back to table · edit · history · where entries came from · files · download
5877 bytes, as of the version from 2026-10-30 13:35 (current). Recorded here, not run.
# Argument selection for T626
This build follows the archived matching local/live skill in
`agents/runs/20261030T125612Z-build-skills/manifest.json`; each copy has SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The attached `skill-manifest.json` preserves that record. This is the builder's
justification, not an independent critique or an acceptance sidecar.
## Quantity and parameters
One dimensionless effective resistance is indexed by the network and its
endpoints. The parameter `lattice` is the genuine discrete network index
`triangular` or `honeycomb`; it does not switch the measured quantity. Together
with the existing square-network table T596, these are the three regular
Euclidean tiling graphs. Both graph names occur in the title and keywords.
The parameter `sublattice` names the destination basis vertex: `A` or `B` on
honeycomb, and the sole label `A` on triangular. The source is always `A_0`.
The integer parameters `m,n` are joint cell coordinates in the specified
120-degree basis. They are not independent real inputs and are not sampled
on a numerical mesh. All four parameters have role `discrete_index`.
Every edge has the same resistance R0; the stored quantity is R_eff/R0.
The basis lengths only locate vertices, and do not change the edge resistance.
The positive combinatorial Laplacian and periodic-quotient limit agree with
T596. Cserti uses the negative Laplacian, so its signs are converted when the
positive operator is stated; its resulting resistance formulas are unchanged.
## Three groups assessed separately
1. **Triangular, A:** all nonzero vertex displacements up to graph distance
12, modulo the twelve rotations/reflections about the source. The unique
representatives satisfy m >= 2n >= 0. They are precisely 1 <= m <= 12,
0 <= n <= floor(m/2), giving 48 rows. This includes axial, reflection-line,
and asymmetric directions.
2. **Honeycomb, A:** all nonzero same-sublattice endpoint pairs up to graph
distance 12. Two honeycomb steps make one triangular step; translation and
endpoint exchange add inversion to source-preserving honeycomb symmetries.
Thus the same wedge applies with 1 <= m <= 6, giving 15 rows. These are
distinct resistance contexts even though their values are three times
triangular values, and both endpoint classes are needed to describe the
honeycomb graph.
3. **Honeycomb, B:** all opposite-sublattice destinations with distance <= 12,
reduced only under the source-preserving group generated by
C(m,n)=(-n-1,m-n) and S(m,n)=(m,m-n). The lexicographically greatest member
of each six-candidate orbit is stored, giving 21 rows. An independent graph
breadth-first traversal checks the graph ball and orbit coverage. Applying
the triangular wedge here would wrongly remove endpoint classes.
The origin has zero resistance and is excluded in the two A groups.
Honeycomb B(0,0) is a distinct adjacent vertex and is included. The distances
are D=max(|m|,|n|,|m-n|), 2D for honeycomb A, and
1+2 min(D(m,n),D(m+1,n),D(m+1,n+1)) for honeycomb B. The joint domain is a
finite graph ball, not a Cartesian product of arbitrary choices. There are
six positive even and six odd honeycomb shells, with 3d vertices in shell d.
## Extent and breadth
The graph-distance cutoff 12 is a noncanonical computational reference bound,
not an intrinsic termination of the infinite family. It provides short-range
network resistances and Green-function differences in every direction and both
honeycomb endpoint classes, before extending one axis more deeply. It is
independent of numeral notation. The same distance bound has a common graph
meaning across these two directly related networks; it is not a universal mesh
applied to unrelated functions. Increasing the radius would deepen the same
contexts, while the selected range already includes all local geometries.
The published sources define resistance for arbitrary vertex pairs; no claim
is made that every pair in this finite ball has its own named application.
This is a discrete indexed family with a practical truncation, not a
continuous exploratory reference sample. No mixed category or reference-policy
object is needed. All 84 sound selected rows are retained. No anisotropy,
edge defects, continuously variable geometry, or statistical quantiles occur.
The completeness note and the default generator both specify these same three
groups and bounds. Entries are sorted by graph distance, then triangular before
honeycomb, then A before B, then m,n.
## Checked sources and corpus
- Cserti, *Application of the lattice Green's function for calculating the
resistance of infinite networks of resistors*, arXiv:cond-mat/9909120v4,
sections IV-V, equations (56), (61), (62), (64), (77), (78): the networks,
coordinates, integral and honeycomb reduction.
- Cserti, Széchenyi and Dávid, *Uniform tiling with electrical resistors*,
arXiv:1102.4537v2, section 2: periodic finite quotients and Green matrices.
- Mathar, *Recurrence for the Atkinson-Steenwijk integrals for resistors in
the infinite triangular lattice*, viXra:2208.0111v1, section 3, equations
(49)-(53): triangular symmetry and harmonic recurrences. The conjectures in
section 2.3 are not used.
- Live T596, T400, T393 and T238 were read, and the addresses of the three
linked companions were obtained from live search. They respectively cover
square pair resistances, finite-graph Kirchhoff sums, three-dimensional
origin Green integrals, and finite-graph Laplacian polynomials. None covers
the proposed two networks. The live re-screen and sample digit lookups are
preserved in `screen-live.json` in the build artifacts.
A separate critique must independently assess this selection, inspect the
attached source, and obtain fresh live fingerprints. This file asserts no
critique result.