Paired Spatial Modes, Color Neutrality and the Mass Count

A paired-mode construction can give exact non-Abelian neutrality. Its useful content is an explicit representation and invariant state, rather than cancellation inferred from a count. The same algebra gives a precise meaning to an interaction that acts on two matched parts of a field pattern. It also exposes the additional selection rule needed to associate that neutral branch with leptons and a colored branch with quarks.

All spatial dimensions below are ordinary dimensions of the nested physical manifold. A tensor product of field-mode spaces does not make an electron a composite of two quarks, identify either factor with an antiparticle, or establish a physical division of its six equivalent spatial dimensions into two preferred triples. A compatible field representation and a literal product geometry are different hypotheses.

1. What Oscillator Tensor Factorization Establishes

For a $d$-direction oscillator, let the shell multiplicity and cumulative count through level $N$ be

$$g_d(N)=\binom{N+d-1}{d-1},\qquad D_d(N)=\sum_{j=0}^Ng_d(j)=\binom{N+d}{d}=S(N+1,d).$$

The symmetric algebra of a direct sum has the graded decomposition

$$\operatorname{Sym}^N(A\oplus B) =\bigoplus_{j=0}^N\operatorname{Sym}^jA\otimes \operatorname{Sym}^{N-j}B.$$

If $\dim A=a$ and $\dim B=b$, dimensions give

$$\sum_{j=0}^N\binom{j+a-1}{a-1} \binom{N-j+b-1}{b-1} =\binom{N+a+b-1}{a+b-1}.$$

The generating-function proof is simply

$$(1-t)^{-a}(1-t)^{-b}=(1-t)^{-(a+b)}.$$

Thus six has a $3+3$ shell factorization and ten has a $5+5$ factorization. Every split $a+b=d$ works, including asymmetric ones. The tensor product acts on oscillator Fock spaces; the underlying oscillator-coordinate spaces combine by direct sum. In particular a tensor product of two three-dimensional vector spaces has dimension nine, not six.

These equations fix counts. They supply neither a color group action nor a neutral occupied state. A dynamical or geometric realization must specify those separately.

2. The Correct Two-Sided Cancellation

Let $V=\mathbb C^k$ have Hermitian generators $T_A$ of $SU(k)$, with

$$\operatorname{tr}(T_AT_B)=\frac12\delta_{AB},\qquad \sum_AT_A^2=C_FI,\qquad C_F=\frac{k^2-1}{2k}.$$

On $V\otimes V^*\cong\operatorname{End}(V)$, the total action is

$$Q_A=T_A\otimes I-I\otimes T_A^T, \qquad Q_A X=[T_A,X].$$

The normalized trace tensor is

$$|\Omega_k\rangle=\frac1{\sqrt{k}}\sum_{i=1}^k |i\rangle\otimes|\bar i\rangle.$$

Every generator annihilates it. The complete quadratic operator is

$$\boxed{\mathcal C=\sum_AQ_A^2 =k\left(I-|\Omega_k\rangle\langle\Omega_k|\right).}$$

One way to derive this identity is the fundamental completeness relation

$$\sum_A(T_A)_{ij}(T_A)_{lm} =\frac12\left(\delta_{im}\delta_{jl} -\frac1k\delta_{ij}\delta_{lm}\right).$$

Equivalently, $V\otimes V^*=\mathbf1\oplus\mathrm{adj}$; the Casimir is zero on the identity and $k$ on traceless matrices. A common positive energy $\kappa\langle\mathcal C\rangle$, with $\kappa>0$, favors the trace state and leaves an adjoint gap $\kappa k$. The coefficient, physical carrier and occupation law are additional premises. A gap in this finite carrier is not a complete localized particle solution.

This is a genuine two-sided cancellation: the action on the left index equals the opposite action on the right index. Equal and opposite generator expectations alone are weaker. An uncorrelated pair is generally not the trace tensor, and $\rho=I_k/k$ has zero mean generators while retaining fundamental Casimir $C_F$.

