An emergent response
The medium does some of the work.
A material can behave simply at large scales even when its constituents obey much richer rules. Could gravity have a similar microscopic origin?
LQ8 makes this question calculable for one proposed ingredient: a quantum layer with eight energy bands. A variable called x changes the way its states mix. We calculate how the energy of its occupied states responds, including the effect of spatial variations.
To use that response in gravity, the proposal identifies x with the component of a gravitational potential gradient along an interface normal. This identification, and the physical origin of the interfaces themselves, remain inputs to the model.
In everyday language
A local approximation lets each point respond on its own. The full calculation lets quantum states connect points across the layer. The difference becomes measurable when the imposed pattern changes over short distances.
The microscopic definition
Eight bands, one specified operator.
The Hamiltonian defines the model rather than being inferred from an astronomical fit. Its state space combines three two-state factors, giving 2 × 2 × 2 = 8 components. The numerical calculation retains all eight bands.
- τ, σ, ρ
- Pauli operators acting on the three two-state factors; ρ here is an internal index, not a matter density.
- PL, PR
- Projectors (1 + τz)/2 and (1 − τz)/2. The state |1⟩ belongs to the auxiliary ρ space.
- M = 1
- Mixing between the left and right sectors.
- Ω = 20
- Energy penalty separating the low-energy and auxiliary states.
- rW = 0.45
- Wilson coefficient selected from eight values in the galaxy comparison, then held fixed for the new interface calculations.
- x
- Dimensionless interface variable; proposed gravitational identification x = n · ∇Φ / gdark.
Occupation, normalization, and the low-energy projection
At a uniform source, the two lowest bands are occupied. The chemical potential is fixed at μD = −2M² / (Ω + √(Ω² + 4M²)). The energy density e is half the sum of their energies, integrated over the Brillouin zone with normalized measure d²q/(2π)². A periodic cell of L sites has 2L occupied levels and energy normalized by 2L.
For large Ω, projection gives an effective mass meff(x) ≈ Mx/√(1 + x²). This helps explain the response: the low-energy gap opens at small x and saturates at large x. The reported equilibrium solutions use the full Hamiltonian rather than this projected expression.
At the critical uniform background, the low-energy crossing occurs at the chemical potential. For the nonuniform equilibria, occupation is checked on the integration grid and at additional momenta; these samples do not prove the filling condition everywhere between them.
From energy to a force law
Why a cubic energy term matters.
Near the critical uniform state, the band energy contains a negative quadratic term and a positive cubic term. If a bare gravitational term cancels the quadratic part, the cubic contribution controls the weak-field response.
With the isotropic local coupling below, this leads to μ(g) ≈ g/a0 at low acceleration. In spherical symmetry, the gravitational acceleration then approaches g ≈ √(a0gN). The corresponding flat-rotation scaling, vflat⁴ = GMba0, is the established MOND limit; recovering it does not by itself validate or establish the novelty of this microscopic model.
For the observed acceleration relation that motivates this limit, see McGaugh, Lelli & Schombert (2016). The present comparison is quantified on the observations page.
The local functional and critical matching
Here g = |∇Φ|, ρb is baryonic matter density, and u averages over interface orientations. The selected fit gives gdark ≈ 1.82955 × 10−10 m s−2 and a0 ≈ 2.22600 × 10−10 m s−2.
The uniform response defines this local model. A varying three-dimensional potential cannot in general be treated by substituting its value independently into a uniform energy table.
The new effective construction
A spacetime action with a constrained response.
The solar study adds an explicit metric and a preferred time foliation to the fitted force law. A constraint fixes the trace of gravitational momentum and removes the extra scalar wave that made the earlier metric attempt unstable. On the regular branch, two tensor gravitational modes remain.
The force response keeps the selected transition A = −0.45, T = 4.25 and the scale gdark = 1.571664 × 10−10 m s−2. An additional term returns the response to Newtonian gravity at extraordinarily small acceleration. The familiar MOND scaling becomes an intermediate regime; the original cubic limit above is no longer the exact zero-field asymptote of this candidate.
A prescribed clock term allows ordinary homogeneous dust and radiation, which the intermediate pure kinetic-conformal action could not support. The high-acceleration branch recovers the GR metric used for Mercury and Cassini’s light delay. These are consequences of the added effective action under its stated approximations.
A target for the microscopic study
The complete constrained action has not been derived from the eight quantum bands. The new construction below supplies additional quantum sectors for parts of its response, while retaining the metric and matter coupling as assumptions.
Read the solar tests · Inspect the complete action and constraints.
6 September 2026 / A partial construction
What could produce these equations?
We can now specify quantum constituents whose solved energies produce parts of the desired response. Making the complete gravitational dynamics emerge is still unfinished.
