# A constrained metric candidate for the solar tests

Working derivation, 5 September 2026. This is an exploratory effective model,
not a derivation of gravity from the eight-band Hamiltonian. Numerical results
and the final acceptance statement will be recorded separately. Earlier
failed candidates remain in their original files.

## Action and its status

Use c=1 and write the ADM variables as lapse N, shift N^i and spatial metric
h_ij. Let K_ij=(dot h_ij-D_i N_j-D_j N_i)/(2N), K=h^ij K_ij,
a_i=D_i ln N, and a=(h^ij a_i a_j)^(1/2). The candidate action is

    S = (16 pi G)^(-1) integral N sqrt(h)
        [K_ij K^ij - K^2/3 + R3 + F(a)
         + (2/3) q(t) K + q(t)^2/6] + S_m[g, matter].

q(t) is spatially uniform. It labels the trace of gravitational momentum,
not a freely propagating local field. A concrete clock prescription is
q(t)=-2/t, t>0; t is a foliation coordinate, not necessarily proper cosmic
time. The homogeneous lapse must be solved. A constant-q approximation is
appropriate to the local solar calculation; setting q=0 gives the previously
tested kinetic-conformal local action.

Equivalently, with a timelike foliation scalar T, unit future normal
n_mu=-partial_mu T/sqrt[-(partial T)^2], and K=div n, the covariant notation is

    S = (16 pi G)^(-1) integral sqrt(-g)
        [R4 + (2/3)(K+q(T)/2)^2 + F(a)] + S_m[g, matter],

up to the usual gravitational boundary term. q(T) remains an explicitly
specified clock function. Universal coupling of ordinary matter to this
metric is assumed. The preferred foliation and its constraint are additions
to the model, not consequences of its bands.

The construction uses known kinetic-conformal and auxiliary-constraint
frameworks: [Bellorin and Restuccia](https://arxiv.org/abs/1612.04414) and
[Yao et al.](https://arxiv.org/abs/2011.00805). No novelty claim is made for
these ingredients, nor for introducing an auxiliary constraint in gravity.
The acceleration-potential matching follows the established
[MOND foliation approach](https://arxiv.org/abs/1107.5264).

## Response at measured and extremely weak accelerations

Let mu_previous be the frozen A=-0.45, T=4.25 transition response, with the
numerical splice repair specified below. Introduce

    mu_C(x) = mu_previous(x)
              + epsilon / [(x+epsilon)(1+(x/ell)^2)],
    x = |grad Phi|/g_dark,
    epsilon = 10^-9, ell = 0.01.

The acceleration scale stays fixed at the previous galaxy fit. The added
term is a new phenomenological assumption. It is not fitted to Mercury or
to a new galaxy sample. Its purpose is to restore mu_C(0)=1 and positive
stiffness at zero field. The MOND-like behavior persists over an intermediate
range; it is no longer the exact x->0 asymptote.

Writing d(x)=epsilon/[(x+epsilon)(1+(x/ell)^2)], its radial contribution is

    d(x)+x d'(x) = d(x)[epsilon/(x+epsilon)
                             -2(x/ell)^2/(1+(x/ell)^2)].

Both the total transverse coefficient mu_C and the total radial coefficient
r_C=mu_C+x mu_C' must stay positive. A decreasing mu by itself is not a
violation of this static condition. The new high-field term is proportional
to epsilon ell^2/x^3, so it has the same leading inverse-power falloff as the
original tail with a very small additional coefficient.

The inherited low/core approximation has a small numerical jump in mu near
x=10^-4. In this new candidate only, replace the force x mu_previous by a
cubic Hermite join on [5*10^-5, 2*10^-4], matching both endpoint forces and
their analytic derivatives. This gives a continuously differentiable force
and removes the jump; the join's derivative stays positive. The interval is
far below the smallest predicted x in the RAR sample (about 0.0765). This
is a numerical repair, not a change fitted to observations. Its small energy
integral is included in the reported vacuum-term convention.

