Generalized sector coupling

Summary: A common interaction on arbitrary nested sectors yields reciprocal responses, normalized vertices, and testable coupling relations, with neutrinos as a channel-specific control.

Physical scope

There is one infinite physical manifold with one nested chain of ordinary macroscopic spatial submanifolds. Particles at different depths are structures of $\Psi_\infty$ on that chain. All directions belonging to a given particle are of the same spatial kind. A decomposition relative to another particle's submanifold describes their relation; it does not give either particle an inherent division into different kinds of space.

The following construction applies to arbitrary finite collections of sectors, including the working depths $2,3,4,5,6,10$. It assumes specified spatial maps, measures, mode subspaces, and an interaction channel. It does not require accepting a particular list of global sphere or projective topologies. Actual embeddings and their domain conditions must be supplied wherever a proposed topology is used.

IDWT has no mediator particles. The source profiles, response modes, and representation indices below describe the one field on this physical space. Coupling starts with its direct interaction terms. Eliminating some of its modes is a conditional reduction of those equations, not a particle-exchange mechanism.

The working coupling hypothesis includes a changing geometric match: each particle can drive the other's spatial pattern toward greater compatibility or toward incompatibility during an encounter. Bilateral polarization coupling separates that physical target from a restricted orientation benchmark whose relative overlap remains constant. Both the change in compatibility and its effect on subsequent interaction must follow from the same dynamics.

The concrete shape calculation is developed in Transition-density coupling, followed by Reciprocal scattering of spatial modes. The complementary route through rotation-neutral modes and their spatial transport tests how one geometric operation can be blocked while another remains active. The constructions below provide their common spatial maps and normalization.

Localized spatial fields and reciprocal feedback

Positive-frequency scalar vertices retain complete incoming bands and the off-diagonal spatial interference that builds the local source. A fixed normalized incoming polarization has massless inclusive squared strength $\sin^2(\theta/2)$; a count of output components supplies no further multiplier. Available channels and the actual coherent contraction are separate parts of the calculation.

Axial spatial Dirac packets give an exact angular family with transverse localization and complete complex polarization vectors in odd-dimensional space. A full Cartesian calculation independently checks its three-dimensional head-on response. Localized scalar encounters also evolve four incoming packets in two chiral patterns, retaining the generated spatial modes beyond the resonant quartet. At the stated three-dimensional benchmark, opposite relative phases reverse an approximately $1.77\%$ classical chiral-norm transfer, with conserved total norm and a continuum reflection law.

Changing spatial compatibility compares the incoming and outgoing full patterns through the same prescribed free encounter, after removing their known free evolution. At fixed five-dimensional preparation rules and one coefficient, different complex phases give either increased or decreased contact. The effects are small and survive positive unit normalization of each outgoing pattern. Time ordering of the source derives the first nonzero contact coefficient from the complete local equation. Its independent coupling variations retain both partners' spatial response; no steering toward alignment is added.

Scalar Born frequency accounting supplies a further constraint. Complete scalar vertices remove the negative-output resonant Born channel for positive incoming scattering data, while a finite-time positive preparation can still have a nonresonant signed response. Those spectral signs have no assigned particle-antiparticle interpretation here.

These are conditional controls of the specified local scalar dynamics on ordinary spatial fields. Selecting stable occupied particle patterns, projecting one physically normalized interaction across distinct depths, and identifying observed electric and weak channels remain necessary. The calculations do not determine the global geometry of the observed three-dimensional space or an observed neutrino interaction rate.

Spatial current, propagation and measurable response

Local polarized Dirac pulses retain the actual scalar cross term of two traveling parts of the same field. They give exact spatial mismatch conditions and reciprocal weak-scattering maps with both complex-linear and conjugate terms. The full nonlinear collision changes normalized spatial compatibility in either direction under the same local equation. These are transversely unbound flat pulse controls, not identified particles or a completed distinct-depth coupling law.

Collective translation waves derives $\mathcal I=M+T/d$ from the complete scalar Dirac field. The two displacement polarizations at depth two propagate at half the underlying Dirac speed on the zero-bare-mass branch. Their count alone does not identify photons.

With an explicitly supplied minimal connection of the same field, Soft spatial current response derives the Thomson current, magnetic moment and uniform polarization correction. Soft polarized scattering retains the photon phase and every target polarization transition. At photon speed one, the forward spin coefficient is $(g_{\rm mag}-2)^2/(8\mathcal I^2)$. Canonical spatial spin response independently relates the electric spin-orbit coefficient to that same magnetic moment: $c_S=g_{\rm mag}-1$. These relations constrain one common current. The conditional six-dimensional scalar ground profile predicts $g_{\rm mag}=2.07691916$, which differs from the measured electron value $2.00231930436$; changing the common scale does not repair this ratio.

