Testing spatial patterns, compatible geometry and the mass law together

A spatial field can nearly solve its standing equation while having the wrong mass ratios or an unbalanced supporting variation. All three must be kept visible. This continuation retains the same finite-current candidate, the same compatible nested conformal-sphere geometry and all six maintained depths. Mass is the selected positive frequency.

The geometry remains one containing metric and its restrictions. Particle patterns vary independently inside their sectors; they are not restrictions of other particles. Global spherical topology is a conditional geometric choice, not compactification.

What the joint test asks

For a canonically normalized field, write

$$E_1[\psi;g]=\langle\psi,F_g[\psi]\rangle, \qquad \omega=E_1[\psi;g],\qquad R=F_g[\psi]-\omega\psi.$$

The inherited standing branch requires $R=0$. It is a sufficient branch of the positive wave action, not every possible solution of its second-order equation. The expectation $E_1$ is the trial eigenvalue in this first-order search; it is not the physical frequency of every other standing branch. An exact extension below makes that distinction explicit while always identifying mass with the actual oscillation frequency. Four necessary support tests vary the common logarithmic radius and the three shared conformal coefficients:

$$f_a=\partial_a E_1\big|_\psi, \qquad a=0,1,2,3.$$

These are derivatives with the canonical field fixed. They test the specific hypothesis that this occupied field and its curvature-dependent interaction supply the support, without an additional geometric action or background stress. An externally supported geometry need not make these derivatives vanish. Neither alternative has been derived from the master field here. Passing these four paths would also not prove stationarity against every tensor-metric variation.

The mass check uses

$$u_{n,d}=\frac{\omega_{n,d}}{S(n,d)},\qquad S(n,d)=\binom{n+d-1}{d}.$$

The units must agree within a depth, with the maintained shared unit also checked between depths six and ten. There are no fitted particle-specific mass coefficients. The two bottom constituent indices remain separate spectral controls; deriving their physical combined branch is still open.

The joint numerical objective sums the squared relative field defects, the squared $f_a/\omega$, and the squared deviations of $\log u$ from the mean within each required group. Equal weights are an explicit numerical search choice. They do not change the physical action or make a nonzero tradeoff minimum a solution.

Efficient full spatial evaluation

For a fixed spatial pattern, the pointwise field has the form

$$F=aU_0+bU_1+\kappa U_2+g(\kappa)U_3, \quad a=\frac{e^{-\sigma}}{R_d},\quad b=-a\sigma'.$$

Here $U_0$ is the reference spatial Dirac or Hodge term, $U_1$ is its conformal-gradient term, $U_2$ is the stated component curvature offset, and $U_3$ is the full local finite-current source per unit coupling. Each contains the actual spatially varying component field. Their values can be retained while the compatible geometry is varied. The residual is evaluated pointwise, including components outside the finite spectral basis. It is not replaced by its projection alone.

The complete real derivative of the current enters the field-residual gradient. Homogeneity and the variational symmetry of $F$ give

$$\delta E_1=2\operatorname{Re}\langle\delta\psi,F\rangle, \qquad \delta f_a=2\operatorname{Re} \langle\delta\psi,\partial_aF\rangle.$$

Canonical normalization projects these derivatives onto the tangent space of the unit sphere of fields. These identities permit support and mass defects to vary along with the field defect, rather than optimizing each condition in isolation. Independent directional differences check the implemented derivatives.

Exact finite-chain drift away from the simplex law

There is a useful exact result for the round reference carrier with saturated negative current. In units with wave speed one, let

$$T_{jj}=\frac{n+j}{R},\qquad T_{j,j-1}=G, \qquad \gamma=GR,$$

where $j=0,\ldots,d-1$. Its smallest singular frequency is $\epsilon=1/\|T^{-1}\|_2$. Signs in the inverse can be removed by diagonal orthogonal transformations. The inverse magnitudes are

$$|(T^{-1})_{ji}|= R\frac{\gamma^{j-i}}{\prod_{k=i}^{j}(n+k)},\qquad j\ge i.$$

Divide this matrix by its lower-left entry $C=R\gamma^{d-1}/(n)_d$, where $(n)_d=n(n+1)\cdots(n+d-1)=d!S(n,d)$. The resulting matrix has

$$M_{ji}= \frac{\prod_{k\notin\{i,\ldots,j\}}(n+k)} {\gamma^{d-1-j+i}},\qquad \boxed{\frac{\epsilon}{S(n,d)} =\frac{d!}{R\gamma^{d-1}}\frac1{\|M(n,\gamma)\|_2}.}$$

The lower-left entry of $M$ is one. Every other nonzero entry strictly increases with $n$. Its positive left and right leading singular vectors therefore make $\|M\|_2$ strictly increase with $n$. Consequently, at a fixed finite $\gamma$, the mass unit $\epsilon/S(n,d)$ strictly decreases with mode index.