For identical factors the total action has a plus sign. Its squared generator is

$$\mathcal C_{VV}=\frac{k^2-2}{k}I+\mathsf S,$$

with eigenvalues

$$\frac{(k-1)(k+2)}k\quad\text{on }\operatorname{Sym}^2V, \qquad \frac{(k+1)(k-2)}k\quad\text{on }\Lambda^2V.$$

At $k=3$, these are $10/3$ and $4/3$. Hence $3\otimes3=6\oplus\bar3$ has no singlet, while $3\otimes\bar3=1\oplus8$ does. At $k=2$, the antisymmetric identical pair is a singlet, an essential lower-rank control. These are standard representation identities; see Beisert's lectures, chapters 6–7. The generator equations above specify their use here.

3. Pairwise Forces Do Not Prevent Three-Part Neutrality

A two-body interaction can also support a three-triplet singlet. For the normalized alternating tensor $\epsilon_{ijk}/\sqrt6$,

$$(T_A^{(1)}+T_A^{(2)}+T_A^{(3)})|\epsilon\rangle=0, \qquad \sum_AT_A^{(i)}T_A^{(j)}|\epsilon\rangle =-\frac23|\epsilon\rangle\quad(i\ne j).$$

Consequently the three pair contractions sum to $-2$, although the complete state is neutral under a uniform total color action. Internal pair interaction and total charge cancellation are different properties. An interaction being pairwise does not by itself prohibit cancellation among three parts. The relevant distinction is which invariant state the field supports and occupies.

The existing volume-to-pair calculation supplies the same antisymmetric pair channel from a geometric trilinear volume. It therefore offers a common comparison between a two-slot trace branch and a three-slot determinant branch without assuming an independent mediator particle.

4. Ordinary Even-Dimensional Space Can Supply Conjugate Mode Variables

Choose an orthogonal complex structure $J$ on an ordinary real tangent space of dimension $2m$. Its complexification decomposes as

$$(T^*Q_{2m})_{\mathbb C}=V\oplus\bar V, \qquad \dim_{\mathbb C}V=m.$$

These are the two eigenspaces of $J$ after complexification, not two preferred ordinary spatial halves. In local complex coordinates the mode variables are $z_i,\bar z_i$. This changes no physical dimension. Existence and compatibility of a global $J$, and the field's use of its subgroup, must be established for the chosen spatial manifold.

For $m\ge2$, the $SU(m)$-invariant polynomials in one vector and its conjugate are powers of

$$q=\sum_{i=1}^m z_i\bar z_i.$$

Indeed $\operatorname{Sym}^pV$ is irreducible and can pair to a scalar with $\operatorname{Sym}^r\bar V$ precisely when $p=r$. Schur's lemma gives one scalar in that case. A complete shell $\operatorname{Sym}^N(V\oplus\bar V)$ therefore has one singlet for even $N$, and none for odd $N$. Its invariant Hilbert series is $1/(1-t^2)$. The cumulative band has

$$R_N=\lfloor N/2\rfloor+1$$

singlets. The rank-one complex case has trivial $SU(1)$, so every polynomial is invariant; it must be treated separately.

For the maintained charged-lepton labels, $N=n-1$:

Thus ten does fit a corrected conjugate-pair construction, and all three maintained charged-lepton levels permit its even-shell singlet. This is compatibility under the specified representation. It does not derive the levels or prove that the same group is observed as color.

5. Apply the Same Rule Across Sectors

Extend the tangent prescription to odd $d=2m+1$ by retaining one real spectator coordinate $u$. Its invariant ring for $m\ge2$ is generated by $q$ of degree two and $u$ of degree one. A shell contains $\lfloor N/2\rfloor+1$ invariants, with Hilbert series

$$\frac1{(1-t)(1-t^2)}.$$

The cumulative count is $\sum_{r=0}^{\lfloor N/2\rfloor}(N-2r+1)$. This supplies invariants at every neutrino label at five. The $SU(2)$ action used there is not the same color $SU(3)$.

At depth four, the same prescription gives an $SU(2)$ singlet in the up-quark's $N=2$ shell, while charm and top have odd $N=19,71$. Thus a universal rule that every pattern occupies this scalar shell would already fail the full up-type family. Depth three has only the trivial $SU(1)$ in this tangent prescription; it does not produce the required color triplet. These controls show why tangent realification alone cannot replace the established relative-mode color construction.

