Searching compatible nested geometries with one field law

A geometry search must retain the nesting of physical sectors. A good independent fit at depth two and another at depth three do not constitute a joint solution unless their metrics can occur in the same containing geometry. This calculation imposes that compatibility before solving for any particle frequency.

The first implemented family consists of nested, conformally deformed spheres. It is a finite-dimensional search within the existing spherical spatial candidate, not an enumeration of all geometries in ten or fewer dimensions. Its global spherical topology is an assumption of this candidate. It does not introduce compactified physical dimensions.

One metric and its compatible restrictions

Start with an intrinsic ten-sphere of radius $R_{10}$. Mathematical Euclidean embedding coordinates $X$ describe nested latitude spheres; they do not introduce an additional physical dimension. Choose

$$0<R_2\le R_3\le R_4\le R_5\le R_6\le R_{10}.$$

Each lower sphere is a latitude sphere of the next containing sphere. For a jump of several dimensions, fix the necessary additional embedding coordinates. The squared coordinate offsets sum to the difference of the squared radii. Equal radii give equatorial inclusions.

On the containing sphere define

$$Q=\frac{X_1^2+X_2^2}{R_{10}^2},\qquad \sigma(Q)=a_1(Q-\tfrac12)+a_2(1-6Q+6Q^2) +a_3(-1+12Q-30Q^2+20Q^3).$$

The containing metric is $g_{10}=e^{2\sigma(Q)}g_{10,0}$. On a depth-$d$ member, let $q=(X_1^2+X_2^2)/R_d^2$. Its metric is

$$\boxed{g_d=e^{2\sigma((R_d/R_{10})^2q)}g_{d,0}.}$$

Every lower metric is therefore the actual pullback of the higher one. Direct restriction and restriction through intermediate depths agree. The radii are parameterized by one outer radius and five ratios in $[0.65,1]$; the three shape coefficients are shared. Different particles at the same depth always see the same candidate radius and metric.

This restriction is particular to the latitude-sphere family. It is not a theorem that all nested physical sectors must have ordered curvature radii or spherical topology. Particle fields are not required to be restrictions of one another.

Curvature and the stated supporting law

For the unit round metric at depth $d$,

$$|\nabla q|^2=4q(1-q),\qquad \Delta_+f=-4q(1-q)f''+[2(d+1)q-4]f',$$

where $\Delta_+$ is the positive Laplacian. The curvature scale of the conformal metric is

$$\kappa_d=\frac{e^{-\sigma}}{R_d} \sqrt{1+\frac2d\Delta_+\sigma -\frac{d-2}{d}|\nabla\sigma|^2}.$$

The square-root argument is a polynomial. Its minimum is evaluated at both endpoints and every real stationary point in $[0,1]$, rather than only at integration nodes. This is a floating-point domain check, not an interval certificate.

The screen retains the stated inverse supporting law

$$g(\kappa)=\frac{100\kappa^4}{1+\kappa^4}$$

in units with the reference length and wave speed equal to one. Its coefficient and functional form are common inputs. They have not been derived from the master field. A negative scalar curvature is a failure of this particular real-curvature law's domain, not a failure of metric compatibility or a proof that such a geometry is physically impossible.

The field and profile tests

The finite calculation uses the same timelike current source

$$F[\psi]=H_0\psi-\frac{g}{s} \left[jK+\eta_d^2(\rho I-z\Sigma)\right]\psi, \qquad s^2=j^2+\eta_d^2(\rho^2-z^2).$$

Here $\rho=\psi^\dagger\psi$, $j=\psi^\dagger K\psi$, and $z=\psi^\dagger\Sigma\psi$, with $\Sigma=\operatorname{diag}(I_d,-I_d)$. The fixed matrix $K$ links adjacent components of the existing finite chain. For spinors, $H_0=\chi D_g+\kappa A$, where $\chi$ exchanges the two component blocks and $A=\chi\operatorname{diag}(a-d/2)$, $a=0,\ldots,d-1$. The depth-two form version uses its actual Hodge operator and parity-dependent curvature offsets. The positive action is $\|\partial_t\psi\|^2-\|F[\psi]\|^2$.

The six values of $\eta_d$ are taken from the preceding pair calculation, with one fixed value per depth. They are not refitted for a particle or geometry. A separate saturated calculation uses $\eta=0$ on the strictly negative-current branch.

All maintained spectral controls enter each joint candidate: W, Z and Higgs; down, strange and the two bottom constituent indices; up, charm and top; the three neutrino indices; electron and muon; and tau. The bottom constituent indices are not two separate particles. The bosonic depth-two calculation retains scalar and differential-form components.

Each reference spinor field has canonical conformal transport

$$\widetilde D_\sigma=e^{-\sigma/2}D e^{-\sigma/2}.$$

Two spatial profiles are tested: the canonically transported reference field and its normalized multiple $e^{\sigma/2}$. The latter cancels the explicit conformal derivative term in this restricted field family. For depth-two forms, the corresponding transport is $E_\sigma D_{\mathrm f}E_\sigma$, where $E_\sigma=e^{-\sigma}P_{\rm even}+P_{\rm odd}$. Its alternative profile multiplies the even component by $e^\sigma$. It is not assigned the spinor weight.

