Photon geometry and physical polarization

IDWT's working photon picture is an extended transverse connection between material patterns of the single field $\Psi_\infty$. It is present along its span, with an advancing phase. The proposed photon is not a point object traveling from source to target. Its two bound spatial directions, its extension through other directions and its completed energy-transfer event are separate parts of the hypothesis.

This volume develops the geometric and algebraic results available for that picture. A transverse plane gives explicit polarization mathematics; it does not by itself derive a finite connection, where that connection ends, observed timing, energy transfer or protected zero rest mass. The earlier separated-wave, Maxwell and finite-matrix calculations remain useful conditional comparisons, with their additional premises stated.

Reviewed Connections as of 5 November 2026

The new vector-clock comparison is deliberately limited. An ordinary real vector has an invariant first-order temporal skew clock only in dimension two; that two-component vector does not supply the same spatial Clifford law as a spinor. The corrected carrier discussion therefore does not derive photon, Higgs or massive-boson equations. The gapped density family also contains a depth-two spinor branch, so it gives no general exclusion of spinors there. Physical transverse polarization, the extended-connection photon hypothesis and its timing/detection questions retain their own scope.

The connection and the event

The proposed connection spreads transversely in a line-like wedge or triangle-shaped structure. Its bound plane belongs to the same shared higher-dimensional physical space as both endpoint patterns. “Free without” refers to its extension along a direction outside that bound plane. It does not compactify a dimension or introduce a mediator particle. Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle.

In the proposed deterministic event, the actual field configuration has one best-matched target. Completing the transfer changes the configuration for later events. Successive connections can differ in their origin, orientation, phase and polarization. Their apparent statistics are to come from these unmeasured real-valued conditions, not a postulated position-probability law. Geometry favors closer suitable targets over otherwise equivalent distant ones; where the connection ends, and its quantitative distance dependence, still require the same-field equations.

The working interpretation assigns one connection to one target, so more light means more connections. It associates the large resonant target area with the extended photon's scale rather than the dense atom's size. Contact phase matters: a crest or trough is proposed to engage differently from a zero crossing. These are physical hypotheses to be tested, not results of the projector calculations below.

A free extension is uniform over its span and ends where it makes contact. It is not infinite, and nothing travels along it: the object is already spread across the distance, so a transfer through free directions has no crossing time. Light rays are not infinite; they end where they hit something. Changes of bound profiles and bound-direction energy transport still take time. The actual equations must still satisfy conservation and observed signalling constraints. Neither a nonzero field tail nor a static overlap proves that a photon event occurs. A coherent model must account for endpoint energy, supporting stress, recoil and the full field balance. It cannot assume those quantities vanish because no traveling particle is introduced.

What phase must explain

A phase difference at a single frequency does not determine a delay. For example, in a linear-response comparison, write a transfer function $H(\omega)=|H(\omega)|e^{i\Phi(\omega)}$ with the time convention $e^{-i\omega t}$. A narrow-band stationary-phase calculation identifies a delay from $\partial_\omega\Phi$, not from $\Phi$ at one chosen frequency alone. The IDWT connection must derive its response and observable timing, including their distance dependence. Naming phase transfer does not yet reproduce the measured timing law.

The same requirement separates this proposal from mathematical models whose retarded and advanced solutions already assume a conventional time-of-flight response. Those may be comparisons, but do not establish the proposed absence of photon-object travel.

The two physical bound directions

At a point on a connection, let $E_\gamma$ be its oriented physical two-plane in a local real Euclidean spatial chart $V$ of dimension $r\ge3$. Then

$$V=E_\gamma\oplus E_\gamma^\perp, \qquad \dim E_\gamma=2, \qquad \dim E_\gamma^\perp=r-2.$$

“Normal” means perpendicular to this particular plane. It is not a different kind of spatial direction. There is one such direction in a three-direction observer chart, four within a six-dimensional comparison and eight within a ten-dimensional comparison. This does not give the electron or tau an inherent physical compartmentalization.

