2.5PN Is the First Dissipative Order Because Mass Conservation Kills n = 1: Where the Even-Order Rule of the Yamagishi Higher-Order Delay Interpolation Breaks. Third edition.
This paper set out that the c^-2 hierarchy of the post-Newtonian (PN) expansion descends from the Taylor expansion in the retarded time t_ret = t – r/c, and showed that the PN expansion branches into a CONSERVATIVE series (integer orders 1PN, 2PN, 3PN, 4PN = c^-2, c^-4, c^-6, c^-8; time-reversal symmetric) and a DISSIPATIVE one (half-integer orders 2.5PN, 3.5PN, 4.5PN = c^-5, c^-7, c^-9; asymmetric, radiation reaction). The second edition recovered that branching correctly and retracted the first edition’s error that “odd-order terms vanish by parity” – a statement which had dropped radiation reaction, the very cause of a binary’s inspiral, out of the series. THAT IS AN IMPORTANT CORRECTION AS PHYSICS AND IS NOT ALTERED HERE. The two other retractions of the second edition – the correspondence of c^-5 with Paper 34’s c^5/5 (“5D bulk capacity”), and the identification of 5PN accuracy with a sub-pixel limit of 1e-10 – also stand and are not reinstated.
ALL NUMBERS WERE RE-VERIFIED AND ALL FOUR AGREE: Mercury’s perihelion advance 42.990 arcsec per century (from delta-phi = 6 pi G M/(c^2 a (1 – e^2)) with a = 5.791e10 m, e = 0.2056, period 87.969 days, 415.20 orbits per century; observed 42.98 +- 0.04 arcsec), the chirp mass of a 1.4 + 1.4 solar-mass binary 1.2188 solar masses, the cycle count from 10 Hz N_cyc = 1.603e4, the required accuracy delta_req = 1/(2 pi N_cyc) = 9.93e-6, and the 5PN relative term (0.3)^10 = 5.905e-6. Every number in the second edition is correct.
What the third edition adds is an answer to one question the second edition left open: WHY DOES THE DISSIPATIVE SERIES START AT 2.5PN, and not at 0.5PN or 1.5PN?
FIRST, THE SECOND EDITION’S REASON IS CORRECT BUT NEARLY DEFINITIONAL. It said “in a conservative system the Lagrangian is invariant under time reversal, so terms containing odd time derivatives cannot contribute to the equations of motion”. True – a time-reversal-symmetric system does not admit a term that breaks time reversal. But that amounts to saying that a conservative system has no dissipative term, which is roughly what “conservative” means. And it leaves out the important part: WHERE THE DISSIPATION BEGINS. Since the expansion advances in units of (1/c) d/dt, an odd term could in principle appear at n = 1, that is at 0.5PN. It does not. Why?
SECOND, THE ANSWER LIES IN THE n = 1 TERM OF THE NEAR-ZONE EXPANSION. Expanding the retarded potential, V = G integral rho(t – r’/c)/r’ d3x’ = G sum_n [(-1)^n/(n! c^n)] d^n/dt^n integral rho r’^(n-1) d3x’, the terms fare as follows. n = 0 is the Newtonian potential and survives (even). n = 1 is -(G/c) Mdot and VANISHES BY CONSERVATION OF MASS. n = 2 is +(G/2c^2) d2/dt2 integral rho r’ and survives (even, 1PN). n = 3 is -(G/6c^3) d3/dt3 integral rho r’^2 and is THE FIRST SURVIVING ODD TERM. Since integral rho r’^2 d3x’ is nothing other than the quadrupole moment, the n = 3 term is THE THIRD TIME DERIVATIVE OF THE QUADRUPOLE – the Burke-Thorne radiation-reaction potential (Burke-Thorne 1970). The c^-3 in the potential becomes c^-5 in the energy loss rate: the reaction force is F ~ c^-3 Qddd, its power is P = F.v, and balancing against the quadrupole formula fixes c^-5. SO 2.5PN IS THE FIRST DISSIPATIVE ORDER BECAUSE n = 1 IS KILLED BY MASS CONSERVATION. Had mass not been conserved, dissipation would have begun at 0.5PN. Time-reversal symmetry says only that odd terms are absent from a conservative system; WHERE THE PROHIBITION LIFTS IS DECIDED BY A CONSERVATION LAW.