The real component vector adjusts through the complete finite chain and, for finite $\eta_d$, its nonlinear current fixed point. Its selected positive frequency remains the mass in frequency units. Profile normalization is not used as mass.

The principal field diagnostic is

$$r_F=\frac{\|F[\phi]-\omega\phi\|}{\omega\|\phi\|}.$$

This tests the inherited first-order standing branch, which supplies a solution of the positive wave action when it holds. It does not exhaust all possible solutions of the second-order equation involving $DF^*F$. The full spatial field is not freely optimized: only the stated profiles and real component vectors are retained.

Necessary supporting tests differentiate the generating energy along a common scale change and the three shared shape changes. For the adapted profiles these paths change both geometry and profile. Vanishing path derivatives would be necessary for a full joint stationary solution; it would not establish stationarity against independent, arbitrary field and tensor-metric variations. A nonzero derivative rules out stationarity of that tested configuration.

A joint score is the worst particle's combined field defect and largest normalized path derivative. Opposing errors from different particles are never summed to cancel one another. Passing thresholds are numerical screening tolerances, not observational error bars or a stability theorem.

Independent orientations inside the same compatible geometry

For spinors at depths three and above, rotate the complex spatial pattern plane through the same angle $\theta$ in two coordinate planes. The geometry stays fixed. For a degree-$n$ field, the normalized marginal law of its geometric coordinate $q$ is

$$\sum_{\ell=0}^{n} \binom n\ell p^\ell(1-p)^{n-\ell} \operatorname{Beta}\!\left(\ell+1,n-\ell+\frac{d-1}{2}\right), \qquad p=\cos^2\theta.$$

This follows by expanding the rotated complex harmonic, averaging its relative phases, and using the sphere's coordinate-square distribution. Its moments independently verify the numerical integration. On a round metric, the frequency is independent of $\theta$.

The orientation refinement retains seven angles and both profiles on selected complete metric chains. It includes the frequency derivative with respect to orientation at interior angles. This is one continuous rotation family, not the entire orientation space. The depth-two form calculation does not use this four-coordinate spinor construction. An axisymmetric marginal must not be mistaken for an axisymmetric full field or an isotropic tensor stress.

Search bounds and interpretation

The initial design contains 1,208 compatible chains: 1,024 reproducible Sobol samples, round boundary families, and explicit small deformations. The outer radius ranges from 0.4 to 2.5; shape coefficients range over $[-0.3,0.3]$, $[-0.06,0.06]$, and $[-0.02,0.02]$. Three starting configurations are refined jointly within the same bounds. Higher-order integration checks the best retained results.

The completed results and verification receipts are recorded below. Even a successful screen would still need unrestricted field adjustment, independent metric-stress tests, stability, mass-ratio checks and physical scale selection. Failure of this screen excludes only the tested configurations and profiles under the stated law.

Why a spatially varying component pattern is necessary in this family

There is a structural limitation in the first profile screen. For a spinor reference eigenfield $Du=\beta u$, put $\psi=f(q)\phi\otimes u$, with one real component vector $\phi$ throughout space. If $f=e^{\alpha\sigma}$, canonical transport gives

$$\widetilde D_\sigma(fu)= f e^{-\sigma}\left[\beta u+ (\alpha-\tfrac12)\operatorname{cl}(d\sigma)u\right].$$

The Clifford term has no real cross term with the retained real component ray. Its squared norm supplies a nonnegative contribution to the field residual. Thus the unchanged profile cannot give an exact standing field on a nonconstant conformal metric. The covariant choice $\alpha=1/2$ removes this particular obstruction.

There is still a second restriction. Define $\chi\phi=(v,u)$ for $\phi=(u,v)$, and let $A=\chi\operatorname{diag}(j-d/2)$. The finite-current contribution on this separated ray is a fixed matrix $J$ multiplied by the local strength $g(q)$. The remaining equation is

$$\left[a(q)\chi+\kappa(q)A+g(q)J\right]\phi=\omega\phi, \qquad a(q)=\frac{\beta e^{-\sigma(q)}}{R_d}.$$

For the retained positive arithmetic-diagonal component chain at $d\ge3$, the vectors $\phi,\chi\phi,A\phi$ are linearly independent. All components of the lowest singular vectors are nonzero. A proposed relation would require $(s+t(j-d/2))^2=r^2$ at three or more distinct values of $j$. This forces $t=0$, leaving a possible $\chi$-eigenvector. That would make the left and right singular vectors equal up to sign. The first row of the lower-triangular chain and the last row of its transpose would then require its distinct endpoint diagonal entries to coincide. They do not.

The averaged eigen-equation consequently makes $\chi\phi,A\phi,J\phi$ independent as well. Subtracting the averaged equation from the pointwise one forces all three scalar coefficients to be constant. In particular, $a(q)$ constant forces $\sigma$ constant.

