Coupling as higher-dimensional polarization

Current physical premise (IDWT): 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. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

The generalized sector coupling calculation develops reciprocal spatial maps, canonical vertex normalization, common field response, and a neutrino channel control for arbitrary nested sectors. IDWT coupling is direct dynamics of $\Psi_\infty$; it assumes no mediator particles. Effective response kernels must be derived from that same field.

Both objects carry spatial polarization. The current bilateral calculation retains their relative patterns and reciprocal response. A descending source calculation starts from the higher-dimensional geometry while retaining the lower mode's propagation. The photon's two bound dimensions need not set the source strength.

The spatial shape vertices now give an explicit higher-order polarization law, and reciprocal scattering connects it to incoming and outgoing wave flux. A complementary neutral-mode construction derives cancellation between equal spatial rotation planes. Its geometric transport allows a distinct nonzero response within the same neutral modes. These are conditional mechanisms; the physical current and occupied particle solutions must still select and normalize them.

The full spatial Dirac calculation now retains the orbital and spin parts of these rotations and the complete scalar source. Its polarized shape response shows how both angular field components respond. A dynamical matching calculation finds shapes with zero static access whose response opens at nonzero frequency, with both objects' polarizations retained.

A nested spatial pair solves both profiles from one common interaction energy. Their reciprocal amplitude coupling then follows from the same energy's second derivatives, including both profiles' deformation. These conditional Dirac models provide calculable mechanisms and their limits: scalar binding does not select the occupied polarization, and some bound branches are unstable. Identifying physical photon and neutrino channels still requires the actual particle solution and current.

The source-frequency calculation tracks which temporal frequency belongs to the interacting spatial pattern. Exact two-frequency currents provide a bound-frequency clock, while their changing size and full response prevent identifying slow motion alone with weak coupling. Incoming shape modes separates an already formed incident pattern from a force that creates it, retaining both objects' response and the helicity cancellation mechanism.

The complete periodic pair response retains both source clocks, both spatial polarizations and scalar-center motion. Its polarization selection test calculates cancelling subspaces for the actual incoming spatial pattern. A checked rest barrier opens linearly in normal momentum, with common physical rotations preserving the result. These are conditional source identities; assigning the occupied photon and neutrino fields and their normalized rates remains a physical derivation.

The complete polarization description now tests which spatial tensors retain every amplitude and phase. A reciprocal six-inside-ten example shows why equal mean currents can conceal different responses of the full polarized fields. Spatial textures add the bound-coordinate kinetic terms needed when polarization varies across a particle. These calculations keep both individual fields and their actual spatial equations in view.

The complete spatial current now connects translation inertia, magnetic moment and low-frequency polarization response. Retaining the photon phase gives the forward scattering coefficient, while the moving spatial frame fixes the spin-orbit coefficient. The conditional scalar profile does not yet match the electron magnetic moment. Its current and charge normalization remain physical inputs to derive.

A separate stationary spatial-wave construction supports exact finite null displacements. Completing its actual spatial profile gives a common interaction and a retarded response with derived outgoing radiation. This is a component-action benchmark, not an identified electromagnetic or weak interaction of the master field.

The corresponding polarized pulse collisions have reciprocal energy-dependent delays. The outgoing spatial source depends on both complex polarizations, allowing a radiation zero while conservative coupling survives in this specified component model.

Current Reading and Scope — 2 November 2026

The current pair target is reciprocal response of two patterns of the same field, with amplitude, phase, shape and orientation retained. Each particle’s portions share its fundamental; different pattern kinds may have different fundamentals. Registered count-scaled mass frequencies must not be substituted for those fundamentals. A ramp in field amplitude is not itself a force potential; the full action or stress must determine the response.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

The polarization and sector connections now connect orientation changes to the invariant clock and spin-commuting charge rule, and derive circulation diagnostics for phase-offset copies of one waveform. Their harmonics describe waveform shape; they do not assign independent natural frequencies to the portions.

The current of the existing localized spinor is explicit. For a chiral seed at even depth it is $j=-2F^2Bx/a$, where B is the seed's spatial two-form. At four, this is the same self-dual tensor entering the Hodge-channel comparison. At six it has three circulation planes; at ten the five-plane result requires the pure-spinor branch. Across all finite-norm depths, including the odd ones, the integrated norm current vanishes exactly despite local circulation. The current is conserved under a specified candidate evolution equation; it has not yet been identified with electric current, energy flow or the common fundamental. That distinction also separates internal circulation from the electron's literal external trajectory.

