tail_index

std.stats.tail_index · Level L4

The least N ≥ 0 with |u_n − L| = |c|/(n + d) < ε for every n ≥ N, for the sequences u_n = L + c/(n + d) (d ≥ 1) and ε = p/q > 0 given by positive integers, in exact integer arithmetic: N = max(0, ⌊|c|q/p⌋ − d + 1). The error never grows along the tail, so this one N works for the whole tail, which a finite sample of terms could not show; equality |c|/(n + d) = ε does not count. Over a finite domain, one c and d per point, it is a pointwise witness at each point, and their maximum is a uniform one.

N = max(0, ⌊|c|q/p⌋ − d + 1)

Signature

tail_index(c: i64[m], d: i64[m], p: i64[], q: i64[]) → i64[m], i64[]

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 integer formula, and its maximum in exact rational arithmetic, on all 40 test cases.
All 206 int64 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 int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Identity

Calls

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Called by

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sha256:b23725913b5c36044436de491fc0dfb35d4d975f502ced88c1217e1d535ab9e8

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

Control handle

Symbol: Ω:std.stats.tail_index · Ω:tail_index
Pins: sha256:2cccf8aa3994665bbd460e3dc50f1e51fc12cc790c6d375679526286492e1268this graph alone
Evidence: sha256:8f509a932b836334f1036bcc74479bd825fb661860a76567631df5bd9c5d2a05the 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.