One can instead retain the same $SU(3)$ on six ordinary directions inside a containing $d\ge6$ space, with $s=d-6$ real singlet spectators. The invariant Hilbert series is

$$H_d(t)=\frac1{(1-t^2)(1-t)^s}.$$

At ten this gives 8,086 shell singlets and 44,408 cumulative singlets through $N=22$. The full $SU(5)$ scalar is among them, but color invariance alone no longer singles it out. The choice of six-plane and its transport are additional geometric data; this is not an inherent six-plus-four division of the tau.

The separate complete middle-Hodge carrier has ranks $1,3,3,10,10,126$ at the maintained depths. Pairing that entire carrier with its dual gives one canonical trace singlet at every rank, including the quark ranks. A common Casimir penalty on all those paired carriers therefore cannot select lepton neutrality while retaining their fundamental quark branches without an additional carrier or occupation rule. The oscillator rank $m$ and Hodge rank $k_m$ describe different objects.

6. Color Cancellation Need Not Cancel Electric Charge

The fully dual $U(k)$ pair $V\otimes V^*$ cancels its common central $U(1)$ too. That version cannot directly describe an electrically charged lepton. However, color conjugation and electric phase need not be the same action.

A coherent candidate is

$$\mathcal F=\operatorname{End}(V)\otimes L, \qquad X\mapsto e^{iq\theta}UXU^{-1}, \quad U\in SU(k),$$

where $L$ is one specified electric line character. The trace branch $I\otimes\ell$ has zero color and electric charge $q$, independent of its oscillator level. For a matrix-valued field on ordinary space, a commutator connection obeys

$$\nabla(sI\otimes\ell) =(ds+iqA s)I\otimes\ell,$$

when the color connection acts through $[\mathcal A,I]=0$. The scalar branch can therefore remain spatially dynamical while color neutral. The electric line and its normalization must come from the same geometric field model. They are not derived merely by writing this tensor product. The existing shared normal determinant-line construction is a place to seek that derivation.

7. Preserve the Mass-Count Normalization When Selecting a Singlet

For a normalized source vector $v$ in the full cumulative band, the maintained coherent contact has

$$T=D_d(N)|v\rangle\langle v|,\qquad T^2=D_d(N)T.$$

If $P$ projects onto invariant modes, then

$$(PTP)^2=D_d(N)\|Pv\|^2(PTP).$$

The nonzero frequency contribution changes unless $v$ was already invariant. Counting only the invariant modes in an equal-weight toy contact replaces the full count by their count. Thus projecting before contact cannot silently preserve the original mass frequency.

There is a compatible alternative to test: an invariant source in the full band retains the full coefficient $D_d(N)$. For a simple non-Abelian group, a commuting rank-one contact has an invariant source line. In a cumulative coherent band $\operatorname{Sym}^N(\mathbb C\oplus V\oplus\bar V)$, the only invariant degree-one vector is the slack direction. At $N>0$ an invariant coherent ray must be that slack ray, since a continuous character of $SU(m)$ is trivial. Higher even-shell singlets are correlated pair polynomials rather than single coherent powers of a nonzero spatial vector.

The actual spatial evaluation construction has different source data and may avoid that coherent-ray restriction. An equivariant evaluation at a subgroup-fixed point can map an invariant fibre vector into an invariant full-band state while retaining the full diagonal-kernel normalization. Its field, kinetic and embedding hypotheses need a direct calculation; this possibility is not established by the oscillator invariant count.

8. Uniform Neutrality Versus Spatial Response

For a source acting with real spatial weights $a,b$ on the paired slots, put

$$J_A=aT_A\otimes I-bI\otimes T_A^T.$$

Its exact action on the trace state gives

$$\boxed{\sum_A\|J_A\Omega_k\|^2 =C_F(a-b)^2.}$$

Matched weights give exact cancellation. Unequal sampling can excite the traceless complement even though the total state remains a singlet under the uniform group action. If the two slots describe spatially separated parts, their size, separation and complement gap determine this residual response. If they instead describe the two indices of a pointwise matrix field with a commutator coupling, the identity commutes locally and this particular residual need not arise. A tensor factorization alone does not choose either physical realization.

This makes the two-sided-force proposal concrete: derive whether the common same-field interaction samples matched local actions or resolves different parts of a correlated pattern, and calculate which representation it excites. Neither an average cancellation nor availability of a singlet is sufficient to claim absence of every strong response.