A reciprocal contact benchmark derives spatial force contributions and orientation torques from one specified quartic density term. Unequal-frequency averaging retains full polarization dependence, with a scalar contribution even at orthogonal polarization. Aligned nested Gaussian controls restrict this contribution to shared bound directions. The trial contact is distinct from the critical spinor profile's equation and does not yet supply complete kinetic dynamics, stable particle binding or the observed interaction strengths. Both notes record the calculations and the scope of their independent checks.

Historical conditional framework: Sections 1–14 below retain earlier projective and prescribed-sector calculations as mathematical comparisons. Their additional geometry is not the current physical premise, and their projection rules do not establish a current IDWT coupling law. The new ordinary-space calculations above are the current continuation. No result is transferred between those actions merely because each uses polarization.

1. Geometric coupling mechanism

Full-field controls extend these comparisons to localized encounters. Complete three-dimensional packets show phase-sensitive reciprocal chiral-norm transfer beyond a four-mode truncation. Five-dimensional spatial patterns can increase or decrease a specified subsequent contact comparison; ordered source variations derive its leading change directly from the same local equation. These are unbound-field benchmarks. Their norm transfer is not an assigned particle count or electric charge, and their scalar coefficient has not been identified with measured coupling strengths.

The weighted oscillator response also shows why source counts require a physical spectrum and normalization. An exact Hockey-Stick sum alone does not determine a finite effective interaction when additional response modes are included.

This module starts from the proposed particle depths, the single Dirac field $\Psi_\infty$, and bound-within/free-without. A particle's sector is the image of a projection

$$\Pi_d:M_\infty\longrightarrow Q_d, \qquad \dim_{\mathbb R}Q_d=d.$$

The particle lives on $Q_d$.

The sectors are nested through embeddings $Q_a\hookrightarrow Q_b$. Where a compatible quotient or submersion $\pi_{ba}:Q_b\to Q_a$ also exists, it supplies a further projection map; a quotient alone does not establish the required inclusion. A chosen metric splits the tangent space along the embedded $Q_a$ into tangent and normal directions, and along a submersion into horizontal and vertical directions.

This split gives an exact mathematical meaning to higher-dimensional polarization:

• the local $Spin(b)$ spinor restricts into lower-depth $Spin(a)$ polarization and vertical or normal $Spin(b-a)$ polarization;
• spatial variation of the projected or normal modes induces a connection over $Q_a$, whose curvature must be calculated; a frame change alone does not establish a physical interaction;
• a cross-depth vertex survives only when the extra polarizations contract to a singlet and the tangential representations admit an invariant pairing.

Particle depth thereby supplies spinor branchings, normal-bundle gauge candidates, and invariant selection rules. To identify electric charge, colour, weak chirality and coupling constants, the physical holonomy, selected modes and complete current must be derived from the field dynamics.

Here $d$ occurs in the base tangent and cotangent spaces, the kinetic principal symbol $\gamma^ap_a$, and the pushforward measure used for cross-depth contact. The projected-symbol theorem is developed in particle depth as spatial geometry.

2. Structures carried by sector depth

A real oriented sector manifold $Q_d$ of depth $d$ supplies at each point:

$$SO(d), \qquad Spin(d), \qquad S^{d-1}\subset\mathbb R^d,$$

together with tensor and spinor representations of those groups. Globally, its topology determines the corresponding spin or $Spin^c$ structure. Complex polarization, a Hermitian form, phase quotient, and connection supply

$$\mathbb{CP}^n, \qquad SU(n), \qquad U(1)_{\rm em}, \qquad SU(2)_L, \qquad SU(3)_{\rm colour}.$$

A complex projective sector is formed from a complex vector space by quotient under common complex scale. A unitary group preserves its Hermitian form, and local frame redundancy supplies the gauge connection.

There are accidental low-dimensional isomorphisms,

$$Spin(2)\cong U(1), \qquad Spin(3)\cong SU(2), \qquad Spin(4)\cong SU(2)\times SU(2), \qquad Spin(5)\cong Sp(2), \qquad Spin(6)\cong SU(4).$$

These exact group identities provide the candidate frame symmetries. The Dirac action makes a physical gauge identification by inducing the observed connection, representations, currents, and selection rules.

