Large-N Gravity Toy — Full Derivation

This page is the mathematical companion to /lab/large-n-gravity. The toy lets you slide N from 1 to 10⁵⁰ and watch G_eff, Hubble H₀, and the gravity-EM hierarchy ratio change. At N ≈ 1.7×10⁴², all benchmarks pass simultaneously — recovering measured Newton's G, the observed H₀, and resolving the hierarchy problem.

SPT mechanism

Formulas

The magic number N ≈ 1.7×10⁴²

The hierarchy ratio between gravitational and electromagnetic forces between two protons is ≈ 10⁻⁴². In SPT, this is not a separate fundamental constant — it is the count of nodes whose phases cancel out gravity. The toy preset 'Recover Newton's G' snaps the slider to N = 10^42.23, at which point all four benchmarks pass simultaneously.

Benchmarks at N = 10^42.23

Mathematical soundness (preserved at all N)

• 1/r² law preserved — comes from 3D geometry of disturbance spreading, independent of N. The N-dependence is in the coupling constant, not the spatial distribution.
• Equivalence principle — universal coupling G_eff applies identically to all matter (no fifth force in this toy).
• Energy conservation — gravitational potential energy U = −G_eff m₁m₂/r is well-defined; gradient gives Newton's force.
• No tachyon — V_grav has positive curvature in attraction direction; small perturbations are stable.

Ab-initio mode — N = 2¹⁴⁰ from cascade depth

The toy ships an Ab-initio toggle in the control panel. Turning it on locks the slider to log₁₀(2¹⁴⁰) ≈ 42.144. The cascade depth 140 has a structural interpretation: 140 = 7 yao × 20 cosmic generations of the Bagua-membrane subdivision. Each yao binary at each generation doubles the count of independent phase-mixed nodes, so N = 2¹⁴⁰ ≈ 1.39×10⁴² is the geometric prediction without any calibration.

Honest 22 % prefactor residual

The calibrated value is N = 1.7×10⁴² (log₁₀ = 42.230); the ab-initio bare value is N = 2¹⁴⁰ = 1.394×10⁴² (log₁₀ = 42.144). The gap of 0.086 in log₁₀ is a factor of 1.222 in N — a 22 % prefactor residual. This residual propagates predictably through the rest of the toy:

Compare with the Higgs ab-initio derivation, where the analogous geometric ratio (factor 12 = 24/2 from the cosine Taylor coefficients) is exact and combined with SM RG running it lands m_H = 125.10 GeV exactly. The Large-N case awaits a similar closed-form prefactor identity.

Conclusion

pip install sympy numpy && python3 scripts/spt_chsh_hierarchy.py