In the four-plus-six spatial reference split, the base has four ordinary spatial dimensions and its perpendicular complement has six. Time is a separate coordinate in the infinite-dimensional physical spacetime. This four-plus-six split concerns ten spatial directions in the matter domain.

The curvature charge construction can be made into an explicit finite source on the existing ten-dimensional spatial spinor. Its transverse normalization can be calculated from the same field instead of assigned independently. A wider class of spatial orientation profiles preserves the same charge weights and permits a finite charge radius. It also permits neutral configurations with nonzero signed internal source distributions.

The important dynamical distinction is now calculable: the topological source term preserves the charge but does not, by itself, restore a preferred size in the fixed-boundary family considered here. A separate metric-dependent energy must provide localization. A bounded trial-energy calculation below shows one possible stabilizing balance, with its changed premises and limited scope explicit.

These results are principal-authored conditional research. They are not independently reviewed, and they do not establish a completed physical electromagnetic interaction, selected particle count or all-family solution.

1. Make the Transverse Distribution from the Same Field

Split ordinary spatial coordinates as x=(u,y), with four u coordinates and six y coordinates, using the existing reference geometry. Suppose the common field has a conserved positive norm current:

∂tρ+∇u⋅ju+∇y⋅jy=0,N=∫ρ d10x<∞. \partial_t\rho+\nabla_u\cdot j_u+\nabla_y\cdot j_y=0, \qquad N=\int\rho\,\mathrm d^{10}x<\infty.

Assume vanishing flux through the relevant spatial infinities. Define

w(u,t)=1N∫ρ(u,y,t) d6y,jˉu(u,t)=1N∫ju(u,y,t) d6y.(1) w(u,t)=\frac1N\int\rho(u,y,t)\,\mathrm d^6y, \qquad \bar j_u(u,t)=\frac1N\int j_u(u,y,t)\,\mathrm d^6y. \tag{1}

Then the four-direction continuity equation holds exactly and the integral of w is one. Let vol4 be the ordinary four-dimensional volume form and let contraction by a vector be denoted by iota. The spacetime four-form

W=w vol4−dt∧ιjˉuvol4(2) W=w\,\mathrm{vol}_4-\mathrm dt\wedge\iota_{\bar j_u}\mathrm{vol}_4 \tag{2}

is closed. Pull it back to the complete physical spacetime. It supplies the closed four-form required by the previous source construction without a second physical field or an independently chosen normalization.

This is an effective description using spatial integrals. It is nonlocal in y, requires finite N and a norm-continuity law, and is not automatically a fundamental local constitutive law. No probability interpretation is attached to the normalized density.

If the curvature six-form has the same integrated charge q on each normal slice, then

J10=e∗W∧Q6,dJ10=0,∫t=constantJ10=e∗q.(3) \mathscr J_{10}=e_* W\wedge\mathscr Q_6, \quad \mathrm d\mathscr J_{10}=0, \quad \int_{t=\mathrm{constant}}\mathscr J_{10}=e_*q. \tag{3}

The overall field amplitude cancels in (1). The common unit e-star remains to be calibrated or derived. Closure of (3) is a current construction, not a derivation of the force that it produces.

2. An Explicit Source on the Known Ten-Dimensional Profile

Use the maintained critical profile in ten spatial directions:

Ψ=A(1+∣x∣2/a2)−5(I+iγ⋅x/a)χ,χ†χ=1, \Psi=A(1+|x|^2/a^2)^{-5}(I+i\gamma\cdot x/a)\chi, \qquad \chi^\dagger\chi=1,

with a product pure seed whose polarization is block diagonal in the four-plus-six spatial split. Its density and total norm are

ρ=A2(1+∣x∣2/a2)−9,N=π56720A2a10.(4) \rho=A^2(1+|x|^2/a^2)^{-9}, \qquad N=\frac{\pi^5}{6720}A^2a^{10}. \tag{4}

At each fixed u the normal phase connection is

Ay=−(Jy)idyia2+∣u∣2+∣y∣2. \mathcal A_y=-\frac{(Jy)_i\mathrm dy^i}{a^2+|u|^2+|y|^2}.

