R.O.M. — machine-checked audit of the closed algebraic system

Companion to the WILL RG Part I audit. Every identity of the R.O.M. equation set is re-expressed as a residual that must vanish, reduced to a single canonical parameterization in the three unitless degrees of freedom (beta, e, O) plus the scale anchor R_s, the unit labels c, G, and the observer angles (i, omega_i, theta_sur).

Result

• 224 checks, 224 with the expected verdict, 0 problems.
• 195 closed symbolically; 29 verified at 50 digits on 24 independent random admissible draws each (the method used in ROM_FULL_TEST.ipynb).
• 8 of the checks are negative controls — identities that must fail. Each one localises a specific discrepancy in the document; they are listed in the assumption ledger below.
• No symbol in any verified identity falls outside the declared registers, and c and G cancel wherever the Factorization Theorem says they must (layer R6: every unit-full quantity divided by its class scale is free of both).

Verification method

simplify() and radsimp() are deliberately not used. On expressions containing symbolic rational powers such as ((1+e)/(1-e))**(1/2) — which pervade the apsidal block — radsimp attempts to rationalize the denominator and blows up (integer explosion → MemoryError). Each check is therefore screened on a relative residual (the absolute residual divided by the largest additive term, since these identities are dimensionful and a true cancellation to 50 significant digits still leaves an absolute residual proportional to the terms being cancelled), and only then passed through a terminating normal-form cascade for symbolic closure.

Where every number in the system comes from

Numerical reproductions (40-digit arithmetic)

Mercury perihelion advance

These reproduce the table in Sec mercury digit for digit.

Sun–Earth L1 from observables only

Neither G nor M enters anywhere in this chain.

Checks by layer

Layer R1 observables (7 checks)

Notes:

• R1.1 — Gravitational redshift is the only input needed for the S^2 amplitude.
• R1.7 — All three forms coincide only because of closure kappa^2 = 2 beta^2.

Layer R10 optics (12 checks)

Notes:

• R10.3 — Confirms the DERIVED deflection denominator beta_p^2(2-kappa_p^2)-…
• R10.4 — NEGATIVE CONTROL / TRANSCRIPTION SLIP: the two printings of the deflection formula differ by the factor kappa_p^2 inside (2-kappa_p^2). The derivation via Gamma(kappa_Xp) supports the second form. residual does not vanish
• R10.5 — Exact leading order in the field strength.
• R10.5b — The printed Rutherford limit silently also assumes beta_p^2 « 1; the exact leading coefficient carries the extra factor (1+beta_p^2).
• R10.7 — Matches the GR photon sphere r = 3GM/c^2 exactly, with no metric.
• R10.8 — Identical to the GR shadow radius 3 sqrt(3) GM/c^2.
• R10.9 — NEGATIVE CONTROL: the two agree only at leading order 2 kappa_p^2 (the correct GR weak-field 4GM/(c^2 b)); they diverge at different radii (kappa_p^2 = 1/2 vs the photon sphere 2/3). The author already flags this section as approximate and under revision. residual does not vanish
• R10.10 — So the disagreement is higher-order, not a weak-field error.

• R10.11 — The four substantival posits (container, autonomous t, coordinate grid, scale c^2dt^2) are what turn this identity into a metric. cascade left 87 ops unresolved; verified numerically instead, max

  residual

  = 1.34e-51

Layer R11 rotation (15 checks)

Notes:

• R11.3 — Exact agreement with the Kerr metric, derived without a metric.
• R11.9 — Part I’s additive S^2 closure capped kappa^2 <= 1; the ROTATING branch beta^2 = 1 is what licenses kappa^2 = 2. This is where the Part I internal boundary (A5) is resolved.
• R11.11 — Isolated by antisymmetrising in beta_spin: matches the published polynomial term for term.
• R11.14 — Matches Omega_dot T = 2GJ/(c^2 a^3) for a circular orbit.
• R11.15 — Not an independent relation: it is closure in legacy dress.

Layer R12 waves (WIP) (5 checks)

Notes:

• R12.1 — Both balance points satisfy r = a, so each horizon contributes two crossings; the source argument 2*theta follows by counting.
• R12.4 — Uses 4 eta <= 1, kappa_obs^2 <= 1 and the STATIC bound beta_src^2 <= 1/2.
• R12.5 — The author attributes the factor 2 to the historical R_s = 2GM/c^2 convention. It is a real factor-2 offset from the quadrupole formula, not an identity – consistent with the WORK IN PROGRESS label.