The transversality module uses a separated classical field comparison

$$\Psi_\gamma(u,y,t)=\chi_0(u)\,\varepsilon e^{i(\mathbf K\cdot y-\omega t)},\qquad u\in E_\gamma,\quad y\in E_\gamma^\perp,$$

with $\varepsilon\in E_\gamma\otimes\mathbb C$ and $\mathbf K\in E_\gamma^\perp$. Complex notation packages two real oscillating quadratures. It does not introduce a complex physical position space or a Born probability. The geometric conclusion is

$$P_{E_\gamma}\mathbf K=0,\qquad \mathbf K\cdot\varepsilon=0.$$

It holds at every containing depth under these assignments. It is a conditional transversality theorem: assigning the phase gradient and polarization to orthogonal subspaces enforces orthogonality. Depth two alone does not derive that allocation or the electromagnetic constraints. The constant-magnitude factor has uniform free-direction support and changing phase; it is not a finite-energy completed connection merely because it illustrates extension along the whole span.

There is a useful constraint check. If $\chi_0\varepsilon$ is identified directly with a source-free electric field, its full divergence additionally requires $\varepsilon\cdot\nabla_u\chi_0=0$. A real scalar envelope with constant circular polarization then has both transverse derivatives zero. That particular factorization cannot also provide transverse localization. Spatially varying polarization, additional components or a different reconstruction from $\Psi_\infty$ may change the result.

Linear, circular and elliptical polarization

Choose real orthonormal vectors $e_1,e_2$ in $E_\gamma$. For $\varepsilon=a+ib$, the real harmonic displacement is

$$\operatorname{Re}(\varepsilon e^{-i\omega t}) =a\cos\omega t+b\sin\omega t.$$

Collinear $a,b$ give linear motion; orthogonal equal-length $a,b$ give circular motion; the general nondegenerate case is elliptical. This is actual real-plane motion. The rank of one ellipse does not determine the support dimension of the complete field pattern.

The circular basis and its physical rotation law are

$$e_\pm=\frac{e_1\pm i e_2}{\sqrt2},\qquad e_\pm\mapsto e^{\mp i\alpha}e_\pm.$$

Thus the bound plane has two orientation weights $\pm1$. In a three-direction observer description, where the plane is the full complement of the phase direction, these reproduce the helicity weights of the usual transverse spin-one comparison. A higher-dimensional massless representation needs its own full symmetry analysis; the two-plane construction does not supply all of it automatically.

A parity-complete solution space permits both circular modes. A particular real field need not occupy both independent positive-frequency modes: reality pairs each positive-frequency component with its negative-frequency conjugate. For instance, $\operatorname{Re}[(e_1+i e_2)e^{i(ky-\omega t)}]$ is real and circular.

For normalized linear preparation $\varepsilon$ and linear analyzer $e$, the conditional squared projection is $|e^\dagger\varepsilon|^2$. It gives $\cos^2\alpha$ for relative linear angle $\alpha$ and $1/2$ for either circular input against a linear analyzer. This is an angular overlap of classical field components. An event rate requires temporal dynamics, the actual incident normalization and available target responses; the overlap alone is not a probability postulate.

Pattern-sensitive coupling in the same field

The plane-coupling calculation retains a target's spatial rotation tensor. Let $k$ be a unit phase direction normal to $E_\gamma$ and write $u\wedge v=uv^T-vu^T$. For complex skew matrices use

$$\langle A,B\rangle=\frac12\operatorname{tr}(A^\dagger B).$$

With $B_a=e_a\wedge k$, the $B_a$ are orthonormal. For a target tensor $M$, the projection onto this photon's two allowed wave amplitudes is

$$\mathcal Q_{E,k}M=(P_EMk)\wedge k,\qquad \|\mathcal Q_{E,k}M\|^2=\|P_EMk\|^2.$$

This projector has rank two. A projector onto every transverse amplitude has rank $r-1$ and agrees with it only at $r=3$. Summing all higher-dimensional transverse directions is therefore a different diagnostic from retaining one physical photon plane.