Conserved spatial charge derives the continuity identity for a proposed charge operator with every spatial derivative retained. A neutral rotation eigenspace by itself is not a conserved local electric-charge sector. A rotation-based source must retain its actual orbital and frame terms. No depth-five neutrality or surviving weak rate has yet followed from the same physical interaction.

A separate declared stationary component action supports exact null spatial pulses, with two normalizable spatial displacements and finite pulse energy. Its spatial displacement action derives the common quartic interaction by completing the full profile. The retarded spatial response then supplies a positive frequency-dependent kernel, a derived soft logarithm and outgoing radiation of the same field. This construction provides a calculable interaction mechanism with both polarizations. Its component action, background and coupling to occupied particle fields must still be derived from the master action before comparison with physical photon or neutrino rates.

Polarized pulse collisions gives exact reciprocal energy-dependent delays within the local displacement action. Polarized spatial radiation separately retains its full spatial continuum: opposite circular senses can cancel the leading outgoing channel while the conservative interaction survives. The same spectral measure fixes the long-pulse emitted-energy coefficient.

Complete polarization before choosing a coupling shape

Complete spatial polarization data retains every amplitude and relative phase in a specified spatial multiplet. Its circulation tensor is complete for one six-dimensional chiral multiplet, while a ten-dimensional field can require four-form information as well. These are field descriptors on the actual spatial manifold.

Equal-current spatial polarizations constructs two complete ten-dimensional bound fields with identical density and Dirac current but different responses to the same polarized lower partner. One reciprocal six-inside-ten rotation-moment energy gives an exact pair solution: both mean tensors stay fixed while one higher field's full polarization changes. The stated interaction is a conditional benchmark; its coefficient and occupied modes are not identified physical couplings.

Spatial polarization textures then allows amplitudes and phases to vary across the actual bound coordinates. It derives both the retained equation and its required Dirac complement. Preserving a scalar source does not by itself make a texture a spatial solution. This supplies the kinetic test needed before treating a richer pattern as a particle's allowed polarization.

Parts of the calculation

1. Nested sector reciprocity derives the upper-to-lower marginal and its reciprocal cylindrical response. Compatible maps compose along the chain without introducing a new coupling factor for every intermediate sector.

2. Canonical coupling normalization derives physical mode vertices from the kinetic metric and shows which charge and transition identities survive projection.

3. Common field response eliminates specified stable response modes of the same field. It derives the resulting correction to direct pair interactions and determines exactly when that correction has rank one.

4. Neutrino channel control tests the same construction against electric neutrality, colour singlet structure, surviving weak vertices, and low-energy response suppression.

5. Nested moment coupling applies the existing dot-square kernel to all distinct working depths. It gives an exact minimum-depth matrix, its full rank, and the trace projection required for a scalar rank-one response.

Results that do not require a parameter for every pair

For a real scalar interaction on $Q_a\subset Q_b$, the reciprocal source responses are

$$V_a=gK R_{ba}j_b,\qquad V_b=gR_{ba}^{*}K^{*}j_a.$$

The same coefficient appears in the action. Differences between the two response maps come from their domains, measures, and transpose relation. For compatible projections through $Q_c$, the source map satisfies $R_{ca}=R_{ba}R_{cb}$.

After canonical matter normalization, a coherent vertex becomes $Z_a^{-1/2}K_{ab}Z_b^{-1/2}$. This removes arbitrary choices of mode amplitude convention. A common conserved generator acting as $q$ on each retained family becomes $qI$ in every normalized family, even when their depths and profiles differ. This preserves a supplied universal charge; the physical charge assignment still has to be derived.

Eliminating a positive response block of the same field gives the correction

$$G_{ab}=B_a^{*}L^{-1}B_b.$$

For fixed source preparations this implies

$$G_{ba}=G_{ab}^{*},\qquad |g_{ab}|^2\le g_{aa}g_{bb}.$$

Rank one occurs exactly when all nonzero profiles $L^{-1/2}B_df_d$ are proportional. The same field can have many independent spatial responses, producing a higher-rank correction. Its direct coupling block remains part of the complete interaction. The inequality is limited to the positive static channel specified in the derivation; it does not automatically govern every physical force.