3. Bound-within/free-without supplies contact

A depth-$a$ field is a field on $Q_a$. Its continuation to the master manifold is the pullback

$$\Psi_a^{\uparrow\infty}=\Pi_a^*\psi_a.$$

It is constant along directions in $\ker d\Pi_a$. This is the coordinate-free form of spanning every direction in which the particle has no localized motion.

When a pair admits a sector projection $\pi_{ba}:Q_b\to Q_a$, the lower field pulls back to $Q_b$ as

$$\psi_a^{\uparrow b}=\pi_{ba}^*\psi_a.$$

A higher-depth current enters the lower interaction through pushforward over the projection fibre:

$$\bar J_{b\to a}=(\pi_{ba})_*J_b.$$

In local product coordinates this is the full fibre integral. The subsequent invariant projection extracts the component of $\bar J_{b\to a}$ carried by the lower interaction.

4. Clifford factorization

At a point where a sector relation supplies a horizontal/vertical or tangent/normal split, let

$$V_b=V_a\oplus N_q, \qquad q=b-a.$$

Over the complex numbers, the Clifford algebra respects an orthogonal direct sum through the graded tensor product:

$$Cl(V_b)_{\mathbb C} \cong Cl(V_a)_{\mathbb C}\,\widehat\otimes\, Cl(N_q)_{\mathbb C}.$$

Accordingly, a higher-depth spinor decomposes under $Spin(a)\times Spin(q)\subset Spin(b)$ into lower-depth spinor factors and normal spinor factors. Schematically,

$$S_b\downarrow_{Spin(a)\times Spin(q)} \sim S_a\,\widehat\otimes\,S_q,$$

with chirality correlated when both factors are even-dimensional.

For $a=2r$ and $q=2s$,

$$S_{a+q}^{+} = (S_a^+\otimes S_q^+) \oplus (S_a^-\otimes S_q^-),$$

$$S_{a+q}^{-} = (S_a^+\otimes S_q^-) \oplus (S_a^-\otimes S_q^+).$$

This is the first robust coupling result. On their shared directions, a higher-dimensional fermion resolves into a lower spinor and a normal polarization label.

The normal label is also what can forbid a coupling. If a lower-depth operator is a scalar under $Spin(q)$, then integration over the normal fibre retains only the $Spin(q)$-singlet part of the higher current.

5. Vertical and normal geometry produce connections

There are two related cases.

For a quotient $\pi_{ba}:Q_b\to Q_a$, the vertical bundle is

$$V_{b/a}=\ker d\pi_{ba}.$$

Choosing a horizontal complement $TQ_b=H\oplus V_{b/a}$ is an Ehresmann connection. When the quotient is a principal $G$ bundle, changing the local fibre frame is a $G$ gauge transformation and the horizontal distribution is a principal gauge connection. The Hopf maps are the special case $G=U(1)$.

For an embedding $Q_a\hookrightarrow Q_b$, the corresponding object is the normal bundle $N_{b/a}$. Choose a local orthonormal frame adapted to the split:

$$\{e_\alpha\}_{\alpha=1}^{a} \cup \{n_i\}_{i=1}^{q}.$$

Along $Q_a$, the higher-dimensional spin connection decomposes as

$$\nabla_\mu^{(b)} = \partial_\mu +\frac14\omega_\mu^{\alpha\beta}\Gamma_{\alpha\beta} +\frac12\omega_\mu^{\alpha i}\Gamma_{\alpha i} +\frac14\omega_\mu^{ij}\Gamma_{ij}.$$

The three nontrivial pieces have distinct meanings:

$$\omega_\mu^{\alpha\beta} =\text{the tangential spin connection},$$

$$\omega_\mu^{\alpha i} =\text{the mixed or extrinsic coupling},$$

$$A_\mu^\perp =\frac14\omega_\mu^{ij}\Gamma_{ij} =\text{the normal-bundle connection}.$$

Under a position-dependent normal-frame rotation

$$h(x)\in Spin(q),$$

the last term transforms as

$$A_\mu^\perp \longmapsto hA_\mu^\perp h^{-1} -(\partial_\mu h)h^{-1}.$$

This is exactly the transformation law of a non-Abelian gauge potential. Its curvature is

$$F^\perp=dA^\perp+A^\perp\wedge A^\perp.$$

The construction uses macroscopic spatial directions of the same $M_\infty$. Relative to a nested lower sector, rotations among its normal spatial directions preserve the lower tangent frame and act on projected field components as gauge transformations. This is the lower observer's description of spatial geometry, not an independently postulated internal space or a new set of physical dimensions.

