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Equation 59 · Part 16 · The Clock That Comes Back Wrong by Exactly Its Mass

Numerator: langle H_rm NRrangle

⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v.\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v.
⟨HNR⟩\langle H_{\rm NR}\rangle

What this part means

The complete quantity above the fraction bar.

Its job in the formula

langle HrH_rm NRrangle occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

…NR} , separating the rest energy from whatever nonrelativistic energy — kinetic, internal — sits on top of it. The relativistic loop phase becomes ⟨H⟩ℏc2 b⋅v=Mℏ b⋅v+⟨HNR⟩ℏc2 b⋅v\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v. Holding b\mathbf b , v\mathbf v , M , and ⟨\langle HNRH_{\rm NR}⟩\rangle fixed and sending c→\to∞\infty , the second term vanishes and the first survives unchanged, converting a state-dependent relativistic phase into a fixed, universal constant multiplying every vector in the fixed- M sector — Bargmann’s central charge, reconstructed as the…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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