The Leedskalnin Equation: Revised and Extended Through the Unified Prime Lattice

Griff gurwell

doi:10.5281/zenodo.20454719

Published May 30, 2026 | Version v6

Preprint Open

The Leedskalnin Equation: Revised and Extended Through the Unified Prime Lattice

Griff gurwell

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Overview

This is a revised and extended edition of the original Leedskalnin Equation paper (Zenodo, March 2026). The original work established four independent derivations of the CTF base frequency $f_{0} = 53 e = 10373/72 = 144.06944 \dots$ Hz, the prime‑swapping control test identifying prime 53 as unique, the micro‑gap $δ = f_{0} - 53 e \approx 0.0005075$ Hz, and a 12‑emitter dodecahedral resonance simulation with watch logic and burst envelope. Those results remain unchanged and are not retracted.

The new contribution of this revision is the full integration of those results into the unified Prime Lattice Coherence Theorem (PLCT) – a mathematical framework built on the prime lattice, the Lock‑Out Theorem, and the Partition Theorem. The lattice was developed independently after the original paper and is now applied retroactively to give every number in the original work an axiom‑level home. No numbers change; two results are promoted from observations to theorems; several new structural arithmetic facts are added.

Key New Results (Not in Original)

Micro‑gap as a theorem, not an observation
The Lock‑Out Theorem proves that $f_{0} = 10373/72$ (denominator $72 = 2^{3} \times 3^{2}$ ) is Tier‑1 (primes ${2, 3}$ ) and therefore maintains zero accumulated drift $D (x, B) = 0$ at all scales. The expression $53 e$ introduces the Tier‑4 prime 53 (outside ${2, 3, 5}$ ), which necessarily produces unbounded logarithmic drift. A Tier‑1 rational cannot equal a Tier‑4 transcendental; hence $δ > 0$ is mathematically forced. The gap is no longer merely a “physical tolerance” – it is a structural necessity of the prime lattice.
Triple lattice lock of prime 53
Prime 53 is shown to be the unique prime satisfying three independent lattice coordinates simultaneously:
- Tier‑4 (prime set ${53}$ outside ${2, 3, 5, 7}$ )
- Temporal zone ( $53 mod 9 = 8 \in {2, 5, 8}$ )
- Prime index $P_{16}$ where $16 = 2^{4}$ is exactly the exponent of prime 2 in the spatial harmonic $Λ = 144 = 2^{4} \times 3^{2}$ .
  The original prime‑swapping control test (primes 41–71) is reinterpreted as the empirical shadow of this triple lock – explaining why 53 is unique and why all other primes miss the fractional signature $1/ (P e) \approx 0.00694$ .
Inscription as PLCT tier map
- Base‑60 = $2^{2} \times 3 \times 5$ – the smallest positive integer whose prime set is exactly ${2, 3, 5}$ (Tier‑2). The Sumerian sexagesimal system is therefore arithmetic at the coherence boundary of the lattice.
- Coefficients 28:15:53:15 from the decomposition $6, 105, 195 = 28 \times 6 0^{3} + 15 \times 6 0^{2} + 53 \times 60 + 15$ map to tiers T3:T2:T4:T2 and zones Hard Wall → Spine → Temporal → Spine. This sequence traces the Lock‑Out Theorem path from the Hard Wall prime $P_{4} = 7$ through the Tier‑2 gateway to the Temporal lock prime 53.
- Prime mirror $7129$ satisfies $7129 mod 144 = 73$ , and $73$ is one of the six Partition Theorem universal lock values $L = {0, 1, 9, 64, 73, 81}$ .
- Primary inscription number $6, 105, 195$ is a Spine element: $mod 9 = 0$ (Spine zone), digital root = 9, and $mod 144 = 27 = 3^{3}$ (pure Tier‑1).
Simulation parameters as Tier‑1
The burst envelope $9$ ON / $27$ OFF cycles are $3^{2}$ and $3^{3}$ ; their sum is $36 = 2^{2} \times 3^{2}$ , and $36 \times 4 = 144 = Λ$ . The ratio $9 : 27 = 1 : 3 = P_{1} : P_{2}$ – the ratio of the two generators of the ${2, 3}$ lattice. The duty cycle $1/4 = 2^{-2}$ is pure Tier‑1.
Prime mirror as Tier‑1/Tier‑2 ratio
$2971 7129 \approx 60 144 = 2 ^{2} \times 3 \times 5 2 ^{4} \times 3 ^{2} = 5 12$ . The mirror approximates the ratio of the spatial harmonic (Tier‑1) to the smallest Tier‑2 base.

What Is New vs. What Is Unchanged

Unchanged: The four independent derivations of $f_{0}$ (recursive lock, constants survey, base‑60 decomposition, prime mirror), the prime‑swapping control test data, the 12‑emitter simulation results (mean $g \approx 0.66$ , min $g \approx 0.21$ ), the hardware specification, and the experimental protocol. The caveat that the inscription mapping is hypothesis‑generating, not proof of intentional design, is preserved.
New (this revision): The micro‑gap theorem, triple lock theorem, base‑60 tier identification, coefficient tier/zone map, lock value verification for 7129, Spine element verification for 6105195, burst envelope tier analysis, and the prime mirror tier interpretation. Also three open research directions (coefficient 28 and fine‑structure screening integer, Hard Wall–Hard Wall prime mirror structure, and the Tier‑2×Tier‑4 factorization of 6105195).

Scope and Honesty

The paper is explicit about what is proved (theorems marked as such) versus what is observed (numerical coincidences that await explanation) versus what is conjectural (the open research directions). No claim is made that the inscription was designed with knowledge of the prime lattice; the mapping shows structural consistency only. No claim of antigravity, time dilation, or real‑world load reduction is made – the simulation remains a toy model with a hypothetical Heaviside coupling.

Reproducibility

All numerical results are verified with a Python script (included in the Appendix) that uses only standard libraries (math, fractions). The script computes the micro‑gap, verifies the triple lock, checks prime sets, computes residues mod 9 and mod 144, and confirms the burst envelope arithmetic. Runtime < 5 seconds.

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