Speculative Numeral Naming

Design: naming the numeral systems

Speculative design. Open, not settled — thinking and proposals, not shipped behavior. See Roadmap for what's in flight.

Moved whole from design/numeral-naming.md.

Status: decided — option A (2026-09-25); the renames are being executed. An audit of the numeral-system heads against prior art, and a proposal. The renames are not made yet — §4 is the fork.

1. The problem

The heads grew one at a time and the suffixes disagree. Some are …Radix, some are …System, some are a bare proper noun:

MixedRadix, BalancedRadix, NegativeRadix, BijectiveRadix, PrimorialRadix, Factoradic, Zeckendorf, Ostrowski, CombinatorialSystem, ResidueSystem.

Three suffix conventions for one concept, and no way to tell from a name whether something is a system, a base, or a function. It also reads badly in the one place it matters — the base slot of IntegerDigits, where every one of these appears as a value.

Related: the guide's heading counted them ("Ten systems in one slot"), which is a number guaranteed to go stale. Now "One slot, many systems".

2. Prior art

What each system is actually called in the literature and in other systems. This is the column that should drive the naming, and it is also what the oracle mappings will need.

Ours	Standard name	Elsewhere
MixedRadix	mixed radix	Wolfram MixedRadix — exact match
Factoradic	factorial number system	the digits are the Lehmer code
PrimorialRadix	primorial number system	—
BalancedRadix	balanced base / balanced ternary	—
NegativeRadix	negative base (negabinary…)	—
BijectiveRadix	bijective numeration	spreadsheet columns are bijective base 26
Zeckendorf	Zeckendorf representation	the α = φ case of Ostrowski
Ostrowski	Ostrowski numeration	—
CombinatorialSystem	combinatorial number system	"combinadics"; colex k-subset unranking
ResidueSystem	residue number system (RNS)	—

Two things fall out. "Radix" is wrong for four of them — a factorial, primorial or Fibonacci system has no radix, it has a sequence of place values — and the literature says "number system" far more often than "radix". And Factoradic is a nickname; the formal name is the factorial number system.

3. What is missing from the set

• PositionalNumerals — there is no head for "ordinary base b" as a system value, only compute-engine's integer base slot. Giving the family a name would let the base slot be uniformly a system rather than sometimes an integer.
• Hensel / p-adic place values — done as AdicNumerals(b, prec?) in the slot and AdicNumeral(b, x, prec?) as the value, with arithmetic, valuation and Hensel lifting. Composite b included. Named with the …Numerals suffix of option A below, so it is the first head on that convention; the rest still wait on the decision.
• BaseForm. Wolfram's BaseForm[n, b] is a display wrapper, not a digit extractor. We have no display side at all; IntegerDigits returns the digits and nothing renders them back as a numeral string. That is a real gap, and it is the natural home for the alphabet question (BijectiveRadix(26) → "AAA" rather than [1,1,1]).

4. The fork

Option A — suffix everything Numerals. FactorialNumerals, BalancedNumerals, BijectiveNumerals, ZeckendorfNumerals, OstrowskiNumerals, ResidueNumerals, CombinatorialNumerals, MixedRadixNumerals, PositionalNumerals.

• Uniform, and reads correctly in the base slot: IntegerDigits(n, ZeckendorfNumerals).
• Loses the Wolfram match on MixedRadix.
• Ten renames, all of them public surface.

Option B — suffix NumberSystem, following the literature. FactorialNumberSystem, ResidueNumberSystem, … Most faithful to prior art, and the longest.

Option C — keep Radix where there is a radix, System where there is not. MixedRadix and BalancedRadix keep their names; Factoradic → FactorialSystem, PrimorialRadix → PrimorialSystem, BijectiveRadix → BijectiveSystem. Smallest change that removes the actual error, but leaves two conventions standing.

Recommendation: A, with the old names kept as aliases. It is the only option that makes the base slot read uniformly, which is the place these names are actually used; "numerals" is accurate for all of them in a way "radix" is not for four; and the MixedRadix Wolfram match is preserved by the alias rather than lost. Aliases also mean the rename is not a breaking change, which is what makes ten of them tolerable.

Whatever is chosen, the aliases should be data — one table mapping alias → canonical — so the reference pages, the oracle mappings and the parser all read the same source.

5. Settled, and done

• Zero in a bijective base is the empty numeral. It used to decline. Bijective numeration is a bijection from the NON-NEGATIVE integers onto finite strings over ${1..k}$, and the empty string is the one left for zero — so declining left the round trip undefined at the only interesting point. Fixed, with the round trip now pinned from 0.

6. A related question: siblings, and namespace size

Raised while auditing the collection heads, and it generalises past numerals.

Tuples and Words are order-isomorphic, and it is tempting to keep one. But a name for a family carries more than its elements: it implies a representation, an ordering, and how much structure is being claimed. Those are real distinctions and enumeratio already models them — sibling collections sharing stats and maps through an order-isomorphism, one carrier and one canonical order each. Collapsing them loses the distinction; keeping both costs almost nothing but namespace.

Which suggests the Wolfram answer: let the namespace be large, and fix the discoverability with tooling rather than by deleting names. Default the reference index to a curated set, let search reach everything, and make the sibling relation itself queryable so Tuples can say what it is isomorphic to and how. The cost is a table, not code.

The open question is whether every collection wants a HEAD, or whether a family is better addressed as a value — Collection("weak_compositions") — with heads reserved for the ones that earn notation. 41 heads is comfortable; the ~270 in enumeratio is not obviously so, and the answer probably differs for collections (many, mechanical) versus stats and maps (fewer, each with real notation).