For U_C'(x)=x mu_C(x) and a_*=g_dark/c^2, define

    F(a) = 2a^2 - 4 a_*^2 U_C(a/a_*) + C,
    chi(a) = F'(a)/(2a) = 2[1-mu_C(a/a_*)].

Then the stationary Newtonian equation is exactly

    div[mu_C(|grad Phi|/g_dark) grad Phi] = 4 pi G rho.

C is an additive gravitational vacuum term. A convenient local convention
is F(infinity)=0, equivalently

    F(a)/a_*^2 = 4 integral_(a/a_*)^infinity x[mu_C(x)-1] dx.

This convention does not solve the cosmological-constant problem. Changing
C changes the cosmological constant without changing the leading AQUAL
response. It must not be hidden when discussing cosmology.

## Why the scalar constraint changes

Factor the common (16 pi G)^(-1) out of the gravitational canonical momentum
and denote the remaining momentum density by P^ij. Direct differentiation
of the action gives

    P^ij = sqrt(h)[K^ij-K h^ij/3+q h^ij/3],
    C_q = h_ij P^ij - sqrt(h) q(t) = 0.

The kinetic Hessian has rank five, with a null trace direction. The canonical
Hamiltonian before imposing C_q can be written

    H = (16 pi G)^(-1) integral {N[(P_ij P^ij-P^2/2)/sqrt(h)
                    -sqrt(h)(R3+F(a))]
                  + N^i H_i + lambda C_q + sigma P_N} + H_m.

Here P=h_ij P^ij and P_N=0 is the lapse momentum constraint. The algebraic
conditions {C_q,C_q}=0 and {P_N,C_q}=0 hold, including the explicit time
dependence of q in its subsequent preservation equation. Preservation gives
two additional scalar constraints. On a regular branch these four are second
class. After the three spatial diffeomorphisms the local count is

    [14 - 2*3 - 4]/2 = 2 gravitational degrees of freedom.

This is a constrained action defined at its degenerate kinetic point. It is
not obtained by assigning an infinite or zero scalar speed to the failed
propagating model. The complete classification is conditional on the
constraint operators being invertible with the chosen boundary conditions.
The local principal checks do not establish global existence on all spacetimes.

In the local stationary limit, N=1+n, h_ij=e^(2 zeta)delta_ij and
N_i=partial_i b give the scalar principal action

    (2/3)(Delta b)^2
      + k^2[2 zeta^2 + 4 n zeta + alpha_direction n^2],
    alpha_perpendicular = 2(1-mu_C),
    alpha_parallel = 2(1-r_C).

There is no scalar time-derivative term. The uneliminated scalar matrix has
determinant (4/3) k^8 (alpha_direction-2). It is invertible at nonzero k when
the corresponding response stiffness is positive. Negative alpha_parallel
therefore does not produce the old propagating instability.

For a spatially varying weak background set
M=mu_C(I-ee^T)+r_C ee^T, H=-div(M grad), L=-Delta. The principal scalar block is

    2L(zeta+n)=s_zeta,
    2L(zeta+n)-2H n=s_n.

Elimination gives H n=(s_zeta-s_n)/2. Its quadratic form is
integral grad(n)^T M grad(n), positive for positive response stiffness.
Numerical block and eliminated solves test this identity in three spatial
dimensions. They are not a full relativistic evolution calculation.

For completeness, the nonlinear scalar secondary constraints on C_q=0 are,
with kappa=16 pi G, s=P_TF,ij P_TF^ij/h, rho=T_nn and S=h^ij T_ij,

    s-q^2/6-R3-F+2 D_i(chi a^i)+2 chi a^2+kappa rho = 0,
    3s+q^2/2+R3+3F-2 chi a^2-4 Delta_h N/N
       +kappa S-2 dot(q)/N = 0.

The first follows by varying N. The second follows by preserving C_q,
including its explicit time derivative. For a local generator C_q[f],
delta h_ij=f h_ij and delta P_TF^ij=-f P_TF^ij. The latter is therefore
also checked by a finite conformal variation of the Hamiltonian, with
canonical matter momenta held fixed. Adding the two constraints gives

    Delta_h N/N - D_i(chi a^i)/2 - F/2
      = s+q^2/12+kappa(rho+S)/4-dot(q)/(2N).