Consequences for the particle hierarchy

The generalized calculation separates three effects that can coexist:

• Equal charge across unequal depths. Consistent normalization preserves a common charge generator. Large differences in mode count need not produce differences in charge.
• Unequal response within one interaction. Spatial profile differences and field dynamics change finite-momentum vertices and contact strengths without changing the conserved zero-momentum charge.
• Absence from a particular channel. A derived generator or source map can annihilate a sector while other channel maps remain nonzero. Active neutrinos are an essential test of this distinction.

The explicit Gaussian benchmark couples equal shared marginals equally at every containing depth. Different widths produce a strict deficit from rank-one saturation. Consequently spatial nesting alone neither suppresses depth five nor guarantees a common rank-one response. These are exact results for the declared model, not a fitted particle coupling table.

The rank-one common-anchor proposal can now be tested at the operator level: derive whether the actual normalized source images are proportional in the field-response metric. Even if they are, converting that response to mass poles requires the reference-band factors and kinetic normalization retained in that proposal.

Result for the existing moment kernel

For centered isotropic profiles on nested flat spatial subspaces, the dot-square kernel gives $G_{de}=u_du_e\min(d,e)$, with $u_d$ containing the one-sector coefficient and second-moment scale. The matrix across $(2,3,4,5,6,10)$ has rank six when every $u_d$ is nonzero. Its normalized pair correlations are $\sqrt{\min(d,e)/\max(d,e)}$, independent of those scales. In particular, depth five has ordinary nonzero entries. This is a result for the specified scalar interaction, not an electric or weak coupling prediction.

Selecting only the trace of the spatial moment tensor would instead give the rank-one geometric factor $de/D$ in a containing depth $D$. The complete kernel also includes a nonzero traceless response. This identifies a definite physical projection to derive if a scalar common response is to follow from this kernel.

Physical inputs still needed for numerical predictions

The common construction fixes relations after its operators are specified. To predict particle couplings, V2 must supply the actual spatial and polarization maps $B_d$, the field's linearized evolution operator, the mode kinetic metrics, and the occupied preparations from its field equations. Depth-dependent parameters hidden inside those maps remain parameters. The electric, weak, and colour identifications also require the correct generators and representations on every sector. For the photon this includes its protected zero mode and transverse propagating field. The positive scalar confinement model developed below has no localized zero bound frequency; its massive normal branch cannot by itself supply that photon mode. This statement concerns the frozen bound eigenproblem. Collective fluctuations also vary the source and can have zero modes; their propagation requires the full fluctuation operator.

The next useful physical calculation is to derive a channel's source maps and evaluate their response Gram matrix across several particle sectors. This would test common-source factorization, channel zeros, and coupling ratios together. The present work supplies the equations and independently checked benchmarks needed to assess that calculation.

Verification

generalized_coupling_checks.py uses weighted spatial marginals, finite differences of reciprocal energies, conservative time evolution, nonorthogonal mode changes, independent response minimization, Gaussian quadrature, and explicit charge and weak-transition matrices. These verify the stated conditional results; they do not assign empirical coupling constants.

• Interaction operator tools — supplies the projected variational and spatial-profile prerequisites.

Coupling between two polarized spatial patterns

Both interacting objects carry spatial polarization. Their relative shapes, orientations, phases, and allowed responses must remain in the calculation. The higher-dimensional field may determine the source that drives a lower response; the lower bound depth need not set its strength. This possibility and photon-led probing are assessed through variations of the same action.

The new calculation proceeds through the following connected steps:

1. Descending photon source selects the photon bound mode while retaining normal propagation coordinates and phase. It distinguishes a higher source from a complete stationary solution.
2. Dirac source linearization differentiates the written nonlinear scalar term, retaining both ordinary and conjugate variations and the cubic spatial/frequency overlap.
3. Bilateral polarization coupling varies both objects. A zero static overlap can coexist with a nonzero mixed response; a solved reciprocal orientation benchmark conserves relative overlap while both objects precess.
4. Coherent spatial overlap retains both shapes and phases in exact Gaussian integrals and a normalized opposing-lobe source whose long-wavelength contribution cancels.
5. Polarized response spectrum combines normalized spatial vertices with the allowed target motions and finite exposure. It separates a protected dark input from a spectral or truncation zero.

Three finite geometric controls make the spatial assumptions testable. Photon plane coupling restricts a specified magnetic two-form term to the actual two photon amplitudes. Isotropic polarization comparison proves identical five- and six-dimensional overlap distributions for the stated uniform tensor ensemble. Spatial source complementarity derives which higher vector and rotational descriptors survive on a lower subspace, including the conditional depth-five joint-source lower bound.

