Spatial Curvature, Persistent Patterns and Observable Projections

One useful way to connect particle properties is to ask which features of a field pattern can change continuously and which remain fixed during that change. Mass, radius and magnetic response may depend on the detailed shape, while another quantity can depend only on how its orientation closes at the boundary. A shared charge-like label would then be compatible with different masses without assigning each particle an unrelated law.

This note constructs such a distinction for an existing IDWT spatial profile. The main result is an explicit phase connection whose curvature is built from the same spinor polarization that already determines the profile's circulating norm current. In a pure-polarization branch, a normalized integral of the highest curvature product equals one at depths four, six and ten. The value is independent of the profile radius and overall amplitude. At depths six and ten it reverses sign when the connection reverses. These are conditional mathematical results, author-checked and awaiting independent review. Electric charge, count selection and physical stability are further identifications to test.

All coordinates below belong to a ten-direction matter reduction within infinitely many ordinary physical spatial dimensions plus time. Restriction to a depth means dependence on a subset of those spatial directions. Spinor coefficients and trajectory covariance matrices are descriptions of this field and its motion, not additional physical manifolds. No mediator or compact spatial dimension is introduced.

1. Ingredients from the Supplied Literature

Kai Lou's Higher form symmetries of axion electrodynamics in 6-dimension with mass, Physics Letters B 882 (2026), 140996, constructs conserved differential forms from field strengths and their products, then studies their coupled symmetries. Its equations (55)–(57) provide the immediate mathematical lead: use the closure of a curvature form to construct several related quantities. The IDWT construction below supplies its own connection directly from its existing field; it does not assume the paper's independent axion and gauge fields. Original article.

Katsushi Arisaka's The Electron as the Ground State of Nature, version 20.3, provides a second lead: connect identity and persistence through a common functional and its distinguished configurations. Sections 11–13 and Appendix D are the portions examined in detail here. Its particular uniqueness and generation claims depend on companion results and a different geometric framework. We use the functional-and-transition question as a constructive lead and formulate it for actual IDWT fields below. The newer version 23.1 was consulted for navigation; version 20.3 is the supplied source used for the comparison. This is a targeted extraction, not a complete audit of either version or its companion series. Supplied record.

Cameron J. R. Duncan's dissertation Ultrafast Electron Diffraction with High Six-Dimensional Brightness was identified, but its full text was not retrieved. A directly accessible related experimental paper by Li, Duncan and collaborators measures beam covariance, emittance and detector resolution. Its section III suggests a useful diagnostic: preserve the distinction between complete deterministic motion and the part registered by a detector. The calculation in section 7 below is our own elementary projection example, not a result attributed to the inaccessible dissertation. Related primary paper.

The incomplete Science DOI fragment /doi/abs/10.1126 does not identify a source and contributes no claim to this note.

2. Construct a Connection from the Existing Field

Start with the maintained spinor profile and its current, equations C1–C5 of Spinor Polarization and Spatial Circulation:

$$\psi(x)=F(r)\left(1+\frac{i x\cdot\gamma}{a}\right)\chi, \quad F(r)=A(1+r^2/a^2)^{-d/2}, \quad s=\chi^\dagger\chi>0.$$

Here a is a positive physical length, A is an uncalibrated field amplitude and the Hermitian gamma matrices encode rotations of the physical spatial directions. Set

$$v_i=\chi^\dagger\gamma_i\chi, \qquad B_{ij}=-i\chi^\dagger\gamma_i\gamma_j\chi\quad(i\ne j), \qquad D=a^2+r^2.$$

Normalize only to extract orientation:

$$z=\frac{\psi}{\sqrt{\psi^\dagger\psi}} =\frac{(a+i x\cdot\gamma)\chi}{\sqrt{sD}}, \qquad \mathcal A=-i z^\dagger\mathrm dz. \tag{1}$$

This normalization assigns no physical amplitude or position probability. Complex notation packages real components of the same field. The real one-form $\mathcal A$ measures local phase variation as a function of physical position; it is not an independent field added to the theory.

