Period-Ten Fixed-Point Decomposition

Abstract

The 123 period-ten fixed-point equations decompose into eight bounded blocks.

The legal three-step prefixes LLL, LLR, LRT, RTL, RTR, TLL, TLR, and TRT partition the symbolic equations. Each block is compared separately with the inherited and primitive orbit phases.

Theorem 1.1 (The eight fixed-point blocks are bounded).

$$\mathrm{periodTenFixedBlockLengths}\left(\right)=\mathrm{list}\left(21,13,21,21,13,13,8,13\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationTenFixed.golden_fixed_point_block_counts_ten (✓ std3). ∎

Source. Repository-derived.

Commentary.

The block sizes are 21, 13, 21, 21, 13, 13, 8, and 13; no arithmetic comparison expands more than twenty-one equations at once.

Theorem 1.2 (Every period-ten fixed point is an enumerated orbit phase).

$$\mathrm{toFinset}\left(\mathrm{goldenFixedPointCodes}\left(10\right)\right)=goldenExpectedPointCodesTen$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationTenFixed.golden_fixed_point_codes_ten_decompose (✓ std3). ∎

Source. Repository-derived.

Commentary.

After the eight exact comparisons are recombined, the fixed-point codes are exactly the inherited divisor-period phases and the eleven new ten-cycles.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenFixed.golden_fixed_point_block_counts_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenFixed.golden_fixed_point_codes_ten_decompose
• Dependency: D5/S0/Tower/GoldenPeriodic/EnumerationTenData