Now in English, French and German, plus it's derrivation from first principles with zero parameters makes for a compelling case Updated the paper, now has numerical confirmation of a photonic prediction. Domain VII is no longer predictions-only — it has data, in both band edge and log transmission ratio, prediction 1 until deposited, the empirical analyses they contain cannot be independently verified. Readers are encouraged to check the mathematics against the data sources cited (NASA Exoplanet Archive, Planck 2018). The Jacobsthal characteristic polynomial r 2 − r − 2 = 0 is derived from Hamiltonian mechanics: it is identically the xed-point equation of the symplectic trace map Tn+1 = T 2 n −2, which follows from the CayleyHamilton theorem for any M ∈ SL(2, R) with det(M) = 1. The period-2 orbits of the same map yield the Fibonacci polynomial r 2+r−1 = 0. Together, the combined quartic (T 2 − T − 2)(T 2 + T − 1) = 0 produces four rootsλ = 2, λ− = −1, 1/φ, −φfrom which every quantity in this paper descends. In the circular restricted three-body problem, the Poincaré return map has trace Tr(M) = 2 cosh(σphys) with σphys ∝ µ 1/2 ; the eigenvalue λ = 2 is the exact zero-mass limit. The Diamond Lattice of Jacobsthal pairs lying on y = λx + det(M) = 2x + 1encodes this structure geometrically. The FSRE and CANLE tools used throughout are proved to be the forward and inverse iterations of the same Möbius map, with the thread τ = z+ − z− = λ − 1/λ = 3/2 as their difference and det(M) = z+ · z− = 1 as their product. The TitiusBode factor (λ = 2) and the Kepler exponent (τ = 3/2) are not independent empirical laws: one is the eigenvalue, the other is its thread, and their equality holds only for Newtonian gravity (α = −1). These eigenvalues generate a thread τ = λ − 1/λ = 3/2selected both algebraically (the eigenvalue-reciprocal gap) and physically (the unique nontrivial Fibonacci convergent resolvable at observed coherence Q ∼ 1030)and are bounded from below by φ = (1+√ 5)/2 (the KAM stability ceiling). All empirical results are dimensionless. In seven domains... (I) pure algebra, (II) the solar system and EarthMoon system, (III) KAM dynamics, (IV) TRAPPIST-1, (V) the exoplanet population, (VI) Kerr black holes, (VII) photonic quasicrystals ( All outlandish claims, but bear with me here and enjoy the ride) ...the same eigenvalue emerges. The exoplanet window mean ⟨R⟩window = 71/35 is validated by an oset-specic test (p = 0.0015), CANLE cluster bootstrap (p = 0.67 vs 71/35, N = 56), and REBOUND N-body simulations clustering at 43/21. In Domain VI, the Möbius xed-point structure at the Jacobsthal trace k = 5/2 selects the Kerr spin aK/M = ρ = 1/λ. The Smarr decomposition at this spin gives angular fraction f+ = (2 − √ 3)/2, and the two-horizon angular sum f+ + f− = 2 = λ is proved for all spins. The product Ωdm = f+ × λ = 2 − √ 3 ≈ 0.268 lies within 0.3σ of the Planck 2018 measurement. Every factor in this product is derived from Hamiltonian mechanics; the physical identication with dark matter density remains a conjecture. A Monte Carlo null test gives p < 3.1 × 10−4 . Falsiable predictions include a hard bound of n ∗ = 7 on compact resonant chain length. A connecting thread runs through Narayana's cows sequence (c. 700 AD): its consecutive ratios walk λ → τ → 4/3 → τ → τ before converging to the supergolden ratio, and its inverted polynomial isolates λ = 2 as the unique nontrivial root when paired with the Jacobsthal polynomial. Note: I imagine this work will be dismissed by the title alone, the problem i am up against is both partly human and partly science, but as i am independent researcher (dog groomer) I refuse to split this paper into smaller pieces to strengthen th individual claims, the bigger picture is the important part, directly diluting the bigger picture reduces the impact and visual representation of the overall claim, so I won't be submitting this for publishing anytime soon, I have had experience with journals in my short tenure of researching and what I found is that you can write the entire mathmatical link but by the time it passes peer review it ends up being trivial, screw that for a lark, I don't want journals telling me what to do.No I just finished putting the jigsaw puzzle together I'm not taking it apart after putting the jigsaw together just because people prefer to look at it that way. That's my thoughts anyway..now keep an open mind, see the flaws but visualize what is being built and happy reading DC 4/03/2026 P.S not bad for someone who left school 40 years ago with no qualifications, maybe not great..but not bad either
David Coates (Fri,) studied this question.