Topological Definition of Orbital Decay — Test and Exact Consequences

Hypothesis (Anton, July 2026): in a relationally closed dyad the contours of the kinematic projections self-intersect after one orbital phase cycle, $\beta_{Ao}(0) = \beta_{Ao}(2\pi)$; orbital decay is the failure of this closure on the $S^1$ carrier, an open boundary $\beta_{Ao}(0) < \beta_{Ao}(2\pi)$; the shed relational capacity per cycle is the discrete shift $\Delta\beta_{Ao}^2$ (or $\Delta\beta_{Ao}$); this mismatch is the direct algebraic measure of the closure defect $\langle\delta\rangle_{\text{cycle}} \neq 1$ and replaces continuous $dt$ evolution with an exact topological discontinuity.

Sources: WILL DATABASE/WILL_RG_I.txt, R.O.M. closed set (all phase functions as published), decay skeleton ROM_RADIATIVE_DECAY.md ($\varepsilon_W$, $\varepsilon_h$). GR quadrupole fluxes appear only as labeled comparison input. Verification: rom_topological_decay.py — 38 symbolic + numeric checks, 38/38 pass in a single run (~40 s). The $N_{\text{ISCO}}$ integral was additionally confirmed standalone with doubled step count (agreement to 14 digits).

0. Status summary

Read this first; it is the honest boundary of the result.

Confirmed: the closed-cycle premise is an exact identity of the closed set; the per-cycle boundary mismatch is a well-posed, exact, first-order meter of openness, independent of how the loss is distributed within the cycle; the quadratic form $\Delta\beta_{Ao}^2$ (not $\Delta\beta_{Ao}$) is the ledger-native variable; the full mismatch function over one cycle carries exactly the two radiative channels — no more, no fewer.

Corrected (two claims fail as literally stated, and the algebra supplies the repair):

1. The strict inequality $\beta_{Ao}(0) < \beta_{Ao}(2\pi)$ is not phase-universal. At the perihelion reference it reverses sign for $e > e^{}$, where $e^{} = 0.5557305673$ is the unique root in $(0,1)$ of $37e^3 - 84e^2 + 208e - 96 = 0$. PSR B1913+16 ($e = 0.617$) sits above the threshold: its perihelion kinematic amplitude decreases per cycle while the binary decays. Sign-definite formulations exist and are given in §2.
2. The identification “mismatch = direct algebraic measure of $\langle\delta\rangle_{\text{cycle}} \neq 1$” runs backwards. The cycle-averaged closure factor is blind at first order to the decay (it is $O(\varepsilon^2)$ for any loss profile symmetric under $O \to 2\pi - O$, and at first order it meters the asymmetry of the within-cycle loss profile, not the loss totals). The boundary mismatch is the correct first-order meter. The topological definition does not measure the old defect — it supersedes it, and repairs a latent gap in cor:condition (§4).

Unchanged: Lemma 1 of the skeleton stands. The mismatch magnitudes are the two fluxes themselves; the closed algebra does not supply them (§6). The topological definition is the framework-native definition and meter of decay, not its source law.

1. Setting and the closed premise

All phase variables of the closed set are functions of $\cos O$, $\sin O$ with the true phase $O$; hence exactly $2\pi$-periodic (verified: $\beta_o$, $\kappa_o$, $\delta_o$, $Q_o$). The boundary match $\beta_{Ao}(0) = \beta_{Ao}(2\pi)$ is an identity, not a condition. Note the precession subtlety: the contour that closes lives on the phase carrier, not in coordinate space — the spatial ellipse precesses ($O = \Omega\,o$), yet the state contour in $(\beta_o, \kappa_o)$ self-intersects after $\Delta O = 2\pi$ exactly. One “cycle” is therefore the anomalistic (perihelion-to-perihelion) cycle, and precession does not contaminate the decay meter.

The open cycle is described by the two exact per-cycle jumps of the skeleton,

with the exact state map (at fixed $R_s$, from $h_W = \frac{R_s c\sqrt{1-e^2}}{2\beta}$):

In the adiabatic reading $J_W = \varepsilon_W$, $J_h = \varepsilon_h$ per cycle, but every boxed statement below is exact at finite jumps unless marked first-order.

2. The mismatch dictionary (exact jump algebra)

The boundary mismatch of $\beta_{Ao}^2$ at a fixed reference phase $O^{*}$, taken between the cycle’s endpoint states, is at first order

with $\Delta e = \frac{1-e^2}{e}\left(J_h - \frac{J_W}{2}\right)$ (the decay-web line, re-derived here from the exact map). The channel content is completely resolved by distinguished readings:

Two structural theorems follow (both machine-verified):

Completeness. The reconstruction determinant for readings at two phases is $-\frac{2\beta^4}{e}(\cos O_1 - \cos O_2)$: mismatches at any two phases with $\cos O_1 \neq \cos O_2$ determine $(J_W, J_h)$ uniquely, and the mismatch function over one cycle spans exactly a 2-dimensional space. The unclosed contour carries the complete radiative state of the pair — precisely the two channels of the skeleton, nothing further. The topological definition and the two-flux skeleton are the same object in dual presentation.