Thus the exact round saturated reference branch does not give the exact simplex mass law at any finite gap. It approaches that law as the corner entry dominates at large $\gamma$. This theorem concerns that reference branch. A finite-current correction, another spatial branch, or a deformed geometry requires its own calculation.

The norm of the first column supplies the quantitative bound

$$\frac{\epsilon}{d!S(n,d)/(R\gamma^{d-1})} \le\frac1{\sqrt{1+[(n+d-1)/\gamma]^2}}.$$

For the current round trial law,

$$G=g(R)=\frac{100}{1+R^4},\qquad \gamma\le\frac{100\,3^{3/4}}4\simeq56.9877.$$

For example, one-percent accuracy to the leading simplex expression for the depth-four index-72 control requires at least $\gamma\ge75/\sqrt{0.99^{-2}-1}$, approximately 526.3. The present strength cannot reach that regime at any round radius. This comparison is to the leading expression, not by itself a fitted within-sector observational error bound.

A rigorous round depth-two radial obstruction

The finite-current round reference state has an effective gap $G\le g$. Let $\epsilon$ be its positive singular frequency, $k<0$ its normalized component current, and

$$C=\frac{g\eta_2^2}{\sqrt{k^2+\eta_2^2}},\qquad \omega=\epsilon-C,\qquad 0\le C\le g\eta_2.$$

Define $A=-G\epsilon_G/\epsilon$ and $q_g=-d\log g/d\log R$. The fixed-point envelope derivative is

$$\frac{d\omega}{d\log R} =-\epsilon[1-(q_g-1)A]+q_g C.$$

For depth two the singular-value formula gives, with $\Gamma=GR$,

$$A=\frac{\Gamma^2} {\sqrt{\Gamma^2+1}\sqrt{\Gamma^2+(2n+1)^2}} <\frac{\Gamma}{2n+1}.$$

For the stated curvature law, $0\le q_g<4$. A direct maximization with $x=R^4$ gives

$$\max_R (q_g-1)_+\,gR=100B,\qquad B=\frac{x_*^{1/4}(3x_*-1)}{(1+x_*)^2} \simeq0.667555,\quad x_*=\frac{11+4\sqrt7}{9}.$$

Let $\gamma_{\max}=100\,3^{3/4}/4$ and $b_n=100B/(2n+1)$. The smallest singular-value perturbation bound gives $\epsilon\ge R^{-1}(n-\gamma_{\max})$. For the three maintained indices, $n>\gamma_{\max}$ and $b_n<1$, hence

$$\boxed{R\frac{d\omega}{d\log R} \le -(1-b_n)(n-\gamma_{\max}) +4\gamma_{\max}\eta_2.}$$

With the unchanged $\eta_2=0.02264240274802975$, the right-hand side is approximately -5.556, -9.017 and -19.565 for indices 76, 81 and 95. It is strictly negative. No radius can balance these round reference branches under this particular supporting law, wherever their retained finite-current branch exists. The statement includes the actual depth-two form carrier. It does not exclude other textured states or another supporting law.

Compatibility also constrains the large-gap radial limit

For the saturated large-gap limit, radial balance under the same law gives

$$R_*(d)=\left(\frac{d}{3d-4}\right)^{1/4}.$$

This decreases strictly with depth. Nested round latitude spheres instead require $R_2\le R_3\le\cdots\le R_{10}$. Thus the leading radial equilibria cannot be assembled into that compatible round family, even though each radius separately solves its leading radial equation. Finite-chain corrections and deformed geometries are distinct tests. There is no conclusion here about arbitrary nested metrics or the measured geometry of ordinary space.