9. An Equivariant Evaluation Can Retain the Full Count

The mass-normalization question has a useful exact answer under specified symmetry assumptions. Let a compact group $G$ act unitarily on the full spatial band $\mathcal H_N$ and on the fibre $F_x$, fix the physical point $x$, and make evaluation equivariant:

$$E_x\rho(g)=\tau(g)E_x.$$

Suppose $B_N=E_xE_x^\dagger=S B_0$, with $B_0>0$ commuting with $\tau(G)$. For a normalized invariant fibre vector $u$, set

$$w=E_x^\dagger B_0^{-1/2}u.$$

Equivariance and the kernel normalization give

$$\rho(g)w=w,\qquad \|w\|^2=S,\qquad E_x^\dagger B_0^{-1}E_xw=Sw.$$

Thus a normalized invariant state $w/\sqrt S$ can retain the full nonzero contact eigenvalue $S$, even if the invariant subspace contains far fewer vectors than the original band. Its spatial evaluation weights need not equal one per surviving vector. Deleting modes from an equal-weight toy source is a different operation.

If the bare spatial kinetic operator commutes with $G$, complete invariant spatial modes form a preserved Hilbert subspace. The contact and kinetic operator can then be treated together, with the bright-to-complement dressing already derived in the spatial-band construction. The finite checker retains that complement under the common diagnostic $R=\mu=1$; its dressed frequencies differ from the isolated contact eigenvalue by the stated kinetic mean and resolvent correction.

There are two material limits. First, a subgroup of spatial rotations yields joint spatial-spin invariance, which is not automatically the independent observed color action. Second, invariance of a complete spatial function is not a pointwise projection of its spinor fibre. The spatial dependence can supply components that a fibre-only truncation discards.

10. Compare the Kinetic Symbols of Neutral Projections

On $\Sigma_{2m}=\Lambda^*\mathbb C^m$, the $SU(m)$-invariant spinors for $m\ge2$ are the vacuum and top exterior form. Let $P_0$ project onto those two fibre vectors. Creation and annihilation change degree by one, so

$$P_0\gamma_aP_0=0\qquad(a=1,\ldots,2m).$$

For the odd extension to $2m+1$, the additional Clifford matrix is exterior parity. Its compression survives, while the other $2m$ directions still vanish. The result is an exact warning against discarding all nonsinglet spinor components at each spatial point:

Each discarded tangential Clifford direction has squared leakage two from the selected two-dimensional fibre. The sum is $4m$. The principal symbol no longer describes the intended depth when this pointwise projector is used at $d\ge4$. This excludes that truncation as a full spatial Dirac carrier, rather than excluding a neutral particle or a complete invariant spatial mode.

A genuine relative-mode multiplicity has a different outcome. If

$$\Sigma_d\otimes\operatorname{End}(V)\otimes L$$

carries Clifford multiplication as $\gamma_d^a\otimes I\otimes I$ and color as $I\otimes[T_A,\cdot]\otimes I$, projecting only the matrix factor onto its identity preserves every $d$-direction Clifford relation. This uses a color multiplicity separate from spin, exactly as required in the relative-frame construction. It still needs its origin in equivalent modes of the same spatial field.

The mass normalization can survive this selection too. With the same multiplicity factor in the physical and reference evaluation bands,

$$B_N=S B_0\otimes I_F.$$

Restricting $F$ to its invariant trace line gives $B_N^{(0)}=S B_0^{(0)}$, rather than replacing $S$ by an oscillator singlet count. Selecting the color component and deleting spatial oscillator modes are therefore mathematically different. This provides a consistent target for a common action: retain the full spatial band and its bare kinetic operator, derive the relative-mode carrier, and select its neutral trace without altering the spatial count. It does not yet derive why the quark depths use a fundamental branch instead.

11. A Geometric Source of Conjugate Partners at Six and Ten

The Hodge channels themselves distinguish four from six and ten. For middle forms in ordinary Euclidean dimension $2m$,

$$*^2=(-1)^m,\qquad P_\pm=\frac12(I\pm\lambda^{-1}*),\qquad \lambda^2=(-1)^m.$$

The Hodge star is real. When $m$ is even, $\lambda=1$ and complex conjugation preserves each real chiral channel. When $m$ is odd, $\lambda=i$ and

$$\overline{P_+F}=P_-\bar F.$$

For a real form the two channels are then conjugate partners. This is standard Hodge geometry; see Viaclovsky, lecture 25, equation 26.1 and following remark.