Channel diagonalization at the apsides. The geometric mean of the apsidal kinematic amplitudes reads the energy channel, their ratio reads the shape channel: $\beta_p\beta_a = \beta^2$ and $\beta_p/\beta_a = e_X$ identically, so $\frac{1}{2}\Delta\ln(\beta_p^2\beta_a^2) = J_W$ and $\frac{1}{2}\Delta\ln(\beta_p^2/\beta_a^2) = \Delta\ln e_X$. The framework’s Structural-Dynamical Equivalence chain ($e_X = \delta_a/\delta_p$) makes the second reading equivalently the per-cycle jump of the apsidal closure-factor contrast $\Delta\ln(\delta_a/\delta_p)$ — the closure factor enters the decay bookkeeping as an apsidal ratio jump, not as a cycle average.

The radiated ledger connects directly: inserting $\Delta\beta^2$ (the balance-point reading) into the skeleton’s exact energy bridge,

so the “relational capacity shed per cycle” of the hypothesis is, in the weak field, $\frac{1}{2}\Delta\beta^2$ per unit $E_0$ — the topological mismatch is the shed binding energy, with the exact strong-field correction factor supplied by the bridge.

3. Sign structure: where the literal inequality holds and where it fails

With the GR-matching fluxes (comparison input), the relative perihelion mismatch is $\Delta\ln\beta_p^2 = J_W\,B(e)$ with $B(e) = 1 + \frac{2}{e}\left(\frac{\varepsilon_h}{\varepsilon_W} - \frac{1}{2}\right)$ and small-$e$ expansion $B(e) = 1 - \frac{19}{6}e + O(e^3)$ (symbolic). $B(e) = 0$ is equivalent to $f_W(e) = (1+e)f_h(e)$, i.e. the cubic

verified independently by bisection on $B$ and by the polynomial root. The full sign field of $\Delta\beta_o^2(O^{})$ is linear in $\cos O^{}$, so a single critical curve $\cos O_c(e)$ separates negative (perihelion side) from positive (aphelion side) mismatch for $e > e^{}$; sample values: $O_c(0.617) = 41.6^\circ$, $O_c(0.7) = 65.0^\circ$, $O_c(0.8) = 88.7^\circ$, $O_c(0.9) = 115.6^\circ$. For $e < e^{}$ the mismatch is positive at every phase. The aphelion mismatch and both invariant readings ($\Delta\beta^2$, $\Delta\ln(\beta_p^2\beta_a^2)$) are positive for all $e \in (0,1)$ during decay.

Physical content: above $e^{*}$, circularization pulls the perihelion speed down faster than deepening pushes it up. The two known double neutron stars fall on opposite sides — a concrete internal contrast:

Consequence for the definition: the open state must be defined by mismatch, $\beta_{Ao}(0) \neq \beta_{Ao}(2\pi)$ at generic reference phase — equivalently by the sign-definite invariant readings $J_W > 0$, $J_h > 0$ — not by the phase-pointwise inequality, which fails at perihelion for $e > e^{*}$.

4. Relation to the closure defect: the meter is superseded, not measured

The closure factor is a pure shape variable: $\delta_o = \frac{1+e\cos O}{1+e^2+2e\cos O}$ contains no $\beta^2$ at all (symbolic check D0). Per def:closure_factor the global amplitudes are the cycle averages of the local ones, so the cycle defect is $\langle\delta\rangle_{\text{cycle}} = \frac{\langle\kappa_o^2\rangle_t}{2\langle\beta_o^2\rangle_t}$, equal to $1$ exactly on every closed cycle for every admissible $(\beta^2, e)$. That universality is precisely what disarms it as a decay meter: a first-order drift along any state direction cannot move a quantity that equals $1$ identically on the state manifold. Machine results over one radiating cycle ($e_0 = 0.5$, $\beta_0^2 = 0.01$, per-cycle flux $\varepsilon = 10^{-3}$, RK4 in true phase with the loss distributed by a within-cycle profile $p(O)$):

Fitted scaling orders in $\varepsilon$: defect $2.002$, boundary mismatch $0.992$. The pattern proves three statements at once. The averaged defect is second order for parity-symmetric profiles (and realistic emission is symmetric about perihelion at leading order); at first order it responds only to the asymmetry of the loss profile — it flips sign under mirror reflection of the profile while the totals and the boundary mismatch are unchanged; the boundary mismatch is identical across all profiles to $10^{-12}$ and equals the exact endpoint algebra of §2. For the circular radiating orbit the failure is complete: exact spiral kinematics give $\langle\delta\rangle - 1 = -\varepsilon_W^2/(4\pi^2)$ (verified at three flux values) while the boundary mismatch reads $\beta^2\varepsilon_W$ at first order — under cor:condition as written, a circular inspiral is “closed” to first order despite radiating.

