02 / OUR INQUIRY · 01
Toward a Theory of
Timeless Gravity.
Our central ambition is to investigate a geometric origin of matter. A separate, calculable study asks how physical clocks describe change from within a system.
Explore the modelEXPLORATORY RESEARCH PROGRAM · GRAVITON
THE AIM / TWO DISTINCT LINES OF WORK
New physics starts
with a precise question.
Graviton’s 2021 manuscript, Particle fields from gravitational waves, explores whether matter and gauge fields can arise from metric perturbations. It is an original research proposal requiring mathematical review, not an established derivation of particle physics.
Relational time is a complementary study. Page–Wootters and later work describe quantum evolution relative to physical clock readings. We use this literature to construct an explicit benchmark. There is no demonstrated mathematical bridge from this benchmark to the manuscript’s geometry–matter proposal.
THE LONG-TERM HORIZON
Seek principles that change what is possible.
Graviton aims to discover new physics: a deeper account of matter, gravity, and time that makes predictions we can test. We also want to investigate whether such an account could open a route to spacetime travel or communication beyond established limits.
There is no demonstrated route from the clock model below to either capability. Under standard quantum mechanics, entanglement alone cannot transmit a controllable faster-than-light message. A known theoretical construction of a traversable wormhole also preserves causality. Sources 8–9.
A proposal that changes these conclusions must identify the assumptions it replaces, recover the observations that existing theories already explain, and make a measurable prediction that distinguishes it. Those questions belong to our separate inquiry into entanglement and spacetime.
GEOMETRIC MATTER / A CONCRETE CONSISTENCY TEST
What survives
the constraints?
Graviton’s central ambition remains a geometric origin of matter. Our immediate target is precise: does the proposed construction contain a physical matter-like excitation after coordinate freedom and the field equations’ constraints are accounted for? A changing formula alone does not establish a new particle.
We have checked two algebraic steps in the 2021 manuscript. These checks use its four-dimensional, linearized setting, constant coefficients, and convention □ = −∂²/∂t² + ∇², with c = 1. Write h = ημνhμν for the perturbation’s trace and ℰ for the effective-source trace.
Two algebraic repairs, with their limits
Retaining the stated relation ℰ = κh, equation 9 requires the factor of one half below. Keeping the printed equation 15 would instead require explicitly redefining the relation as ℰ = 2κh. Neither convention derives that assumed source relation or establishes a physical scalar.
For equation 22, set a = 3μ²/2 − Λ. In a static spherical region r > 0, (∇² − a)h = −4Λ has the particular solution
The printed constant −4 leaves the residual 6μ². These repairs follow by substitution, using ∇²r² = 6. Homogeneous solutions and boundary conditions must also be supplied; solving this scalar equation does not solve the complete tensor system.
The full tensor equation is more restrictive.
Equation 18 has an identically divergence-free Einstein derivative operator. If the effective source is conserved, then, for μ² ≠ 0, its divergence and trace require:
Adding the manuscript’s condition ∂μhμν = 0 forces h to be constant. The current nonconstant radial trace family is therefore excluded under these combined assumptions.
For the cosmological-source specialization, ℰ = −2Λ(4 + h), the constant agrees with 4Λ/a. When a = 0 and Λ ≠ 0, the full algebraic trace relation has no solution. Dropping source conservation requires a justified alternative source model. Consistency of a flat background with nonzero Λ also remains unresolved.
A reproducible null benchmark: an oscillation with zero curvature
In the massless linearized metric theory, take equation 30 with Ax = (ε/k) cos[k(t − z)], all other components zero, k > 0, and small dimensionless ε. Then htx = −ε sin[k(t − z)] and hzx = +ε sin[k(t − z)], together with their symmetric partners.
All 256 index combinations vanish by exact cancellation of commuting derivatives. Although its components oscillate, this metric perturbation is removable locally by ξμ = −Aμ. It does not demonstrate a matter field. A modified theory making Aμ physical requires its own justified symmetries and dynamics; coordinate gauge freedom cannot simply be assumed after adding a mass deformation. Deser’s linear spin-2 construction provides the gauge-theory context.
The next model has a clear acceptance test.
Supply an action, consistent background and source, reduced quadratic dynamics, and a physical mode count. A proposed matter-like excitation must produce a gauge-invariant response to a specified probe. Reject a candidate that violates its constraints, disappears under an allowed coordinate transformation, or has an unstable physical quadratic Hamiltonian within its stated regime.
