KNOTT-4.3 — Bergman’s Base-Φ (‘Phigit’) Numeral System

Under the carry law Φⁿ = Φⁿ⁻¹ + Φⁿ⁻² (100_Φ = 011_Φ), every positive integer N ≥ 1 admits a unique finite minimal non-adjacent {0, 1} expansion in powers of Φ; excluding infinite 011-tails yields a unique 00-free maximal form and exactly Fₖ valid finite phigit forms for every Fibonacci number Fₖ.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-4.3
Curated Source	Dr. Ron Knott, Phigits and the Base Phi System (#canonical)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.phigits.construct_phigit_scale_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/divine

Constructive Synthesis & Exact Proof

Discovered in 1957 by fourteen-year-old George Bergman (Bergman1957) and extensively developed by Dr. Ron Knott on phigits.html (KnottPhigits), the base-Φ (“phigit”) numeral system represents every positive integer uniquely as a finite non-adjacent binary sum of powers of the irrational radix \(\Phi\).

• Carry Law & Minimal Phigit Scale: Along base-\(\Phi\) axis OP3 carrying ticks Pm2 (\(\Phi^{-2}\)), Pm1 (\(\Phi^{-1}\)), P0 (\(\Phi^0 = 1\)), P1 (\(\Phi^1\)), P2 (\(\Phi^2\)), and P3 (\(\Phi^3\)), the identity \(\Phi + \Phi^{-2} = 2\) decomposes \(2 = 10.01_\Phi\) into segments A2M2 and M2B2 (aligned to P1 via P1M2), \(3 = 100.01_\Phi\) into A3M3 and M3B3, and \(4 = 101.01_\Phi\) into A4M4 (framing OP2M4A4 via P2M4), M4N4, and N4B4.

• Maximal Phigit Form & Fibonacci Multiplicity: Applying \(100_\Phi \to 011_\Phi\) (while forbidding non-terminating \(\dots 011\) tails via \(101_\Phi \to 01111_\Phi\)) produces a unique 00-free maximal phigit expansion, and every Fibonacci integer \(F_k\) admits exactly \(F_k\) valid finite phigit representations while Édouard Lucas numbers \(L_{2m}\) achieve local minima.

Canonical Minimal, Maximal, & Multiplicity Phigit Table

Base-\(\Phi\) (‘Phigit’) Expansions for \(N = 1\dots 12\)

\(N\)	Minimal Form (11-free)	Maximal Form (00-free interior)	Valid Finite Phigit Forms
1	1	1	1
2	10.01	1.11	2
3	100.01	10.1111	3
4	101.01	11.1111	3
5	1000.1001	101.1111	5
6	1010.0001	111.0111	5
7	10000.0001	1010.101111	5
8	10001.0001	1011.101111	8
9	10010.0101	1101.101111	8
10	10100.0101	1110.111111	8
11	10101.0101	1111.111111	5
12	100000.101001	10101.111111	10

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
phigits 2 decoded	\(2\)
phigits 5 form count	\(5\)
phigits 8 form count	\(8\)
phigits 13 form count	\(13\)
phigits 21 form count	\(21\)
lucas 7 form count	\(5\)
lucas 11 form count	\(5\)