Multiplication of the matrices in (1), using their Clifford relation, gives

$$\mathcal A_i=\frac{a v_i-(Bx)_i}{sD}. \tag{2}$$

For an even-depth chiral seed, v vanishes. Write J=B/s. Then

$$\mathcal A_i=-\frac{(Jx)_i}{D}, \qquad \mathcal A_i=\frac{j_i}{2a\rho}, \quad \rho=\psi^\dagger\psi, \quad j_i=\psi^\dagger\gamma_i\psi. \tag{3}$$

Equation (3) is the first bridge: the phase connection is proportional to the circulating norm current divided by density. The proportionality follows for this actual profile, rather than being imposed as an electromagnetic source law.

Its curvature $\mathcal F=\mathrm d\mathcal A$, with components defined by $\mathcal F=\tfrac12\mathcal F_{ij}\mathrm dx^i\wedge\mathrm dx^j$, is

$$\mathcal F_{ij}=\frac{2J_{ij}}{D} +\frac{2[x_i(Jx)_j-(Jx)_i x_j]}{D^2}. \tag{4}$$

In particular, $\mathcal F(0)=2J/a^2$. Polarization planes become curvature planes in ordinary space. No separate axion is needed to obtain this mathematical object.

3. One Curvature Gives Several Closed Forms

Where the normalized field is smooth,

$$\mathrm d\mathcal F=0, \qquad \mathrm d(\mathcal F^{\wedge k})=0. \tag{5}$$

The first equality follows from taking an exterior derivative twice; the second follows from the product rule. These are kinematic identities. They do not yet say that a particular integral is a physical Noether charge or that a detector measures it.

On a spatial d-dimensional restriction, a two-form has at most floor(d/2) independent plane blocks. Its kth exterior power must vanish when 2k>d. For maximal-rank preparations:

These are products of oriented area forms, not additional oscillators. A particle may retain one common temporal fundamental while carrying several such spatial quantities. Actual rank can be lower than the dimensional maximum, and derivatives in containing directions require a full ambient extension.

The absence of a top-degree even product in odd spatial dimension does not establish neutrality. Odd-depth fields still have lower-degree closed forms and can carry boundary data. The maintained depth-five charge construction must remain a separate source until an explicit map connects it to these quantities.

4. A Size-Independent Integral at Six and Ten

Take d=2m and choose the pure-polarization branch with J consisting of m equally normalized oriented two-plane blocks. In coordinates aligned with them, each block is

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad J^2=-I.$$

This branch is compatible with the maintained six-dimensional chiral polarization construction. At ten dimensions purity is an additional restriction on the seed, not automatic for every chiral seed.

Rotational symmetry preserving J lets us place x in the first two-plane. Equation (4) then has first block coefficient $2a^2/D^2$ and all remaining block coefficients $2/D$. Therefore

$$\operatorname{Pf}(\mathcal F)=\frac{2^m a^2}{(a^2+r^2)^{m+1}}, \qquad \mathcal F^{\wedge m}=m!\frac{2^m a^2}{(a^2+r^2)^{m+1}}\mathrm d^{2m}x. \tag{6}$$

The Pfaffian is the signed product of the independent plane coefficients; its square is the determinant. Using the sphere area $2\pi^m/\Gamma(m)$ and

$$\int_0^\infty\frac{r^{2m-1}}{(a^2+r^2)^{m+1}}\mathrm dr =\frac{1}{2ma^2},$$

we obtain the dimensionless quantity

$$\mathcal Q_m=\frac{1}{(2\pi)^m}\int_{\mathbb R^{2m}}\mathcal F^{\wedge m}=1. \tag{7}$$

This normalization is stated explicitly. It is not a measurement of elementary electric charge and is not the factorial convention sometimes used for a Chern character.

Inside a ball of radius R the same calculation gives

$$\mathcal Q_m(R)=\left(\frac{R^2}{a^2+R^2}\right)^m. \tag{8}$$

Thus the spatial distribution changes with a and depth while the full integral remains one. A cancels before differentiation. The mathematical result also holds at d=2, where the original profile's norm is not finite; that endpoint is an algebraic control, not a localized particle claim.

Under $\mathcal A\mapsto-\mathcal A$, the integral becomes $(-1)^m\mathcal Q_m$. Complex conjugation of the normalized coefficient field produces this reversal of its phase connection. For six and ten dimensions, m is odd and the sign reverses. At four dimensions it does not. Whether that coefficient conjugation is the physical antiparticle operation requires the actual common dynamical law; the sign calculation alone does not establish it.

