action_axioms

std.algebra.action_axioms · Level L3

The axioms of a finite group action, A[g, x] = g·x for a group with Cayley table T acting on the points 0 … k − 1, each checked over the whole table, as 1 or 0: every g·x is a point; e·x = x for every x; and (gh)·x = g·(h·x) for every g, h and x. A valid action need not be faithful, free or transitive.

e·x = x; (gh)·x = g·(h·x)

Signature

action_axioms(T: i64[n, n], A: i64[n, k]) → f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference every entry, the identity's row, and every g, h, x in exact rational arithmetic, on all 40 test cases.
All 120 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

group_axioms

Called by

—

sha256:c2e5e6f55cd51a8d61ec2222c240b975af4fe14259fe5ece79c8e1ae38940171

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.action_axioms · Ω:action_axioms
Pins: sha256:7e5f39c4995c476c599ceb0a2dc2f5ccd977214fadba71c6457212509f93f130this graph and the 1 it reaches through calls
Evidence: sha256:30c3ab2b363e8f3ae40284b67866d2b0fdea5aefa4f3fb55f5321eeb260eb38fthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.