For a real target tensor $J$, put $h_a=e_a^TJk$ and $C=h^\dagger h=\|P_EJk\|^2$. A normalized input $c\in\mathbb C^2$ has squared overlap $|c^\dagger h|^2$. Its maximum is $C$; either circular input gives $C/2$ for real $J$. The sum over two orthonormal inputs is $C$, whereas their equal average is $C/2$. Every input is dark to this tensor precisely when $P_EJk=0$. Sharing spatial directions does not decide that equation.

Several independently distinguishable target responses produce

$$W=\sum_\lambda w_\lambda h_\lambda h_\lambda^\dagger, \qquad w_\lambda\ge0,\qquad c^\dagger Wc\ge0.$$

A dark input must be orthogonal to every response with positive weight. Two independent response vectors can remove the dark direction present for a single tensor. Coherent contributions to the same output instead add as amplitudes before squaring. Spatial profile integrals, phases and canonical normalization remain necessary for a physical transition vertex. A static overlap zero also does not prove that its mixed response derivative vanishes when both patterns deform.

Source-led projection and reciprocity

The descending-source construction starts from the higher-dimensional pattern and projects the resulting same-field source. In a fixed product chart, let $\chi_0(u)$ be a normalized mathematical bound profile and $S_\gamma$ an isometric map from the two amplitudes to retained components of the one field. Define

$$(I_\gamma a)(u,y)=\chi_0(u)S_\gamma a(y),\qquad (P_\gamma\Psi)(y)=\int\overline{\chi_0(u)} S_\gamma^\dagger\Psi(u,y)\,du.$$

Then $P_\gamma I_\gamma=I$. Integrating only the bound coordinates retains the normal-coordinate phase. This norm is a mode normalization, not a physical position probability. The map, product measure and field identification are premises; moving or curved frames require their derivatives and Jacobians.

For $\Psi=I_\gamma a+h$ and $P_\gamma h=0$, a specified field equation $L\Psi+\mathcal N[\Psi]=0$ gives the exact row

$$P_\gamma L I_\gamma a =-P_\gamma Lh-P_\gamma\mathcal N[I_\gamma a+h].$$

The complementary linear term vanishes only if the operator preserves the two subspaces. A nonzero projected static source is not automatically radiation or a completed photon event. The complete background and its temporal spectrum must be solved.

If $H$ is the real self-adjoint energy Hessian at a valid background, the cross response is

$$B_{\gamma b}=I_\gamma^*HI_b, \qquad B_{b\gamma}=I_b^*HI_\gamma=B_{\gamma b}^*.$$

One action supplies both directions. Nonorthogonal retained families need the kinetic Gram normalization, and field-dependent inclusions must also be varied. These are conditional same-field reduction identities, not independent photon and target fields or a calibrated absorption law.

Connection geometry and masslessness

The following orientation and bundle constructions are mathematical comparisons. Their parameter spaces are not additional physical manifolds. A physical use must express the relevant quantities as algebraic functions of the single field on the same 10+1 spacetime. The newer particle-geometry account examines that distinction in its full-spatial charge calculation.

The orientation space of the plane in a real $r$-dimensional chart is $\operatorname{Gr}_2^+(\mathbb R^r)$. At observer depth three, it is $S^2$: an oriented plane determines an oriented unit normal. The equivalent $\mathbb{CP}^1$ description is an orientation space here, not a compact physical extra dimension. The two helicity-line bundles have Chern numbers $\pm2$, up to orientation convention, and correspond to $\mathcal O(\pm2)$ rather than the basic Hopf line $\mathcal O(1)$.

A smooth single-valued scalar phase cannot by itself supply a nontrivial electromagnetic curvature:

$$A=d\theta\quad\Longrightarrow\quad F=dA=d^2\theta=0.$$

A normalized representative $z\in\mathbb C^2$, $z^\dagger z=1$, instead gives the conditional Hopf connection

$$A=-iz^\dagger dz,\qquad F=-i\,dz^\dagger\wedge dz, \qquad A\mapsto A+d\alpha\quad(z\mapsto e^{i\alpha}z).$$

Its curvature can be nonzero. For one map into the two-real-dimensional target $\mathbb{CP}^1$, however, $F\wedge F=0$. This particular map does not establish a general higher-dimensional connection. The desired connection and its dynamics must be derived from retained and eliminated modes of $\Psi_\infty$, rather than added as an independent mediator.

