# Argument selection for T658 The single free parameter is the real coefficient k in P_k=(1+x)(1+y)(x+y)-kxy. Natural logarithms and normalized Haar measure are fixed. The polynomial is not rescaled, and the sign of k is retained. The integral is the ordinary logarithmic Mahler measure at torus zeros as well as away from them. Integration variables x,y are not table parameters. This build follows the archived coordinated skill manifest `agents/runs/20261101T190803Z-build-skills/manifest.json`. Both skill-local.md and skill-live.md have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`. The selection must receive a separate, independent critique before the controller offers the draft. This note is the builder's justification, not independent acceptance evidence. ## Mixed assessment, with exhaustive disjoint membership 1. **intrinsic**: `-1`, `0`, `8`. These are exactly the real singular fibres, by Mellit, Proposition 2.1. The first two admit factorizations and the last is the torus endpoint giving triangular and honeycomb spanning-tree entropies. 2. **documented_application**: `-8`, `-2`, `1`, `2`, `4`, `7`, `4+2*sqrt(5)`, `8+4*sqrt(5)`. Mellit, Section 1.2 and Section 5, proves the conductor-14 identities for -8,1,7. Rogers and Zudilin, Theorem 5 and its corollary, treats conductor 20 at 4,-2 and the positive quadratic pair. Its Theorem 7 treats 2 and conductor 36. Unlike the queued proposal, this assessment puts -2 here: the corollary explicitly establishes its L-value formula. No input is added or removed by that correction of role. 3. **arithmetic_structure**: `4-2*sqrt(5)`, `8-4*sqrt(5)`. These complete the real embeddings of the selected quadratic arguments. Their minimal polynomials are X^2-8X-4 and X^2-16X-16, respectively. The positive embedding identities motivate the orbits; no identical L-value formula is claimed for the negative embeddings. Both embeddings are recomputed from exact radical identities at each working precision. 4. **reference_sample**: `-16`, `-15`, `-14`, `-13`, `-12`, `-11`, `-10`, `-9`, `-7`, `-6`, `-5`, `-4`, `-3`, `3`, `5`, `6`, `9`, `10`, `11`, `12`, `13`, `14`, `15`, `16`. These 24 remaining integers supply bounded exploratory reference coverage. Their individual arithmetic applications are not asserted. The groups have sizes 3,8,2,24 and total 37. Their union is all integers -16 <= k <= 16 plus the four stated quadratic inputs. Integer entries are ordered by absolute value, positive before negative; adjacent conjugate pairs follow. The mathematical parameter domain remains all real k. ## Reference policy - **purpose**: recognize integrals of small integral polynomials in this reciprocal-cubic normalization, including arithmetic-period relations not preselected by a successful integer-relation search. - **bounds**: exactly the 24 reference integers listed in group 4. Combined with the distinguished integers, the integer range is [-16,16]. There are no extra rational grids or Cartesian products. - **coverage_rationale**: both exterior regimes and the torus-zero interval [-1,8] are represented. The absolute coefficient bound 16 supplies a short extension past the positive boundary 8 while treating the negative side on equal terms. The cutoff is a practical editorial choice, not a theorem or an entry-count target. The conjugate pairs cover nonintegral arithmetic inputs before increasing the depth of the integer range. This is meaningful breadth without copying the CM catalogue of the different Hessian family. - **noncanonical_acknowledged**: true. - **notation_independent**: true. Integral coefficients and an absolute bound do not privilege decimal notation. The coefficient normalization is fixed. - **sources_checked**: true for the family, normalization, torus interval, singular fibres and claimed arithmetic identities. No publication is claimed to use every exploratory input. ## Checked sources and family alignment - Rogers and Zudilin, *From L-series of elliptic curves to Mahler measures*, version 3: https://arxiv.org/html/1012.3036v3 . Introduction (definition and equations 7,8), Theorems 4,5,7 and the corollary to Theorem 5. Equation 7 is used only outside [-4,2], never blindly continued into that interval. - Mellit, *Elliptic dilogarithms and parallel lines*, version 1: https://arxiv.org/html/1207.4722v1 . Section 1.2, Propositions 2.1,2.3, and Section 5. - Guttmann and Rogers, *Spanning tree generating functions and Mahler measures*, version 2: https://arxiv.org/html/1207.2815v2 . Section 5.1, equation 27 and its endpoint. The displayed calculation following (27) contains a factor-two inconsistency: the correct identities are g(8)=5*m(1+x+y) and T_hex(1)=g(8)/2. These follow from the first hypergeometric expression, the endpoint equation, and independent numerical checks. No ambiguous factor from that chain is adopted. The family issue #290 and the live documents of T145,T153,T282,T283,T292, T521,T522,T647,T648,T649,T650,T651 were inspected. T649 uses the parameter cube: its N(t) is the source's n(cuberoot(t)). We retain k itself here. T650 is a modified period and T651 a determinant; neither is an ordinary measure nor a selector value of this table. There is no exponential convention, derivative quantity or second polynomial family in the entries. Public searches, already_here, already_asked, sample value lookups, and tag checks are saved in screening.json and cheap-checks.json. No duplicate family was found. T649 was read through its authenticated API; its public page is not yet accessible. Its stable T-number is used as a cross-reference. The existing tags number theory, special values, elliptic curves, integral, and statistical mechanics all exist. No Mahler measure tag exists. That tag would fit at least T282,T283,T649,T658 and is recommended without creating it.