Together the quotient and embedding cases give the cleanest available realization of coupling as higher-dimensional polarization:

$$\boxed{ \text{vertical or normal rotation of the higher mode} \quad\longrightarrow\quad \text{gauge action induced by spatial rotations seen on the lower sector} }.$$

It remains conditional on the normal connection being dynamical and on its holonomy matching an observed interaction group.

6. The invariant coupling criterion

Let a lower carrier transform in $F_a$, and let two matter sectors restrict to representations $R_b$ and $R_c$ of

$$Spin(a)\times Spin(q).$$

For the specified derivative-free trilinear channel, the possible vertices form the invariant space

$$\boxed{ \mathcal V_{abc} = \operatorname{Inv}_{Spin(a)\times Spin(q)} \left(F_a\otimes R_b^*\otimes R_c\right) }.$$

The dimension counts vertices in this particular algebraic channel. A zero excludes that channel; a nonzero space leaves its coefficient to the dynamics. Derivatives, further field factors and geometric tensors carry additional representations and require their own invariant-space test. For example, a vector and two scalars admit no derivative-free scalar contraction under spatial rotations, but the derivative interaction

$$A_i\,i(\phi^*\partial_i\phi-\phi\partial_i\phi^*)$$

is a rotational scalar. Here the vector and scalar may be components of the same field. This is a counterexample to the unrestricted selection rule, not a derived IDWT interaction. A full channel exclusion must specify the derivative order, field order and available geometric tensors.

For a diagonal current, the tangential test can be written

$$\operatorname{Hom}_{Spin(a)} \left( (R_b^*\otimes R_b)\downarrow_{Spin(a)},F_a^* \right)\ne0,$$

while the normal factors must contract to a singlet. A schematic action is

$$S_{\rm int} =g_{ab}\int_{Q_a} \left\langle A_a,(\pi_{ba})_*J_b \right\rangle.$$

The pushforward implements full cross-depth contact. The invariant brackets implement polarization selection. The constant $g_{ab}$ and the field equations implement dynamics.

7. What the proposed depths actually carry

The complex spatial spinors fixed by real depth are:

At depth six, the proposed projective sector and the chiral spinor ray space coincide:

$$Spin(6)\cong SU(4), \qquad \mathbb P(S_6^+)\cong\mathbb{CP}^3.$$

The projected Dirac dynamics selects the relevant chiral component or order parameter for the electron.

8. Exact branchings along the particle-depth chain

The adjacent restrictions alternate in a simple way:

$$S_3\downarrow_{Spin(2)} =S_2^+\oplus S_2^-,$$

$$S_4^+\downarrow_{Spin(3)} \cong S_3, \qquad S_4^-\downarrow_{Spin(3)} \cong S_3,$$

$$S_5\downarrow_{Spin(4)} =S_4^+\oplus S_4^-,$$

$$S_6^+\downarrow_{Spin(5)} \cong S_5, \qquad S_6^-\downarrow_{Spin(5)} \cong S_5.$$

Therefore the proposed depths $2,3,4,5,6$ form a coherent chain of spatial spinor restrictions. Every step carries the lower-depth spin polarization inside the higher-depth one.

The jump from depth six to depth ten also has an exact decomposition. Under

$$Spin(10)\supset Spin(6)\times Spin(4),$$

one chiral spinor branches as

$$S_{10}^+ = (S_6^+\otimes S_4^+) \oplus (S_6^-\otimes S_4^-).$$

Using the low-dimensional isomorphisms,

$$\mathbf{16} =(\mathbf4,\mathbf2,\mathbf1) \oplus (\overline{\mathbf4},\mathbf1,\mathbf2).$$

This makes the $6\leftrightarrow10$ relation mathematically substantial: the ten-dimensional polarization contains a six-dimensional chiral spinor together with a two-state normal polarization. The tau projector and electromagnetic generator are the physical closure data.

9. Normal groups implied by selected depth differences

For a lower sector embedded in a higher one, the natural normal-frame group is $Spin(b-a)$. Some relevant cases are:

These are structure groups produced by the depth differences. The physical gauge group is the normal holonomy

$$Hol(N_{b/a})\subseteq Spin(b-a),$$

which can be smaller than the full structure group. Its value must follow from the connection and the background state.

