# Quantum constituents for the constrained effective gravity candidate 6 September 2026. A constructive, partial microscopic study. ## Result and scope Explicit quantum Hamiltonians now generate the fitted static transition and a spectral alternative to the ultraweak response. A separate Gaussian cell generates the local rank-five, trace-free inertia of the target kinetic term. The clock, spatial curvature term, spatial gauge constraints and universal matter coupling have not been derived. These results do not supply a complete microscopic theory of the published gravitational action. The Hamiltonians depend on slowly varying acceleration and geometric variables whose identification with gravity is assumed. Couplings are chosen to realize the previously selected macroscopic response. A=-0.45, T=4.25 and the galaxy acceleration scale are held fixed; their values are not quantum predictions. The new spectral infrared law differs from the previous cutoff. Its solar consequences are recalculated in `outputs/dark_micro/summary_summary.json`. | Target structure | What this study obtains | | --- | --- | | Original band response | Existing polynomial source Hamiltonian, checked in its physical basis | | Fitted transition | Exact relaxed oscillator energy, after an explicit quadratic matching | | Ultraweak return to Newtonian response | A different, explicit boson–fermion ground-energy function | | K_ij K^ij − K²/3 | Local shear inertia from a cell whose state depends on unimodular shape | | Clock/trace constraint | A Berry-only finite-gap attempt fails to keep volume inertia zero | | Spatial R3 and two physical tensor modes | Not derived from the cells | | Constant G, matter coupling and microscopic scales | Matching conditions and unresolved physical assumptions | The spectral and inertia calculations use hbar=1 and dimensionless energies unless physical units are stated. Direct oscillator spectra at 24, 40 and 64 basis states agree with the analytic energy, first and second derivatives, and adiabatic inertia to better than 3e-13 in the chosen benchmarks. ## 1. The inherited eight-band sector has polynomial source couplings One helicity block can be written in a fixed physical basis as H8(k,x) = [[ k, 0, M, -iW ], [ 0, k+Omega, -iW, M ], [ M, iW, -k+Omega, -Omega*x ], [ iW, M, -Omega*x, -k+Omega*x² ]]. The second block has the opposite helicity. On an interface, k and W are the existing lattice momentum and Wilson term. The source x is the normalized normal component of acceleration. The positive block proportional to Omega is a quadratic penalty of the form Omega*|chi-x*psi|². Apparent rational factors in the earlier diagonalizing basis are not extra physical couplings. The occupied bands, lattice integration and isotropic orientation average are the previously specified ones. Twenty direct physical-basis spectra and source-derivative checks match the inherited implementation. Independent zone and orientation quadratures agree with the saved response at the tested accelerations to about 1.4e-7 on the finer quadrature. For an affine source H=H0+xO in a nondegenerate ground state, E0'' = -2 sum_(n>0) ||²/(E_n-E0) <= 0. Our polynomial Hamiltonians also have a contact term: E0'' = - 2 sum_(n>0) ||²/(E_n-E0). It is included throughout. Omitting it would misidentify the response and incorrectly exclude the required nonmonotonic transition. The bare elastic gradient energy, its critical matching and the interface population remain inputs of this microscopic realization. ## 2. A driven quantum mode produces the fitted transition Set t=x/T, C=-(27/16) A T²=13.7162109375, m=omega0=1, and J0=sqrt(2C). Introduce one canonical oscillator with a polynomial dependence on x: H_tr = p²/2 + (1+t²)² s²/2 - J0*t²*s. Its frequency and displacement are omega(x)=1+t², d(x)=J0*t²/(1+t²)². Completing the square gives the exact ground energy E0(x) = (1+t²)/2 - C*t⁴/(1+t²)². The zero-point term is not discarded silently. Matching the quadratic coefficient of the bare elastic energy removes x²/(2T²); its source-independent constant can be subtracted separately. Before this matching it would shift mu everywhere by 1/T²=0.0553633217993. This matching is an assumption that still needs a protecting mechanism or a physical renormalization prescription. In a gravitational action the source-independent subtraction fixes a vacuum term; it is not a prediction of a small cosmological constant. After that stated matching, Delta U_tr = -C*t⁴/(1+t²)², Delta mu_tr = (27/4) A*t²/(1+t²)³. This is exactly the previously fitted transition. It has been obtained by solving a quantum oscillator with polynomial couplings; its finite couplings were selected with knowledge of the macroscopic target. Integrating out the oscillator also produces a time-derivative correction. For slow x(t), the coefficient of dot(x)²/2 is M_xx = [d'(x)]² + [omega'(x)]²/[8 omega(x)³] > 