This gives a natural source of paired data at six and ten without imposing a spatial three-plus-three or five-plus-five product. Applying the field reality condition is an additional premise; the fundamental IDWT spinor has not been assumed to be this real middle form. Restriction to the odd shadow also needs its declared field map, rather than treating the two depths as two particle constituents.

Conjugate data alone does not produce a singlet current. Consider positive second-order kinetic energy for $z$ and $y=\bar z$ in conjugate representations, with generators $T$ and $-T^T$. The algebraic density cancels:

$$z^\dagger Tz+y^\dagger(-T^T)y=0.$$

But the Noether current of their kinetic norm is

$$j_T=2\operatorname{Im}\left(z^\dagger T\dot z +y^\dagger(-T^T)\dot y\right) =4\operatorname{Im}(z^\dagger T\dot z).$$

For $z=e^{-i\omega t}(1,0,0)^T$ and $T=\operatorname{diag}(1/2,-1/2,0)$, this is $-2\omega$, while the kinetic norm is $2\omega^2$. A same-sign first-order Schrödinger kinetic form instead cancels when this reality relation is imposed, so its vanishing density cannot be offered as a nondegenerate first-order dynamics for the realified pair.

The geometric conjugation pattern is therefore a useful candidate for supplying paired field data, with a specific missing step: a common action must produce the tensor trace or another genuinely invariant occupied pattern. A direct sum of conjugate components with a reality relation is not the tensor singlet, and cancellation must be checked in the current derived from the actual action.

12. Paired Curvature Cancels a Velocity Channel, Not Every Interaction

The positive mixed-curvature action gives a precise dynamical test of oppositely oriented responses. Retain two independent fast modes of the same field, with equal positive gaps and opposite spatial-pattern maps:

$$H_f=\frac\Delta2\sum_{s=\pm1} [(x_s-Aq)^2+(y_s-sB\lambda)^2].$$

Each $(x_s,y_s)$ is a canonical pair. Positive bare kinetic and restoring terms for $q,\lambda$ make the complete eight-variable Hamiltonian positive. This is a four-oscillator mode reduction. It does not impose $c_- =\bar c_+$ on a same-sign first-order kinetic action; that different constraint would cancel the symplectic form.

The two cross-response terms are

$$g_s(\Omega)=s\frac{\Delta^2AB}{\Delta^2-\Omega^2}, \qquad g_++g_-=0.$$

Thus equal conjugate pattern maps cancel the reciprocal velocity cross-response exactly at all frequencies, rather than only in an adiabatic expansion. The diagonal terms add:

$$\delta M_q(\Omega)=\frac{2\Delta A^2}{\Delta^2-\Omega^2},\qquad \delta M_\lambda(\Omega)=\frac{2\Delta B^2}{\Delta^2-\Omega^2}.$$

With $\lambda$ prescribed, the surviving source response is

$$R_{\rm paired}(\Omega)=\frac{2\Delta B^2\Omega^2}{\Delta^2-\Omega^2}.$$

It has a real pole in this conservative source problem at the bare gap and tends to $-2\Delta B^2$ at high frequency. Pairing therefore suppresses one cross-channel while retaining induced inertia and direct response. It does not eliminate all interactions. The full autonomous system has its complete shifted normal frequencies; a prescribed-source susceptibility and the autonomous spectrum are different calculations.

Equality is material. With equal weights but unequal gaps,

$$g_{\rm residual}(\Omega)=AB\left[ \frac{\Delta_+^2}{\Delta_+^2-\Omega^2} -\frac{\Delta_-^2}{\Delta_-^2-\Omega^2}\right] =AB\Omega^2\left(\frac1{\Delta_+^2}-\frac1{\Delta_-^2}\right) +O(\Omega^4).$$

The leading adiabatic gyro term still cancels, but the finite-frequency response generally does not. Unequal occupation weights also leave a nonzero leading gyro term. A physical pairing explanation must derive the gaps, weights and actual pattern maps under one law.