10. Interaction audit

Electromagnetism

Every Dirac spinor admits a vector current of the form

$$J^A=\overline\Psi\Gamma^A\Psi.$$

In a specified lower-base reduction, the tangential component enters the reduced interaction. Identifying its representation weight with electric charge and its projected current with the measured electromagnetic response requires the same-field derivation. A proposed electromagnetic generator must give the appropriate neutral response on the neutrino projector; depth alone does not establish that generator or the projection.

The electromagnetic identification maps the $Spin(2)\cong U(1)$ frame connection to the phase connection acting with the observed charges.

Weak interaction

$Spin(3)\cong SU(2)$ appears naturally as the normal group for $2\subset5$ and $3\subset6$, while $Spin(4)$ supplies two $SU(2)$ factors for $2\subset6$ and $6\subset10$. Thus the proposed depths contain genuine two-state normal polarizations that could support weak doublets.

The weak-sector projector must select the left-chiral component, leave the right component as a singlet, generate hypercharge, and identify one normal $SU(2)$ universally across the fermion sectors.

Colour

The direct $3\subset4$ relation has a one-dimensional normal space. The colour candidate therefore comes from the non-adjacent normal geometry $4\subset10$:

An $SU(3)$ candidate can sit inside

$$Spin(6)\cong SU(4),$$

which occurs as the normal group for $4\subset10$. A fixed nonzero chiral normal spinor has stabilizer $SU(3)$ inside $Spin(6)\cong SU(4)$. Fixing only its ray leaves $S(U(1)\times U(3))\cong U(3)$, whose derived subgroup is $SU(3)$; selecting the spinor phase removes the additional continuous factor. A nonzero real vector in the six-dimensional normal tangent representation instead has stabilizer $Spin(5)$ and does not implement this reduction. These are different representations, even though a chiral spinor is also a vector in its own complex four-dimensional representation space. The sector projectors must still place both quark depths in triplets and the lepton depths in singlets, and those subspaces must be preserved by the complete spatial operator. A group embedding alone does not establish that selection.

Gravity

Scalar and stress-energy bilinears contain normal singlets, making gravity the least representation-sensitive cross-depth channel. Full marginal contact therefore supports universal gravitational response; the full-manifold dynamics supplies its force law and normalization.

Six-to-ten charged-lepton relation

The $Spin(10)\to Spin(6)\times Spin(4)$ branching is the strongest direct higher-polarization relation after the six-dimensional electron itself. It naturally adds a two-state left or right normal spinor to a six-dimensional chiral spinor. A projector choosing the appropriate summand could relate the tau to the electron--muon sector. The projector, mass splitting, and universal electromagnetic generator remain open.

11. Gauge fields as one-forms

Higher-dimensional polarization enlarges the representation and supplies more rotation planes. The gauge connection and curvature remain

$$A\in\Omega^1(Q_a,\mathfrak g), \qquad F=dA+A\wedge A\in\Omega^2(Q_a,\mathfrak g).$$

The higher spatial dimension appears through additional connection components and through the Lie algebra $\mathfrak{spin}(q)$.

For any actual $p$-form coupling, pullback to $Q_a$ retains only tangential components. A component with a normal form index vanishes on the lower base; normal polarization indices carried by the representation are instead contracted through the singlet rule.

13. The decisive calculation

Begin with a Dirac action on the full manifold,

$$S_\Psi = \int_{M_\infty} \overline\Psi_\infty \left(i\Gamma^A\nabla_A-\mathcal M\right) \Psi_\infty\,d\mu.$$

Let $P_d$ denote the dynamically defined projection onto fields living on $Q_d$. The spatial sector operator and a cross-sector block are

$$D_d=P_dD_\infty P_d, \qquad D_{de}=P_dD_\infty P_e.$$

The diagonal operator must be the appropriate spin or $Spin^c$ Dirac operator on the actual manifold $Q_d$. For each quotient or embedding relation, decompose the cross-sector operator into tangential, mixed, and vertical or normal pieces and calculate

$$P_d\Gamma^A\nabla_A P_e.$$

The calculation must answer four questions:

1. Which vertical- or normal-bundle holonomy groups actually occur?
2. Which sector spinors form singlets, doublets, or higher multiplets under those groups?
3. Which diagonal and transition blocks survive?
4. Are their normalizations universal where observation requires them to be?

