Quantum Consepts - that can be wrong-but Usually ain't

  A Unified Extension of Noether’s Theorem: From Pythagorean Commas to Cyclic Drifts and the Conservation of Effective Time Abstract Emmy Noether’s seminal 1918 result established that continuous symmetries of a physical system’s action entail corresponding conservation laws. Most famously, time-translation symmetry yields conservation of energy. In this paper, we extend Noether’s conceptual framework to include systems where a small, systematic “shift” in the time variable accumulates cycle by cycle. By treating this accumulated shift Δ t \Delta t on equal footing with the usual time parameter t t , we introduce an effective time t e f f = t + Δ t t_{\mathrm{eff}} = t + \Delta t . We show that if the total system exhibits invariance under a combined transformation t → t + ϵ t \to t + \epsilon and Δ t → Δ t − ϵ , \Delta t \to \Delta t - \epsilon, the quantity t e f f t_{\mathrm{eff}} remains invariant—what we call a “conservation of time.” To illustrate this principle, w...

I have been trying to hone the actual equation down and I think I finally have the complete version.  While a2+b2+h(a2b2/R2)=c2 works great... it's not as easy to use... this new theory fixes some of the ambiguity.  Table of Contents Introduction Spherical and Hyperbolic Pythagorean Theorems Spherical: Exact Law of Cosines Spherical: Taylor Expansion Hyperbolic: Exact Law of Cosines Hyperbolic: Taylor Expansion Summarizing the Curvature‐Corrected Formulas Rigorous Checks of the Expansion Order Estimates and Neglected Terms Consistency with Euclidean Limit Geometry of “Proper Triangles” vs. “Right Triangles” Gauss–Bonnet and Angle/Area Proofs Gauss–Bonnet in 2D Relationship to the Sides a , b , c a,b,c Scissor Congruences and Area Comparisons Connecting to Spin‐ 1 2 \tfrac12 The Rotation Group S O ( 3 ) SO(3) vs. Its Double Cover S U ( 2 ) SU(2) Sign Flip for a 2 π 2\pi Rotation: Proof from Group Theory Geometric Interpretation: S 2 S^2 (Bloch...