This provides a concrete candidate reason for different charged-lepton shapes to retain an equal signed geometric label. It neither forces the pure ten-dimensional branch nor incorporates the occupied simplex counts. Its value can coexist with varying mass because it is insensitive to a continuous size change and an overall amplitude change. Agreement with electric charge would require a common physical source law and the existing charge-generator normalization, not a relabeling of (7).

5. Boundary Conditions Are Part of the Mechanism

There is no contradiction between $\mathcal F=\mathrm d\mathcal A$ globally on ordinary space and the nonzero integral in (7). Stokes' theorem gives

$$\int_{B_R}\mathcal F^{\wedge m} =\int_{\partial B_R}\mathcal A\wedge\mathcal F^{\wedge(m-1)}. \tag{9}$$

The normalized orientation has nontrivial angular dependence at large radius, even though the unnormalized field decays. The boundary term survives. No compactification is being used.

For a smooth variation of the connection,

$$\delta\mathcal Q_m=\frac{m}{(2\pi)^m} \int_{\partial V}\delta\mathcal A\wedge\mathcal F^{\wedge(m-1)}. \tag{10}$$

Consequently, compactly supported smooth variations preserve the full integral. Unrestricted boundary changes, zeros at which normalization fails, or changes of seed class need separate treatment. The result is a boundary-controlled invariant for this class, not automatic integer quantization for all fields.

The normalized orientation also cannot be assigned an independent Maxwell energy without checking the tail. For m greater than one, the tangential curvature falls as r to the minus two, so the unweighted spatial integral of its squared magnitude diverges at d at least four. A constructive finite diagnostic is instead the density-weighted integral of rho times the squared curvature: since rho falls as r to the power 2 minus 2d, its radial tail falls as r to the power minus d minus three and converges. Such weighting would be a new constitutive choice if added to the action. At fixed total norm its value scales as a to the minus four, so this term alone does not select a finite size. Moreover, the original field becomes small at infinity: changing normalized boundary orientation need not have a finite lower energy cost. A conserved boundary label is not by itself energetic protection against decay.

A smooth phase change $z\mapsto e^{i\theta(x,t)}z$ gives $\mathcal A\mapsto\mathcal A+\mathrm d\theta$, so curvature is unchanged. This establishes insensitivity to an overall temporal phase and to a common smooth local phase factor. It does not establish insensitivity to arbitrary changes in relative phases of the constituent portions: those can change z itself, introduce zeros or change boundary orientation. Nor does invariance of this diagnostic make arbitrary local phase changes symmetries of the original field equation.

6. Join the Invariant to the Persistent-Configuration Idea

The constructive adaptation of the geometric-electron paper is to search for distinguished actual field patterns at fixed conserved data. A possible variational question is

$$\inf E[\Psi]\quad\text{subject to the same physical charges, prescribed boundary class, and permitted spatial support.} \tag{11}$$

The energy and constraints must come from one action. Equation (7) supplies a concrete candidate geometric constraint to investigate. It does not supply E, nor does it establish that Q equals electric charge.

Three separate checks then become explicit. First, does a minimizer exist rather than spreading out or collapsing? Second, is its shape unique after allowing translations, rotations and the common temporal phase? Third, are there higher persistent branches with the same physical charge but allowed lower-energy final configurations and outgoing field flux?

A constrained minimum can support persistence; uniqueness can support identical intrinsic properties. These are different conclusions and must be checked separately. In particular, a topological label alone does not choose a size: (7) is identical for every a. The existing critical profile's scale and clock issues therefore remain active. Our new invariant gives a sharper constrained problem rather than silently resolving those issues.

The analogy with a descending functional must also retain total energy conservation. In a closed finite Hamiltonian system, a generic volume of initial states cannot simply contract into an attracting minimum. A localized core can instead lose energy to other regions of the same field. Then a reduced core description can relax while the full field conserves energy. A local minimum without an accessible escape process does not yield a finite decay rate merely because another lower minimum exists. One must derive outgoing channels, an instability or an allowed transfer mechanism; no noise or tunneling law is inserted here.

For a periodic field, a constrained functional such as E minus omega times a genuine conserved phase charge can locate relative equilibria if that phase symmetry exists. The charge in that multiplier is not automatically (7), the simplex count, or electric charge. Their identification would need a derivation. Likewise, counting critical points of a chosen auxiliary geometry does not establish three physical generations. The IDWT question is the number of persistent solutions of its actual common equations.