The most informative first tests are

$$Hol(N_{5/2})\subseteq Spin(3)\cong SU(2),$$

for a possible weak doublet structure,

$$Hol(N_{10/4})\subseteq Spin(6)\cong SU(4),$$

for a possible $SU(3)$ colour reduction, and

$$S_{10}^+\downarrow_{Spin(6)\times Spin(4)},$$

for the charged-lepton relation.

Success requires more than recovering familiar group names. The same calculation must prove that the neutrino electromagnetic generator and lepton colour generators vanish, that both quark depths receive the same colour representation, and that unwanted off-diagonal transitions are absent or massively suppressed.

14. Status

Derived from dimensionality and spin geometry

• the sector-projection ontology and horizontal/vertical or tangent/normal splits for related particle sectors;
• complex Clifford factorization and every displayed spinor branching;
• the vertical/normal-frame connection and its gauge transformation law;
• the invariant/singlet criterion for a cross-depth vertex;
• the exceptional six-dimensional projective spinor orientation;
• the $Spin(10)\to Spin(6)\times Spin(4)$ chiral decomposition;
• conditional uniqueness of the $\mathbb{CP}^n/S^{2n+1}$ sector list from depth, minimal complex amplitudes, normalization, and the phase rule;
• the spin/$Spin^c$ viability test for every candidate sector manifold.

Physically promising but conditional

• observed forces as vertical- or normal-bundle holonomies;
• the reconstructed homogeneous spaces as the actual particle-sector projections $Q_d$;
• weak $SU(2)$ from a three-dimensional normal bundle;
• colour $SU(3)$ as a reduced subgroup of a six-dimensional normal spin group;
• the tau’s ten-dimensional spatial field described relative to the electron’s six-dimensional submanifold through the corresponding $Spin(6)\times Spin(4)$ representation; this does not divide the tau’s spatial directions into different physical kinds.

Open physical derivations

• any Standard Model gauge identification;
• electric charge or hypercharge assignments;
• colour triplet and singlet assignments;
• parity violation;
• coupling constants;
• the particle-depth spectrum itself;
• the dynamical choice of complex-minimal polarization and phase quotient in each sector.

The concise result is:

$$\boxed{ \text{higher-dimensional polarization supplies candidate geometric channels;} \quad \text{the projected holonomies must reproduce the observed gauge theory.} }$$

A specified reciprocal spatial benchmark

Local polarized Dirac pulses also give a direct autonomous encounter under the written local scalar term. The nonlinear pulse comparison shows both stronger and weaker normalized pattern match after reciprocal response. Transverse binding, particle identification and different-depth source maps remain required before this control describes observed coupling.

Band Geometry and Explicit Response Operators

The spatial-band construction connects several of this module's ingredients. Its normalized evaluation frame $U$ supplies the occupied projector $P=UU^\dagger$, connection $A=U^\dagger dU$, curvature and shape-response tensor

$$F=dU^\dagger(I-P)\wedge dU,\qquad \mathcal Q_{ab}=(\partial_aU)^\dagger(I-P)(\partial_bU).$$

The same frames define a geometric overlap vertex. With explicit common field embeddings, $K=U_a^\dagger U_b$ has defect $I-K^\dagger K=U_b^\dagger(I-P_a)U_b$. Thus contact, transport and access to omitted channels constrain one another once the actual modes are supplied.

Two applications retain the physical choices needed to use this calculus. The angular-kernel energy has exact simplex and cross-polytope minima and shape spectra. The color volume contraction gives $I-\mathsf S$ on two triplet slots after a stated isotropic third-mode contraction. Its scalar and Casimir parts follow explicitly. The same color rank also enters conditional charge and weak-anomaly tests. These are candidate reductions of same-field spatial data; the source, occupation, interaction sign and observed chiral spectrum remain to derive.

Sources and companion modules

Primary mathematical cross-checks:

• Ivan Todorov, Clifford Algebras and Spinors — complex Clifford algebras, spinor dimensions, and low-dimensional spin group isomorphisms.
• Nathan Berkovits and Sergey A. Rodrigues Jr. and Samuel Wainer, A Clifford Bundle Approach to the Differential Geometry of Branes — tangent and normal Clifford geometry for submanifolds and adapted connections.

Companion V2 modules:

• Bound within, free without
• Photon transversality and the two helicities
• Conditional electron–nucleus frame model