=0. There are no free radial
coordinates, central densities, gravitational constants, masses or length
scales in the table. These scale out. The parameter selects a background
model, not a second quantity.
The actual enumeration is 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, in this order.
The assessment is mixed, with disjoint exhaustive groups:
* intrinsic: 1. The linear Lane–Emden solution sin(xi)/xi gives N_1=2/pi.
* documented_application: 3/2 and 3. These are idealized stellar models with
nonrelativistic and ultrarelativistic pressure, respectively. The index 3/2
also describes a convective envelope. Their mass-radius coefficients are
explicitly tabulated by the Princeton notes and Pols.
* reference_sample: 1/2, 2, 5/2, 7/2, 4, 9/2. These are exploratory reference
coefficients for low-complexity algebraic density–enthalpy laws
rho/rho_c=theta^n. No application is asserted for every selected exponent.
The reference_policy is:
* purpose: identify mass-radius coefficients from the simplest nonlinear
algebraic polytropic laws beyond the distinguished stellar and linear cases.
* bounds: positive integer and half-integer n<=9/2, excluding 1, 3/2 and 3
assigned to the other groups; no other arguments.
* coverage_rationale: span stiff through centrally concentrated finite models,
including n>3, with exponents that become integral monomials under theta=t^2.
This covers integral and square-root algebraic laws before higher
denominators or a closer approach to the infinite-radius endpoint at five.
It is a sparse recognition reference, not an interpolation mesh or a size
quota. Breadth across these laws is retained rather than reducing the table
to the stellar application indices alone.
* noncanonical_acknowledged: true. Denominator two and the 9/2 cutoff are
practical, noncanonical choices, not a theorem about useful equations of state.
* notation_independent: true. The criterion is algebraic exponent complexity,
independent of decimal spelling.
* sources_checked: true for the model, normalization, stellar applications and
endpoint behavior, not for a purported publication using every reference n.
Compared with family #246's S, n=0 is excluded because the defining EOS
coefficient contains 1/n. The finite-radius condition excludes n=5. n=3
remains well defined although the mass is independent of radius at fixed K.
Models above three are mathematical equilibria without a stability claim.
The same indices in companion tables refer to invariants of the same models;
this is not a mesh imposed on unrelated functions. There are no interactions
between independently sampled parameters.
Checked sources:
* https://www.astro.princeton.edu/~gk/A403/polytrop.pdf, equations (poly.3),
(poly.5)–(poly.11b), and p.3's application discussion and (poly.17a,b).
* https://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/chapter1-4.pdf,
Table 4.1 and equations (4.11)–(4.17), particularly (4.15).
* https://wenbinlu.github.io/files/LaneEmden.pdf, equations (1)–(7) and
the p.2 discussion of nonrelativistic degeneracy and convective envelopes.
The current archived canonical skill is recorded by
agents/runs/20261030T044705Z-build-skills/manifest.json. Both local and live
versions have SHA-256 28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The attached skill-manifest.json and skill-evidence.json preserve this evidence.
This is the builder's justification. A separate critique must evaluate it and
bind its own assessment to the live selection and all source dependencies.