Changing the curvature normalization does not suffice

The previous supporting law uses the normalized curvature scale $\kappa^2=\mathrm{Scal}/[d(d-1)]$. An alternative common law can use scalar curvature itself:

$$g=G_0\frac{\mathrm{Scal}^2}{1+\mathrm{Scal}^2},\qquad \mathrm{Scal}=\frac{d(d-1)}{R_d^2}.$$

Lengths and curvature are expressed in the same fixed reference units at every depth. This is a different trial law, not a correction of an incorrect formula. It introduces no particle-specific coefficient. Its leading stationary radii are

$$R_d=\sqrt{d(d-1)}\left(\frac{d}{3d-4}\right)^{1/4}.$$

They increase with depth, so this entire round chain is compatible before any field test. Nine controls use this chain, three common radius multipliers and three common strengths. Improving the radius ordering does not restore the required shared electron–tau unit.

There is an exact bound for that shared unit on the chain at its leading stationary radii, using the saturated finite matrices without a large-gap approximation. Let $I_d(h)=|T_d(h)^{-1}|$, where $h>0$ is the common strength. Then

$$\frac{\epsilon_{23,10}/S(23,10)} {\epsilon_{13,6}/S(13,6)} =\frac{S(13,6)}{S(23,10)} \frac{\|I_6(h)\|_2}{\|I_{10}(h)\|_2}.$$

Each inverse entry is a positive monomial in $h$, of degree at most $d-1$. On an interval $[h_-,h_+]$, bound the numerator norm by the sum of its entries at $h_+$, and the denominator norm by its largest entry at $h_-$. Rational upper and lower enclosures of the two radii make this an exact rational inequality. At small strength, the denominator's diagonal entries give a nonzero bound. At large strength, the numerator grows at most as $h^5$, while the denominator's corner grows as $h^9$.

Fifty-nine intervals, including both unbounded end ranges, give

$$\boxed{\frac{\epsilon_{23,10}/S(23,10)} {\epsilon_{13,6}/S(13,6)}<0.008 \quad\text{for every }h>0.}$$

The largest recorded upper bound is approximately 0.00765128. The certificate uses rational arithmetic, not floating-point sampling of strengths. Thus this compatible round chain cannot supply the shared unit simply by choosing another common saturated coupling strength. It is not a theorem about other metrics or finite-current corrections.

An exact family of full positive-wave states

The positive wave action also has standing states outside the first-order eigenfield ansatz. This extension follows from the structure of the candidate itself.

Write the two component blocks as $\psi=(u,v)$. The spatial linear operator and $K$ are off diagonal between these blocks. For positive real numbers $a,b$, the finite-current norm scales as $s[au,bv]=ab\,s[u,v]$, including for spatially varying fields. Consequently,

$$\boxed{F[au,bv]=(bF_u[u,v],\ aF_v[u,v]).}$$

Suppose $F[u,v]=\Omega(u,v)$, with $\Omega>0$, and $\|u\|^2+\|v\|^2=1$. The generating energy is homogeneous of degree one in each block separately. Its variational identity gives $\|u\|^2=\|v\|^2=1/2$. Choose

$$a=\cos\theta+\sin\theta,\quad b=\cos\theta-\sin\theta,\quad |\theta|<\frac\pi4.$$

Then $\psi_\theta=(au,bv)$ stays normalized, and

$$F[\psi_\theta]=\Omega(bu,av),\qquad DF[\psi_\theta]^*F[\psi_\theta]=\Omega^2\psi_\theta.$$

To obtain the second identity, differentiate the boxed block-scaling identity in the direction $(bu,av)$, and use the real variational self-adjointness of $DF$. Thus $e^{-i\Omega t}\psi_\theta$ solves the complete positive wave equation whenever the starting field does. The differentiated identities apply on the smooth timelike-current domain. Spatial nodal sets retain the same weak or limiting interpretation required by the starting field; this construction adds no new regularity claim there.

Every member has the same oscillation frequency and the same mass. Its first-order expectation instead equals $\langle\psi_\theta,F[\psi_\theta]\rangle =\Omega\cos(2\theta)$. That expectation is not its oscillation frequency unless the state belongs to the first-order eigenbranch. The difference does not separate mass from frequency; it identifies which spectral equation supplies the frequency. Overall field magnitude is not being used as mass.

The positive block factors redistribute the component pattern. They do not derive physical polarization counts, rotations of spatial directions, antiparticles or photon behavior. The endpoint where one block disappears is excluded because the current norm reaches its singular boundary.

