Acceleration is not an aging term, and the twin comparison is not symmetric: scope caveats

This remark delimits the interpretation of the proper-time comparison of the twin reunion (The piecewise inertial twin reunion: conditional differential aging) in the adopted model, with the ideal clock assumptions of Adopted ideal-clock and ideal-vacuum-light interpretation and Adopted free trajectories of massive particles and the clock hypothesis and the SI conventions of SI units and coordinate conventions of the special-relativity framework. It proves nothing new; it records what the earlier items do and do not assert.

In the model the reading of an ideal clock between two events of its history is the proper-time integral of its worldline, by clause (i) of Adopted ideal-clock and ideal-vacuum-light interpretation for $C^1$ histories and by clause (ii) of Adopted free trajectories of massive particles and the clock hypothesis for the weak piecewise $C^1$ class (Total proper time of a piecewise C^1 future timelike worldline); no term depending on the four-acceleration (Four-velocity and four-acceleration of a future timelike worldline) or on any higher derivative of the history appears, and the clock hypothesis is precisely the exclusion of such terms. The corner of the piecewise inertial history is an idealization: a physical turnaround occupies a finite parameter interval, the worldline is then $C^1$ or smooth there, and its proper time is still the integral. The endpoint bound and strictness clause of The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space apply to every weak piecewise future timelike history with the same endpoints $p,q$. Thus any smoothing of the corner that remains in that class, preserves $p,q$, and retains at least one event outside the straight segment still gives a strictly smaller reading than the stay-at-home clock. No existence or convergence claim for a smoothing family, or equality with the sharp-turn value $2T\mathrm{/}\gamma$, is asserted.

One history of the reunion is the straight segment from $p$ to $q$; the other contains the corner event $r$ of The piecewise inertial twin reunion: conditional differential aging, and the three events $p,r,q$ are not collinear: in the chart of that item their coordinate columns are $(0,0,0,0)$, $(cT,vT,0,0)$ and $(2cT,0,0,0)$ with $vT\mathrm{\ne }0$. Every change of inertial chart is an affine bijection (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition) and maps collinear triples to collinear triples, so no change of inertial frame maps the image of the stay-at-home history onto the image of the traveler's history, nor fixes both endpoints and exchanges the two histories. The two twins are therefore not related by an inertial change of description, and the difference of readings is a model consequence of the different histories, not a consequence of who is 'really moving': the symmetric time-dilation relation of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation (i) concerns the two frames, while the reunion comparison concerns the two clock readings at the single common event $q$.

In each momentary inertial rest frame of the traveler the stay-at-home clock runs slow by the factor $1\mathrm{/}\gamma$, and the traveler may legitimately apply the time-dilation relation using successive pairs of simultaneous events, one on each clock history in that momentary frame (Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation (i), Simultaneity slices and rest distance relative to a future unit timelike vector). But the two legs use different inertial frames. With $\gamma =(1-v^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$ and the turn event $r$ of The piecewise inertial twin reunion: conditional differential aging, choose both rest-frame charts with origin $p$ as in The piecewise inertial twin reunion: conditional differential aging; the outbound chart $x^{\mathrm{\prime }}$ then satisfies $x^{\mathrm{\prime }0}=\gamma (x^0-\frac{v}{c}{x}^1)$, so its slice through $r$ has $x^{\mathrm{\prime }0}=\gamma cT(1-v^2\mathrm{/}{c}^2)=cT\mathrm{/}\gamma$ and meets the stay-at-home worldline $x^1=0$ at $x^0=\frac{1}{\gamma }\cdot \frac{cT}{\gamma }=cT(1-v^2\mathrm{/}{c}^2)$, that is at coordinate time $t=T(1-v^2\mathrm{/}{c}^2)=T\mathrm{/}{\gamma }^2$; the inbound rest frame's chart $x^{\mathrm{\prime }\mathrm{\prime }}$ satisfies $x^{\mathrm{\prime }\mathrm{\prime }0}=\gamma (x^0+\frac{v}{c}{x}^1)$, and its slice through $r$ meets the same worldline at $x^0=cT(1+v^2\mathrm{/}{c}^2)$, that is at coordinate time $t=T(1+v^2\mathrm{/}{c}^2)$. The two slices differ by $2T{v}^2\mathrm{/}{c}^2$ in coordinate time, and the identity $2T=\frac{T}{{\gamma }^2}+\frac{2T{v}^2}{c^2}+\frac{T}{{\gamma }^2},$ which only uses $1\mathrm{/}{\gamma }^2=1-v^2\mathrm{/}{c}^2$, inserts this gap into the traveler's two leg-by-leg accounts and reproduces the stay-at-home reading $2T$ at the reunion, in agreement with the conditional comparison result Same-endpoint clock comparison: the inertial clock ages most. The naive reciprocity statement 'each twin judges the other's clock to run slow' therefore compares different pairs of events for the two travelers — events simultaneous in different momentary frames — and does not apply to the reunion comparison, which concerns the two clock readings at the single common event $q$.

Everything above is a conditional statement about ideal clocks in the adopted model: the clocks are ideal with no acceleration-dependent reading term, the histories are prescribed, spacetime is flat and $c$ is fixed (Adopted free trajectories of massive particles and the clock hypothesis, Adopted ideal-clock and ideal-vacuum-light interpretation). No measurement, apparatus, calibration, sampling or uncertainty model is involved, and no claim is made about real clocks or about nature; real clock comparisons would additionally require apparatus and uncertainty analysis and are outside this remark.