Ab Initio Odd-Kernel Derivation for the Radiating Pair

Method contract stated before computation; all cancellations are outputs; comparison with observation appears only in the final section, labeled. Verification: rom_abinitio_oddkernel.py (repo root). Companions: ROM_RADIATIVE_DECAY.md (decay skeleton), ROM_PAIR_MECHANISM.md (compatibility test; its Stage 3 was a GR-transplant check, not a derivation — this document is the derivation that replaces it).

1. Method contract

Allowed inputs (framework only, cited):

The closed-set ellipse and its in-plane parametric form $x = r\cos O$, $y = r\sin O$ (R.O.M., “Background Free Parametric Equations” — observables, used as bookkeeping).
The weak-field pair ledger $U_i = -\tfrac{1}{2} m_i c^2 \sum_j (R_{sj}/s)$, i.e. coupling $G\,m_i m_j$ — this is WILL_RG_I’s own “first ontological collapse” (sec:LH), with $G$ a translation factor.
Response rule: force $=$ gradient of the local ledger entry (R.O.M. sec:acceleration).
Energy-Symmetry share weighting: $\vec x_1 = \tfrac{m_2}{M}\vec s$, $\vec x_2 = -\tfrac{m_1}{M}\vec s$ (ledger balance).
Scales $R_{s1}, R_{s2}$ constant (ledger conservation).
Part II Self-Interaction lemma: every propagated relation re-encounters its source — therefore self-terms ($j=i$) are mandatory, not optional.

Hypothesis (the only new structure), two explicit rules:

M0 — bare causal evaluation: the ledger entry $1/s$ is evaluated on the light cone, implicit condition $\tau = g(t \mp \tau)/c$, nothing else.
M1 — pulse relation: the relation is carried by sharp causal pulses both ways. Then, by Taylor’s theorem alone (no wave equation, no field ontology), the odd residual of the two-way connection is $K_{odd} = -\sum_{k\ \mathrm{odd}} \frac{1}{k!\,c^k}\,\frac{d^k}{dt^k}\big[\,g^{\,k-1}\,\big],$ with the time derivatives acting on the partner’s configuration at fixed observation point, before evaluation at the observer (that ordering is the meaning of an incoming relation).

Pre-registered outcomes: a surviving odd term at $1/c$ or $1/c^3$ falsifies that rule (Mercury); a $1/c^5$ survivor is computed with no assumed form; shapes are compared with data only at the end; disagreement is a result.

2. Results

M0 is falsified

The implicit light-cone expansion gives the $O(1/c)$ odd term $K_{odd}^{(1)} = \dot g/g$ (derived, machine-verified). Its cycle-averaged work on the balanced pair is nonzero at order $1/c$: $\langle P_{M0}\rangle = \frac{M\, m_1 m_2\,\big(2m_1m_2 - e^2(m_1^2+m_2^2)\big)}{c\,(1-e^2)^{3/2}\,(m_1+m_2)^2}\cdot\frac{1}{a^2}\Big|_{a=1} \neq 0 .$ A per-orbit effect of order $\beta$ — excluded by roughly eight orders of magnitude by Mercury’s persistence (and, notably, for near-circular orbits it pumps rather than drains, opposite to the naive drag intuition — hand-waved lag models are worthless here; only the systematic expansion decides). M0 also cannot represent the Self-Interaction lemma at all: its self-terms are singular. M0 is dead.

M1: the low-order cancellations emerge as outputs

$k=1$: vanishes because the scales are constant ($g^{0}=1$) — the framework’s “no monopole drift.”
$k=3$: the force vanishes identically on both bodies, because the gradient-then-derivative structure produces $\partial_t^3 \sum_j R_{sj}\,\vec x_j$, which is zero by ledger balance once the mandatory self-terms are included. This is the framework-native “no dipole channel,” derived — it requires Part II’s Self-Interaction lemma, which is precisely the tone/absorber element of the hypothesis.