These results supply explicit response operators and shape integrals, rather than a particle coupling table. Predicting distinctive five-dimensional suppression still requires the actual selected spatial solution, its allowed variations, and the physical source/output map. A chosen zero projection, an averaged density, or one missing response in a truncated basis does not replace that derivation.

polarization_coupling_checks.py checks the geometric projectors and distributions, full nonlinear source derivatives, reciprocal evolution, spectral response, and complex spatial integrals by independent methods.

From spatial shape to an actual response

Transition-density coupling derives the complete first and second variations of the nonlocal projected-density energy. Both changing spatial patterns enter its coherent quartic vertex; the frozen-background potential is retained alongside the product of source variations. For normalized quadrupole modes, the dot-square kernel yields

$$T_{0A;0C}=\frac{g}{2\omega_a\omega_b} \operatorname{tr}(A B C B^T).$$

Here $A$ and $C$ describe both shapes and $B$ their specified geometric comparison. Two traceless planar quadrupoles with relative angle $\theta$ have an amplitude proportional to $\cos2\theta$. At $45^\circ$ their coupling vanishes; at $90^\circ$ its sign reverses. A breathing mode and a traceless shape have a different exact selection rule. These are derived higher-order polarization examples on actual spatial profiles, with no particle-specific zero imposed.

Reciprocal scattering of spatial modes then solves incoming propagation and the target's response together in a conservative model of the same field. Orthogonal polarized inputs leave the target unexcited. Matched inputs reflect according to the frequency response, and a direct spatial wavepacket reproduces the result. A small nonzero vertex can still reflect fully at an isolated narrow resonance; a rare interaction rate requires the spectrum and incoming flux, not just a small static overlap.

The target's complete allowed variations also matter for photons. Polarized photon transition strength derives the full two-polarization response matrix for the specified simple spatial Pauli term. One incident polarization can be dark, but its summed response has a positive bound at every tested depth. This particular probe therefore supplies no automatic five-dimensional suppression.

A higher-source route to distinct channels

A more specific mechanism candidate starts with a six-dimensional spatial complex structure. Its compression to any five-dimensional hyperplane has two equal rotation planes and one null axis. In the spatial spin action, rotation-neutral modes then satisfy

$$Q_5=\frac{H_1+H_2}{2},\qquad P_0=\frac{I-H_1H_2}{2},\qquad Q_5P_0=0.$$

Three equal planes in six dimensions and five in ten have no zero spin weight. The cancellation is an operator identity for the specified probe, not merely zero mean source. The five-dimensional particle still has five ordinary spatial dimensions. Nonneutral polarizations also exist there, and the full higher source has a mixed term that acts on the neutral modes. A physical photon projection must determine which terms it samples. The equal-rate guarantee is specific to the six-to-five compression; direct ten-to-five examples can have unequal rates and no neutral vector.

Spatial transport of the neutral doublet provides a constructive second channel. The neutral projector is $P_0=(I-n\cdot\gamma)/2$, where $n$ is the actual spatial null-axis direction. As this direction varies over physical positions, the projected kinetic energy acquires a nonzero $\mathfrak{su}(2)$ curvature and the fixed positive term $|\nabla n|^2|b|^2/4$. Its determinant curvature is zero. Thus neutrality under the balanced rotation can coexist with response to changing spatial geometry, all within the one field.

This does not yet identify electromagnetic and weak interactions. Selection of the occupied neutral band, the complete physical current, reciprocal source motion and its normalization remain essential. Scalar self-binding alone leaves neutral and nonneutral constant polarizations degenerate. The neutral projection also does not close the first-order spatial Dirac symbol: perpendicular momenta still act through the omitted modes, whose contribution must be retained in deriving an effective kinetic term. Rotations of nested sectors further shows that the equal-plane rotations move some members of a fixed chain. They can preserve the entire nesting by transporting it together, but the moving-embedding and omitted-mode terms must then be included.

The decisive next calculation is the same-field source projection on a selected spatial particle solution: retain both partners' variations, identify the physical photon current, and check whether it annihilates the five-dimensional mode while another derived response remains. The current notes supply explicit shape vertices, a concrete neutral candidate and its geometric transport, together with the equations needed to test that identification.

spatial_coupling_mechanism_checks.py verifies these results with full Clifford representations, nonlocal finite differences, spatial quadrature, exact moving-mode derivatives, conservative scattering, direct wavepacket evolution, and independent projector transport.

