Real Spinor Clocks, Depth Selection and the Common-Field Test

The dimension-property comparison has a constructive dynamical extension. The non-real spinor types are precisely the minimal real spin modules that admit a nondegenerate rotation-invariant alternating kinetic form. Such a form generates internal classical oscillation. Within dimensions one through ten, it exists at exactly two, three, four, five, six and ten.

This is a conditional mechanism, not an established particle selector. Restriction of one common ten-dimensional spin carrier supplies repeated modules and restores the missing forms at seven, eight and nine. A successful selection mechanism therefore needs a dynamical reason for a selected minimal spinor fibre, not just the Bott-periodic table. The complete finite spatial-band extension below retains that fibre premise while allowing arbitrarily many spatial modes. Neither representation reality nor this kinetic criterion identifies antiparticles.

All matrices below act on real field components. Positions remain ordinary coordinates in shared flat spatial space. No complex spatial dimensions, mediator particles or position probabilities enter. Spatial-spin carriers are hypotheses about field transformation, not derived observed particle spin.

1. From a Representation Type to a Kinetic Equation

Let V be a real irreducible representation of Spin(d), with positive invariant inner product. The algebra of equivariant real endomorphisms is a division algebra F, equal to R, C or H. For the minimal spin modules the periodic classification follows from the even real Clifford algebra. See Baez, The Octonions, section 2.4 and Table 4 for that external representation-theory input. The dynamical construction and common-carrier test below are additional calculations.

Write an invariant alternating form as omega(u,v) = u^T B v. Invariance and antisymmetry are exactly

$$[B,R(g)]=0,\qquad B^T=-B.$$

The invariant metric makes transposition conjugation on F. Hence the skew part has real dimension zero, one or three. In the latter two cases every nonzero skew element satisfies B squared equal to a negative scalar identity. Normalize its magnitude to obtain J squared equal to minus the identity.

For a fixed J, one classical candidate action is

$$S=\int dt\,d^{10}x\left[\tfrac12\psi^T J\partial_t\psi -\mathcal H(\psi,\nabla\psi,\ldots)\right],\qquad J\partial_t\psi=\frac{\delta H}{\delta\psi}.$$

This uses a field and its own real internal components. It is first order in time and requires no separate mediator. In a finite internal-mode reduction put I = psi^T psi/2 and H = h(I). Then

$$\dot\psi=-h'(I)J\psi,\qquad \dot I=0,\qquad \psi(t)=[\cos(\Omega t)I_V-\sin(\Omega t)J]\psi(0), \quad \Omega=h'(I).$$

For positive quadratic energy h(I) = omega I this is an exact constant-norm oscillator of rate omega. The construction exhibits a clock; it derives neither localization nor the IDOS value of omega. Its rotation covariance does not establish Lorentz covariance or a physical spin-statistics rule. For d below ten, a selected active plane and its spatial realization still need to be specified covariantly in the containing space.

The selection statement is now exact: one irreducible real spin module, a rotation-invariant nondegenerate first-order kinetic form, and no additional canonical copy select the six occupied depths within 1 through 10. Each italicized restriction is a physical hypothesis to justify. Above ten the same rule allows eleven and twelve; it does not bound the containing dimension.

2. One Common Ambient Carrier Changes the Answer

Choose a complex half-spin module of Spin(10), and regard all its components as a real 32-dimensional vector. Its constant complex-structure matrix J0 commutes with Spin(10). It therefore commutes with every subgroup Spin(d). This elementary containment already proves that restriction alone cannot remove clocks at depths seven, eight or nine.

An independent exact Clifford computation gives the full real commutants:

The checker constructs ten Clifford generators on five two-state tensor factors, restricts their even products to one chirality, then converts them to real 32 by 32 matrices. The 1,024 real Pauli tensor words form a basis of all real endomorphisms. Each word commutes or anticommutes with every adjacent Clifford-pair generator. Their conjugations are diagonal on this basis, so counting simultaneous commuting words gives the entire commutant, not a search for selected examples. Transpose parity counts its skew part.