Thus the normal calculation uses the effective squared radius a squared plus the squared u radius. Integrating (4) over y gives the normalized marginal

w(u)=20a8π2(a2+∣u∣2)6.(5) w(u)=\frac{20a^8}{\pi^2(a^2+|u|^2)^6}. \tag{5}

The normalized cubic-curvature density on that normal slice is

h(y;u)=6(a2+∣u∣2)π3(a2+∣u∣2+∣y∣2)4,∫h d6y=1.(6) h(y;u)=\frac{6(a^2+|u|^2)}{\pi^3(a^2+|u|^2+|y|^2)^4}, \quad \int h\,\mathrm d^6y=1. \tag{6}

For matched reference and particle radial functions, the mixed-curvature identity holds pointwise, not only after integration. Hence the spatial source density is explicitly

ρQ(u,y)=e∗q 120a8π5(a2+∣u∣2)5(a2+∣u∣2+∣y∣2)4.(7) \boxed{\rho_Q(u,y)=e_*q\, \frac{120a^8}{\pi^5(a^2+|u|^2)^5 (a^2+|u|^2+|y|^2)^4}.} \tag{7}

Its integral is e-star times q for q equal to zero, minus one, two thirds or minus one third under the existing grade/branch choices. It is finite at the origin, absolutely integrable and independent of A. The normal and marginal integrations each normalize to one, so no arbitrary division by an infinite free volume occurs.

The underlying spatial field solves the known critical Dirac equation, with the stated amplitude–radius relation, and has finite norm and finite critical Hamiltonian energy. For the canonical energy convention D Psi minus the critical density nonlinearity, direct integration gives

EN=49a.(8) \frac EN=\frac{4}{9a}. \tag{8}

This is the critical static branch. Its previously documented nonzero-clock and scale issues remain; (7) does not turn it into a tau or an occupied simplex-count solution. The use of the existing carrier weights also does not select which seed the physical dynamics occupies.

The normal mean-square radius of (6) diverges logarithmically. That is a property of this source ansatz, even though its total charge and the field energy are finite. The next construction repairs that particular tail without changing the integrated charges.

3. Charge Does Not Require the Original Radial Tail

For a normal six-dimensional orientation field take

z(y)=cos⁡β(r) χ+isin⁡β(r)(y^⋅γ)χ,f(r)=sin⁡2β(r), z(y)=\cos\beta(r)\,\chi+i\sin\beta(r)(\widehat y\cdot\gamma)\chi, \qquad f(r)=\sin^2\beta(r),

with a chiral unit seed, beta zero at the origin and beta approaching pi over two at infinity. Its connection is

AJ=−f(r)r2(Jy)idyi.(9) \mathcal A_J=-\frac{f(r)}{r^2}(Jy)_i\mathrm dy^i. \tag{9}

A sufficient smooth-origin condition is f(r)=r squared times h(r squared), with smooth h and compatible smooth square roots for cos beta and sin beta divided by r. The explicit profile below satisfies it; a quadratic leading order alone is insufficient for an arbitrary profile. Write g=f/r squared. Direct differentiation gives

(FJ)ij=2gJij−g′r[yi(Jy)j−(Jy)iyj].(10) (\mathcal F_J)_{ij}=2gJ_{ij} -\frac{g'}r\big[y_i(Jy)_j-(Jy)_i y_j\big]. \tag{10}

In an aligned reference plane, its radial block is f-prime divided by r, while its two other plane blocks are 2f divided by r squared. The common congruence of these matrices preserves the mixed-alignment relation. The enclosed cubic integral is simply

1(2π)3∫∣y∣<RF03=f(R)3.(11) \frac1{(2\pi)^3}\int_{|y|<R}\mathcal F_0^3=f(R)^3. \tag{11}

With f zero at the centre and one at infinity, the full integral is one. The charge weights therefore survive a broad class of smooth radial deformations. No separate frequency is introduced by f: it describes spatial orientation.