Layer R13 numerical (10 checks)

Notes:

• R13.1 — per orbit: RG = 5.01867565337e-7 rad, GR = 5.01867573869e-7 rad, difference = -8.53143e-15, predicted = -8.53143e-15, relative agreement = 1.51e-33
• R13.2 — RG = 42.9807844914’’ , GR = 42.980785222’’ , difference = -7.30646e-7’’ , omitted/total = 1.7e-8
• R13.3 — from observables: 2954.79122 m vs 2GM/c^2 = 2953.25008 m (relative gap 0.000522); G and M never used.
• R13.4 — R_E = 1.49622005e+11 m; mu = R_sE/R_sSun = 3.03059e-6
• R13.5 — quintic root: 1.49625e+9 m; cube-root approximation: 1.501288e+9 m; published 1.498e9 m
• R13.6 — Closed form not given in the paper: the quintic factors completely into a scale ratio against a cubic in the separation. Both sides are polynomials, so this is an exact identity in (alpha, mu).
• R13.7 — The x and x^2 terms on the right are exactly the corrections the cube-root approximation drops; this is the relational analogue of the Hill-radius scaling.
• R13.9 — Not claimed in the paper. The R.O.M. quintic is therefore the classical L1 quintic to O(mu gamma^3) ~ 1e-12 for the Sun-Earth system.
• R13.10 — R.O.M. root = 1.49625e+9 m, classical CR3BP root = 1.49624951e+9 m, relative gap = 3.322e-7. The paper’s ‘better than 0.3%’ claim is limited by the input observables, not by the quintic: the cube-root approximation (1.501288e+9 m) is itself 0.00337 away from the exact root.

Layer R2 global chains (47 checks)

Notes:

• R2.2 — R.O.M. fixes rho_max = c^2/(8 pi G R_s^2) AT THE HORIZON, so the ratio is kappa^6, not kappa^2. G and M still cancel.
• R2.2b — NEGATIVE CONTROL / NORMALISATION CLASH: the printed square-root form is Part I’s, where rho_max = c^2/(8 pi G r^2) is evaluated at the SAME radius as rho. R.O.M. re-defines rho_max at the horizon instead, which turns the ratio into kappa^6. Either the chain entry needs the sixth root or rho_max needs Part I’s same-radius definition. residual does not vanish
• R2.2c — Confirms the two conventions are each internally consistent; only the cross-document reuse of the symbol rho_max is not.
• R2.4 — Holds at EVERY phase O: the vis-viva invariant.

• R2.9 — At the balance point tau_o = tau, so one spectroscopic reading gives kappa^2. max

  residual

  over 24 random draws = 1.78e-51

• R2.22 — True at every phase – the field identity is phase-local.
• R2.27 — The GLOBAL omega is required here, exactly as printed: the phase factors reduce beta_o back to beta.
• R2.27b — NEGATIVE CONTROL: confirms the printed omega is correct and the entry is not a typo for the phase-local omega_o. residual does not vanish
• R2.38 — Carries the Part I 4pi (S^2 surface) normalisation, not Newton’s 4pi/3.
• R2.40b — Three forms of the surface gravity label; G and M cancel in the first two.
• R2.43 — Energy block is Part I’s, rescaled by the anchor R_s instead of E_0.

Layer R3 eccentricity (24 checks)

Notes:

• R3.7 — Equivalent to the phase invariant beta_T^2 eta_o^2 = beta^2(1-e^2).
• R3.8 — This is the content of the inversion; the printed formula is its quadratic-formula root.

• R3.8b — Valid on the half of the orbit nearer periapsis. max

  residual

  over 24 random draws = 2e-51

• R3.8c — NEGATIVE CONTROL / BRANCH RESTRICTION: the second root is (eta_o-1)/e, also positive there, and e becomes the SMALLER of the two. The entry as printed needs a +- with a selection rule, like the ones already stated for Methods B and C. residual does not vanish

• R3.8d — Completes the branch structure the printed entry is missing. max

  residual

  over 24 random draws = 6.68e-51

• R3.8e — So the two roots coincide at O = arccos(-2e/(1+e^2)); the printed + sign is correct before that phase and the - sign after it.

