Summary. A relational reformulation in which the fundamental invariant is a “count” — a reparametrization-invariant phase functional over histories — is developed into a candidate gravity theory. With stated postulates it recovers Newtonian gravity, the full weak-field limit of GR with PPN β = γ = 1, and passes all 1PN tests identically to GR. It departs from GR at 2PN with exactly computable coefficients, and its static solution is the horizonless exponential (Yilmaz-type) metric. Two quantitative tests follow. Test I (new): the conservative inspiral tower shifts by Δα₂ = −10 and Δα₆ = +229.8 (flux fixed, test-mass numerators). Because the theory predicts approximately absolute coefficient shifts while δφ̂ᵢ is fractional and η-dependent (α₆ changes sign at η ≈ 0.053), the meaningful comparison is in absolute units at the catalogue’s mass ratios: the GWTC-4.0 restricted bounds correspond to ≈ 7.2 (2PN) and ≈ 39 (3PN), so the excesses are 1.4× and 5.9× — pressed, with the single coupled escape Δf₂ = −1 zeroing 2PN and halving 3PN to ≈ 2.8×. No verdict is drawn from single-coefficient readings; the likelihood-level comparison is the decisive open computation. Test II: black-hole shadows +4.63% larger than Schwarzschild, opposite in sign to Kerr spin corrections; conditional on the not-yet-derived rotating solution. A named consistency obstruction (the Gupta–Feynman–Deser bootstrap; Misner’s Yilmaz critique) gates the whole construction and is unresolved.
State: histories γ in a configuration space carry a phase z[γ] = e2πiN[γ].
Worldline theorem (conditional). Given A1–A4 and a quadratic form, N[γ] = ν₀∫dτ, with ν₀ a system constant. Operationally ν₀ = mc²/h (an identification, not a derivation of mass). Note N = S/h: the quantum sector is the path integral in changed variables and claims nothing beyond it. Only count differences are observable. The count vanishes on null curves: photon predictions below are carried by the metric, not by the count.
Stationary count (stationary phase of SUM) applied to N ≈ ν₀∫(1 + Φ/c² − v²/2c²)dt gives
Source law: shift symmetry (only rate ratios observable) + exact superposition (linearity) + locality/isotropy at lowest derivative order make the field density uniquely |∇Φ|²; stationary total count then yields ∇²Φ = 4πGρ, G empirical. Open: ρ is mass density only — pressure and radiation sources are undefined (no TOV sector yet).
Expansion gives PPN β = γ = 1: light deflection 1.75″, Cassini Shapiro, Mercury 42.98″/cy, Nordtvedt η = 0 — all identical to GR at 1PN (verified numerically: ray-traced deflection → 4GM/c²b with a 1/b residual that is the metric’s own second-order bending term, exactly 4πM²/b² — 16/15 of GR’s 15π/4; perihelion quadrature matches 1PN at the 10⁻³–10⁻⁴ level over its converged range a ≤ 10⁴ M). Tensor completion (“sharpened” form): Kμν obeys linearized Einstein equations; observable metric g = η e2K. Then frame dragging and quadrupole radiation match linearized GR (GP-B/LARES, double pulsar pass at current precision), conditional on §4.
| 1 | u | u² | u³ | |
|---|---|---|---|---|
| g₀₀ count / Schwarzschild (isotropic) | 1 / 1 | −2 / −2 | +2 / +2 | −4/3 / −3/2 |
| gsp count / Schwarzschild | 1 / 1 | +2 / +2 | +2 / +3/2 | — / — |
Identical through 1PN; first departure at 2PN (u = GM/c²r). The 2PN family is exact and extends to light: the second-order bending coefficient is 4π M²/b² against GR’s 15π/4 — ratio 16/15 exactly (verified by 30-digit quadrature of the Fermat integral) — unmeasurable at the solar limb but part of the same tower as Δe₂ = −1/3 and Δe₃ = −35/18. Strong field: no horizon (g₀₀ = e−2u > 0); areal-radius minimum (throat) at r = GM/c², 𝓡 = e·GM/c²; coordinate light-travel time to r→0 diverges (no echoes), proper infall time finite (≈3.53 GM/c³ from 3GM/c²). Compatibility with surface-emission constraints on horizonless objects (Broderick–Narayan) is argued, not shown: those constraints address steady-state thermal re-emission, and neither formation from collapse nor the long-time fate of accreted matter near the throat has been computed.