A curvature action such as $-\tfrac14\int F_{AB}F^{AB}$ is an explicitly specified effective comparison. Its null plane-wave equation is compatible with transversality and a gauge equivalence removing the pure-gauge direction. That comparison does not derive the required photon-plane restriction or the complete connection event.

The rest-count convention $S(0,2)=0$ must be distinguished from a protected dynamical zero. For a complete effective mass operator the required identity is

$$P_\gamma\mathcal M P_\gamma=0$$

through the allowed nonlinear and corrected dynamics. A symmetry or constraint must explain it. Naming the zero-count branch supplies no such protection and does not set the connection's oscillation frequency to zero.

Energy, stress and momentum comparisons

The completed-transfer hypothesis must derive the full same-field energy and recoil balance. It currently locates transferred energy at the material endpoints; whether a proposed connection also has energetic support or stress must follow from its actual action. The following Maxwell identities are empirical and mathematical comparison constraints.

For a massless four-momentum with positive energy, $p=E/c$. In the source-free Maxwell comparison, let $u=\epsilon_0(|\mathbf E|^2+c^2|\mathbf B|^2)/2$ denote energy density and $\mathbf S=\epsilon_0c^2\mathbf E\times\mathbf B$ its energy flux. Here $u$ is a scalar density, distinct from the earlier bound-coordinate label. Momentum density is $\mathbf S/c^2$, and the real fields obey

$$u^2-\frac{|\mathbf S|^2}{c^2} =\frac{\epsilon_0^2}{4} \left[(|\mathbf E|^2-c^2|\mathbf B|^2)^2 +4c^2(\mathbf E\cdot\mathbf B)^2\right]\ge0.$$

Hence an integrated finite-energy field has $|\mathbf P|\le U/c$. Equality requires both locally null fields and aligned flux wherever energy is present. Two disjoint equal-energy contributions at angles $\pm\alpha$ give $|\mathbf P|=(U/c)\cos\alpha$. Thus one cannot infer an integrated null four-vector from a local plane-mode identity when the complete field has directional spread. This is especially relevant to a finite extended connection and its endpoint stress.

If a one-quantum normalization is adopted in a comparison, $E=\hbar\omega$ and $\mathbf p=\hbar\mathbf K$ are additional energy–frequency identifications. The classical phase-gradient ratio does not independently derive that normalization. The same distinction between supplied action scale and complete physical energy appears in volume 12.

The massive depth-two comparison

The depth-two control tests W, Z and Higgs labels $(76,81,95)$ within a prescribed two-mode normal-carrier calculation. It does not assume that their physical polarization is identical to the photon. In particular a massive vector's longitudinal channel and the Higgs scalar comparison require their own field realization. A two-component matrix degeneracy does not establish either physical mode content or helicity.

For the prescribed hopping $g=\gamma h_0$ with $\gamma=10000$ and $h_0=0.4773956986626318$, the matrix is

$$T_n=\begin{pmatrix}n&0\\g&n+1\end{pmatrix},\quad A=g^2+n^2+(n+1)^2, \quad s_+^2=\frac{A+\sqrt{A^2-4n^2(n+1)^2}}2, \quad s_- =\frac{n(n+1)}{s_+}.$$

The numerator is $2S(n,2)$. The finite correction to its leading count law satisfies

$$\frac1{\sqrt{1+[n^2+(n+1)^2]/g^2}} \le\frac{s_-}{n(n+1)/g}\le1.$$

For an endpoint deformation $T'_{01}=\gamma\kappa_2$, the exact normalized response is

$$\frac{|s_-'|}{|\gamma\kappa_2|} =\frac{g s_+}{s_+^2-s_-^2}.$$

These are candidate finite-hopping corrections, not residuals against measured masses. The normal construction's depth-two round feature is zero-dimensional and does not supply its assigned spatial harmonic tower; the surface parameter is prescribed. Its positive constrained shape curvatures do not solve that physical carrier problem. Two opposite polarization blocks can cancel the specified centered two-band beat source, but those different frequencies are not the two equal-frequency polarization components of ordinary light, nor is their supporting energy thereby removed.