Full spatial Dirac sources and reciprocal deformation

Rotation-neutral Dirac fields retains the full ordinary spatial field, including its angular lower components. The total rotation is orbital plus spin: a five-dimensional neutral field can occupy all five spatial directions even when its lower components are not neutral under the spin operator alone. Its local spatial current need not vanish.

The scalar moment branch solves a complete first-order spatial Dirac equation with its own scalar source. Its bound frequencies, dilation law and radial fluctuations are computed together. This is a conditional source model. Several massless-background branches have growing radial modes, and the scalar source leaves rest polarizations degenerate. It does not supply the actual Coulomb-plus-precession electron orbit or a selection of the occupied five-dimensional neutral mode.

Dressed source vertices reconstruct the eliminated spatial components and their kinetic norm. If $\Omega$ lifts a retained amplitude into the full field, a source $V$ has canonical vertex

$$V_{\rm can}=Z^{-1/2}\Omega^\dagger V\Omega Z^{-1/2}, \qquad Z=\Omega^\dagger\Omega.$$

A zero bare projection can hide a nonzero full response. Conversely, a supplied universal charge $qI$ remains $qI$ after consistent normalization; discarding part of the norm cannot derive neutrality.

The polarized shape response differentiates the actual spatial profiles in every quadrupole direction and retains both ordinary and conjugate fluctuations. The translation response includes the scalar center variation, recovers the zero translation mode and derives its inertial pole. A separately centered shape pairing alone has no separation-dependent force; that requires the pair's spatial contact or propagation dependence.

Dynamical shape opening then gives a concrete matching barrier. A higher shape can have zero static restriction into the lower sector, yet the higher field's polarized response rotates it into an accessible shape. Its leading low-frequency transfer retains both objects:

$$\delta M_a=\epsilon\, \chi_a(z,J_a)\mathcal R\chi_b(z,J_b)h_b+O(\epsilon^2).$$

The calculated opening amplitude can be linear in frequency without a special small coefficient assigned to one depth. A rate additionally requires the physical incident mode, occupied target and flux.

The self-consistent nested pair goes beyond fixed isolated profiles. One common scalar energy deforms both partners, with the same source coefficient in their shared directions. Their reciprocal scalar envelopes consequently have a symmetric coupling matrix $G_{ij}=\partial^2U/\partial s_i\partial s_j$. Its entries obey $Gs=M/5$ in the massless-background model; they cannot be fitted independently for each pair. Several nested depths have a common amplitude mode and a softer relative-amplitude mode, derived from their solved profiles with the same interaction coefficient.

Normal Dirac propagation retains the partner which carries motion through an ordinary containing direction. Exact uniform boosts preserve the scalar binding source. The full moment-response equation also retains source variation, spatial center motion and both ordinary and conjugate normal fluctuations. It provides a route to collective dispersion without fixing the bound profiles inside a small envelope ansatz. The scalar envelope derives the accompanying density-dependent profile response and also gives its explicit full-field residual when the density varies. Stability of those restricted envelope equations does not establish stability of the spatial bound profiles or decoupling of their center and polarization motions.

These results supply actual spatial source and response calculations for the proposed matching mechanism. The remaining physical identification is the occupied particle geometry and its photon current, followed by the surviving interaction channels and their rates. No neutral projector, shape label or relative-amplitude eigenvalue by itself completes that step.

Persistent spatial shape barriers classifies the complete geometric kernel across nested depths for an orthogonal higher circulation. Three restriction maps determine whether a higher pattern has direct access, opens linearly or quadratically in frequency, or stays invisible to the specified interaction. The visible ranks change with the higher circulation relative to the actual lower inclusion. A common-coefficient ten-to-five Dirac example verifies the quadratic opening with both responses and the lower breathing mode. The classification holds a specified source pattern fixed as frequency changes. A free spatial mode must also satisfy its own eigenproblem; a pure shape helicity with zero restriction remains inaccessible.

Spatial Dirac polarization waves restores the rest polarizations omitted by a single scalar envelope. Full linear polarization waves are gapless, and exact finite polarized planes can change the bound spatial current while preserving the scalar source. This shows directly why scalar binding alone does not select an occupied polarization or identify neutrality.

Reflection symmetry of scalar Dirac sources extends that boundary beyond radial confinement. Reflection-preserving scalar sources retain at least four-mode multiplets at depths five and six, even with unequal axis strengths. Their spatial shape response can depend on polarization while the source still lacks an isolated selected neutral doublet.