Obstruction (unresolved): a free spin-2 field's linearized Bianchi identity forces η-conservation of the source, but matter moves on g = ηe2K; iterating the repair is the bootstrap that terminates in Einstein's equations (Gupta 1954; Feynman; Deser 1970; cf. Wald). Misner (gr-qc/9904052) argues the Yilmaz-type vacuum is inconsistent without a distributional source at the origin. Candidate escape: the composition postulate refuses the bootstrap premise (K stays linear; observable metric is the exponential). Not demonstrated. This gates every result below. Also noted: Φ requires an asymptotically flat quasi-static reference — the theory as formulated does not extend to cosmology — and the fixed background η is in tension with axiom A3.
Circular test-mass orbits in the count metric have the closed form
Gauge-invariant expansion E(x) = 1 − (x/2)(1 + e₁x + e₂x² + …), x = (MΩ)2/3:
Stationary-phase transfer with the GR flux held fixed: the SPA decomposition α₂ = 10(3e₂ − 2e₁f₁ + f₁² − f₂) (verified against TaylorF2 exactly) gives ∂α₂/∂e₂ = +30, ∂α₂/∂f₂ = −10, so Δα₂ = 10(3Δe₂ − Δf₂) = −10 − 10Δf₂, hence
Comparator (GWTC-4.0, arXiv:2603.19020 v2, Table 2; 42 O4a + 49 earlier events; PN bounds improved 1.2–5.5× over GWTC-3). δφ̂₄ at 90%: hierarchical −0.02+0.45/−0.43 (FTI·EOB), +0.07±0.45 (TIGER·Phenom); shared-value “restricted” −0.03+0.16/−0.15 and +0.06+0.24/−0.25; δφ̂₆ (non-log row): restricted ±0.06 (FTI) / ±0.13 (TIGER), hierarchical ±0.18 / ±0.29. At the catalogue’s mass ratios (η ≈ ¼) the prediction is δφ̂₄ = −10/α₂(¼) = −0.216 — on the restricted edges (−0.18/−0.19), with the ±18% numerator caveat straddling both. No verdict is drawn from either reading: the theory predicts approximately absolute Δαᵢ, which maps to per-event-varying fractional δφ̂ᵢ — a pattern the shared-value analysis forbids and the hierarchical one merely permits. Neither tests it.
The full tower (corrected in this revision — the earlier 3PN transfer was incomplete). e₃ = −7195/576 vs GR −675/64 ⇒ Δe₃ = −35/18 exactly. The 3PN SPA coefficient takes contributions from both energy shifts: ∂α₆/∂e₃ = −80 and ∂α₆/∂e₂ = 60f₁ = −6235/28, giving the flux-fixed Δα₆ = +155.6 + 74.2 = +229.8. Fractional comparison at 3PN is ill-posed — α₆(η) crosses zero at η ≈ 0.053 (q ≈ 17), inside the catalogue — so the meaningful units are absolute coefficient shifts at η = ¼: restricted bounds ≈ 7.2 (2PN) and ≈ 39 (3PN) against |Δα₂| = 10 and |Δα₆| = 230, excesses of 1.4× and 5.9×: 3PN is the harder constraint. (i) Since 3Δe₂ = −1, Δf₂ = −1 sets Δα₂ = 0 identically. (ii) ∂α₆/∂f₂ = 4975/42 = +118.5, so the same Δf₂ = −1 takes Δα₆ to +111.3 (≈ 2.8× the bound): the escape and the escalation are one unknown. Decisive next computations: Δf₂ and Δf₃ (one problem), the two-body dynamics (η-dependence of Δeₙ), and the likelihood-level comparison against public posteriors. ISCO for reference: uisco = (3−√5)/4, xisco = 0.1589 (Schwarzschild 1/6) — a ~5% earlier termination in x and a shifted merger frequency (a prediction; not checked against observed peak frequencies here); inspiral energy release 5.479% vs 5.719% of Mc².
