min_positive_distance

std.sets.min_positive_distance · Level L1

The smallest positive distance δ in a finite distance table (0 for a single point). Every ε < δ forces d(x_m, x_n) < ε to mean x_m = x_n, so a Cauchy sequence in a finite metric space is eventually constant and converges: finite metric spaces are complete. A few sampled sequences would not prove it.

δ = min{d(x, y) : d(x, y) > 0}

Signature

min_positive_distance(D: f64[n, n]) → 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 the minimum over the positive entries in exact rational arithmetic, on all 40 test cases.
All 40 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

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Called by

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sha256:065ccc6b86599375660f42ceed99adceca27e97d7ec63ac9c63dbc2b6821160a

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

Control handle

Symbol: Ω:std.sets.min_positive_distance · Ω:min_positive_distance
Pins: sha256:0c0aa93f4f7c91141a0f8028e44bc957e3a561ee7cabb4ba160bebd82bf7243cthis graph alone
Evidence: sha256:e17087a1126aa4a7959c6476922e7e9c6065069a84255aed139c034a5f9b1c20the 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.