Relational Decay Law for Radiating Binaries (R.O.M.)

Sources: WILL DATABASE/WILL_RG_I.txt, WILL DATABASE/R.O.M..txt (read in full; treated as ground hypothesis). Verification: rom_decay_verify.py — 20/20 symbolic + numeric checks pass.

0. Status summary

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

Forced by the closed system (given assumptions A1–A3 below): the complete kinematic web of the inspiral — every decay rate (semi-major axis, period, eccentricity, precession, radiated ledger) is algebraically locked to exactly two per-cycle flux numbers; the exact energy bridge between period decay and radiated power; one exact constraint tying the two fluxes together at e = 0; the domain endpoint of the adiabatic law at the closed set’s own marginal-stability invariant Q² = 1/2.

Provably NOT determined by the closed system: the two flux functions themselves — how much ledger leaves per cycle. Lemma 1 (§6) shows no function assembled from closed-set quantities can be the flux. This is not a defect of the derivation; it coincides with the framework’s own statement (WILL_RG_I, sec:challenges): “Gravitational waves and the full radiative two-body problem remain to be formulated… requires a relational description of propagating degrees of freedom on the carriers and remains an open technical task.”

Where the closed structure does not determine a step, this document states so and stops. The GR comparison appears only in §9.

1. Framework resources used

From the two source documents, exactly these elements are used:

Closed-set identities (R.O.M., section “Closed Algebraical System”): kappa² = 2 beta² (Closure Theorem); a = R_s/kappa² = R_s/(2 beta²); beta² = R_s/(2a) (energy invariant / binding energy); T = 2 pi a/(beta c) = pi R_s/(beta³ c); h_W = R_s c sqrt(1−e²)/(2 beta) (invariant angular momentum); Delta_phi = 2 pi (3 beta² − 2 beta⁴)/(1−e²) (precession per orbit); E_loc = E_0 kappa_X/beta_Y (Energy Symmetry Law scaling); the critical-state table (dynamic ISCO state: beta² = 1/6, kappa² = 1/3, Q² = 1/2, r = 3 R_s).

Openness bookkeeping (WILL_RG_I, def:closure_factor and cor:condition): delta ≡ kappa_o²/(2 beta_o²); closed systems satisfy ⟨delta⟩_cycle = 1; “Open systems display ⟨delta⟩_cycle ≠ 1, the magnitude of which quantifies the energy flow through unaccounted channels.” The Radiating Binary bullet: “The closure defect delta quantifies the relational ledger lost to gravitational radiation. Including all channels restores closure.”

A point the documents leave implicit, made precise and verified here (script Part B): the cycle average in ⟨delta⟩_cycle = 1 is the time average. Verified exactly: ⟨kappa_o²⟩_t = 2 beta² = kappa² and ⟨beta_o²⟩_t = beta² over one conservative cycle, for all e in (0,1). Phase-averaging does not close (it leaves a residue −2 beta² e²/(1−e²)); time-averaging closes identically. This fixes the measure in which the closure defect of a radiating binary is defined.

2. New symbols (all definitions, no physics added)

Symbol Definition Meaning
N cycle count evolution parameter: number of completed orbital cycles. Chosen instead of an external time coordinate — cycle counting is relational chronometry, consistent with the framework’s refusal of background time.
epsilon_W d ln(beta²)/dN per-cycle fractional deepening of the energy invariant beta². The energy-channel flux. Dimensionless.
epsilon_h −d ln(h_W)/dN per-cycle fractional loss of the angular-momentum invariant h_W. The angular-momentum-channel flux. Dimensionless.
Tdot dT/dt_obs = d ln T/dN dimensionless period drift (observer counts cycles with the drifting period: dt_obs = T dN). This is the binary-pulsar Pb-dot observable.
R_s1, R_s2 scales of the two bodies needed only in §7/§9. R_s ≡ R_s1 + R_s2.
eta R_s1 R_s2 / (R_s1+R_s2)² symmetric scale ratio (0 < eta ≤ 1/4). Framework-compatible: the L1 section of R.O.M. already operates with two scales (R_sE/R_s☉).

3. Assumptions (complete list, each labeled)

Given A1–A3, the following is forced, not assumed: the bound-sector state at fixed R_s is the pair (beta, e). Therefore the entire radiative drift per cycle is exhausted by exactly two numbers — the drifts of the two invariants beta² and h_W. Two channels, no more, no fewer (spin/orientation excluded by A3). This is the two-channel completeness property.