This is a restriction of the separated component-ray ansatz, not an exclusion of deformed geometries in IDWT. A meaningful extension must allow the component pattern itself to vary with position. The follow-up calculation therefore retains an axial spatial basis in every component, including both spatial spectral signs. Its local nonlinear source is integrated before projection; the norm of the residual outside the retained basis is included. Every such field is tested on the same compatible metric chain. This is a larger conditional search, not yet a complete spatial or geometric variational solution.

A second compatible family: complete flat sectors

The flat chain $\mathbb R^2\subset\mathbb R^3\subset\cdots \subset\mathbb R^{10}$ is compatible without selecting radii or shape coefficients. Under the particular curvature-dependent candidate tested here, $\kappa=0$ gives $g(0)=0$, and the curvature offsets vanish. The remaining spatial operator is the free Dirac operator, or the actual free Hodge operator for the depth-two forms. Its square is $c^2(-\Delta)$ on the appropriate components.

A square-integrable standing field on a complete flat sector would obey

$$(-c^2\Delta-\omega^2)\psi=0.$$

Fourier transformation requires its square-integrable transform to be supported on $|k|^2=\omega^2/c^2$. This set has Lebesgue measure zero, including the zero-frequency case. The only such square-integrable field is zero. Thus this supporting-law candidate cannot provide a localized standing particle anywhere in the complete flat chain.

This is an analytic exclusion under the stated law, with no artificial boundary, external potential or altered asymptotic conditions. It does not exclude locally flat regions of a curved geometry, confinement from a different same-field law, or flat directions outside the bound sector. It also does not assert that ordinary physical space has measured nonzero curvature. No empirical geometric conclusion follows from this conditional calculation.

Spatial-pattern refinement on one compatible chain

All sixteen controls were relaxed on the same chain with every radius 0.8 and shared coefficients (0.003, -0.0002, 0.0001). The calculation retained a spatial basis through cutoff two, then expanded it to cutoff three for the first maintained control at each depth. The normalized pointwise first-order field residuals were:

Residuals are fractions, not percentages. These are computed field defects, not discrepancies from measured particle masses. The cutoff-three fits except the depth-two W control reached their iteration limits. Their remaining residuals must not be reported as converged minima or exact solutions. The cutoff-two fields agree under 100- and 200-point integration; that does not prove convergence with respect to the spatial basis.

The results demonstrate that spatially varying component patterns can substantially reduce the first screen's defects without changing the nested metric independently for each particle. They do not eliminate the supporting forces. None of these configurations supplies a fully self-supported particle geometry.

The raw minimum-current diagnostic for the high-index depth-two Higgs control includes extremely small polar tails, where normalized-current diagnostics are set to zero below the numerical density cutoff. This is not evidence of a physical null-current boundary. The weighted residuals are checked by integration refinement; a pointwise domain certificate for every tail is not supplied.

Common-strength controls

A separate control changes the common coefficient in the curvature law for the whole chain, using 10, 100, 1000 and 10000 at each of four common radii (0.4, 0.8, 1.2, 2.5). No particle-specific strength is fitted. None of these sixteen complete round chains has zero radial force for every maintained control. At the larger strengths, the previously fixed finite regularizers also prevent the algorithm from retaining a positive branch for some modes. That is a domain limitation of this candidate and branch procedure, not a proof against every nonlinear branch or supporting law.

Completed joint scan

Both current-law versions completed the same 1,208 compatible metric configurations. Each retained 1,010 configurations in its tested curvature and component-branch domain; 198 were outside that domain. These are not 198 failures of nesting compatibility: every generated chain was compatible by construction. Three joint refinements were completed for each law.

No sampled or optimized candidate passed the simultaneous field tolerance of 1e-6 and path-force tolerance of 1e-5. The score only ranks unsuccessful tradeoffs; its minimum is not an approximate physical solution. The best points also approach the lower radius bound, so the finite search does not establish a global minimum over the metric family.

The orientation refinement improves some necessary force diagnostics, but none of its five metric chains passes either. The strongest statement is therefore a completed conditional screen, together with a concrete reason to enlarge the spatial field ansatz. The scan does not establish which geometry nature realizes or finish the mass operator.

Reproducible verification

The six identity and implementation reports pass 1,489 checks in total. They cover nested metric restriction, curvature, derivative identities, normalization, finite-current fixed points, orientation moments, actual bosonic form transport, the spatial residual gradient and agreement between the screening formulas and an independently assembled pointwise spatial operator. Completed numerical screens are recorded separately from passing identity checks. Their failures, iteration limits, coefficient arrays and integration refinements are preserved with the source hashes.

The main entry points are joint_geometry_fast_scan.py and joint_geometry_finite_scan.py for the joint metric search, joint_orientation_screen.py for independent pattern headings, joint_spatial_screen.py for component textures, and joint_strength_audit.py for common-strength controls. The checkpoint manifest identifies their exact archived versions and the raw numerical arrays.