This freedom does not remove the supporting stress. For a parameter variation preserving the stated block structure, the real matrix elements $\langle u,\partial F_u\rangle$ and $\langle v,\partial F_v\rangle$ are equal by the same separate homogeneity. Since $a^2+b^2=2$,

$$\partial\|F[\psi_\theta]\|^2 =\partial\|F[\psi_0]\|^2.$$

The normalized wave family therefore has the same metric source as its starting eigenfield. It extends the solution space of the candidate, but cannot repair an unbalanced radius by choosing different block weights. The resulting neutral direction is not a proof of stability against all other perturbations.

Free-direction factorization at the action level

There is a stronger propagation statement than constructing a particular first-order moving eigenfield. Retain the previously stated product extension, a parallel Hermitian normal symbol $\Sigma$, $\Sigma^2=I$, and the anticommutators $\{H_0,\Sigma\}=\{K,\Sigma\}=0$. For a constant normal momentum $p$ in frequency units, write

$$F_p[\psi]=F_0[\psi]+p\Sigma\psi.$$

Self-adjointness and anticommutation give $\operatorname{Re}\langle H_0\psi,\Sigma\psi\rangle=0$. For the current term, the corresponding real inner product vanishes pointwise: the $jK$ term has zero real part, and the other two terms cancel as $\rho z-z\rho=0$. Expanding the norm therefore gives the exact functional identity

$$\boxed{\|F_p[\psi]\|^2 =\|F_0[\psi]\|^2+p^2\|\psi\|^2.}$$

Variation of this identity yields

$$DF_p[\psi]^*F_p[\psi] =DF_0[\psi]^*F_0[\psi]+p^2\psi.$$

Every valid bound standing field of the full positive wave equation, not just a first-order eigenfield, consequently has a constant-momentum product extension with

$$\boxed{\Omega(p)^2=\omega^2+p^2.}$$

Its bound profile can remain unchanged. At zero net normal momentum, $\partial_p^2\Omega=1/\omega>0$. The bound field still oscillates. This preserves mass as its selected frequency and extends the previous free-direction inertial calculation to all valid standing branches of this candidate.

This is an action identity under explicit product and domain assumptions. It does not establish nonlinear superposition of different normal momenta, a localized particle packet, arbitrary bound-direction motion, or a globally free normal geometry. In particular, the nested latitude metrics tested in the joint search have not been shown to realize this product extension. The propagation lemma and that global geometry must not be combined into a claimed physical solution without establishing their compatibility.

Numerical results

The alternating search completed three cycles of shared metric adjustment and spatial field relaxation. Its final largest relative field defect was 0.4613 and its largest normalized metric-path force was 30.063. Doubling the quadrature changed its field defects by at most 2.14e-8. Neither condition is close to a solution, and the required mass units also remain inconsistent.

The simultaneous search varied all sixteen spatial patterns and the nine compatible metric parameters in one objective. It stopped after 49 iterations at its specified computation limit. Its endpoint is not converged, and six controls failed the subsequent integration-refinement check. The original reports retain those failures.

The problem is especially clear where the optimized local current reverses sign. Its numerator is a polynomial of degree at most twice the retained axial cutoff after removing the common harmonic density factor. Locating all its interior roots allows adaptive integration to split around the finite-current transition widths. For the depth-five index-22 control, the relative field defect rises from the optimizer-grid value 8.5414 to 9.8044. For depth-ten index 23 it rises from 19.0004 to 21.7907. These are large failed field equations, not small corrections to particle masses. The apparent objective reduction must not be interpreted as a physical improvement. The full positive-wave derivative also fails at these endpoints; its projected residual already suffices to exclude them.

This search retains a finite axial basis with a fixed spatial heading, including both spectral signs and complex component coefficients. It is not unrestricted angular or orientation variation. Further basis refinement cannot be replaced by quadrature refinement. No global minimum or exhaustive exclusion of the conformal family is claimed.

Eight identity and implementation reports pass 1,505 checks. A separate 32-check endpoint integration audit contains six failures, which motivated the adaptive investigation rather than being discarded. Four completed numerical screens record the alternating search, simultaneous search, scalar-curvature controls and adaptive audit. These categories are kept distinct from a successful physical solution.

The clearest positive result is the larger full-wave family and the action-level propagation identity. The clearest restriction is that the tested scalar-curvature support constructions do not simultaneously produce supported compatible geometry and the maintained mass units. Developing the supporting term from the same field remains the central open step; none of these calculations completes the mass operator.