4. Derivation

Step 1 — Object [framework definition]. A radiating binary is a bound configuration whose orbital sector is relationally open (WILL_RG_I, Radiating Binary bullet). By cor:condition its per-cycle time-averaged closure defect measures the ledger leaving through the unaccounted channel. Under A1 it is a drifting sequence of closed states.

Step 2 — Channels [forced under A1–A3]. State = (beta, e) at fixed R_s ⇒ the drift is fully specified by epsilon_W and epsilon_h (definitions in §2).

Step 3 — Kinematic web [forced; each line an exact identity of the closed set, differentiated at fixed R_s; verified symbolically]:

Note what the last line says: to leading order the drift of the periastron advance is controlled by the angular-momentum channel (12 pi beta² epsilon_h/(1−e²)), not the energy channel — an internal, testable structural statement.

Step 4 — Energy bridge [forced]. The framework’s exact state energy is E_loc/E_0 = kappa_X/beta_Y; on the closure line kappa² = 2 beta² this is sqrt(1−2 beta²)/sqrt(1−beta²). Differentiating:

The bridge diverges as beta² → 1/2 (kappa → 1, dynamic-horizon state): deepening the orbit costs unboundedly much ledger per unit of beta². The decay drives the state along the closure line toward saturation.

Step 5 — Domain and endpoint [forced]. The closed set’s own critical table places marginal orbital stability at the dynamic ISCO state Q² = Q_Y² = 1/2, i.e. beta² = 1/6, kappa² = 1/3, a = 3 R_s. There A1 fails by the framework’s own classification (marginal stability = no adiabatic neighbor states). The decay law’s domain is 0 < beta² < 1/6; the adiabatic inspiral terminates at the framework’s ISCO invariant.

Step 6 — Circularity lock [forced constraint on the unknown fluxes]. In the web, de/dN = ((1−e²)/e)(epsilon_h − epsilon_W/2). For any smooth flux law, finiteness of de/dN at e = 0 forces

epsilon_h(beta, 0, eta) = epsilon_W(beta, 0, eta)/2.

The closed structure thus fixes one exact relation between the two missing functions on the circular boundary, without knowing either. (Whether e grows or decays at e > 0 — circularization — is not forced; it depends on the sign of epsilon_h − epsilon_W/2 away from e = 0.)

5. Result — the relational decay law (SymPy, R.O.M. notation)

from sympy import symbols, Function, sqrt, pi, Rational

# --- evolution parameter and state (new symbols defined in section 2) ---
N          = symbols('N', real=True)            # cycle count
beta       = Function('beta')(N)                # global kinetic projection
e          = Function('e')(N)                   # eccentricity
R_s, c, E_0 = symbols('R_s c E_0', positive=True)
eps_W      = symbols('varepsilon_W', real=True) # := d ln(beta**2)/dN   (energy channel)
eps_h      = symbols('varepsilon_h', real=True) # := -d ln(h_W)/dN      (ang.-mom. channel)

# --- FORCED kinematic web (exact identities of the closed set, A1-A3) ---
dln_beta2_dN  = eps_W                                        # definition
dln_a_dN      = -eps_W                                       # a = R_s/(2 beta**2)
dln_T_dN      = -Rational(3, 2)*eps_W                        # T = pi R_s/(beta**3 c)
Tdot          = -Rational(3, 2)*eps_W                        # dT/dt_obs, dimensionless
de_dN         = ((1 - e**2)/e)*(eps_h - eps_W/2)             # h_W = R_s c sqrt(1-e**2)/(2 beta)
dDelta_phi_dN = (4*pi*beta**2/(1 - e**2)) \
                * (3*eps_h - beta**2*eps_W - 2*beta**2*eps_h)  # chain rule on Delta_phi

# --- FORCED energy bridge (Energy Symmetry Law, closure inserted) ---
P_cycle_over_E0 = (beta**2*eps_W/2) \
                  / ((1 - beta**2)**Rational(3, 2)*sqrt(1 - 2*beta**2))
# = radiated ledger per cycle per unit E_0;  weak field -> (beta**2/2) eps_W

# --- FORCED constraint on the unknown fluxes at the circular boundary ---
# eps_h(beta, 0, eta) == eps_W(beta, 0, eta)/2

# --- domain of validity (closed set's own critical table) ---
# 0 < beta**2 < 1/6      (terminates at dynamic ISCO invariant Q**2 = 1/2)

# --- NOT determined by the closed system (proof in section 6) ---
# eps_W = F_W(beta, e, eta),  eps_h = F_h(beta, e, eta):  functional forms unknown.