THIRD, THIS GIVES A REASON FOR THE AGREEMENT WITH THE QUADRUPOLE FORMULA. The second edition correctly said that “the c^-5 of the quadrupole formula and the entry of radiation reaction at 2.5PN necessarily agree as a matter of energy balance”. The third edition names the source of that agreement: BOTH COME FROM THE SAME n = 3 TERM. The quadrupole formula (G/5c^5) avg[Qddd Qddd] is far-field radiation, the square of the quadrupole’s third derivative; the 2.5PN reaction force is the near-zone n = 3 term, the quadrupole’s third derivative. The two are not “coincidentally the same order”; they are two appearances of one quantity – one counting from outside the energy the system loses, the other counting from inside the force the system feels. This is the reverse of a judgement this series has often passed: Paper 86 warned that “a common ancestor is not a common root” and Paper 100 that “a resonance is a signature, not a proof”. BUT HERE THE COMMON SOURCE CAN ACTUALLY BE POINTED AT: one place, the n = 3 term. The discipline of distinguishing distinct roots from a common root does not forbid a common root; it only demands that they be distinguished.
FOURTH, THE LAYER-3 VOCABULARY IS REMOVED FROM THE BODY. “Latency correction”, “frame interpolation”, “discharge”, “sub-pixel” and the like were limited by the second edition itself at the outset as “vocabulary this series supplies for a reading, and not the content of a theorem”. That limitation is right. But the second edition left the vocabulary standing throughout the body – in Section 5 (“the required accuracy is the tolerance of each pipeline stage so that the accumulated frame drift does not reach one radian”), in Section 7 (“a standard computational algorithm interpolating the positions of dynamic objects by Taylor expansion in a rendering system of finite propagation speed”), and in Corollary 4.2 (“local frame interpolation”, “discharge out of the system”). The third edition carries the limitation out and removes the vocabulary from the body. Nothing is lost: that integer-order PN is conserved within the system while half-integer PN radiates out of it is PHYSICS, resting on time-reversal symmetry and mass conservation, and does not depend on the vocabulary. The “discharge into a higher dimension” retracted in the second edition is likewise not reinstated – the energy flows out as a gravitational wave propagating in ordinary three-dimensional space.
In addition, “the 5PN correction sits exactly at the threshold” was weakened to “the same order of magnitude”. The agreement of orders was confirmed in the re-verification (9.93e-6 against 5.905e-6), but delta_req is an order-of-magnitude estimate from the rule of thumb that the accumulated phase error should stay under one radian, and (v/c)^10 is the size of the relative term at the worst point, just before merger; since the phase accumulates mostly at low frequency, the actual phase contribution of a high PN order is smaller. The second edition’s observation that this agrees with the actual methodology – analytic waveforms to 3.5PN-4PN, matched to numerical relativity near merger – remains correct. And the second edition’s note on Mercury – “this agreement is a standard result of general relativity and not support for this paper’s picture; the same number is obtained without adopting that reading” – is the most disciplined sentence in the paper and is kept unchanged.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (the near-zone expansion of the retarded potential, Burke-Thorne 1970 radiation reaction, the PN theory of Blanchet 2014 and Maggiore 2007, Einstein’s 1915 perihelion result, the phase condition of matched filtering). NO NEW THEOREM, NO PREDICTION, AND NOTHING NEW TO PHYSICS – what is new here is internal to this series. What is not claimed: no prediction of general relativity is altered / the Mercury agreement is not treated as support for this series’ reading / c^-5 is not matched to a 5D bulk capacity (c^5/5 is retracted; Papers 34, 42) / the destination of the outflowing energy is not a higher dimension / 5PN accuracy is not identified with a sub-pixel limit / the phase-coherence condition is not read as a sharp threshold / the content of Section 4 is not claimed as new to physics (Blanchet 2014, Burke-Thorne 1970) / Layer-3 vocabulary is not placed in the body (quarantined under a non-scientific label).
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). 2.5PN Is the First Dissipative Order Because Mass Conservation Kills $n=1$ — 3rd ed.: Where the Even-Order Rule of the Yamagishi Higher-Order Delay Interpolation Breaks — [Paper 49]. Zenodo.
DOI: 10.5281/zenodo.21602658
ORCID: 0009-0006-8587-3450