Radix & Positional Numeral Notation Registry

A canonical mathematical, computational, and governance radix registry mapping 43 distinct radices and positional notations (Base 1 through Base 360), with native transcoding and first-class support for HUMMBL Base120 (PIN-CODE-RESY coordinates).

Governed by: scripts/radix.py (stdlib-only, Python 3.11+, zero external dependencies).

Usage

[radix] list                               # List all 43 registered radices
[radix] list --category binary_to_text     # Filter by category
[radix] info base120                       # Inspect metadata, divisors, totient, and alphabet
[radix] convert 255 --from dec --to hex    # Transcode decimal 255 to hex (ff)
[radix] convert 255 --from dec --to b120   # Transcode decimal 255 to Base120 (P3.P16)
[radix] convert 11111111 --from bin --to dna # Transcode binary to DNA Base-4 (TTTT)
[radix] convert 703 --from dec --to bijective_base26 # Spreadsheet column (AAA)
[radix] b120 12345                         # Encode decimal 12345 to Base120 (SY3.SY6)
[radix] b120 SY3.SY6                       # Decode Base120 coordinates to decimal 12345
[radix] identify P1.CO5                    # Heuristically detect radix and decode value

Categories Mapped (43 Radices)

1. Standard Positional: Unary (1), Binary (2), Ternary (3), Quaternary (4), Quinary (5), Senary (6), Septenary (7), Octal (8), Nonary (9), Decimal (10), Undecimal (11), Duodecimal/Dozenal (12), Tetradecimal (14), Hexadecimal (16), Vigesimal (20), Tetravigesimal (24), Base26 (26), Trigesimal (30), Base36 (36), Sexagesimal (60), Base94 (94), Base256 (256).
2. Binary-to-Text & Data Encoding: DNA Base-4 (ACGT), RFC 4648 Base32, Base32 Hex, Crockford Base32, z-base-32, Geohash Base32, RFC 9285 Base45, Bitcoin Base58, Flickr Base58, Ripple Base58, Base62, RFC 4648 Base64, URL-Safe Base64, Adobe Ascii85, RFC 1924 IPv6 Base85, ZeroMQ Z85, basE91.
3. Bijective Numeral Systems: Bijective Base-26 (Spreadsheet column notation 1->A, 26->Z, 27->AA, 702->ZZ; zero-less positional mapping).
4. Non-Standard & Signed: Balanced Ternary ({-1, 0, +1} mapped to {T, 0, 1}, Setun computer architecture).
5. Geometric / Astronomical: Base 360 ($360 = 2^3 \cdot 3^2 \cdot 5$, 24 divisors, circular rotational degree space).
6. HUMMBL Governed Radix: Base120 ($5! = 120 = 2^3 \cdot 3 \cdot 5$, 16 divisors, full PIN-CODE-RESY coordinate mapping + 120-symbol deterministic alphabet).

The HUMMBL Base120 Coordinate Basis

Base120 functions as the mathematical radix for HUMMBL intelligence and agent reasoning. It organizes 120 discrete operator positions into 6 transformations (20 operators each):

$$\mathbf{PIN-CODE-RESY} = 6 \text{ Families} \times 20 \text{ Operators} = 120 \text{ Positions}$$

Prefix	Family	Focus	Operator Range	Radix Range
P	Perspective	Point of view, framing, assumptions	P1 – P20	0 – 19
IN	Inversion	Counterfactuals, edge cases, premortems	IN1 – IN20	20 – 39
CO	Composition	Integration, synthesis, combining parts	CO1 – CO20	40 – 59
DE	Decomposition	Analysis, orthogonal factor isolation	DE1 – DE20	60 – 79
RE	Recursion	Self-reference, loops, compounding	RE1 – RE20	80 – 99
SY	Systems	Macro constraints, emergent behavior	SY1 – SY20	100 – 119

Multi-Digit Positional Coordinates

Any integer $N \ge 0$ is uniquely expressed in positional Base120 notation as dot-delimited operator coordinates: $$N = \sum_{k=0}^{M} d_k \cdot 120^k \quad \text{where } d_k \in [0, 119]$$

• Decimal 0 $\to$ P1
• Decimal 44 $\to$ CO5
• Decimal 120 $\to$ P2.P1 ($1 \cdot 120^1 + 0 \cdot 120^0$)
• Decimal 255 $\to$ P3.P16 ($2 \cdot 120^1 + 15 \cdot 120^0$)
• Decimal 12345 $\to$ SY3.SY6 ($102 \cdot 120^1 + 105 \cdot 120^0$)

Execution Mechanics

This skill invokes scripts/radix.py via Python:

python scripts/radix.py $ARGUMENTS

All operations are stdlib-only, non-mutating (advisory), and execute in under 10ms.