For a concrete fast approach to the asymptotic orientation, choose

f(r)=1−e−r2/a2.(12) f(r)=1-e^{-r^2/a^2}. \tag{12}

The corresponding normalized charge density is

ha(r)=6e−r2/a2(1−e−r2/a2)2π3a2r4,ha(0)=6π3a6.(13) h_a(r)=\frac{6e^{-r^2/a^2}(1-e^{-r^2/a^2})^2}{\pi^3 a^2 r^4}, \qquad h_a(0)=\frac6{\pi^3a^6}. \tag{13}

It is smooth, positive and integrates to one. If u=r squared divided by a squared, its radial measure is 3 exp(-u) times (1-exp(-u)) squared du. Consequently

⟨r2⟩=116a2,⟨r4⟩=8518a4.(14) \langle r^2\rangle=\frac{11}{6}a^2, \qquad \langle r^4\rangle=\frac{85}{18}a^4. \tag{14}

These are moments of the proposed source distribution, not measured particle radii. All its nonnegative polynomial radial moments are finite. Although the orientation curvature still has angular tails, its top radial charge density now decays rapidly.

An explicit finite field carrying (12) is

Ψ(u,y)=Aexp⁡[−∣u∣22b2−∣y∣22a2]z(y),N=A2π5b4a6.(15) \Psi(u,y)=A\exp\left[-\frac{|u|^2}{2b^2}-\frac{|y|^2}{2a^2}\right]z(y), \qquad N=A^2\pi^5b^4a^6. \tag{15}

Here b is a width, unrelated to the earlier determinant-branch bit. The latter is denoted epsilon below when needed. A fixed base spinor and the normal spinor embed in the ordinary ten-dimensional Clifford carrier. The apparent unit-vector singularity is removable because sin beta vanishes linearly. This is a prescribed smooth field, not a demonstrated stationary solution of the critical Dirac equation.

Its marginal is w=exp(-|u| squared/b squared)/(pi squared b to the fourth), and the source is e-star times q times w times (13). Thus both the source and its supporting trial field are finite, with their normalization calculated explicitly.

4. Neutrality Can Survive a Change of Shape

Reference and particle connections need not share the same radial function to preserve the total charge. Boundary integration gives the enclosed mixed integral

1(2π)3∫∣y∣<RFJ∧F02=ℓfJ(R)f0(R)2.(16) \frac1{(2\pi)^3}\int_{|y|<R}\mathcal F_J\wedge\mathcal F_0^2 =\ell f_J(R)f_0(R)^2. \tag{16}

The enclosed charge in dimensionless units is therefore

Q(R)=(12−ϵ)f0(R)3−ℓ2fJ(R)f0(R)2.(17) Q(R)=\left(\frac12-\epsilon\right)f_0(R)^3 -\frac\ell2 f_J(R)f_0(R)^2. \tag{17}

If both functions approach one, the total charge remains the existing value. In the neutral vacuum branch, ell is one and epsilon is zero. Equal functions give pointwise neutrality. Unequal functions give a signed distribution with exactly zero total charge.

For f0=1-exp(-r squared/a0 squared) and fJ=1-exp(-r squared/aJ squared), set lambda=a0 squared/aJ squared. Integration by parts gives the signed second moment

∫r2 dQ=a02[16−1λ(λ+1)(λ+2)].(18) \boxed{\int r^2\,\mathrm dQ =a_0^2\left[\frac16-\frac1{\lambda(\lambda+1)(\lambda+2)}\right].} \tag{18}

At equal widths it vanishes. Near lambda equal to one its slope is 11a0 squared over 36. A six-dimensional isotropic Fourier probe of this candidate density has a leading response equal to minus k squared times (18) divided by twelve. This is the geometric form factor of a signed neutral distribution; it is not yet an electromagnetic neutrino-radius prediction or an interaction-rate calculation.

5. Conserved Transport and the Undetermined Circulation Law

The closed-form current (3) accommodates rigid transport through the translated full spacetime forms. More generally the marginal construction (1) retains the base current supplied by the same field. Charge conservation does not uniquely determine the divergence-free circulating part of the normal current.

For illustration, the normalized norm current of the chiral normal texture is

jyρ=−sin⁡(2β)Jy^. \frac{j_y}{\rho}=-\sin(2\beta)J\widehat y.

