Study 39 Actuarial Domain

Study 39 — The actuarial domain, exact against float

Study 38 measured property-and-casualty reserving. This covers the rest of the domain — life, pensions, multi-state health, and aggregation at the discrete cuts a capital regime is written on — on two public archives whose every published value is fixed-decimal, and therefore an exact rational before any arithmetic runs. Across 10,878 measured quantities the exact and floating-point answers agree to at least fourteen significant digits, in every arm. The reporting units the rules are written on sit eleven digits coarser than that.

Note

CORRECTED 2026-09-08, the same day it was sealed — the first version of this page reported a tightest margin of eight significant digits and read something into it. That was our instrument, not the arithmetic. The first comparator rounded both values at each digit and asked whether the rounded results matched. An exact value whose decimal expansion terminates in a 5 sits exactly on a rounding tie, so the exact arm rounds half-away up while the float sits infinitesimally below and rounds down — and the comparator called that a disagreement. It reported a parting at the 8th digit for two values agreeing to the 16th. The tie is a property of the decimal expansion, not of the arithmetic. The comparator now measures the exact relative difference — a Double is a binary rational exactly, from its own bit pattern, so |exact − float| / |exact| is a ratio of two integers and nothing is rounded at all. The corrected reading is 14 significant digits, uniform across all three arms, and the paragraph that read meaning into the shortest product is retired below. A 21st control arm now pins that specific tie case at 15 digits or more, so the defect cannot return unnoticed.

Status: RESULTS SEALED 2026-09-08 · marker STUDY39_ACTUARIAL_DOMAIN_EXACT_VS_FLOAT — two public archives, digest-pinned, no authentication; 21 published life tables (16,128 values, zero cells the archive does not serve) and 295 monthly statutory rate triples; 21 control arms passing in both directions; every figure printed by reproduce/actuarial-domain-exact-vs-float.swift. The subject under grading is the arithmetic and the instrument. No person, insurer, pension scheme or country is assessed, and nothing here is advice.

The exact form of the domain

Actuarial inputs are published as decimal strings with a fixed number of places. A fixed-decimal string is an exact rational: 0.00023 is 23/100000 and 4.42 percent is 442/10000. Nothing about these values is approximate until someone chooses to approximate them.

Every published input in this domain is an exact rational, and every quantity built from them by the standard formulae — annuity, assurance, net premium, deferred pension value, n-step transition — is a ratio of two integers.

The programs below never parse a published decimal into a binary float in the exact arm. Values are read as digit strings and carried as integers over a stated power of ten, and the present values are built over a common denominator so that no rational addition is ever needed and nothing is reduced or rounded until the reporting unit.

The archives

Both measured SERVED 2026-09-08, HTTP 200, no key. Provenance and the two reading traps.

• Eurostat demo_mlifetable — 21 life tables (DE, FR, IT, ES, PL, NL, SE × 2019, 2021, 2022), eight published indicators over the single-year age ladder. 16,128 values, three distinct published precisions, and zero cells the archive does not serve.
• IRS minimum-present-value segment rates, IRC §417(e)(3)(D) — 295 monthly triples, each rate to two decimal places. The most recent published month serves 442, 547, 631 basis points, exactly.

Two things in the life-table archive are not what they look like, and both were measured rather than assumed. The age dimension carries 97 categories of which the single-year ladder is 96: Y_GE85 is an aggregate published beside the ladder, and reading it as an age makes it look like a missing cell that deletes every younger age depending on it. And the terminal open interval carries q = 1.0 exactly, so any chain running through it terminates with probability exactly zero — true, and a measurement of nothing. The ladder is therefore built by name and the chains end at the last closed age.

1. The published tables against their own integer identities

A life table asserts arithmetic about itself. Each identity is checked exactly, in the published units, with no tolerance; a near-miss is counted separately from a pass, because they are different answers.

identity	holds exactly	does not	of which off by exactly one published unit
q(x) + p(x) = 1	2,016	0	—
l(x+1) = l(x) − d(x)	1,510	485	485
T(x) = T(x+1) + L(x)	1,469	526	526

In 3,990 identity tests there is no residual larger than one unit in the last published place.

The complementary probabilities close exactly, everywhere. The two accumulation identities close to the limit of the precision the publisher chose to print, and no further — which is what a rounded table is, stated as a measurement rather than assumed.

(For contrast, and it is a contrast rather than a comparison: the Schedule P filings in Study 38 carry 2,390 cells where a cumulative total falls, by arbitrary amounts. Two archives, two different shapes, both measured the same way.)

2. Life assurance — the net premium

Whole-life assurance over a whole-life annuity-due, per 100,000 sum assured, at each of the three published segment rates. The denominators cancel exactly, so the exact premium is a ratio of two integers and nothing is rounded until the reporting unit.

premiums computed	6,048 (21 tables × 3 rates × 96 ages)
deepest term reached	95 discount multiplications
verdicts differing at the reporting unit	0
fewest agreeing significant digits	14

The distribution of leading significant digits on which the two arithmetics agree: 14 in 1,158 cases, 15 in 3,845, 16 in 922, 17 in 111, 18 in 11, and 19 in one.

This is the depth question Study 38 could not reach. That study measured nine multiplications and said so; this one measures ninety-five, and the reading holds.

