back to table · edit · history · where entries came from · files · download
7071 bytes, as of the version from 2026-11-02 11:04 (current). Recorded here, not run.
# T658 build evidence
Draft: https://numberdb.org/T658
Title: Mahler measures of $(1+x)(1+y)(x+y)-k xy$.
## Governing instructions
Read the family issue #290 before computations. The coordinated skill manifest
is `agents/runs/20261101T190803Z-build-skills/manifest.json`, archived here and
attached as `skill-manifest.json`. Both archived versions, `skill-local.md`
and `skill-live.md`, hash to
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
`skill-evidence.json` records the per-file hashes and manifest path. Their
bytes match. No later or remembered skill was substituted.
## Object, selection and precision
Ordinary logarithmic Mahler measure, real k, normalized Haar measure, natural
logarithm. One named quantity g(k), no modified period. The exact radical
identities are rebuilt at every precision. Integer inputs -16 through 16 are
ordered by magnitude, positive before negative, followed by adjacent pairs
4 +/- 2*sqrt(5) and 8 +/- 4*sqrt(5). The selection comprises 3 intrinsic,
8 documented-application, 2 conjugate-arithmetic and 24 exploratory-reference
arguments. Compared with the proposal, -2 moves to the application group
because Rogers and Zudilin's corollary to Theorem 5 proves its L-value formula.
No arguments were added or removed. See `argument-selection.md` for exhaustive
memberships, bounds and the reference policy.
There are 37 entries: one exact zero, and 36 values at 100 significant digits.
The table's rigour is `heuristic (agreement-checked)`. Numerical quadrature
error estimates and agreement at different precisions are not certified bounds.
The environment reports Sage 10.9, mpmath 1.4.1 and numberdb 0.1.10 in the
first line of `numerics.jsonl`. All Sage work used `agents/sage.sh`.
## Numerical methods and outcomes
- The generator integrates a real Jensen formula with every torus crossing
made an endpoint. Tanh-sinh quadrature is repeated at 140 and 190 working
decimal digits. All nonzero values have at least 139.5173 digits of agreement
relative to max(1,abs(value)); the minimum is at k=6.
- A distinct complex quadratic-root implementation integrates the original
polynomial in y at 125 digits for all 37 arguments. The maximum discrepancy
relative to max(1,abs(value)) is 4.1599009e-126. This implementation shares
Jensen's theorem with the generator, so additional external checks matter.
- The direct-root routine first passes the nonzero k=-1 control, computed
separately using Hurwitz zeta functions. PARI's lfun first reproduces zeta(2).
- PARI independently evaluates elliptic L-functions at s=2 for conductors
14,20,36. The models are [1,0,1,4,-6], [0,0,0,-108,297], [0,0,0,0,1].
Their conductors are checked, and exact Fourier coefficients through q^80
match the eta products in the cited papers. All six applicable integer
evaluations pass at 120 digits.
- Rogers and Zudilin's equation (7) passes for all 27 selected arguments
outside [-4,2], excluding k=8, which has its own degeneration control.
The maximum relative discrepancy is 5.4004336e-125. Inside the unit disk,
the hypergeometric series is summed by a constant-memory recurrence with
term-ratio bound |z|; outside, mpmath supplies analytic continuation. The
prohibited real interval is never evaluated with this identity.
- The exact factorizations at k=0,-1 are verified over QQ[x,y]. The two
quadratic minimal-polynomial identities are checked modulo z^2-5.
- The identities g(-1)=2*mu_3, g(8)=5*mu_3,
2*g(4+2*sqrt(5))-g(8+4*sqrt(5))=g(4), and g(4)=3*N(32)/4 pass.
N uses the Hessian parameter cube, without silently cubing this table's k.
- The displayed standalone Program passes on k=4,-1,8 and 4-2*sqrt(5).
- The repository's unchanged dry_run.py and check.py pass: 37 entries,
longest value representation 146 characters, block 6.7 KB, no exactness,
disguised-exactness or entry-prose findings. The serialization also uses
numberdb's rigour and precision checks and keeps exactly 100 digits for
every nonzero row.
- `verify(sample=None)` after filling reports 37/37 matched, zero differing,
missing or extra. `verify.log` preserves the answer.
- Readback checks inspect the actual decimal intervals, not just their
midpoints: all contain the 190-digit results, with maximum displacement
0.469745835119 units in the last place. The elliptic identities, the
quadratic identity, singular values, and ratios pass again on stored rows.
Checks against authenticated companion values T283#3, T145#3,2,2,
T153 spanning-tree entropies and T649#-1 pass within their recorded
precisions. A perturbation of 1e-90 is rejected both before and after upload.
The checked source HTML and SHA-256 records are retained locally; the URL and
hash inventory is attached as `sources.json`. Primary sources are RZ
(arXiv:1012.3036v3), Mellit (1207.4722v1) and GR (1207.2815v2), with exact
locators in the argument-selection note and table references.
## Resolved source inconsistency
The chain after equation (27) of GR mixes factors appropriate to g(8) and
g(8)/2. Its endpoint formula, first hypergeometric expression, and the
independent Hurwitz-zeta/integral checks fix the normalization unambiguously:
g(8)=5*mu_3=15*sqrt(3)*L(2,chi_-3)/(4*pi), and the honeycomb entropy is g(8)/2.
The table and header consistently store g(k). This was resolved before upload.
## Audit and remaining review work
The live audit has one finding: the four radical inputs carry `param-latex`
labels despite the real parameter type. These labels are exactly the arguments,
for example `$4+2\sqrt5$`; they are not formulas for g(k) or quantity selectors.
The archived skill explicitly allows this case in section 4, paragraphs beginning
“A symbolic identity for a real argument is different” and “An audit warning”.
The labels and stable keys are retained, and the independent critic should
assess this documented advisory. No other audit findings were reported.
The family, companion documents and tags were read. Existing companions are
linked by verified corpus slugs or the stable T649 identifier; that companion
is still inaccessible publicly but accessible through the authenticated API.
No Mahler measure tag exists. Recommend it for T282,T283,T649,T658 without
creating it here. No published table was edited.
The build is complete as an unoffered draft. The controller must obtain the
independent critique and its argument-selection sidecar; this build creates
no acceptance sidecar, offer, review or publication action.
## Reproduction from attachments
With SageMath, numberdb and mpmath available, place the attachments together:
sage -python generate.py
sage -python checks.py
sage -python supplemental.py
sage -python program.py
sage -python readback_checks.py
The first command needs authentication while T658 is private. `checks.py`
uses `numerics.jsonl`; `readback_checks.py` uses `readback.json`,
`numerics.jsonl`, `checks.jsonl`, and `companion-controls.json`.
The generator does not depend on saved numerical results or check outputs.