The Fibonacci–Screening Identity: A Connection Between the Fibonacci Sequence, the Pisano Period of 144, and the Electromagnetic Screening Charge

This paper proves an exact arithmetic identity linking three independently derived constants from the Prime Lattice Coherence Framework (PLCT): the spatial harmonic $\Lambda =144$, its Pisano period $\pi (144)=24$, and the electromagnetic screening integer $\Delta =7$ (which gives $\lfloor 1/\alpha \rfloor =144-7=137$). The identity is:

∑k=124(Fk mod 144)=7×144=1008,k=1∑24(Fkmod144)=7×144=1008,

where FkFk is the $k$-th Fibonacci number. The total residue mass of the Fibonacci sequence modulo the spatial harmonic, summed over exactly one Pisano synchronisation period, equals the screening charge times the spatial harmonic.

Beyond the central identity, the paper presents two additional results:

1. Rank-of-apparition of 23=P923=P9: The prime $23$ – the second temporal prime factor of the CTF frequency identity f0=(1442+10)/144=10373/72f0=(1442+10)/144=10373/72 – first divides the Fibonacci sequence at index $24$, which is exactly the Pisano period of $144$. Consequently, F24/Λ=322=2×7×23=P1×Δ×P9F24/Λ=322=2×7×23=P1×Δ×P9.

2. Cosmological constant residual factor (Gemini, June 2026): The residual factor by which the CTF 134‑layer suppression mechanism exceeds the observed cosmological constant, $\approx 1.4583$, factorises as 2×36/1032×36/103. The exponent $6=\Lambda /\pi (\Lambda )$ appears in both the Fibonacci–Screening Identity (average residue $=6\times \Delta$) and this residual.

All three results share the common thread $\Lambda /\pi (\Lambda )=6=2\times 3$, a pure Tier‑1 number. The paper also reviews Carmichael’s theorem (1913) to prove that F12=144F12=144 is the last pure Tier‑1 Fibonacci number, and documents the unique “prime sandwich” structure (F11=89F11=89, F12=144F12=144, F13=233F13=233) where the spatial harmonic sits at the Spine (ground state) flanked by Temporal‑zone primes.

Scope and honesty:
The central identity is proved by direct computation and is exact. The rank-of-apparition results are exact arithmetic facts. The cosmological constant residual factorisation is a candidate form (accurate to 0.02%) for Open Problem V.2 of the master PLCT document; it is not a derived result. The paper does not claim that the Fibonacci–Screening Identity causes $\lfloor 1/\alpha \rfloor =137$ – causal direction is unknown. The paper is self‑contained for readers new to the CTF Framework, with a comprehensive introduction.

Reproducibility: All computations are verifiable with standard Python (code available in the CTF Framework master document, Appendix F). Verification of the identity, the Pisano period, and the rank-of-apparition values can be completed in under one minute.