Deep Dives Exact Arithmetic

Exact arithmetic

Status: IN PLACE (descriptive layer). Where exactness lives, what it buys, and the line it must not cross. Contracts: the repository's docs/statistics/numeric-contracts.md; code: src/analysis/numeric_policy.jl, src/analysis/exact_summaries.jl.

The three kinds of number

Most statistics are not exact, and no policy makes them so. What a policy can do is stop three different claims from wearing one representation:

Kind	What it means	May be called a fact?
exact	an integer count, or a rational built from counts; no rounding in its history	yes — as a count or proportion
approximate	any floating value, including arbitrary precision	no — it is a number
rounded	an approximation cut to N digits for display	no — it is a rendering

The sentence the whole layer turns on: higher precision is not exactness. BigFloat at 4096 bits rounds; it merely rounds further away. The only exact arithmetic here is integer and rational.

The policy as a type

numeric_policy(; mode, precision_bits, max_denominator_bits, round_digits) builds an immutable NumericPolicySpec, validated at construction:

Field	Modes / limits	Default
mode	:ordinary, :exact_counts, :high_precision	:ordinary
precision_bits	2 … 1 000 000	53 (Float64's significand)
max_denominator_bits	≥ 32	4096
round_digits	≥ 0	6

:ordinary is Float64 throughout and is the only mode under which previously saved analyses are unchanged (the backward-compatibility line). The load-bearing design: nothing switches mode on its own. A caller that needs :exact_counts and is handed :ordinary fails through assert_mode — it does not receive a Float64 that looks like the exact answer. That is the "no coercion Approximate → Exact" rule from Type Theory Meets Statistics made executable.

What exactness buys in practice

exact_summaries.jl (catalogue item 1) computes counts and proportions at exact precision:

• Counts as integers without a ceiling. Read counts beyond 2⁵³−1 — where every float silently loses integers — stay exact. (This was not hypothetical: the boundary audit in issue #52 measured where Float64 stops carrying consecutive integers.)
• Proportions as rationals of counts. 2/3 stays 2/3. A proportion supplied as text ("2/3", "4/6") is exact; one supplied as a float is accepted only as an approximation and labelled as one wherever shown.
• Refusals as facts. A float claiming exactness is refused; a negative count is refused; a total-zero proportion is refused with a name, not rendered as 0.

Independent reference: the suite compares value-by-value against Python's fractions.Fraction (skipping loudly by name if python3 is absent) and plants hand-derived known answers.

Two defects the conditions document caught

Worth recording because they show why conditions precede implementation:

1. to_display printed a rendering labelled 6dp — and then printed eighty digits after the point. That is the rounded/exact boundary leaking; the document ruled, the code changed.
2. Exact rationals rendered as Julia's 2//3 — implementation syntax leaking into a human's display. Again: document right, file wrong.

The line exactness must not cross

No inference is exact just because its inputs are. The descriptive layer claims facts; a p-value, an interval, or an MLE is approximate by construction (a tail mass or an optimiser output), and the policy does not pretend otherwise. What does not exist yet, and is easy to overread into this layer:

• Exact statistical tests (Fisher's exact, exact NB, permutation PERMANOVA) — issue #3, COMING. "Exact tests" there will mean the tail mass is enumerated/permuted rather than asymptotically approximated — a claim about the reference distribution, still living in the approximate/reporting column for display purposes. The naming collision is deliberate and worth meditating on: "exact test" ≠ "exact number".
• Small-n validity: today's honest default for tiny samples is the exact descriptive summary plus "no valid inferential test computed" — the catalogue endorses exactly this default.

Why rationals and not decimals

Decimals (fixed-point, arbitrary or not) are exact only for dyadic-friendly fractions and become a new rounding story otherwise; rationals of integers are the free field over ℤ and carry no rounding history at all. The denominator budget (max_denominator_bits) exists so that accumulated products cannot grow unbounded — at the budget the policy refuses or degrades explicitly, never silently simplifies.