The coefficient repairs and the local gauge identity are established checks. A viable matter-emergence mechanism remains the research objective. The original manuscript is preserved unchanged, and the relational-clock studies below remain separate supporting work.
The reproduction script uses Python’s standard library and exact symbolic bookkeeping. It verifies the radial substitutions and local pure-gauge identity. The tensor-divergence constraint is an analytical deduction above, and the full manuscript is not validated by these checks.
A FEW TERMS BEFORE WE BEGIN
What the equations mean.
Open the concept guide · 10 short definitions
- Relational time
- Describing one system’s change in relation to another physical system used as a clock. The clock is part of the model, rather than an external timekeeper.
- Hamiltonian, H
- The operator representing a system’s energy and generating its quantum evolution. Here its energy levels are evenly spaced by ℏω.
- Constraint, 𝒞
- A rule selecting allowed joint states. In this example, 𝒞|Ψ⟩ = 0 means every occupied clock–system pair has the specified total energy.
- History state, |Ψ⟩
- A joint quantum state that correlates clock readings with system states. The word “history” refers to these correlations, not a stored classical film.
- Conditional state
- The system state assigned after specifying a clock reading and a measurement rule. Without conditioning, the system in the first model is the equal mixture I/d.
- Phase, θ
- A position in a repeating cycle, measured in radians. Here θ = ωt, where ω is an angular frequency. The clock repeats after 2π and does not label successive cycles.
- Qubit / qutrit
- A quantum system with two / three orthogonal basis states. The three-state example has enough distinct energy differences to probe a second harmonic.
- Coherence
- The relative-phase structure between basis states that can contribute to interference. It depends on the chosen basis; a visible fringe alone does not establish entanglement.
- Visibility, V
- Here, the contrast of the two-state return fringe: (Pmax − Pmin)/(Pmax + Pmin). Different phase-error distributions can have the same contrast.
- Residual norm
- The size of 𝒞|Ψ⟩. It is zero when the state satisfies the constraint. A nonzero result identifies a failure of that proposed state, not an experimental discovery.
Notation: ℏ is the reduced Planck constant; I is the identity operator; ⊗ joins clock and system spaces. The notation |ψ⟩ represents a state vector, and d counts the energy levels.
01 / EXACT FINITE-DIMENSIONAL MODEL
One constraint.
A family of clock readings.
Occupied energy pairs satisfy the same total-energy constraint. Choose a clock phase to inspect the conditional system state. The joint history does not change when you select a different reading.
The return probability is the chance of obtaining the initial equal-superposition state in a projection measurement. It can change even though the energy populations do not.
Swipe across an equation to see the full expression.
Both Hamiltonians have equally spaced energies En = nℏω. The subtracted total energy fixes a reference sector; it is not a claim that the universe has zero energy.
How conditioning produces the displayed state
The clock uses normalized phase states |θ⟩C = d−½ Σn=0d−1e−inθ|n⟩. Its phase measurement is the POVM EC(θ) = d|θ⟩⟨θ|/(2π), whose integral from 0 to 2π is the identity. “POVM” names a general quantum measurement; the clock states need not be orthogonal.
This follows by applying ⟨θ|C to |Ψ⟩, normalizing, and removing a global phase. With θ = ωt it obeys Schrödinger evolution under HS. The clock is periodic: it does not count elapsed cycles.
The displayed return probability is |d−1Σme−imθ|². Changing d changes the occupied energy bandwidth (d−1)ℏω, so dimension comparisons do not hold resources constant. Sources 1–3.
02 / IDENTIFIABILITY BENCHMARK
One visibility.
Two different explanations.
A two-state probe cannot distinguish these two distributions of clock-readout error. A three-state probe can access a second harmonic and separate their predictions.
Wrapped Gaussian
A continuous phase-error distribution wrapped around a circle. At V = 0 we use its uniform limiting distribution.
Equal random offsets
Two possible phase errors, +a and −a, each with probability ½. These are discrete offsets, not two Gaussian peaks.
wk = ∫−ππw(δ)cos(kδ)dδ. Here w1 = V for both models; w2,G = V⁴ and w2,B = 2V² − 1. This is an exact consequence of the specified phase averaging, not a new physical law.