• R3.8f — Identifies the spurious branch explicitly. max

  residual

  over 24 random draws = 6.95e-50

• R3.10 — Structural (kappa on S^2) and dynamical (beta on S^1) ratios coincide. cascade left 31 ops unresolved; verified numerically instead, max

  residual

  = 4.46e-51

• R3.17 — The balance point is defined by r = a, i.e. kappa_o = kappa.
• R3.19 — So delta_o(B_a) = 1: the balance point is the local circular-equivalent state.

Layer R4 phase (18 checks)

Notes:

• R4.6 — Pythagorean replacement for vis-viva: no scalar potential subtraction. cascade left 20 ops unresolved; verified numerically instead, max

  residual

  = 8.35e-53

• R4.15 — Verifies the closed form against the mean-anomaly integral by differentiating instead of integrating. cascade left 549 ops unresolved; verified numerically instead, max

  residual

  = 5.44e-51

Layer R5 invariants (15 checks)

Notes:

• R5.8 — Uses the descending branch zeta(>pi) = zeta_closed + 2pi. max

  residual

  over 24 random draws = 0

• R5.9 — The line-of-sight factor is exactly what the phase product strips off.
• R5.10 — Vanishes identically in e, omega_i, i and beta: the inclination and argument of periapsis cancel completely. Genuine invariant.

Layer R6 factorization (28 checks)

Notes:

• R6.a — hat_a = 1/(2*beta**2)
• R6.r_p — hat_r_p = (1 - e)/(2*beta**2)
• R6.r_a — hat_r_a = (e + 1)/(2*beta**2)
• R6.r_o — hat_r_o = (1 - e*2)/(2beta*2(e*cos(O) + 1))
• R6.R_sur — hat_R_sur = sin(theta_sur)/(2*beta**2)
• R6.T — hat_T = pi/beta**3
• R6.t — hat_t = 1/(2*beta**2)
• R6.tau_D — hat_tau_D = pi*sin(theta_sur)(3/2)/beta3
• R6.Delta_to — hat_Delta_to = (-esqrt(1 - e2)*sin(O)/2 + (e*cos(O) + 1)*atan(tan(O/2)/(sqrt(-1/(e - 1))*sqrt(e + 1))))/(beta3(e*cos(O) + 1))
• R6.omega — hat_omega = 2*beta**3
• R6.h — hat_h = sqrt(1 - e*2)/(2beta)
• R6.M — hat_M = 1/2
• R6.rho — hat_rho = beta**6/pi
• R6.Mhat — Independent of beta, e, O: the mass label carries no relational information beyond the scale anchor R_s.
• R6.tauDhat — kappa_sur^2 = 1 means sin(theta_sur) = kappa^2 = 2 beta^2.
• R6.rho.control — NEGATIVE CONTROL: hat_rho would be beta*6/(piR_s), which still carries R_s. The first printing of the table (Historical Units Scale) gives c^2/(G R_s^2) and is the dimensionally correct one. residual does not vanish

Layer R7 input channels (12 checks)

Notes:

• R7.E1 — Uses only a ratio of two angular speeds – no length, no time unit. cascade left 14 ops unresolved; verified numerically instead, max

  residual

  = 1.34e-51

• R7.B3 — This is exactly the Part I rotating-globes inversion beta^2 = (3-sqrt(1+8 tau^2))/4, re-derived here as an orbital channel. max

  residual

  over 24 random draws = 1.34e-51

• R7.B5 — Independent of both r_1 and r_2: any two epochs give the same beta.
• R7.indep — E5 reduces to an identity in e alone with beta cancelling; the beta-menu is independent of e except through explicit e_X factors.

Layer R8 horizon scale (16 checks)

Notes:

• R8.A — Two astrometric radii plus two de-projected speeds. No period, no mass.