3. Pensions — the deferred annuity

The lump sum per 1,000 of annual pension deferred to age 65 and paid for life — the shape the minimum-present-value rule is written on — at the same three published rates.

entry ages scored	2,835
deepest term reached	75 discount multiplications
verdicts differing at the reporting unit	0
fewest agreeing significant digits	14 — 1,028 cases at 14, 1,729 at 15, 71 at 16

4. Multi-state — the n-step survival chain

A life table is a two-state Markov chain and its n-step transition is an exact product of published rationals, ending at the last closed age.

chains scored	1,995
longest chain	95 steps
verdicts differing at parts per billion	0
fewest agreeing significant digits	14 — IT 2022, from age 9, 86 steps

Sixteen chains agree to 14 digits, 1,495 to 15, 442 to 16, and the rest higher. The fewest-agreeing case is a long product — 86 steps — which is the direction accumulated rounding actually runs.

The first version of this page reported this arm's tightest margin as 8 digits in a two-step chain and built a paragraph on the reversal. That number was an artifact of the comparator, as the note at the top of this page records; the corrected measurement removes both the number and the reading.

A three-state morbidity chain is not measured here and is not claimed. The transition rates it needs are not served by an open archive: the SOA MORT tables sit behind a postback application that returns HTML to a direct request, and the Human Mortality Database returns a login page. Both measured 2026-09-08. That is ABSENT — a fact about the archives, not a result about morbidity.

5. Aggregation — does the order of the sum change the total

A capital figure is a sum over many positions, and floating-point addition is not associative. The same law, the same data, three orderings, on the 779 filed integers of Study 38's corpus.

values summed	779 filed integers
the exact total, one value	22,973,085
distinct float totals across three orderings	1
spread between orderings	0
largest single value	11,561,327
the exactly-representable ceiling of a Double	9,007,199,254,740,992

Every value and every running total sit far below that ceiling, so each addition is exact and the ordering cannot matter. That is a statement about magnitude, not about floating point being associative — and the control arm below runs the same code past the ceiling and watches the two orderings part.

6. What this means at a discrete cut

Capital and solvency regimes are written on integer cuts — the US risk-based-capital action levels are published as 200, 150, 100 and 70 percent, and a verdict is which side of a cut a figure falls. The measurement above answers that question in the form it is actually asked:

A verdict rendered at a cut expressed to fewer than fourteen significant digits cannot flip between the exact and the floating-point form of these quantities, anywhere in the 10,878 measured here. The cuts are written to three.

That is the transferable statement, and it is a bound rather than a reassurance: it holds for the quantities and depths measured on this page and transfers to another depth only by being measured there.

Controls

Twenty-one arms, run every time, in both directions — the harness fails if any fails.

• Exact machinery on answers known in advance. 10^18 / 7 = 142857142857142857 remainder 1; 10^19 has twenty characters; half-away rounding on 3/2 and 1/2.
• The fixed-decimal reader keeps digits and refuses what is not a decimal. 0.00023 → 23 at scale 5; 4.42 → 442 at scale 2; a bare integer at scale 0; abc, 1.2.3 and the empty string refused; scaled() pads and never truncates.
• The age ladder excludes an aggregate rather than counting it absent — contiguous from 0, one terminal open interval, 96 steps from the 97 categories served.
• Always-green. A table nobody dies in has an assurance numerator of exactly zero and an annuity numerator that is not zero.
• Always-red, four ways. Agreement is finite and not identical on inputs past 2⁵³; exact 1/3 against Double(1/3) agrees to 15–17 digits and no more; exact 1/3 against 0.34 agrees to at most one significant digit; and 201 values summed both ways differ once past 2⁵³.
• Always-green on the same instrument. Exact 1/4 against 0.25 is the identical rational — so the detector is not simply reporting a difference every time.
• The tie that broke the first comparator is pinned. The exact value 6360389250/10^10 against its floating-point product must agree to 15 digits or more. Under the first instrument this case read 8.
• Zero is a distinct answer. A value of exactly zero returns the not-measurable sentinel, never the "they agree everywhere" code. Without this arm the survival chains reported perfect agreement for the reason that both sides had rounded to nothing.
• The identity arms are testing something — they ran on every published age, and they do not all return the same census.

control arms run    = 21
control arms failed = 0
SELFTEST PASS

What this page measures, and what it does not

• Measured: every published input in this domain is an exact rational; the published tables satisfy their own identities with no residual above one unit in the last printed place; across 10,878 quantities at depths to 95 multiplications, exact and float agree to at least 14 significant digits in every arm; and a sum of filed integers below 2⁵³ is order-independent because each addition is exact.
• Not measured, and therefore not claimed: a three-state morbidity chain, for want of an open archive. Recorded ABSENT with both archives named and their responses dated.
• Not measured: stochastic capital modelling. A Monte Carlo capital figure's reproducibility is a property of a particular firm's model, and constructing one here would be measuring our own construction rather than the field's.
• Two archives, seven countries, three years, one statutory rate series. A different reporting basis is a different measurement.
• The margin is uniform at 14 digits across all three arms, and the fewest-agreeing case in each is a deep product rather than a shallow one. The first version of this page said otherwise on both counts; the correction is recorded at the top rather than swapped in silently.

Reproduce

No account, no key, no network at run time. Both corpora are pinned in this repository.

cd corpus/actuarial && shasum -a 256 -c SHA256SUMS
xcrun swiftc -O -swift-version 5 ../../reproduce/actuarial-domain-exact-vs-float.swift -o /tmp/run && /tmp/run

bash reproduce/validate.sh runs it inside the full harness, which verifies the digests, runs the program, and checks that every figure quoted on this page appears in its output.

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Test claim, not efficacy. Nothing on this page is a statement about any insurer's solvency, any pension scheme's funding, any country's mortality experience, or compliance with any requirement. It is a statement about arithmetic performed on published tables.