7. Use Beam Mathematics to Separate Motion from Registration

The related electron-beam paper reconstructs covariance from position and momentum measurements. That is useful even though its phase-space dimensional count is different from IDWT's physical spatial count. The applicable ingredient is a diagnostic for literal trajectories.

Let q(t), p(t) be deterministic physical trajectories and momenta. A finite observation record or ensemble of specified initial conditions supplies a covariance matrix Sigma of their coordinate list. These averages introduce no Born law. For a linear transfer M and detector projection P,

$$\Sigma_{\rm out}=M\Sigma_{\rm in}M^T, \qquad \Sigma_{\rm detector}=P\Sigma_{\rm out}P^T. \tag{12}$$

Here is an exact control. In coordinate order $(q_1,p_1,q_2,p_2)$, take unit input covariance and the canonical shear

$$q'_1=q_1,\quad p'_1=p_1-tp_2, \quad q'_2=q_2+tq_1,\quad p'_2=p_2. \tag{13}$$

The map preserves the symplectic form and the determinant of the full covariance, which remains one. But the first observed pair has covariance $\operatorname{diag}(1,1+t^2)$, with determinant $1+t^2$. An apparently broader registered distribution can therefore result from deterministic coupling to unobserved directions without an increase in the complete covariance volume. The t here is a dimensionless transfer parameter, not a fitted IDWT evolution time.

For the electron's six spatial coordinates, the complete canonical list would have twelve entries if all six have independent conjugate momenta. For a ten-spatial-dimensional model it would have twenty before any constraints. These lists are bookkeeping for trajectories in the single physical space. They are not additional spatial arenas.

This gives a concrete analysis to apply to the maintained orbit-plus-precession trajectory: compare the complete trajectory covariance, the detector's projection and polarization-dependent changes of that projection. It can distinguish a change of observed shape from a change of the underlying field structure. Equation (13) is the completed diagnostic example, not an assertion that this transfer already describes the electron or explains diffraction.

8. Combined Consequences and Next Calculation

The three useful ingredients fit a common sequence: extract spatial orientation from the field; build its curvature and boundary data; constrain persistent configurations by the quantities the common dynamics actually conserves; compute how those configurations appear in a specified detector projection.

The strongest immediate calculation is to put the existing charge generator and the new connection on the same spatial profile. Test whether one physical source law relates the signed integral to electric charge while retaining the known fractional weights and neutral sector. The four-dimensional sign parity and the ten-dimensional purity condition are informative controls, not details to hide. Then vary the count-dependent shape while testing whether charge stays fixed and magnetic response changes consistently with the supplied mass.

There is also a direct coupling lead: for two spatial patterns, compare their connections, curvature ranks and boundary compatibility under the same action. Arbitrarily multiplying the separate invariants would not derive an interaction. A cross term must follow from the combined field, and it must account for both particles' responses and their actual relative phases and frequencies.

The current calculation establishes a concrete geometric quantity built from an existing IDWT field, not merely a resemblance to another theory. It also shows exactly why that quantity can stay the same while other geometric features vary. Its physical interpretation and compatibility with complete dynamics on the infinite-dimensional physical spacetime remain the next derivation.

Reproduction and Source Status

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. The bounded script boundary checker and its exact results contain 62 passing author checks: exact sampled Pfaffians in depths 2, 4, 6 and 10; analytic radial normalization and finite-ball differentiation; an explicit four-coordinate Clifford realization of the connection; representative closure; exterior-rank controls at 2, 3, 4, 5, 6 and 10; and the deterministic covariance example. These are checks of the stated formulas, not 62 independent physical predictions. The general Pfaffian argument is given in section 4.

Execution used the shared 25-percent-core runner, unit idwt-compute-e80e3e36fad44d4b9e9b9973a6223f93.service, exit zero. The result records the executed code SHA-256. No fitted amplitude, new agents or publication was involved.

The maintained source is content/notes/topics/coupling/spinor-polarization-and-spatial-current.md, snapshot 4081, SHA-256 bc3506a1b78129fe645e092a69f78a58e111dc514bbace7829eedc24aa9998e9. Its attached clock qualification is the critical-clock dilation analysis, SHA-256 1c67bdff98f8815f555b18afe2760f71a0e09e81fc6cdb0709445a59fefad3d8. This work uses the spatial profile and current, and does not claim that their critical temporal completion already supports the physical clock. The case account includes this construction with its independently assessed mathematical scope.