Refinement forced on the framework: cor:condition’s meter $\langle\delta\rangle_{\text{cycle}} \neq 1$ does not quantify the energy flow through unaccounted channels at leading order; the topological boundary mismatch does. The hypothesis’s own construction is the repair. The correct statement of its third claim: the discrete kinematic shift constitutes the direct algebraic measure of relational openness, replacing $\langle\delta\rangle_{\text{cycle}}$ as the first-order meter.

The profile-independence is also what justifies calling the discontinuity exact: the mismatch depends only on the totals $(J_W, J_h)$, not on any within-cycle time resolution — no $dt$-level information survives in it, by construction and by machine check.

5. $\Delta\beta_{Ao}^2$ versus $\Delta\beta_{Ao}$

The quadratic form wins on three independent grounds. The Conservation theorem admits amplitudes into the ledger only in quadratic form ($\text{Amplitude}^2 + \text{Phase}^2 = 1$), and DOF weighting in the Closure theorem is per quadratic DOF, so $\beta_o^2$ is the ledger entry while $\beta_o$ is its square root with no independent ontological standing. The exact jump algebra of §2 is native to $\ln\beta_o^2$ (log-quadratic): the channel readings $2J_W$ and $J_W - 2J_h$ are exact at finite jumps only in this variable. The linear shift $\Delta\beta_o = \frac{\Delta\beta_o^2}{2\beta_o}$ carries the same zero set but distorts the channel weights by the phase-dependent factor $\frac{1}{2\beta_o(O^{*})}$ and breaks the direct energy-bridge reading. Recommended canonical form: the relative quadratic mismatch $\Delta\ln\beta_{Ao}^2$, which is additionally component-independent — with Energy-Symmetry share weighting $\beta_{Ao} = \frac{m_B}{M}\beta_o$, constant shares cancel in the log, so $\Delta\ln\beta_{Ao}^2 = \Delta\ln\beta_{Bo}^2 = \Delta\ln\beta_o^2$ (verified): both components register the same relative defect. The mismatch is a property of the relation, not of either component — reciprocity extends to the open state.

6. Discrete chronometry: the $dt$-replacement claim

The per-cycle map $\beta^2 \to \beta^2 e^{\varepsilon_W}$, $(1-e^2) \to (1-e^2)e^{\varepsilon_W - 2\varepsilon_h}$ iterated as a pure sequence (cycle count $N$, no time coordinate) was compared against the continuous flow: the cycle counts to the ISCO invariant $\beta^2 = \frac{1}{6}$ agree up to the Euler–Maclaurin offset $\approx \frac{1}{2}\ln\frac{\varepsilon_{\text{end}}}{\varepsilon_{\text{start}}} + O(1)$ — a flux-scale-independent handful of cycles (script: $+3.9$ and $+3.7$ at two flux scales against predicted $+2.7$; development runs over an eightfold flux range gave $+3.9 \to +3.4$, decreasing toward the prediction), so the relative difference vanishes as $\varepsilon \to 0$. For PSR B1913+16 the discrete endpoint exists and is finite: $N_{\text{ISCO}} = 7.79\times10^{11}$ cycles from the current state, with $e \to 2.6\times10^{-8}$ (circularized) at arrival. The $dt$-free formulation is well-posed, observationally equivalent to the continuous one at current precision (differences $O(\varepsilon) \sim 10^{-12}$ per cycle), and terminates at the framework’s own marginal-stability invariant rather than at a coordinate singularity.

Consistency with observation, through the topological readout: extracting $\varepsilon_W$ from the apsidal-product jump and inserting it into the skeleton’s forced line $\dot T = -\frac{3}{2}\varepsilon_W$ returns $\dot P_b = -2.4031\times10^{-12}$ for B1913+16 (literature GR $-2.40263\times10^{-12}$) and $-1.2483\times10^{-12}$ for J0737$-$3039 (literature $-1.24792\times10^{-12}$), within input-mass rounding. Nothing new is predicted here — the fluxes were imported — but the topological pipeline reproduces the observables without any $dt$-resolved intermediary.

