group_axioms

std.algebra.group_axioms · Level L3

The group axioms of a finite Cayley table T (T[a, b] = a·b on the elements 0 … n − 1), each checked over the whole table, as 1 or 0: closure (every entry is an element), associativity over all n³ triples, a two-sided identity, and a two-sided inverse for every element. Also the identity e, the first that works (0 when there is none). Associativity checked on some triples only would not be a proof.

(ab)c = a(bc); ea = ae = a; ∀a ∃b: ab = ba = e

Signature

group_axioms(T: i64[n, n]) → f64[], f64[], 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, every triple, every candidate identity and inverse, in Python in exact rational arithmetic, on all 40 test cases.
All 200 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

—

Called by

action_axioms element_orders group_inverses hom_kernel_image is_group is_ideal is_subgroup ring_axioms ring_poly_mul

sha256:1c794bf484208afe9ae9c1856cfc0fb069adc1169ca23d0d1fd2c92818a62191

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.group_axioms · Ω:group_axioms
Pins: sha256:d8ef60086181278bda182d7af06fc01a2c97068af48f73868e69dd2e6351c6efthis graph alone
Evidence: sha256:da7ce1eb7477ff037ec9052514c825d8ed426b99166a6763dbd84356e229b284the 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.