Curvature Cancellation Is Weaker Than Color Neutrality

A useful exact control makes the distinction sharp. For any normalized complex ray $u$, the product ray $v=u\otimes\bar u$ has

$$i v^\dagger dv=0,\qquad g_v=2g_u,$$

where $g$ is the real quantum metric. Its Berry curvature vanishes while its deformation inertia need not vanish. Nevertheless,

$$|\langle\Omega_k|v\rangle|^2=\frac1k,\qquad \langle v|\mathcal C|v\rangle=k-1.$$

At the Hodge ranks $1,3,10,35,126$, this gives quadratic generator expectations $0,2,9,34,125$. A flat product ray remains non-Abelian charged at every nontrivial rank. In contrast the actual trace tensor is invariant under simultaneous $U\otimes\bar U$ frame rotation; that frame motion changes neither the trace state nor its metric. Curvature cancellation and the invariant occupation in section 2 are therefore distinct conditions.

What the Hodge Geometry Supplies Across All Sectors

Complex conjugation exchanges the complete channels at depths $2$, $5/6$ and $9/10$ in the stated even-carrier/shadow prescription. It preserves each channel at $3/4$ and $7/8$. This is a natural geometric source of conjugate data compatible with the desired maintained color-neutral depths. It does not force those data to be two independently occupied positive-frequency modes with equal weights.

The Hodge star also commutes with physical oriented rotations. The complete chiral projector does not vary merely because a spatial coordinate frame rotates; changing an individual occupied spatial mode is additional physical information. A real time-dependent form supplies a conjugate relation between positive- and negative-frequency coefficients, which does not by itself identify two equally occupied positive-frequency bands. The odd shadows require the previously stated restriction map.

These are useful constraints on a possible common cancellation mechanism. The geometry supplies the relevant conjugation pattern, and a complete positive action can cancel an opposite cross-response. Deriving a true occupied color singlet, and identifying the canceled channel with the empirical strong response, remain separate physical steps.

The paired-curvature checker passes 1,485 conditions across 64 cases. Its report preserves exact action reduction, positive complete dynamics under matched and mismatched gaps, metric and Casimir controls, and every maintained depth with neighboring controls through twelve. No particle simulation is used.

Reproducibility and Next Constraints

The single checker verifies every oscillator split for depths two through twelve through shell forty; generator metrics, identical and conjugate pair Casimirs through rank five; three-triplet pair contractions; polynomial and independent Weyl-character invariant counts; all maintained sector assignments; and contact projection and electric-character controls. Its report records the exact executed source hash. No particle simulation is needed for these tests.

The productive completion question is now specific: derive a common geometric field carrier and invariant interaction that selects the charged trace branch at six and ten, leaves the three/four relative-mode triplets active, and retains the simplex frequency and physical electric line. The tensor and oscillator equations above give exact compatibility tests for that construction.

The kinetic and evaluation checker passes 548 conditions across depths two through ten, including neighboring controls, with its report. The Hodge-reality checker passes 1,915 exact conditions through depth twelve, with its report. The latter includes a counterexample to neutrality inferred from conjugate reality alone. These checks retain the distinction between exact algebra, conditional spatial evaluation and an actual physical color mechanism.

13. A Separate Operator-Cancellation Candidate

The symmetric Hodge contact cancels a specified interaction operator at the nonquark maintained depths, rather than making an arbitrary state a singlet. Its complete middle-channel volume action is active at three/four, while the skew Hodge part can supply a velocity response at the complementary depths. This offers a common kernel hypothesis to test alongside invariant occupation. The higher seven/eight control, allowed complementary interactions and noncommuting lower-sector projections prevent treating it as an already derived universal absence of higher forces.

A Free Pair Phase Retains Canonical Dynamics

The explicit phase reduction distinguishes the fixed reality relation above from a zero-moment pair with an independent phase. Its quotient has canonical one-form minus R dphi and a positive cone metric. The bilinear carries determinant weight three in the exterior U3 action but zero in the actual Spin6 lift. Retaining that shift, the doubled bilinear clock and the full added-direction currents prevents transferring this local reduction directly to a charged lepton. It supplies a precise nondegenerate alternative to test under the common physical action.