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# Mass constants of Lane–Emden polytropes (T614)
This build follows family issue numberdb-data#246 and the coordinated skill archived in `agents/runs/20261030T013608Z-build-skills/manifest.json`. Both archived skill files have SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`. This is the builder's justification, not the required independent critique or its acceptance sidecar.
## Quantity and normalization
There is one free parameter, the rational polytropic index `n`, in the mathematical domain 0 <= n <= 5. The regular three-dimensional Lane–Emden solution satisfies theta(0)=1, theta'(0)=0 and (xi² theta')'=-xi² theta^n. Follow its positive real branch to the first zero for n<5. The one stored quantity is mu_n = integral xi² theta^n dxi = -xi_1² theta'(xi_1), with no factor 4 pi. At n=5 the upper limit is infinity and the integral converges to sqrt(3). These conventions agree with the radius companion T613. Its finite-radius selection is extended by the analytic finite-mass endpoint, rather than inserting a fictitious finite radius.
No radial sampling point, central density, length scale, equation-of-state scale, or second named quantity is a parameter. `n` is not classified as a discrete index simply because this selection is finite.
## Disjoint and exhaustive groups (category: mixed)
* **Intrinsic:** `0`, `1`, `5`. The incompressible, linear and analytic infinite-radius solutions give exact profiles and masses 2 sqrt(6), pi and sqrt(3), respectively. Yip, Chan and Leung (2017), section 2 equations (8)–(10) and section 6.2 equation (84), and Princeton's equations (poly.9) and (poly.10b), fix these cases.
* **Documented application:** `3/2`, `3`. These describe idealized nonrelativistic/convective and ultrarelativistic/radiative stellar equations of state. Princeton's Polytropes notes, p. 3, and Wenbin Lu's Polytropic Star, pp. 1–2 and the radiative-envelope discussion, establish the applications. They are model indices, not precision fits to observed stars.
* **Reference sample:** `1/2`, `2`, `5/2`, `7/2`, `4`, `9/2`. These are explicitly exploratory identification references for simple nonlinear algebraic laws rho/rho_c=theta^n. No documented stellar application is asserted for every exponent.
### Reference policy
* **Purpose:** identify mass scales of simple algebraic Lane–Emden models beyond the analytic and distinguished stellar cases.
* **Bounds:** integer and half-integer exponents in 0 <= n <= 9/2, excluding the intrinsic and application groups, with no other model inputs. After theta=t², the selected density powers become integer monomials in t.
* **Coverage rationale:** integral and square-root algebraic laws span stiff through centrally concentrated finite models, including the mathematical equilibria beyond n=3. This gives breadth before higher denominators or closer approaches to the infinite-radius endpoint. The endpoint n=5 is included separately for its analytic finite mass. It does not justify a dense approach to five.
* **Noncanonical acknowledged:** true. Denominator two and the cutoff 9/2 are practical reference choices, not a theorem selecting all useful equations of state, a target number of rows, or an interpolation mesh.
* **Notation independent:** true. Algebraic exponent complexity selects the arguments, independently of decimal representation.
* **Sources checked:** true. The sources establish the function, normalization, exact cases, applications and endpoint behavior. They need not use each exploratory reference exponent.
The default enumeration is `0,1/2,1,3/2,2,5/2,3,7/2,4,9/2,5`, in ascending exact reduced-rational order. It agrees with the completeness note and the intended eleven stored rows. The practical extent follows the family-specific coverage decision; storage measurements do not select a new range. None of the values is an exactly known rational; the three closed forms are irrational and are evaluated as Arb balls.
## Checked sources and corpus links
* 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 2, 5 and 6.2. Their `m(n)` is 4 pi times this table's mu_n; their scaled-coordinate second moment also uses the letter mu and is a different normalization.
* https://www.astro.princeton.edu/~gk/A403/polytrop.pdf : equations (poly.6)–(poly.10b), analytic models and stellar applications.
* https://wenbinlu.github.io/files/LaneEmden.pdf : equations (1)–(4), convective-envelope and nonrelativistic-degeneracy discussion, radiative-envelope discussion.
* T613 was read through `api/table?id=T613`; the actual slug `Radius_of_Lane_Emden_polytropes` was resolved from its website attachment page, not derived from its title. It is currently a draft, so it is available to the account but absent from public search. T7 was read and its actual `Pi` address confirmed by text search. The pi table has no parameter, so its table address identifies its sole value.
The existing tags `physics` and `dimensionless` were confirmed through `api/tag`. `astrophysics` does not exist. Recommend it in the family review for the radius, mass, density ratio, inertia, pressure, mass–radius and tidal Love tables in issue #246; this build does not create it.
## Reproduction and independent review
The generator and its imported `solver.py` are both selection-relevant source dependencies. The solver is copied from T613's live attached source, and this run recomputes the mass rather than importing its previous numerical evidence. `compute.py` varies precision, order, step cap and central matching point. `independent_checks.py` uses a separate radial-coordinate implementation and an integral identity, without importing `solver.py`. The controller must commission a separate critique of these arguments and the numerical evidence before offering the draft. That critic must hash the current attached sources; this document is not an independent verdict.