• R8.B — Negative root. One geometric scale plus one spectroscopic reading. max

  residual

  over 24 random draws = 4.74e-50

• R8.B.pos — NEGATIVE CONTROL: documents why the negative root is selected. residual does not vanish

• R8.C — Holds at every phase O, so a single epoch suffices when a is known. max

  residual

  over 24 random draws = 8.67e-49

• R8.D2 — G cancels between rho_sur and the prefactor: tau_D is purely observational. cascade left 22 ops unresolved; verified numerically instead, max

  residual

  = 1.57e-51

• R8.D4 — At kappa_sur^2 = 1 closure fixes beta_sur^2 = 1/2; G and M are absent.
• R8.D6 — Density of the central body from period and angular radius alone.

• R8.D8 — So the printed entry theta_sur = cbrt(pi/(rho G T^2)) is missing an arcsin; numerically harmless for small angular radii but formally a slip. cascade left 15 ops unresolved; verified numerically instead, max

  residual

  = 6.68e-52

• R8.D9 — NEGATIVE CONTROL: the cube root returns sin(theta_sur), not theta_sur. residual does not vanish

Layer R9 legacy laws (15 checks)

Notes:

• R9.1 — Force is not introduced; dt enters only through the chain rule.
• R9.3 — Its O-derivative vanishes identically – that IS the conservation law.
• R9.5 — Follows from substituting beta = 2pi a/(cT) and kappa^2 = R_s/a into closure; no mass and no G appear.
• R9.8 — Exact, not expanded: substituting closure and kappa^2 = R_s/a suffices.
• R9.9 — An algebraic identity in (R_s, a, e), not a numerical near-match.
• R9.12 — Delta_phi is exactly quadratic in R_s/a, so ‘GR 1PN as first-order RG’ is the removal of ONE term, not the truncation of a series.

Assumption ledger

Every place the chain rests on a stipulation rather than a derivation, plus every confirmed discrepancy. None of these is a fitted parameter.

Corollaries the formalization makes explicit

• The Part I internal boundary is resolved here: kappa^2 = 2 is reachable only on the KINEMATICALLY closed branch (beta^2 = 1), which is the rotating Kerr case. Part I’s r_min = R_s/2 is therefore a Kerr statement, and the static bound remains r >= R_s (R11.9, Sec Contextual Bounds).
• The R.O.M. Kerr horizons and ergosurface are not approximations of the Kerr metric – they are algebraically identical to it (R11.3, R11.4, R11.6), as are the photon sphere r = 1.5 R_s and shadow b = 1.5 sqrt(3) R_s (R10.7, R10.8).
• The precession expansion TERMINATES at second order (R9.12): Delta_phi is exactly quadratic in R_s/a. So ‘GR 1PN as first-order RG’ is not a truncation of an infinite series but the removal of one explicit term.
• The Holographic Decryption Invariant equals 2 identically in e, omega_i, i and beta (R5.10) – inclination and argument of periapsis cancel completely. This is the one genuinely new closed-form invariant in R.O.M.
• Angular-momentum conservation is not a separate law: d/dO of beta_T/kappa_o^2 vanishes identically (R9.3), so h = R_s c e_Y/(2 beta) is a restatement of the phase parameterization.
• At the balance point B_a = arccos(-e) the measured tau equals the global tau, which is what makes Method B a single-reading determination of R_s (R3.17, R8.B) and reproduces Part I’s rotating-globes inversion exactly (R7.B3).
• Eleven input channels (five for e, six for beta) all reduce to the same two numbers (layer R7), so the closed system really is basis-independent as claimed.
• The L1 quintic has an exact closed form not stated in the paper: mu alpha^2 = (1-alpha)^3 (1+alpha+alpha^2) (R13.6). Written as mu(1-x)^2 = x^3(3-3x+x^2) it differs from the CLASSICAL restricted three-body L1 quintic by exactly one term, mu x^3(2-x) (R13.9) – so the two roots agree to 3e-7 relative (R13.10). The relational derivation reproduces the classical three-body result, not merely its Hill-radius leading order.

Reproducing this

python build_rom_report.py   # writes will_rom_core_checks.csv and this report
python will_rom_core.py      # prints the pass/fail summary only

will_rom_core.py has no side effects on import. run_all() returns the records, audit_zero_parameters() returns the symbol and method census, NUM holds the numerical tables, and CAN is the canonical parameterization every check is reduced against. Adding a claim means adding one eq(...) call; set VERBOSE = True for per-check progress.