This is the complete decay law the closed structure supports: a two-flux skeleton in which every observable decay rate is algebraically locked, with the fluxes themselves left as the framework’s acknowledged open element.

6. Insufficiency: proof that the closed system cannot supply the fluxes

Lemma 1 (no internal flux function). Every expression in the closed set is an identity among state quantities (beta, e, R_s and their phase functions). Every admissible state (beta, e, R_s) is realized by an exactly closed conservative system — for which the leak is zero by cor:condition. If some function F of closed-set quantities equaled the flux of a radiating binary, the same state would give F = 0 (closed realization) and F ≠ 0 (radiating realization). Contradiction. Hence epsilon_W, epsilon_h are not elements of the closed algebra; whether and how strongly a state leaks is not encoded in the state. ∎

Candidates examined and rejected (script Part E): tau_Y² is nonzero on closed orbits — it is the conservative phase divergence, stored as precession, not dissipated; Delta_Q(O) and delta_o − 1 are the closed ellipse’s internal breathing, time-averaging to zero; ⟨delta⟩_cycle − 1 is precisely the meter of the leak (it is defined by the flux), so using it as the source law is circular.

Lemma 2 (missing parameter). The empirical decay of binary pulsars depends on both masses separately (equivalently on eta). The single-scale closed set has no such parameter: its state (beta, e, R_s) is invariant under exchanging the bodies’ individual scales at fixed sum. Hence no flux law over the published closed set can even host the observed dependence. The minimal repair is the two-scale state (beta, e, R_s, eta) — already implicit in R.O.M.’s L1 section, but not part of the closed system. ∎

Conclusion. The missing object is the coupling of the orbital sector to a propagating channel — in the framework’s own words (WILL_RG_I, sec:challenges), “a relational description of propagating degrees of freedom on the carriers,” which “remains an open technical task.” The closed structure does not determine epsilon_W and epsilon_h. This derivation stops here.

7. What the framework’s principles force about the missing law

Under A4 + A5, the principles constrain — but do not fix — the fluxes:

The magnitude and the (beta, e, eta)-dependence remain undetermined by the closed structure. Stop.

8. Internal-consistency check

rom_decay_verify.py (repo root; requires sympy + mpmath; runtime ~1 min) verifies 20 checks, all passing:

9. GR-consistency note (comparison only — nothing here is derived from WILL)

The unique flux pair that makes the relational web reproduce the general-relativistic quadrupole (Peters 1964) is, in R.O.M. variables:

# GR-matching fluxes (imported for comparison, NOT derived from the framework)
eps_W_GR = Rational(128,5)*pi*eta*beta**5 \
           * (1 + Rational(73,24)*e**2 + Rational(37,96)*e**4) / (1 - e**2)**Rational(7,2)
eps_h_GR = Rational(64,5)*pi*eta*beta**5 \
           * (1 + Rational(7,8)*e**2) / (1 - e**2)**Rational(5,2)

Both satisfy every constraint C1–C5 (C3: 128/2 = 64 at e = 0). Inserted into the forced web they yield, as verified exact identities: Peters’ de/dt; the standard Pb-dot formula via Tdot = −(3/2) eps_W; and numerically for PSR B1913+16 (m1 = 1.438, m2 = 1.390 M☉, e = 0.6171340, Pb = 27906.98 s): Tdot = −2.4021×10⁻¹² against the literature GR value −2.40263×10⁻¹² (0.02%, within the rounding of the input masses). The observed β⁵ scaling (“v⁵”) and the η-dependence confirm Lemma 2: the flux lives outside the single-scale closed set.

One structural difference survives independently of the flux choice: the energy bookkeeping. Per unit orbiter rest energy, circular-orbit energy in x = beta²:

First order identical, second order different — the same 1st-order-agreement / 2nd-order-departure pattern the source documents establish for precession and time dilation. Consequence: even with identical fluxes, RG predicts a different relation between radiated power and period decay at O(beta⁴), i.e. in late inspiral. This is a falsifiable statement of the forced part of the law, requiring no knowledge of the missing flux functions.

Derived strictly from WILL_RG_I.txt and R.O.M..txt; every forced step machine-verified; every assumption listed in §3; the undetermined element identified, proven undetermined (§6), and left open in agreement with the source framework’s own assessment.