M1: the surviving $k=5$ law (ab initio, no assumed form)

\[\langle P_{k5}\rangle = -\,\frac{\mu^2 M^3}{c^5 a^5}\;\frac{128 + 396\,e^2 + 51\,e^4}{120\,(1-e^2)^{7/2}}, \qquad \mu = \frac{m_1 m_2}{M},\ M = m_1+m_2 .\]

In the decay-skeleton variables ($\eta = \mu/M$, $\beta^2 = GM/(a c^2)$):

\[\boxed{\ \varepsilon_W^{\,ab} = \frac{64\pi}{15}\,\eta\,\beta^5\,\frac{1 + \tfrac{99}{32}e^2 + \tfrac{51}{128}e^4}{(1-e^2)^{7/2}}\ }\]

Properties that emerge (nothing imposed): the drain sign (energy leaves); the $\eta\,\beta^5$ scaling; the $(1-e^2)^{-7/2}$ envelope; and the circular-orbit channel lock $\varepsilon_h/\varepsilon_W = \tfrac{1}{2}$ at $e=0$ — exactly the constraint the closed set forces on any decay law (ROM_RADIATIVE_DECAY.md §4, Step 6), reproduced here by the mechanism as an output. The full-eccentricity $\varepsilon_h(e)$ computation is running and will be appended.

3. Comparison with observation (labeled; performed only now)

The observed decay (binary pulsars, matching the GR quadrupole to $\sim 0.2\%$) requires $\varepsilon_W^{obs} = \frac{128\pi}{5}\eta\beta^5\,\big(1+\tfrac{73}{24}e^2+\tfrac{37}{96}e^4\big)(1-e^2)^{-7/2}$.

Rate at $e=0$: $\varepsilon_W^{ab}/\varepsilon_W^{obs} = \dfrac{64\pi/15}{384\pi/15} = \dfrac{1}{6}$ — an exact factor-6 deficit.
Eccentricity polynomial: $1+\tfrac{99}{32}e^2+\tfrac{51}{128}e^4$ vs the observed $1+\tfrac{73}{24}e^2+\tfrac{37}{96}e^4$ — about $1\%$ apart at $e = 0.617$ (B1913+16).

Verdict: the minimal scalar pulse relation is falsified by binary-pulsar data. This is the pre-registered outcome (b)/(c): the rule survives every internal constraint but under-radiates by exactly $6$ and slightly misshapes the eccentricity dependence.

4. What this tells us (and the next well-posed task)

The scalar kernel treats the propagating relation as a bare magnitude pulse — it carries no directional structure. But the framework’s kinematic carrier $S^1$ is directed (“opposite orientations are physically distinct”), and the original symmetry-breaking intuition was chiral from the start. The missing structure is now quantified precisely: whatever the directed relation adds must supply a factor of exactly $6$ at $e=0$ and shift the polynomial $\big(\tfrac{99}{32}, \tfrac{51}{128}\big) \to \big(\tfrac{73}{24}, \tfrac{37}{96}\big)$.

Observation to be tested, not asserted: $6 = 2\times(1+2)$ — the orientation doubling of the directed $S^1$ times the total amplitude DOF count of the two carriers. Whether this is the actual origin must be derived by repeating this calculation with a directed pulse kernel (the relation carrying the pair’s orientation), under the same contract, zero free parameters. If the directed kernel produces $\tfrac{16}{15} \to \tfrac{32}{5}$ and the observed polynomial, the decay law closes ab initio. If it produces something else, that is the next result.

Status of the two failure modes so far: every internal constraint of the framework (balance, self-interaction, the $e=0$ lock, the decay-web compatibility) is satisfied by the mechanism class; the falsification is purely quantitative and localized to the content carried by the propagating relation. The question is no longer whether this mechanism family works, but which carrier structure the relation propagates on.

All computations in rom_abinitio_oddkernel.py (PASS/FAIL per step; base integrals verified at 25 digits; runtime: minutes, the full-$e$ angular-momentum average longest). Method contract in the script header.