Null orbits: effective potential e−4u/r² ⇒ photon sphere r = 2GM/c², critical impact parameter b = 2e·GM/c² = 5.4366 GM/c² (Schwarzschild 3√3 = 5.1962):
Sign-opposite to Kerr spin (0 to −7.5%, always smaller). Sgr A* (θg = 5.125 μas from GRAVITY orbits): predicted 55.7 μas vs Schwarzschild 53.3; EHT deviation δ = −0.08 ± 0.09 (VLTI) / −0.04 +0.09/−0.10 (Keck): prediction sits ~1σ–1.4σ outside, not excluded. Caveats: EHT δ is Kerr-calibrated (assumed insensitive to the exterior difference — unproven); M87* is not used (ring ≠ shadow; mass prior not independent); the rotating count solution is underived, so the sign argument is a conjecture. Next-generation EHT (few-% shadows) discriminates — if the theory survives §5.
| Observation | GR | Count gravity |
|---|---|---|
| Clocks vs height, to 10⁻¹⁹/mm (Bothwell 2022) | ✓ | ✓ (RATE directly) |
| 1PN classical tests (deflection, Shapiro, perihelion, LLR) | ✓ | ✓ identical (β=γ=1) |
| Frame dragging (GP-B, LARES) | ✓ | ✓ conditional (linear tensor sector; §4 obstruction) |
| Binary pulsar decay (6×10⁻⁵) | ✓ | ✓ conditional (quadrupole from linear sector) |
| GW speed (GW170817) | ✓ | ✓ |
| GW 2PN/3PN phasing (GWTC-4.0) | ✓ | |Δα₂| = 10 (1.4× absolute restricted bound at η=¼), |Δα₆| = 230 (5.9×); Δf₂ = −1 zeroes 2PN and halves 3PN — flux tower and likelihood test undetermined |
| EHT Sgr A* shadow | ✓ | compatible within ~1.4σ; predicts +4.63% (conditional) |
| BH thermodynamics (area law) | ✓ | no object — unaddressed |
| Neutron-star structure (pressure sources) | ✓ | undefined |
| Cosmology / CMB | ✓ | outside the theory's domain as formulated |
Abandonment criterion, on the record: if (1) resolves against the exponential map and (4) closes onto the quadratic form, the axioms' only consistent completion is GR, and the program ends rather than retreats to interpretation.
The static metric is Yilmaz's exponential metric (1958); its wormhole/throat structure is analyzed in Boonserm–Ngampitipan–Simpson–Visser (2018). No priority is claimed for the metric — only for the counting route to it, the exact Δe₂ = −1/3 inspiral coefficient, and the two-kill-test framing. The consistency critique inherited from the Yilmaz literature is adopted here as the program's own gating problem, not disputed away.
Plain Node.js, no dependencies, shipped beside this note:
verify/verify-math.js · sha256 276dd7a70a4e595beb9c088e5c5c4d5814d16b8d1b9834e7d208554d51584fd6
verify/gw-2pn.js · sha256 e9a7f92f3c6544c5969dc1e204769ed2d0f1c89a30ff9a6489cac79f66ca6096
verify/pn-tower.js · sha256 29e49fda14469183da3dba3e035e3d5e9970820bcba3ffc1fbf3a0470653f234
A hash without the file is a commitment device, not reproducibility: the scripts travel in the same folder as this note. Re-issued after an external audit found two fitting bugs in gw-2pn.js (an under-resolved E(x) fit and a wrong phase-alignment basis) that made the shipped script disagree with the — correct — published −0.33 by 7.6%; the transfer self-check now asserts rather than prints. No published physics claim was affected.
The first reproduces the series table, photon sphere, deflection ray-trace, perihelion quadrature, throat integrals and shadow numbers; the second reproduces E(x) coefficients, the phase-transfer factor, δφ̂₄ and the ISCO. The closed form E(u) was verified against direct geodesic computation to 4×10⁻¹⁶; e₂ = −89/24 confirmed by Richardson extrapolation.