The construction adds a driven oscillator to the original quantum bands. Its relaxed energy gives exactly the selected force transition, after an explicit quadratic energy matching. Two bosonic modes and a filled fermionic doublet generate a spectral alternative to the ultraweak cutoff. A separate three-mode Gaussian cell responds to changes of shape with the local shear inertia required by the effective action.
| Target | Result in this study |
|---|---|
| Selected force transition | Exact oscillator energy; couplings and quadratic matching chosen as inputs |
| Ultraweak response | New boson–fermion spectral law; solar screens recalculated |
| Local shear inertia | Five positive shape directions and one volume null direction |
| Clock and full constraints | Not derived; a passive Berry-clock attempt fails |
| Spatial curvature and matter coupling | Still assumptions |
The oscillator Hamiltonian and exact transition
In dimensionless cell units, set t = x/T, C = −(27/16)AT² = 13.7162109375, A = −0.45 and T = 4.25. The canonical oscillator is
Obtaining ΔU requires subtracting the zero-point quadratic contribution through the bare elastic coefficient. Without this matching, μ would shift by 1/T² ≈ 0.05536. The matching has no demonstrated protection here. Twenty-seven diagonalizations, using 24, 40 and 64 basis states, check energy, both source derivatives and adiabatic inertia; the largest absolute discrepancy is 2.95 × 10−13 in the stated cell units.
The same calculation gives a positive coefficient of time-dependent source motion. With x identified with the lapse gradient, this introduces terms absent from the target constrained action. Their low-frequency suppression and effect on the constraints need to be established.
The spectral infrared response and quantum shape cell
Two oscillators contribute +B√(x² + ε²) in ground energy. Two occupied negative levels of a four-component Clifford Hamiltonian contribute −B√(x² + ℓ²). Here ε = 10−9, ℓ = 0.01 and B = εℓ/(ℓ − ε). The resulting response correction is
This replaces the previous cutoff; it is not algebraically the same function. Spectral weights and the gap hierarchy remain inputs. Both sectors add positive adiabatic inertia, despite their opposite ground-energy signs.
For the shape sector, choose a positive matrix Q with det Q = 1 and HQ = (pᵀQ−1p + ωs²yᵀQy)/2. Its ground state has quantum metric tr[(Q−1dQ)²]/8. If Q is identified with h/(det h)1/3, the local slow-motion action is
This verifies a local rank-five inertia. It does not supply the spatial gauge constraints that reduce gravitational motion to two physical polarizations, or the spatial curvature term R(3). The geometry identification, proper time and proper cell density are assumptions.
Two restrictions on completing the model.
The clock term is not produced by the simplest quantum geometric phase. For a passive, nondegenerate ground state with a finite gap, the clock’s required Berry curvature would also bring nonzero volume inertia. That conflicts with the exact volume constraint of the target action. A pure phase change gives only a boundary term and does not fix this.
The physical scales also resist the simplest identification. If the shape and infrared cells have the same number density, matching the measured gravitational coefficient and the chosen force scale leaves at least one characteristic frequency below 5.24 × 10−28 rad/s. Both sectors cannot then be treated as fast quantum modes. Different sectors could evade this restriction, but their density hierarchy needs a physical explanation.
The solar result survives conditionally
All 17 sampled Galactic fields remain inside the quoted Cassini ±2σ quadrupole band, with central Q₂ = -0.44352 × 10−27 s−2. The calculated additional Mercury advance remains below 1.93 × 10−6 arcseconds per century under the retained metric assumptions. These static checks do not establish the missing microscopic dynamics.
The next construction needs an exact constrained phase space or a controlled mechanism preserving the trace and lapse constraints, together with spatial interactions that produce the curvature action and universal coupling to matter.
Complete derivation and restrictions · Code, data and reproduction.
Quantum adiabatic inertia: D’Alessio & Polkovnikov. Constraint protection in an existing geometric theory: Restuccia & Tello-Ortiz. Related emergent tensor constructions: Gu & Wen. None supplies the missing complete derivation for these cells.
The quantum-interface benchmark
A point cannot always respond alone.
The full eight-band energy depends on the entire profile xj. In this benchmark, a sinusoidal generalized source is applied to a periodic cell. Every degree of freedom that preserves the chosen mean is allowed to relax.
At a critical uniform background, the computed spatial stiffness begins in |Q| rather than Q². This provides a mechanism for a weaker response to short-period forcing. The numerical results show how that mechanism behaves beyond infinitesimal perturbations.
What is still an input
The gap between an interface and gravity.
The model assumes a lattice Hamiltonian, a state occupation, a coupling to the potential gradient, and a critical quadratic cancellation. The physical interface geometry and its microscopic scale have not been derived.
The constrained candidate now postulates universal coupling to matter and light and an explicit metric action. Deriving those ingredients, establishing global dynamics, and obtaining controlled predictions for lensing, nonlinear cosmology, and strong-field systems remain open. None follows automatically from a successful static band calculation.
The decisive next step
Derive a common three-dimensional gravitational functional, including the spatial response, and solve it with the same parameters for galaxies and the Solar System. The research programme specifies what that requires.
Related relativistic frameworks are prior art, including Blanchet & Marsat (2011). Their existence does not supply the missing relativistic completion of LQ8.