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# Hessian arguments and reproducibility
This revision replaces the narrow degree-two selection by a source-backed
arithmetic catalogue. It does not use a decimal grid or a target table size.
The scalar argument is `t`, not its real cube root `k`.
## Scalar coverage
- All 177 real CM arguments of degree at most four from the classification
and supplement of Samart and Tao, arXiv:2409.14714v3: 3 rational,
30 quadratic, 4 cubic and 140 quartic arguments.
- Of the quartic arguments, 96 belong to 24 totally real biquadratic
orbits. The other 44 are real embeddings of non-totally-real fields.
Real embeddings are included even when other embeddings are complex.
- The singular/boundary controls `t=-1,0,27` are retained.
- The integer family `1 <= t <= 26` is studied for the modified measure in
Samart, arXiv:2301.05390v3, Table 2. This is an arithmetic reference family,
not a claim that all its conjectural special-value identities are proved.
Ordinary measures at these arguments are included as companion values.
- The exterior values `t=-27,-8,32,125` occur in the proved evaluations in
that paper's Table 1. In particular the newly included ordinary `N(32)`
connects to `g(4)=3N(32)/4` in the other Mahler-measure family T658.
After deduplication this gives **209 ordinary arguments**. The modified
selection is exactly its intersection with `-1 < t < 27`: **83 arguments**,
including both negative interior CM arguments and the exact modified zero.
The function domains themselves are not exhausted by these finite lists.
## Readable, exact CM arguments
For a primitive positive definite integral form `[A,B,C]`, put
tau = (-B + i sqrt(4AC-B^2))/(2A),
t_Q([A,B,C]) = 27 + (eta(tau)/eta(3tau))^12.
The order discriminant is `Delta=B^2-4AC`. It is not the discriminant of
the argument's real number field. The form specifies the level-three
position and embedding; a discriminant alone does not. Do not replace
these representatives by SL(2,Z)-reduced forms, which may change `t_Q`.
The new higher-degree entries display `t_Q([A,B,C])`. Their stable keys,
minimal polynomials, exact rational isolating intervals, source rows and
forms are in `hessian-catalogue.json`. The generator reconstructs each
algebraic root at its working precision; no stored decimal argument is used.
All old keys and numeric payloads remain unchanged, including their order
relative to one another. The new generator verifies these retained numbers;
it is not claimed to have originally produced them.
## Determinants are separate families
T651 retains the fifteen prescribed quadratic determinants, one per oriented
orbit in Samart and Tao, Table 1, rows 13--27. Merely adding different
coefficients or the opposite orientation would not provide another natural
member of this fixed-normalization family.
The quartic table instead contains 24 determinants, for the totally real
orbits in Table 2, rows 52--75, with Table 3's fixed coefficients and sign
variants. Its conjugates are ordered `t1<t2<t3<t4`; it is indexed by `t1`.
The matrix weights are the positive square roots of the listed coefficients,
not the coefficients themselves. The signed determinant is retained without
absolute value or rescaling. Each scalar input is the modified period inside
`(-1,27)` and the ordinary measure outside. The other real quartic and cubic
embeddings do not define members of this totally real determinant family.
## Computation and its limitations
`generate.py`, `hessian_generator.py`, `hessian_periods.py`,
`hessian-catalogue.json` and `entry-metadata.json` form the current generator.
The last file contains only labels and explanations, never numeric answers.
The default `sage -python generate.py` verifies every current entry; it does
not write to the website. Publishing requires the explicit `--publish` flag.
`--snapshot table.json` verifies an offline JSON table export instead.
The main method is adaptive tanh-sinh quadrature of the quadratic Jensen
integral. It splits at the torus crossing and around the narrow unit-modulus
transition near `t=27`. A cancellation-resistant real expression evaluates
the root-modulus integrand. For 100-digit output the nominal precisions are
140 and 190 decimal digits, with 40 extra integration digits (180 and 230
effective working digits); intermediate outputs retain 135 and 185 digits.
Agreement, not the requested precision alone, is tested before storing a
value. The modified value at zero is stored as an exact integer zero.
Independent ordinary checks integrate the original cubic in the second
variable, with geometric subdivision near `t=27`. At nominal precisions of
125 and 145 digits they use 40 extra arithmetic digits (165 and 185 working
digits). Independent modified checks use hypergeometric continuation and,
near `t=27`, an integrated logarithmic series with 20 extra working digits.
Negative arguments use additional working digits recorded in the per-entry
evidence. Both methods require over 110 digits of agreement with the main
computation. Determinants are also compared with separate
signed polynomial expansions. See `verification.json` for completed counts,
measured agreement, cancellation and source hashes for this revision.
These are heuristic agreement checks, **not certified numerical enclosures**.
The separate attached verifier can repeat the full independent comparison:
sage -python independent-verify.py check --catalogue hessian-catalogue.json \
--values independent-values.jsonl --report checks/report.json
Its JSONL input is comparison evidence, not an input to the main generator.
This command includes all scalar arguments and quartic determinants; it may
take substantially longer than checking a few examples.
Exact catalogue checks use irreducible polynomials, root isolation, class
polynomial relations and eta-function enclosures. All 66 class-polynomial
discriminants were recomputed with Sage/Arb, PARI and a direct ball product.
Only the 49 discriminants overlapping T129 were compared with that table;
the remaining 17 are outside its present range. These exact argument checks
do not certify the approximate Mahler integrals.
## Source caveats and historical files
For negative interior arguments, the post-Theorem-1 negative `3F2` prefactor
in Samart's paper yields one quarter of the defining period in the checked
examples. The defining integral agrees instead with the general Theorem 1
`4F3` formula, including the independent `t=-1/8` control. The positive `3F2`
formula remains restricted to `0<t<27`. This is an observed inconsistency,
not an author-confirmed erratum. No quarter-factor is applied here.
The printed fourth-derivative multiplier `s` for quartic row 54 is not used
to adjust any determinant sign. The table defines the signed matrix and cites
the `r/pi^8 * L(C_t/K,2)` identity; it does not assert that fourth-derivative
identity or claim an independent computation of all quartic L-functions.
Earlier attachments are retained for provenance. Their old counts, generators
and checks describe the previous 36/10/15-entry selections unless a file was
explicitly replaced in this revision. They are not evidence of full expanded
coverage. Previous versions of replaced files remain in revision history.
Sources:
- https://arxiv.org/html/2409.14714v3
- https://github.com/petesamart/Data-file-for-det-of-MM/blob/main/data.pdf
- https://arxiv.org/html/2301.05390v3
- https://arxiv.org/html/1012.3036v3