# Build handoff: T616 Built **Moment of inertia $I/(MR^2)$ of polytropic stars**, family issue #246, as [T616](https://numberdb.org/T616), slug `Moment_of_inertia_of_polytropic_stars`. The draft is populated, private and has not been offered, reviewed or published. The controller must run its separate critique and repair stages before offering. No independent argument-review acceptance or sidecar has been created here. ## Instructions and scope The canonical local and live skill copies were read from `agents/runs/20261030T025528Z-build-skills/manifest.json`. Both archived bytes match SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`. The manifest is also cited in numerical-evidence.json, independent-evidence.json, readback-evidence.json and the attached skill-evidence.json. The family was read first with `python3 agents/queue.py show 246`; its text is in family-246.md. The campaign argument gate and argument_review.py were read. All work remained in this run's artifact directory; no source, skill or pipeline edits, commits, pushes or deployment were performed. `screen.already_here`, `screen.already_asked`, additional text searches, value lookups through the client's api/lookup path and tag queries found no matching inertia family. Broad 0.4 matches were inspected as unrelated numerical contexts. Early manually constructed lookup requests were rejected for wire-format errors; cheap-checks.json records the successful client lookups that supersede them. The title claim created T616 (claim-result.json); no collision was bypassed. Read live T613, T614, T615 and T7 and their actual addresses. The table links specifically to the radius, mass and density-concentration companions. Their normalizations are preserved. The tidal Love companion is not built yet; a later family pass can add its I–Love relation link. No extra table was built. ## Quantity and selection The quantity is the axial moment about a diameter, normalized by total mass and squared finite surface radius: I/(MR^2) = 2 J_n / (3 mu_n xi_1(n)^2), J_n = integral_0^xi_1 xi^4 theta_n(xi)^n dxi. The model is spherical, nonrotating, three-dimensional and Newtonian, with the positive real Lane–Emden branch. Zero means incompressible interior density and a surface jump. Five has infinite radius and divergent axial inertia and is excluded from this finite-radius normalization. Models above three are included as mathematical equilibria without a stability claim. There are ten ordered exact rational arguments: `0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2`. The sole parameter is n, type Q, with mathematical domain 0 <= n < 5. The intrinsic group is {0,1}; documented stellar applications select {3/2,3}; the exhaustive reference_sample group is {1/2,2,5/2,7/2,4,9/2}. The attached argument-selection.md gives the complete reference_policy, sources and bounds. Denominator two and cutoff 9/2 are explicitly noncanonical reference choices, selected by algebraic exponent complexity rather than decimal spelling. The default enumeration, stored rows and completeness note agree. ## Computation and checks actually run * Extended the solver downloaded from live T613 to accumulate J and the independent enthalpy integral H = integral xi^2 theta dxi. Full source is attached as solver.py, with its origin recorded. * Computed every selected model at 150 and 210 decimal working precision, orders 160/210, step divisors 12/18 and matching radii 1/2 and 2/3. The worst relative difference is less than 3.25e-154. * Checked J = mu*xi_1^2 - 6H throughout the selection. The n=0 and n=1 positive controls agree with 2/5 and 2/3-4/pi^2, and deliberate displacements are rejected. The output stores exactly 2/5 for n=0 and an Arb-evaluated closed form for n=1. * Independently adapted live T614's radial-coordinate implementation, which imports no production solver, uses exact central coefficients and its own density-moment quadrature. All ten comparisons pass; the worst relative difference is 1.87e-60 (the incompressible endpoint extrapolation). Other selected comparisons are below 7e-74 or zero at the check's working precision. * Checked the Yip–Leung Padé approximation over the seven selected indices from zero through three. At n=3 its measured relative error is 0.002141340817788398..., or 0.2141340818 percent. The initial test mistakenly treated the paper's 0.21 percent statement as a strict bound and failed. This error rounds to 0.21 percent; the final test uses the 0.215 percent rounding boundary and records the interpretation. No fitted approximation supplies any stored digits. The original failed log is preserved. * Ran the repository's unchanged dry_run.py and check.py through agents/sage.sh. It passed exactness, prose and size checks. The serialized entries contain one exact rational and nine 100-significant-digit decimals; their longest spelling is 104 characters and the entries block is approximately 1.43 KiB. * Read all values back through numberdb.table. Checked each stored decimal interval against both moment-integral forms, both exact-profile anchors, and the live radius/mass/density companions. Deliberately displaced values failed containment. Two initial readback-helper runs hit Sage method/coercion errors; the corrected endpoint comparisons pass. Their logs are retained. * Ran PolytropicInertia().verify(sample=None): **10/10 matched, 0 differing, 0 missing, 0 extra**. * The final site audit reports **clean: true, findings: []**. The optional moments tag was removed after the audit found it had no other current destination, despite a stale-looking tag endpoint listing. Current tags are physics and dimensionless. The missing astrophysics tag is proposed for T613, T614, T615, T616 and the later family companions. Rigour is **heuristic (agreement-checked)**. Neither the Taylor truncation, global ODE propagation nor radial endpoint approximation is rigorously bounded. The precision evidence supports the stored digits heuristically; it is not an interval-ODE proof. Runtime versions are Sage 10.9, numberdb 0.1.10, mpmath 1.4.1 and Python 3.12.3. ## Reproduction and critique All reproducible generator/check sources and numerical evidence are attached to T616. The core sources are generate.py and solver.py. Other code attachments are compute.py, independent_checks.py, readback_checks.py, validate.py, dry_run.py and check.py. Source-manifest.json records downloaded sources and hashes. No shared solver dependency outside the attachments is required. After downloading the attachments beside one another, use SageMath: sage -python generate.py sage -python validate.py sage -python independent_checks.py sage -python readback_checks.py Draft reads require the authorized key; in this environment use agents/sage.sh and pipe NUMBERDB_KEY_FILE on stdin as documented in generate.py. No offers or publication actions occur in those commands. The default generator verifies; its --fill flag writes entries to the existing draft only. The separate critic must independently assess the three argument groups and perform numerical checks. Its fingerprint should include at least generate.py and solver.py and all other source/evidence dependencies it reviews, using the current website attachments. It must create its own required -arguments.json sidecar with its NUMBERDB_RUN_ID. This build report is not an acceptance. The final controller handoff is `agents/runs/20261030T025528Z-build.table.json` and names only T616.