One possible constitutive completion is to multiply this velocity by the source density and a common velocity scale c-star. It is divergence-free for a radial source and constant J; its speed is bounded by c-star. For (12), its magnetic moment tensor would be

Mij=e∗qc∗a MJij,M=∫0∞u e−3u/2(1−e−u)5/2 du≈0.14295428919041045.(19) M_{ij}=e_*q c_* a\,\mathcal M J_{ij}, \qquad \mathcal M=\int_0^\infty \sqrt u\,e^{-3u/2}(1-e^{-u})^{5/2}\,\mathrm du \approx0.14295428919041045. \tag{19}

This is a checked conditional integral. The closed-form source law and the trial action have not selected this circulation completion. It must not be called a derived physical magnetic moment. In particular, copied electron and muon shapes with equal a and equal circulation would again give equal moments; count-dependent geometry or circulation is still needed for their different observed mass-scaled responses.

6. What the Topological Action Actually Does on These Fields

The previous trial action contains the terms A wedge F0 cubed and A wedge F wedge F0 squared, multiplied by W. On the static spatial textures with constant temporal phase rates, its radial integrals depend only on the boundary values appearing in (11) and (16).

Changing a, b or the radial interpolation while retaining those boundary values does not change this part of the action. It therefore gives no restoring force for these size variations. A pure common-phase variation is also a boundary term against the closed source. This does not prove that the functional has no effect on more general time-dependent orientation histories, but it rules out claiming that it already sustains these profiles.

The distinction is constructive: geometric charge can stay fixed while metric-dependent energy selects shape. The same charge-preserving family is available to test candidate stabilizing terms without retuning charge to compensate for an energy change.

Because A, its reference and W are induced by the same field, their variations must still be retained together. The partial connection variation in the preceding note is not an independent Maxwell equation. The current has been constructed; the complete response equation remains to be selected and solved.

There is an explicit test of the induced connection contribution. For a normalized coefficient field z and a closed source kernel K, variation of A=minus i z-dagger dz gives

δA=2Im⁡(δz†dz)−id(z†δz). \delta A=2\operatorname{Im}(\delta z^\dagger\mathrm dz) -i\mathrm d(z^\dagger\delta z).

The last term contributes only a boundary term against K. Writing the dual kernel as a vector k-mu, the non-phase variation therefore depends on the projected derivative (I minus zz-dagger) times k-mu partial-mu z. If the kernel is purely temporal and the texture only changes by a common phase, that expression vanishes. Thus a nonzero charge-weighted kernel does not guarantee a nonzero field-shape force. Spatial orientation transport or further time-dependent shape motion can make the projected derivative nonzero; the reference and marginal variations remain additional terms. Once a common action is changed, the conserved current used to define W must also be checked for that action rather than silently borrowed from its predecessor.

7. A Bounded Stabilizing-Energy Comparison

To test whether charge preservation leaves room for a finite size, consider the explicitly new classical comparison energy

E[Ψ]=∫[κ∣∇Ψ∣2−gρ2+hρ3]d10x,κ,g,h>0,(20) E[\Psi]=\int\left[\kappa|\nabla\Psi|^2-g\rho^2+h\rho^3\right]\mathrm d^{10}x, \qquad \kappa,g,h>0, \tag{20}

at fixed conserved norm N. This is not asserted to be the IDWT master action or a preservation of its first-order spatial Dirac law. Its purpose is a bounded common-coefficient test of attraction balanced by short-range density cost, on the explicitly constructed charge-carrying field. It introduces no fitted particle amplitude or mass.

The energy is bounded below at fixed N, since minus g rho squared plus h rho cubed is at least minus g squared rho divided by 4h. Substituting (15), including the derivative of its orientation, gives

E(a,b)=κN(αa2+2b2)−gN232π5a6b4+hN3243π10a12b8,α=π430+158.(21) E(a,b)=\kappa N\left(\frac\alpha{a^2}+\frac2{b^2}\right) -\frac{gN^2}{32\pi^5a^6b^4} +\frac{hN^3}{243\pi^{10}a^{12}b^8}, \quad \alpha=\frac{\pi^4}{30}+\frac{15}{8}. \tag{21}

The orientation contribution to alpha is 3 zeta(4) minus 9/8; the Gaussian amplitude gradient adds three. No spinor-gradient contribution has been dropped.