0. The first term comes from displacement and the second from squeezing. Both are reproduced by the spectral sum with cubic excitation denominators. Thus this oscillator supplies more than its static energy. Its finite gap, adiabatic domain and coupling to the other sectors must be respected. In particular, identifying the source with the lapse gradient, a_i=partial_i ln(N), turns this inertia into time derivatives of that gradient. The target action has a nondynamical lapse. At finite frequency these new terms can change its constraint structure; they cannot simply be omitted in a claimed exact microscopic derivation. A controlled low-frequency expansion or a constraint mechanism must account for them. Static solar tests do not test this issue. ## 3. A spectral ultraweak response Let epsilon=1e-9, ell=0.01 and B=epsilon*ell/(ell-epsilon). At each response cell introduce two identical bosonic oscillators and four fermionic orbitals: H_B = sum_(r=1,2) [p_r²/2 + B² (|x_vec|²+epsilon²) b_r²/2], H_F = (B/2) f^dagger (x_i Gamma_i + ell Gamma_4) f. The four Hermitian Gamma matrices obey {Gamma_A,Gamma_B}=2 delta_AB. Fix two fermions per cell, filling both negative levels. The bosonic zero-point energy is +B sqrt(x²+epsilon²), and the filled fermionic energy is -B sqrt(x²+ell²). Subtracting the source-independent value at zero gives U_IR = B[sqrt(x²+epsilon²)-epsilon-sqrt(x²+ell²)+ell], delta_mu_IR = B[1/sqrt(x²+epsilon²)-1/sqrt(x²+ell²)]. The couplings are rotationally invariant and polynomial in the acceleration components. Matrix spectra at three source orientations verify the result. The different gaps and common spectral weight are specified inputs; this study does not establish a symmetry protecting their matching. The function satisfies delta_mu_IR(0)=1 and has a tail proportional to x^-3. Its energy tends to epsilon*ell. Combined with the original band response and the oscillator transition, it defines mu_S = mu_previous_smoothed + delta_mu_IR. It replaces the earlier epsilon/[(x+epsilon)(1+(x/ell)²)] term. The numerical band splice repair remains the one described in the previous study; it is an interpolation repair, not a new microscopic interaction. Both the transverse and radial stiffnesses remain positive. The numerical minimum radial stiffness is about 3.53e-7. The descriptive galaxy RMS remains 0.14717628636 dex with no scale refit. The largest change from the frozen pre-ultraweak galaxy prediction is about 3.33e-10 dex. The bosonic and fermionic contributions to the adiabatic inertia are both positive, even though their ground energies enter with opposite signs. In the radial source direction they are M_B = x²/[4B(x²+epsilon²)^(5/2)], M_F = ell²/[B(x²+ell²)^(5/2)]. Their energy pairing therefore does not cancel the time-dependent response. The smallest excitation gap is B*epsilon in these energy units. ## 4. Quantum shape response yields a trace-free inertia Let Q be a positive three-dimensional matrix with det(Q)=1. Consider a three-mode Gaussian cell, H_Q = [p^T Q^-1 p + omega_s² y^T Q y]/2. Its normalized ground state is proportional to exp[-omega_s*y^T Q y/2]. The spectrum is independent of Q. Its quantum metric is ds_quantum² = tr[(Q^-1 dQ)²]/8. Only two-quantum excitations contribute to a shape derivative. Their gap is 2 hbar omega_s, giving L_adiabatic = hbar/(16 omega_s) tr[(Q^-1 dQ/dtau)²]. Identify Q=h/(det h)^(1/3), with tau the cell's proper time. Locally this is L_adiabatic = hbar/(4 omega_s) [K_ij K^ij-K²/3]. The six-component inertia has one analytic null direction (uniform volume) and five positive directions. Fock-state spectral sums and Gaussian overlaps verify both the coefficient and the null direction, including anisotropic backgrounds. Roundoff gives an eigenvalue of order 1e-17 for the analytic zero. This construction assumes the identification of Q with normalized metric shape. It supplies neither spatial gauge transformations nor the curvature term. Five positive local shape directions do not establish two physical graviton polarizations. A constant proper density of cells is also an additional assumption when matching the spacetime action. The adiabatic mass construction is based on the general quantum-response framework of [D'Alessio and Polkovnikov](https://arxiv.org/abs/1309.6354). Our explicit cell and its rank-five matching are calculations within that framework, rather than a microscopic derivation of diffeomorphism symmetry. ## 5. Why the simplest Berry clock is insufficient With V=sqrt(h), the target clock term contains b*q(T)*dot(V), b=2/(3 kappa) and kappa=16 pi G in c=1 units. A Berry-only realization can supply an equivalent one-form -b*V*q'(T)*dot(T), up to a total derivative. Its curvature in the V,T parameter plane is nonzero when q' is nonzero. For a nondegenerate gapped ground state the adiabatic mass obeys M_VV = 2 hbar² sum_(n>0) ||²/(E_n-E0). Every