CentralDigitWords

Abstract

Direction-major and scale-minor digits build literal positive words.

This module defines the base-2^r digit arrays, literal powered blocks, complete direction-major and scale-minor words, their all-zero reference word, and the fixed base-length coefficient. These are repository constructions built from the selected actual directions recorded by the preceding owner.

Definition 1.1 (Degree-dependent digit base).

$$\forall r,\mathrm{digitBase}\left(r\right)=\mathrm{pow}\left(2,r\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitBase (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

digitBase r is 2^r.

Definition 1.2 (Complete direction-scale digit arrays).

$$\forall A,r,t,\mathrm{DigitArray}\left(A,r,t\right)=\mathrm{functions}\left(\mathrm{Fin}\left(\mathrm{actualLyndonCount}\left(A,r\right)\right),\mathrm{Fin}\left(t\right),\mathrm{Fin}\left(\mathrm{digitBase}\left(r\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.DigitArray (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

DigitArray A r t is the function space assigning a base-2^r digit to every selected actual direction and every scale below t.

Definition 1.3 (A direction’s encoded natural).

$$\forall r,t,digits,direction,\mathrm{digitValue}\left(r,t,digits,direction\right)=\mathrm{NatOfDigits}\left(\mathrm{digitBase}\left(r\right),\mathrm{ofFn}\left(\mathrm{Fin}\left(t\right),\mathrm{lambda}\left(scale,\mathrm{nat}\left(\mathrm{apply}\left(digits,direction,scale\right)\right)\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitValue (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

digitValue reads the t digits of one direction in base digitBase r using Nat.ofDigits.

Definition 1.4 (An actual positive-word digit block).

$$\forall r,direction,scale,digit,\mathrm{digitBlock}\left(r,direction,scale,digit\right)=\mathrm{append}\left(\mathrm{repeatedWord}\left(digit,\mathrm{literalPowerWord}\left(\mathrm{pow}\left(2,scale\right),\mathrm{left}\left(\mathrm{positivePairWords}\left(r,\mathrm{selectedDirection}\left(r,direction\right)\right)\right)\right)\right),\mathrm{repeatedWord}\left(\mathrm{digitBase}\left(r\right)-1-digit,\mathrm{literalPowerWord}\left(\mathrm{pow}\left(2,scale\right),\mathrm{right}\left(\mathrm{positivePairWords}\left(r,\mathrm{selectedDirection}\left(r,direction\right)\right)\right)\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitBlock (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

For a selected pair (u,v), scale s, and digit j, digitBlock concatenates j copies of the 2^s literal power of u and digitBase r-1-j copies of the matching power of v.

Definition 1.5 (The complete multi-scale positive word).

$$\forall r,t,digits,\mathrm{multiScaleWord}\left(r,t,digits\right)=\mathrm{flatten}\left(\mathrm{ofFn}\left(\mathrm{Fin}\left(\mathrm{actualLyndonCount}\left(A,r\right)\right),\mathrm{lambda}\left(direction,\mathrm{flatten}\left(\mathrm{ofFn}\left(\mathrm{Fin}\left(t\right),\mathrm{lambda}\left(scale,\mathrm{digitBlock}\left(r,direction,scale,\mathrm{apply}\left(digits,direction,scale\right)\right)\right)\right)\right)\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.multiScaleWord (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

multiScaleWord concatenates every digitBlock in direction-major, scale-minor order; at t=0 the result is the empty word.

Definition 1.6 (The zero-digit reference word).

$$\forall r,t,\mathrm{referenceWord}\left(r,t\right)=\mathrm{multiScaleWord}\left(r,t,\mathrm{lambda}\left(direction,\mathrm{lambda}\left(scale,0\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.referenceWord (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

referenceWord A r t is multiScaleWord at the all-zero digit array, hence uses the powered v-side in every block.

Definition 1.7 (Common-length coefficient).

$$\forall A,r,\mathrm{baseLength}\left(A,r\right)=\left(\mathrm{digitBase}\left(r\right)-1\right)\cdot \mathrm{sum}\left(\mathrm{Fin}\left(\mathrm{actualLyndonCount}\left(A,r\right)\right),\mathrm{lambda}\left(direction,\mathrm{length}\left(\mathrm{left}\left(\mathrm{positivePairWords}\left(r,\mathrm{selectedDirection}\left(r,direction\right)\right)\right)\right)\right)\right)$$

Formalization. D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.baseLength (✓ std3).

Source. Repository-derived.

Acknowledgement. Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Commentary.

baseLength A r is (2^r-1) times the sum of the selected u-word lengths and is the fixed coefficient in the multi-scale length law.

References

• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.DigitArray
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.baseLength
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitBase
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitBlock
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.digitValue
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.multiScaleWord
• Truth anchor: D5/S1/Words/Complexity/PositivePairs/Digits/CentralDigitWords.referenceWord
• Dependency: D5/S1/Words/Complexity/PositivePairs/Digits/ActualLyndonDirections
• Dependency: D5/S1/Words/Complexity/PositivePairs/Span/LiteralPowerSubstitution