7. What the topological definition does not do

Lemma 1 of ROM_RADIATIVE_DECAY.md survives untouched and the present construction makes it visible directly: the mismatch function’s two coefficients are $(\varepsilon_W, \varepsilon_h)$. Defining decay as boundary mismatch determines the meter exactly, and determines the magnitudes not at all — every admissible $(\beta^2, e, R_s)$ is still realized by an exactly closed dyad with zero mismatch. The magnitudes remain the business of the propagating-channel program (ROM_ABINITIO_ODDKERNEL.md: scalar kernel falsified at the exact factor $6$; directed kernel pending). One speculative bridge, flagged as outside the present sources: if the Part II phase-closure argument (whole-number windings neither grow nor decay) could be made quantitative on the $S^1$ carrier, a winding-defect condition might constrain the admissible mismatch per cycle; nothing in the two source papers forces this, and it is recorded only as a candidate direction.

8. Suggested revision of the subsubsection

Changes from the draft: mismatch (not strict inequality) defines the open state, with the sign threshold stated; the quadratic log form is canonical; the closure-defect sentence is reversed to “supersedes”; the exactness claim is grounded in profile independence.

\subsubsection{Topological Definition of Orbital Decay}
\hypertarget{sec:topological_decay}{}\label{sec:topological_decay}
In a relationally closed dyad the geometric contours of the kinematic
projections self-intersect after one anomalistic phase cycle. For any
component $A$ the boundary states are identical as an algebraic identity
of the closed set:
\begin{equation}
\beta_{Ao}(O^{*}) = \beta_{Ao}(O^{*} + 2\pi) \quad \forall\, O^{*}.
\end{equation}
Orbital decay is geometrically defined as the failure of this
self-intersection on the $S^1$ carrier: the contour becomes an open
spiral, and the per-cycle mismatch
\begin{equation}
\Delta\ln\beta_{Ao}^2(O^{*}) \neq 0
\end{equation}
defines the open state. The mismatch is exact and topological: it depends
only on the per-cycle totals of the two radiative channels, not on the
within-cycle distribution of the loss. Its channel content is resolved by
two exact readings,
\begin{equation}
\tfrac{1}{2}\,\Delta\ln\!\big(\beta_{Ap}^2\,\beta_{Aa}^2\big)
  = \Delta\ln\beta^2 \equiv \varepsilon_W,
\qquad
\Delta\ln\!\big(1-e^2\big) = \varepsilon_W - 2\varepsilon_h,
\end{equation}
so the unclosed boundary carries the complete radiative state of the
dyad. The sign of the single-phase mismatch is phase dependent: for
$e > e^{*}$, with $e^{*}$ the root of $37e^3 - 84e^2 + 208e - 96 = 0$
($e^{*} \approx 0.5557$), the perihelion reading is negative while the
invariant readings remain positive throughout decay. The discrete
mismatch supersedes the cycle-averaged closure factor as the meter of
openness: $\langle\delta\rangle_{\text{cycle}} - 1$ is second order in
the per-cycle flux (and, at first order, responds only to the asymmetry
of the within-cycle loss profile), whereas $\Delta\ln\beta_{Ao}^2$ is
first order and profile independent. Continuous differential time
evolution ($dt$) is thereby replaced by an exact topological
discontinuity per cycle, with the cycle count $N$ as relational
chronometry.

9. Verification manifest

rom_topological_decay.py (this folder; requires sympy + numpy; ~1 min): Part A (4) closed-cycle periodicity identities; Part B (9) exact jump algebra — decay-web $\Delta e$, apsidal product/ratio, phase-independent vis-viva mismatch, general first-order mismatch formula, pure-channel readings at $B_a$ and $O_\dagger$, non-perturbative balance-point identity, 2D completeness determinant; Part C (7) sign structure under GR fluxes — circularization, $e^{}$ by bisection and by the exact cubic, symbolic small-$e$ slope $-\frac{19}{6}$, aphelion positivity, critical-phase table, B1913+16 sign reversal; Part D (12) defect comparison — $\delta_o$ independence of $\beta^2$, circular spiral defect $-\varepsilon_W^2/(4\pi^2)$ at three fluxes, closed-cycle integrator sanity, profile independence of the mismatch vs profile dependence and $\varepsilon^2$ scaling of the averaged defect, mirror sign flip; Part E (1) component reciprocity; Part F (6) pulsar numbers, topological $\dot P_b$ readout, sign contrast B1913+16 vs J0737$-$3039, discrete-vs-continuous Euler–Maclaurin agreement, finite $N_{\text{ISCO}}$. During development the harness caught two genuine errors in my own first-pass analytics: the leading-order estimate $e^{} \approx \frac{6}{19}$ (superseded by the exact cubic root $0.5557$) and an average-of-ratio misreading of $\langle\delta\rangle_{\text{cycle}}$ (corrected to the ratio-of-averages per def:closure_factor before any conclusion was drawn).

Derived strictly from the two source papers and the prior skeleton; every claim above is a PASS line in the script; the two corrections to the hypothesis are stated as corrections, not reinterpretations; the undetermined element (flux magnitudes) remains undetermined, in agreement with Lemma 1.