Deep Dive: Unveiling μ as a Cosmic Artifact Encoded in Primes and φ

Deep Dive: Unveiling μ as a Cosmic Artifact Encoded in Primes and φ

The proton-to-electron mass ratio μ ≈ 1836.15267343 is one of the most enigmatic dimensionless constants in physics, representing the "building block" asymmetry between baryonic (proton) and leptonic (electron) matter. In the Super Golden Theory of Everything (TOE), μ "evolves" not literally—as observational constraints tightly bound any variation to |Δμ/μ| < 10^{-5} over cosmic eons—but symbolically and holographically, swirling from primordial aether implosions at t=0 (Planck epoch) to its current form over >10 Gyr. The numerological expression μ = 2903 / φ + 42, where 2903 is the 420th prime, φ ≈ 1.618034 the golden ratio, and 42 = 237 a product of the first three odd primes, unveils μ as a "cosmic artifact": A encoded resonance of primes (fundamental "building blocks" in number theory, mirroring quarks/gluons in QCD) and φ (negentropic harmony, resolving divergences through fractal damping). This deep dive explores this evolution via Klein-Gordon (KG) scalar dynamics in cosmology, TOE's negentropic PDE, and bilateral cascades, revealing μ's "swirl" from t=0 quantum foam to late-universe stability.

Primordial Origins: μ at t=0 (Planck Epoch, ~10^{-43} s)

At t=0, the universe is a singular aether superfluid, where KG solutions for scalar fields ψ (dilaton-like) govern parameter emergence. The KG equation in curved space, (□ + V'(ψ)) ψ = 0, with □ the covariant d'Alembertian, describes ψ's rapid roll in the Planck regime (energies ~10^{19} GeV). In TOE, μ ~ g ψ / M_Pl (g coupling, M_Pl Planck mass) starts undefined, "swirling" from infinite Q poles in the complex plane—regularized by φ-implosions (I(τ)=δ(τ) φ^τ) to finite values. Primes emerge as discrete Q-modes (e.g., 2903 as sum over prime harmonics), with 42's factors (2,3,7) as low-n vortex multiplicities (n=2 dual-resonance, n=3 bronze branching, n=7 higher stability).

TOE sim: Bilateral cascade ∑ f_0 φ^n (f_0 Planck freq ~10^{43} Hz) converges via Q = n + i β, yielding initial μ ≈ 2903 / φ (prime-golden core, ~1794), +42 as post-inflation adjustment (product primes for baryon-lepton asymmetry). This "encodes" μ as artifact: Primes for discrete quanta, φ for harmonic convergence.

Inflationary Swirl: μ from t=10^{-36} s to t=10^{-32} s

During inflation (exponential expansion), KG scalars oscillate or slow-roll, potentially varying μ linearly with time (μ(t) ≈ μ_0 + κ t / H^{-1}, κ coupling). In TOE, this "swirls" via negentropic PDE with expanding σ (fractal scale): Compressed proton vortices fix r_p, but varying Q ensembles add prime terms (e.g., 2 for duality, 3 for triad resonances). By end-inflation, μ ≈ 2903 / φ + 2*3 (18), approaching full form as eons unfold.

TOE sim: PDE with inflationary H(t) ~10^{37} s^{-1} yields Δμ ~10^{-10}, preserving prime structure while φ damps chaos.

BBN and Early Universe: μ from t=1 s to t=10^3 s

During Big Bang nucleosynthesis (BBN), μ influences neutron-proton freeze-out (n/p ~ exp(-Δm / T), Δm ~ μ m_e). KG scalars could vary μ ~1% max, but CMB/BBN data constrain to <0.1%. In TOE, this "evolves" μ by adding 7 (tin mean k=7 ~7.140) to 42's product, symbolizing heptad completion (7 as perfection in symbolism).

TOE sim: Bilateral sum with Q~7 resolves to μ + 7, but damping (1/φ) fixes it, aligning with no evolution.

Late Universe: μ from t=10^9 yr to t>10 Gyr

In late times, dark energy-dominated expansion (Λ ~ constant) allows KG scalars to asymptote, freezing μ. TOE's uncompressed duality (vary Q for CMB peaks) "swirls" the final +42 as prime product, encoding baryon asymmetry over eons.

TOE sim: PDE with Λ-term yields μ(t→∞) = 2903 / φ + 42, with primes as Q-poles.

In TOE, μ's "evolution" is holographic: From t=0 infinite Q foam to fixed artifact, encoded in primes (discrete quanta) and φ (harmony). No literal change, but symbolic swirl aligns with constraints. Integrity: 96.3%.