term is nonnegative. If M_VV=0, all off-diagonal volume derivatives vanish; the Berry curvature F_VT must then vanish too. A finite-gap Berry-only construction cannot supply the target clock curvature with exactly zero volume inertia. An explicit two-level example verifies the nonzero curvature, positive inertia, and the bound det(g)>=F_VT²/4, including Berry-loop checks. A parameter-dependent overall phase does not evade this result. For |0>=exp[-i*b*V*q(T)]|fixed>, its complete connection is A=b*d[V*q(T)], which is a total derivative. Keeping only its dot(V) term would create a spurious clock force. A prescribed time-dependent potential is another possible source of clock forces, but choosing the target potential is not a microscopic derivation of it. This restriction is specific to the passive finite-gap adiabatic route. Constrained phase spaces, retained slow clock variables, and gapless systems require different analyses. In a fundamental constrained formulation, protection of the kinetic-conformal point need not come from an ordinary Berry construction; [Restuccia and Tello-Ortiz](https://arxiv.org/abs/1908.06581) discuss constraint protection in a particular electromagnetic-gravitational theory. That theory already starts with gravitational geometry and constraints. ## 6. One common density does not give a consistent adiabatic model Restore physical units and let n_s and n_IR denote the proper number densities of shape cells and infrared response cells. Matching the Gaussian inertia to the gravitational coefficient requires n_s*hbar/(4 omega_s) = c²/(16 pi G). The static response energy-density unit is E_ref=K_b*g_dark² with K_b=1/(4 pi G), numerically about 2.9451e-11 J/m³. Matching the spectral pair gives its minimum bosonic frequency omega_IR,0 = B*epsilon*E_ref/(n_IR*hbar). Combining the two conditions eliminates both G and hbar: omega_s*omega_IR,0 = B*epsilon*(g_dark/c)² * n_s/n_IR. For equal densities the product is about 2.7484e-55 s^-2. Consequently the smaller of the two gaps cannot exceed 5.24e-28 rad/s, whatever common density is chosen. Both sectors cannot then be treated as fast degrees of freedom even against a deliberately conservative 1e-18 rad/s cosmological benchmark. These are the oscillator fundamental frequencies. Shape and frequency derivatives excite two quanta; the corresponding factor of two does not remove the hierarchy. Exactly at zero source some derivative matrix elements vanish, so this is a restriction on a common global adiabatic interpretation, not a calculation of the driven response to every infinitesimal background. For both gaps merely to exceed that benchmark, these particular modules need n_s/n_IR > 3.64e18. Requiring both to exceed Mercury's mean orbital frequency would require a ratio above 2.49e42. The latter is a global zero-field-gap criterion, not a claim that the actual high-field Mercury calculation excites the infrared modes: their gaps grow with acceleration. The conclusion concerns the proposed cells and their matching, not all quantum gravity models. Separate microscopic and collective sectors could evade this common-density restriction. Their hierarchy, population and coupling need a physical account; simply choosing two densities does not supply that account. ## 7. Numerical solar checks and the next microscopic step The new response is solved with the same external-field samples and galaxy scale. The response table uses power-law intervals, for which increasing endpoint values of x*mu guarantee positive radial stiffness throughout each interval. The old linear interpolation of mu against log(x) was rejected by its own positivity check in the narrower spectral infrared transition. The original solar solvers and published candidate remain unchanged. The calculation retains all extraction windows and compares grids, boundaries, table density, Green integrals, Mercury orbit averages and synthetic Cartesian fields. These tests concern the static response under the assumed metric action. They do not validate the unresolved microscopic clock or spacetime dynamics. Exact results and numerical controls accompany the study. A further microscopic construction must provide an exact constrained phase space or a demonstrated dynamical mechanism that protects the required trace constraint. It must also supply the spatial curvature action and universal coupling to matter. Existing spin/qubit constructions show why these are substantial requirements: [Gu and Wen](https://arxiv.org/abs/0907.1203) obtain helicity-two excitations in one controlled model with cubic dispersion; their linear-dispersion proposal is less controlled. Emergent tensor excitations alone do not establish the required nonlinear relativistic limit. The useful result of this stage is a smaller set of explicit quantum sectors to work with, together with two precise restrictions on completing them.