These identities specify more than the constant-background principal
symbol. Their numerical variation check is an off-shell check of the
equations, not a construction of a full evolving spacetime solution.

## The high-acceleration metric limit

As the nonconstant part of F becomes negligible, the canonical Hamiltonian
is the GR Hamiltonian with C_q as a slicing constraint. On the regular GR
branch its multiplier vanishes, K=-q/2, and the added square in the covariant
action vanishes with its first variation. The limiting PPN parameters are
beta=gamma=1 and alpha_1=alpha_2=0; tensor waves have speed c.

In the local static limit, Schwarzschild in isotropic coordinates is an
explicit check: K_ij=0, h_ij=(1+m/(2r))^4 delta_ij and
N=(1-m/(2r))/(1+m/(2r)). Its spatial scalar curvature and spatial Laplacian
of N both vanish. Thus the relativistic perihelion and light-delay benchmarks
come from an explicit metric branch. Small response-tail and tidal terms
must still be assessed separately. A global relativistic Solar/Galactic
matching solution and a tracking-data ephemeris fit are not supplied by this
argument.

## Homogeneous cosmology and its limits

For h_ij=A(t)^2 delta_ij, the trace-free kinetic term vanishes. After
integrating qK by parts the gravitational homogeneous Lagrangian is

    L = A^3 [N(q^2/6+F(0)) - (2/3) dot(q)]/(16 pi G).

Lapse and scale-factor variations, with minimally coupled matter, give

    q^2/6 + F(0) = 16 pi G rho,
    dot(q) = 24 pi G N(rho+p).

Matter conservation gives H=dot(A)/(NA)=-q/6 whenever rho+p is nonzero.
Therefore

    3 H^2 = 8 pi G rho - F(0)/2,
    dH/dtau = -4 pi G(rho+p).

These are the GR homogeneous equations with Lambda=-F(0)/2. In contrast,
the earlier pure kinetic-conformal action with q=0 requires rho+p=0 and
cannot describe ordinary homogeneous dust/radiation. The clock term repairs
that specific obstruction.

Because mu_C(0)=1, F has no quadratic acceleration term at zero field. The
linear homogeneous-background gravitational/matter problem reduces to the
corresponding GR constrained problem, for the specified matter content.
This is not a fit to the CMB, structure formation or expansion data, and does
not replace the cosmological dark-matter component of a successful standard
fit. Nonlinear cosmological perturbations and the finite range of the linear
approximation need separate study. Homogeneous multiplier ambiguity is a
known issue in more general auxiliary-constraint constructions
([2026 analysis](https://arxiv.org/abs/2607.26031)); it cannot be dismissed
by counting degrees of freedom alone.

For this specific action the homogeneous tensor result can also be checked
directly. Write h_ij=A^2(exp gamma)_ij with tr gamma=0 and transverse gamma.
Then sqrt(h)=A^3 and K=3H, independently of gamma, at homogeneous lapse and
zero shift. The qK, q^2 and F(0) terms do not contribute to the tensor
quadratic action. The remaining shear and curvature terms give

    S_tensor^(2) = (64 pi G)^(-1) integral N A^3
        [(dot gamma_ij/N)^2-(partial_k gamma_ij/A)^2].

The kinetic coefficient is positive and the tensor characteristic speed is
c. This calculation fixes the tensor sector on this homogeneous background;
it is not a binary-inspiral waveform or a strong-field stability analysis.

## What would constitute a solar result

Use one action and one response for the conditional Cassini Q2/gamma and
MESSENGER beta comparison. State the prior correlation and numerical error
controls. Retain the original failed scalar model, the stable-envelope
preferred-frame failure, and the pure kinetic-conformal cosmological
obstruction. Solar compatibility would be an effective-model result, not a
quantum derivation, an independent prediction of fitted data, or a validated
theory of the Universe.