At $n=0$ the formal matrix has right and left zero vectors

$$u_0=\frac{(1,-g)^T}{\sqrt{1+g^2}},\qquad v_0=(1,0)^T.$$

An allowed centered deformation lifts the small singular frequency as

$$s_-(\alpha_2)= \frac{|g\gamma\kappa_2|}{\sqrt{1+g^2}} |\alpha_2|+O(|\alpha_2|^2).$$

The reference zero is therefore not protected by that matrix alone. A specified free phase law can coexist with a zero rest eigenvalue, but does not derive photon masslessness or the completed-transfer dynamics.

The current useful results are a physical-plane polarization interface, explicit orientation-dependent response and reciprocal projection identities, together with definite constraints on a protected zero and a finite energetic connection. A common same-field realization must join those results with the connection's actual endpoints, timing and full energy–stress balance. Those bridges remain open.

The frequency of a spectral line

The light of a spectral line has the frequency of the orbit's precession, the motion along the fourth direction. The in-plane orbital frequency matches no line. Because the up quarks act through four directions and the down quark through three, $\omega_w^2/\Omega^2=4/3$ in every atom. With a nucleus of finite width, the fourth-direction frequency at one Bohr radius is 97.08 nm against Lyman-γ at 97.20 nm (volume 18).

Coupling of a light pattern to an electron

Polarization is by definition a spatial direction. Take a transverse pattern $\phi$ of two bound directions and an electron field $\Psi_e$ under the density law, with $n=\Psi_e/\sqrt{\rho_e}$. Expanding the law's energy in $\phi$ gives a first-order and a second-order term. The photon's profile is a trial here, and its rate is not derived.

Second order. The term

$$E_2=-\delta_c\,\rho_e^{1/9}\left[\lvert\phi\rvert^2+\tfrac29\left(\mathrm{Re}\,n^\dagger\phi\right)^2\right]$$

is phase-blind after averaging over the beat, with weight $1+\lvert z\rvert^2/9$, where $z$ is the overlap of the two polarizations. An orthogonal polarization lowers the coupling by at most the factor $10/9$, so orthogonality does not switch it off. $E_2$ is lowest where the light's intensity overlaps $\rho_e^{1/9}$, giving a gradient force toward intensity maxima.

First order. The term $E_1=-2\delta_c\,\mathrm{Re}[z\,e^{-iDt}G(R)]$, with $D$ the frequency difference, is coherent. Its force lies along the phase gradient and depends on relative phase. It is the term that can transfer momentum along the direction of propagation. On an orbit with frequencies $\nu_1,\nu_2$, the phase expands by the Jacobi-Anger identity, and a secular force appears only at

$$D=m\nu_1+n\nu_2,$$

with weights $J_m(a_1)J_n(a_2)$. Torque resonances lie one orbital quantum away, at integer $p=D/\nu$, and tilting the orbit requires a wave vector out of the orbital plane. This algebra has been verified numerically. It is a statement about the stated model; deriving the actual pattern of light and its absorption rate remains open.

Local Polarization and the Complete Spatial Pattern

The photon's two physical polarizations provide one comparison target. The broader common-fundamental texture construction allows phase-offset portions and spatially varying orientations to describe richer matter patterns without adding independently resonating higher modes. A vector may trace one circle at each point while the spatially integrated circulation occupies five ordinary planes. The independently assessed normal-form account supplies both that smooth ten-direction example and the exact local Weyl singular-value classification.

Local vector motion, a spinor coefficient bilinear, whole-field circulation and a particle's centre trajectory remain distinct observables. The example establishes geometric compatibility, not a new photon degree count, a longitudinal photon or a solved material polarization law. Portion-event registration and a waveform's Fourier period likewise require separate observables.