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# Central-to-mean density ratio of Lane–Emden polytropes (T615)
This is the builder's justification, not the independent critique. It follows family issue numberdb-data#246 and the coordinated skill archived in `agents/runs/20261030T020254Z-build-skills/manifest.json`. Both archived skill versions and the snapshot canonical skill have SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`. The archived files were read and compared byte for byte; see skill-evidence.json.
## Quantity and conventions
The sole free parameter is the rational polytropic index n, with mathematical domain 0 <= n < 5. The three-dimensional, spherical, nonrotating Newtonian model has normalized regular solution (xi² theta')' = -xi² theta^n, theta(0)=1, theta'(0)=0. It follows the positive real branch to its first zero xi_1; fractional powers are never continued through negative theta. Density is rho=rho_c theta^n. With R=a xi_1 and M=4 pi a³ rho_c mu_n, the central-to-volume-mean density ratio is C_n=rho_c/(3M/(4 pi R³))=xi_1³/(3 mu_n). The mass constant omits 4 pi.
At n=0 the incompressible interior density is constant, giving exactly C_0=1, independently of any printed digits. The index-one profile gives C_1=pi²/3, evaluated by Arb. Index five has no finite surface and is excluded. The finite-radius normalization is the same as T613 and T614; the latter additionally includes the convergent mass at n=5, which does not define a finite density ratio. There are no extra parameters for radius, central density, EOS scale, root number, or named quantity. The ratio is distinct from radius and mass, so it is not folded into those companions under a quantity parameter.
## Exhaustive parameter assessment: mixed
The ordered enumeration is `0,1/2,1,3/2,2,5/2,3,7/2,4,9/2`. These exact reduced-rational identities form three disjoint groups:
* **Intrinsic:** `0`, `1`. The incompressible and linear Lane–Emden equations have elementary normalized profiles, yielding the analytic controls 1 and pi²/3. Princeton equations (poly.9), (poly.12)–(poly.13), and Yip, Chan and Leung section 2 establish the conventions.
* **Documented application:** `3/2`, `3`. They give idealized nonrelativistic and ultrarelativistic adiabatic stellar models, respectively (Princeton p. 3). A convective envelope also motivates n=3/2 (Wenbin Lu pp. 1–2). These are model applications, not fitted descriptions of observed stars.
* **Reference sample:** `1/2`, `2`, `5/2`, `7/2`, `4`, `9/2`. These are exploratory identification targets for central concentration in simple algebraic density–enthalpy laws rho/rho_c=theta^n. No particular stellar application is claimed for every exponent. The defining family, stiffness interpretation and finite-radius boundary are checked against Yip, Chan and Leung sections 1–2 and Princeton pp. 1–2.
### Reference policy
* **purpose:** recognize central concentration constants of simple nonlinear algebraic Lane–Emden models beyond the exact and distinguished stellar cases.
* **bounds:** integer and half-integer indices with 0 <= n <= 9/2, excluding the four indices in the other two groups; exactly the six reference identities listed above. No radial argument or other independent model input is sampled.
* **coverage_rationale:** integral and square-root powers span stiff through centrally concentrated finite models, including mathematical equilibria with n>3, before higher denominators or closer approaches to five. The identity rho/rho_c=t^(2n) after theta=t² describes exponent complexity independently of decimal spelling. The cutoff avoids the divergent-radius endpoint. It is a sparse identification reference, not a table designed for interpolation or a row-count target.
* **noncanonical_acknowledged:** true. Denominator two and the cutoff 9/2 are practical choices, not a theorem distinguishing every useful EOS. Solver convenience supports feasibility but is not the reason for including this family.
* **notation_independent:** true. Selection is by integral and square-root algebraic exponent complexity, independently of numeral base.
* **sources_checked:** true. Checked sources support the function, normalization, analytic cases, application groups and limiting behavior; they need not document use of each exploratory exponent.
Breadth was reviewed before depth. Keeping only n=0,1,3/2,3 would omit the simplest exploratory nonlinear laws and the strongly concentrated finite models. Increasing denominator bounds or taking indices arbitrarily close to five would enlarge a continuous sampling problem without an additional identification case. The current ten-model selection is therefore retained. Sharing it with structural companions is justified by invariants of the same normalized models, rather than a mesh copied between unrelated functions. No radial stability is asserted for n>3. The parameter is type Q, not a discrete index, and the finite extent belongs only in the completeness note and enumeration.
## Sources and live corpus
* Princeton, *Polytropes*, https://www.astro.princeton.edu/~gk/A403/polytrop.pdf. Equations (poly.6)–(poly.10b) fix the background; (poly.12)–(poly.13) explicitly define this central-to-mean ratio. Page 3 supplies application discussion and the low-precision checks 5.99 and 54.18.
* Wenbin Lu, *Polytropic Star*, https://wenbinlu.github.io/files/LaneEmden.pdf. Equations (1)–(5) give the normalization and mass/radius scale; pp. 1–2 discuss the nonrelativistic and convective models. Equations (16)–(18) support the independent virial check.
* K. L. S. Yip, T. K. Chan and P. T. Leung, *Perturbative solution to the Lane–Emden equation: an eigenvalue approach*, MNRAS 465 (2017), 4265–4280. DOI 10.1093/mnras/stw3041, arXiv:1611.07202. Sections 1–2 discuss the family, exact profiles, surface and center-series limitations. The low-order approximants are not used for high-precision computation.
The live documents for T613 and T614 were read, including all formulas and their attached solver. The actual website links resolve `Radius_of_Lane_Emden_polytropes` and `Mass_constants_of_Lane_Emden_polytropes`; they were not guessed from titles. Both are currently drafts. T7 was read and resolves `Pi`. The public search found no existing density-ratio family; numeric lookup found pi²/3 in an Epstein zeta context, which is a different mathematical meaning and does not duplicate this family. No `repeats` declaration is appropriate for that coincidental theorem-level equality.
Existing tags physics and dimensionless were confirmed. Astrophysics is absent; propose it for the family review, with T613, T614 and T615 as concrete destinations and four further proposed companions. This build does not create a singleton tag.
## Numerical implementation and review boundary
The attached solver.py is copied unchanged from the live T614 attachment, identical to T613's, with its hash recorded in source-manifest.json. The generator freshly recomputes both structural quantities with two precision/order/step/matching-point configurations; it does not import stored rounded values or past numerical evidence. The independent radial-coordinate checks are adapted from T614's live independent_checks.py and do not import the production solver. Exact controls precede comparisons and deliberately perturbed values are rejected. The standard dry_run.py checks are run, not reimplemented.
The generator, solver, argument-selection document and validation source dependencies must be included in the later critic's fingerprint. The controller commissions a separate critique and owns the offer transition. No acceptance sidecar or review action is produced by this build.