Inference Counter – Decoder

Inference Counter Manual

1. Overview

The Inference Counter is a multi-format number system consisting of three parallel representations: Laegna, Waveweaver, and Decimal. Each representation is independent and never normalized across types. Digits define everything: Num, Digits, Hex, Bin, ordering, and classification.

2. Digits and Letters

Laegna digits:

0 → I
1 → O
2 → A
3 → E

Wave digits: identical digits, but Waveweaver uses a different ordering function. Digits themselves are never mutated.

Binary digits:

O = binary digit 1
A = binary digit 2

3. Binary Mapping (Digitwise)

Binary is the simplest part of the system. It is always digitwise, never calculated, never normalized, and never influenced by Waveweaver logic.

Digit  Letter  Binary letters  Binary digitlist
0      I       OO              [1,1]
1      O       OA              [1,2]
2      A       AO              [2,1]
3      E       AA              [2,2]

Examples:

Digits [1,1] → O,O → OA OA → "OAOA"
Digits [1,2] → O,A → OA AO → "OAAO"
Digits [3,0] → E,I → AA OO → "AAOO"

4. Hex Mapping

Hex is computed from digit pairs using a 4×4 table. Hex is defined only when the number of digits is even. Hex is per-type: Laegna Hex uses Laegna digits, Wave Hex uses Wave digits, Decimal Hex uses Decimal integers.

5. Extreme Values

W = −∞
V = −0
U = +0
∩ = +∞

Binary for extremes:

W → WW
V → VV
U → UU
∩ → ∩∩

Hex for extremes: if R is even, the extreme string is halved; otherwise Hex = null.

6. Decimal Representation

Decimal has Num, Signed, Hex, and Bin. Decimal binary is standard base‑2 and only exists for discrete integers. Decimal does not influence Laegna or Wave.

7. Ordering and Indexing

Laegna ordering: lexicographic in base‑4 (I < O < A < E).

Waveweaver ordering: uses WaveweaverValueBase4 to compute index, but digits remain unchanged.

Decimal ordering: symmetric around zero:

−∞, negative numbers, −0, +0, positive numbers, +∞

8. JSON Export Structure

{
  "Decimal": { Num, Signed, Hex, Bin },
  "Wave":    { Num, Digits, Hex, Bin },
  "Laegna":  { Num, Digits, Hex, Bin }
}

Digits are never shared across types. Num is derived from Digits. Hex and Bin are derived from Digits. Decimal has its own Hex/Bin rules. Wave and Laegna use the same formulas but different Digits.

9. R-Level Structure

{
  "meta": { ... },
  "chapters": [
    { "R": 1, "base": 4, "numbers": [...] },
    { "R": 2, ... }
  ]
}

10. Algorithms (High-Level)

Digit → Num:

Num = join(LET[d] for d in Digits)

Digit → Binary:

pair = BIN_TABLE[d]
binary = (O if pair[0]==1 else A) + (O if pair[1]==1 else A)

Digit → Hex:

hex = HEX_TABLE[d0][d1]

Wave ordering:

index = WaveweaverValueBase4(Digits)

Laegna ordering:

index = LaegnaValueBase4(Digits)

11. Guarantees

• No digit mutation
• No normalization
• No reinterpretation
• No cross-type contamination
• Binary is always digitwise
• Hex is always pairwise
• Extremes are always symmetric
• Decimal is always independent

12. Manual Completed

This manual describes the structure, rules, and algorithms of the Inference Counter system. It is now ready for use in the documentation page.