Bound frequency and the actual coupling source

Sawtooth harmonics in geometric coupling supplies a finite harmonic preparation ending at the maintained particle mass-frequency. Its adjacent-mode density beat has coefficient $1-1/S(n,d)$ at the sector fundamental. The same preparation produces reciprocal phase forces, preserves the spatial color-channel decomposition and splits angular shape modes without changing their zero-energy geometry. Its common receiver comparison covers all maintained depths and distinguishes direct density response from a derivative action. Near-equal charged-lepton responses and small derivative-source depth-five values remain conditional on the stated spatial readout, normalization and receiver; they are not derived charges or interaction rates.

Source frequencies in spatial coupling distinguishes the stationary carrier, spatial transition frequencies and complete nonlinear source. Small inter-mode clocks can survive large normal momentum, but the source matrix elements and response scale must accompany them. A common dilation of the scalar bound profile leaves its dimensionless shape response unchanged at a fixed frequency ratio.

Two-frequency spatial Dirac currents gives an exact current oscillating at twice the bound frequency, with fixed canonical norm and one common interaction coefficient. A slower member is broader and can retain a substantial moving current. Its small mean scalar moment does not eliminate relative-component response, and its full linearization has periodic coefficients.

Incoming spatial shape modes shows that an already specified incoming moment enters the reciprocal pair equation without another application of its own susceptibility. Individual shape helicities retain a nonzero hyperplane restriction. Low-frequency opening of a blocked applied force instead arises from changing cancellation between several components. Applying this mechanism to neutrinos therefore requires the selected occupied spatial mode and photon source map, followed by the normalized response of both partners.

Periodic fields, geometry and the full pair

Periodic spatial source response now derives the complete ordinary/conjugate field equation and its frequency sidebands, including scalar-center motion. A scalar rest cancellation can coexist with an oscillating spatial current response. Radial dynamics of two-frequency fields separates the number norm from the kinetic weight and transfers an existing radial instability into the full periodic field.

Two-frequency fields in a readjusted nested pair keeps both deformed profiles and their occupied polarizations. A containing field can maintain a finite lower frequency as the lower mean scalar balance approaches zero. The reciprocal frequency derivative remains finite, while the scalar-center curvature becomes singular. The exactly zero-balance center is undefined, so that endpoint is a limit of positive-balance fields, not an already established nonlinear evolution at zero scalar normalization.

The slow shape poles have frequencies and residues that decrease together, but the tested positive-frequency member has negative canonical spectral weight. The total scalar response remains finite. This calculation does not identify a stable weakly coupled particle from a soft pole alone.

Reciprocal response with two spatial clocks retains both backgrounds and the curvature of their common energy. Deformation can open a shape–center sideband which cancels in a central radial example. Normal propagation can also generate responses involving both clocks. The complete polarization test calculates exactly which second occupied polarizations cancel a selected sideband and how that kernel changes with the incoming lower spatial pattern. For one explicit lower traceless pattern, every finite spectral residue cancels at rest; normal momentum opens its amplitude linearly and its squared response quadratically. Common rotation of the fields and incoming pattern preserves the result. These are source-channel identities with the same coefficient across depths, not a depth-five interaction zero or an interaction rate.

A global antilinear reduction has separate matrix, scalar-interference and nested-projection requirements. It cannot be supplied by assuming that a two-component envelope symmetry acts on every higher-dimensional spatial perturbation. The full response equations remain available when that shortcut does not apply. The physical photon current, selected particle geometry and stability must still connect these conditional mechanisms to observed coupling.

A common spatial pair with changing compatibility

Mode counts and a finite response

Spectral weights in oscillator response shows why exact Hockey-Stick source counts still need the response spectrum. For the stated cylindrical scalar-source map, stiffness growing as the paired normal level to power $s$ gives a finite infinite-band response precisely when $s>q/2$. An affine spectrum in four normal directions leaves shared-mode-dependent logarithmic growth after its common linear growth cancels. The result constrains that source and spectrum; it is not a depth-five suppression or an observed coupling law.

Radial polarization at even depths

Complete radial polarization fields provides an exact spherical family in every dimension from two onward, including six and ten, with independent complex radial polarization vectors. Its full Cartesian current can have tangential components despite a radial density, and its complete integrated polarization matrix is conserved. The radial fields share one center; this supplies an even-depth spatial control while general separated-center encounters remain open.