Reproduce the calculation and interpret its limits
Start with |ψ₀⟩ = (|0⟩ + |1⟩ + |2⟩)/√3. Average |ψ(θ+δ)⟩⟨ψ(θ+δ)| over a normalized symmetric error distribution w(δ), then evaluate ⟨ψ₀|ρw(θ)|ψ₀⟩. Expanding the squared sum of three complex amplitudes gives the expression above.
For the Gaussian, σ = √(−2 ln V) and wk = exp(−k²σ²/2). For the binary model, a = arccos V and wk = cos(ka). At V = 0.8 and θ = 0, the calculated probabilities differ by 0.0288. At V = 1 the models coincide. At θ = 45°, their qutrit predictions coincide even when their second harmonics differ.
These figures model uncertain readouts, not a clock operating a detector. A sinusoidal fringe alone does not certify entanglement. Physical tests would require independent state preparation and measurement checks. Phase-noise inference and multilevel probing have established precedents; no novelty is claimed here.
03 / CONSISTENCY DIAGNOSTIC
Change the interaction.
Check the state again.
Add an energy–energy coupling while deliberately keeping the old three-level history. The figure shows exactly when that state stops satisfying the constraint. It does not pretend the old state remains a valid solution.
𝒞λ = HC⊗I + I⊗HS − λHC⊗HS − 2ℏωI. The history is (|2,0⟩ + |1,1⟩ + |0,2⟩)/√3. λ has inverse-energy units.
The zero-residual state must be solved again after changing the model. This diagnostic assumes a coupling; it neither derives gravity nor maps the slider to a real gravitational field. Published interacting-clock models are a next reproduction target. Source 4.
RESEARCH PLAN
What these calculations
let us do next.
- Audit the original geometry–matter proposal.
The coefficient and gauge checks now identify which assumptions fail together. A revised construction must derive its action and constraints, then establish a physical matter-like mode before assigning particle content.
- Reproduce operational clock models.
Add explicit preparation, readout, and detector assumptions. Compare uncertainty in a recorded phase with a physical clock triggering a measurement. They need not describe the same experiment.
- Recover known gravitational benchmarks.
Test relativistic clock predictions with clearly specified backgrounds and interactions. Recovery is a benchmark, not evidence that a new gravitational theory has been established.
- Formulate a distinguishing prediction.
Only a defined model with consistent dynamics, appropriate classical limits, and an observable difference can support a claim beyond the existing frameworks.
METHODS / REPRODUCIBILITY
Equations you can inspect.
Results you can reproduce.
All curves are calculated from the formulas on this page. They contain no experimental data or fitted parameters. Independent matrix and numerical-integration checks of the baseline model gave maximum discrepancies below 4 × 10⁻¹⁵ in double precision.
The checks include constraint satisfaction, conditional states, phase-measurement completeness, density-matrix positivity, and the Gaussian-versus-binary benchmark. They verify the implementation of these finite models, not a theory of gravity.
Python reproduction requires NumPy. Units in the script are ℏ = ω = 1. Near-zero negative eigenvalues of order 10⁻¹⁶ reflect floating-point rounding.
- Page & Wootters (1983) — Evolution without evolution
Foundational conditional-time construction.
- Giovannetti, Lloyd & Maccone (2015) — Quantum Time
Constraints, conditioning, and measurement statistics.
- Chataignier et al. (2026) — Relational Dynamics with Periodic Clocks
Periodic-clock domains and limitations; first preprint 2024.
- Smith & Ahmadi (2019) — Quantizing time: Interacting clocks and systems
Explicit interactions require different conditional dynamics.
- Höhn, Smith & Lock (2021) — The Trinity of Relational Quantum Dynamics
Physical observables and changes of temporal reference frame.
- Hausmann, Schmidhuber & Castro-Ruiz (2025) — Measurement events relative to temporal quantum reference frames
Nonideal clocks and operationally different measurement procedures.
- Ali, Alshal & Haq (2021) — Particle fields from gravitational waves
Graviton’s geometry–matter manuscript. Separate proposal under mathematical review.
- Horodecki et al. (2009) — Quantum entanglement
Quantum correlations and their operational uses within established theory.
- Gao, Jafferis & Wall (2017) — Traversable Wormholes via a Double Trace Deformation
A specific coupled-boundary construction; traversability without a causality violation.
Updated
FIRST INQUIRY / OPEN RESEARCH
The next step is a revised geometric construction with consistent dynamics and a gauge-invariant observable. The completed consistency checks define what that construction must overcome.
Return to the curious reader brief