At fixed volume a to the sixth times b to the fourth, the optimum aspect ratio is b squared/a squared=3/alpha. Write b=eta a with that ratio. The remaining energy has the form

E(a)=Ka2−Ga10+Ha20,K,G,H>0,(22) E(a)=\frac K{a^2}-\frac G{a^{10}}+\frac H{a^{20}},\qquad K,G,H>0, \tag{22}

where K, G and H are the combinations given by (21), not three additional adjustable particle constants. Its stationary equation is

Ka18−5Ga10+10H=0.(23) Ka^{18}-5Ga^{10}+10H=0. \tag{23}

Two positive stationary radii exist when

H<2G9(25G9K)5/4.(24) H<\frac{2G}{9}\left(\frac{25G}{9K}\right)^{5/4}. \tag{24}

The smaller is a radial minimum and the larger a radial maximum. At a stationary radius the second derivative is 100G/a to the twelfth minus 36K/a to the fourth. The aspect direction is also minimized by the ratio above. Thus the smaller branch is a minimum within this two-width family. A dimensionless control K=G=1, H=0.01 verifies a negative-energy trial minimum and the second unstable stationary point; these are not particle parameter fits.

This calculation establishes compatibility between finite-size variational stabilization and the unchanged charge weights. It does not prove a full field solution or stability against every deformation. In fact, a Gaussian cannot be the exact far tail of a stationary solution of the local energy (20): the Laplacian-to-field ratio grows quadratically with radius while the nonlinear coefficients vanish. A localized stationary branch of that comparison requires its own decaying Helmholtz-type outer solution and a solved core-to-tail boundary problem. That tail replacement can retain the orientation boundary data responsible for charge, but its equations must be solved rather than inferred from the trial minimum.

8. Physical Scope and Next Closure

The exact ten-dimensional critical source in section 2 and the fast-tail trial in section 3 are two distinct constructions. The first retains the known spatial Dirac solution and its existing clock limitation. The second repairs the charge moments but changes the spatial profile and requires new dynamics. Their successful properties cannot be combined as if one solved field already possessed them all.

A finite texture varying through all ten spatial directions also does not automatically preserve the literal lower-depth particle assignments. To interpret it as a lower-depth core with an orientation tail, one must derive that distinction from the common dynamics. Taking transverse widths to infinity does not solve the problem: the normalized marginal spreads without a localized limit, and the unnormalized uniform extension has infinite full-space norm.

The next closure is therefore an explicit common field equation admitting a persistent core, the required orientation tail, its temporal clock and a reciprocal response. The current work supplies normalized source formulas, charge-preserving shape freedom, a neutral-response diagnostic and energy conditions that such an equation can be tested against. It leaves the simplex count and the family support selection open.

Verification and Preservation

Independent analytical assessment now covers the stated connection, boundary integral, inherited charge weights, both source-action variations, finite marginal, radial moments and restricted trial-energy minimum. The assessment derives these identities without numerical replay; the successful checker runs remain author execution. Exact assessment SHA-256: e08b7131c1985c29ed09d9ad25fc6937165a8761165ea5e4342cd3c7251899bf. A physical source law, common charge unit, self-consistent clocked particle and complete core/tail dynamics remain open. Executable evidence is available as the finite-source checker, neutral-shape checker and ten-direction normalization checker. Their exact outputs are finite source, neutral shape and normalization.

The corrected main checker passed 31 conditions, including radial moments, generic curvature, finite orientation energy, conditional circulation, marginal continuity and trial-energy stationary points. The neutral-shape checker passed eight conditions. The ten-dimensional checker verifies the norm, marginal, curvature normalization, energy ratio and logarithmic charge-radius boundary. The first main attempt contained a wrong derivative sign in its new generic curvature formula; it was caught, corrected and rerun, with the failed code retained. It does not change the preceding rational-profile formulas.

All scientific executions used the shared 25-percent-core runner. Checks are author validation, not independent review or physical identification. The partial source action and its static response limits were assessed analytically. The maintained charge-curvature note at snapshot 4084, SHA-256 f07deee8a399c1410662b9de43ca814b56890a922a41c435fda35036dc29eb06, and the critical-profile/current source with its clock qualification remain the inputs. The maintained case incorporates the independently assessed mathematical construction, with the physical closure questions retained.