The positive finding and its limitation concern different representation choices. At depth nine, for example, the ambient restriction has two copies of the real 16-dimensional spin module. Their relative rotation supplies J0. An individual copy has no invariant J. At eight the same mechanism acts in each repeated chirality. At seven four copies provide six skew generators.

More generally, m identical real-type copies have commutant M_m(R). An invertible skew form exists exactly when m is even. Complex- and quaternionic-type copies admit one at every positive multiplicity. This is a multiplicity-sensitive selector. Spatial mode counts are not automatically these multiplicities: spatial modes themselves transform, so the complete tensor representation must first be decomposed.

3. Why Making a Partner Expensive Does Not Simply Remove It

A common law might try to isolate one irreducible module energetically. There is a concrete diagnostic for that proposal. For two real-type copies q,p with canonical action, take positive constants Delta and kappa:

$$L=p^T\dot q-\tfrac12\Delta\|p\|^2-\tfrac12\kappa\|q\|^2.$$

Eliminating p gives

$$p=\dot q/\Delta,\qquad L_{\rm eff}=\frac{\|\dot q\|^2}{2\Delta} -\frac\kappa2\|q\|^2,\qquad \Omega^2=\kappa\Delta.$$

Thus a finite stiff partner produces an ordinary second-order oscillation; it does not forbid a pattern. An exact single-copy first-order restriction and an approximate energetic isolation are physically different. A common gap could move these modes above an accessible frequency window, but its size and the window must follow from the field action. No gap value has been fitted here. The existing two-real-component entropy action is already a canonical doubling and therefore does not inherit the minimal-carrier exclusion.

4. A New Response Geometry: Discrete and Continuous Clock Axes

For a complex-type minimal module the normalized choices are J and minus J. For a quaternionic module write

$$J(a)=a_1J_1+a_2J_2+a_3J_3,\quad |a|=1,\qquad J_iJ_j+J_jJ_i=-2\delta_{ij}I_V.$$

The clock choices form a two-sphere. This is a sphere of real field operators, not extra physical positions. Quaternionic unit transformations act transitively on it, with a circle stabilizer. In particular J and minus J are connected by these internal transformations, unlike the two isolated choices in the complex-type commutant.

For two clock matrices in an identified copy of the same carrier,

$$-\frac1{\dim V}\operatorname{tr}[J(a)J(b)]=a\cdot b.$$

Consequently the simplest bilinear trace contact

$$V_{\rm contact}=\frac{g(R)}{\dim V}\operatorname{tr}[J(a)J(b)] =-g(R)a\cdot b$$

has projected force in clock-orientation space

$$-\nabla_{S^2,a}V=g(R)[b-(a\cdot b)a].$$

For positive g, parallel axes minimize this conditional energy; negative g favors antiparallel axes. This is a continuous orientation response for H and a two-sign comparison for C. A response rate requires a kinetic law for the axes. Promoting a to a time-dependent field also changes the kinetic variation above; it must not be done while keeping J artificially fixed. No universal cross-depth identification, overlap g(R), or physical force has been supplied by this algebraic contact.

This predicts a specific distinction to test in a completed model: minimal H-type carriers have two continuous transverse clock-axis variations, whereas minimal C-type carriers have none at fixed kinetic scale. It does not mean that C-type particles lack ordinary spatial polarization. Depth five has exactly the same H-type axis freedom as three and four. A universal quaternionic-clock coupling therefore cannot alone explain quark-only color or neutrino suppression. Three clock generators are not three colors.

An explicit reduced response model makes this distinction quantitative. Introduce positive collective inertias iota_a and iota_b and the constrained action

$$L_{\rm axes}=\tfrac12\iota_a|\dot a|^2+ \tfrac12\iota_b|\dot b|^2+g\,a\cdot b,\qquad |a|=|b|=1.$$

This is a separate effective candidate for pattern variables; its inertias and contact must be obtained from the same field before physical use. It is not an assertion that the preceding fixed-J action supplies this term. Variation gives

$$\iota_a(\ddot a+|\dot a|^2a)=g[b-(a\cdot b)a]$$

and its reciprocal b equation. At constant positive g and aligned axes, the two transverse components of delta = delta_a minus delta_b obey

$$\ddot\delta+\nu_{\rm axes}^2\delta=0,\qquad \nu_{\rm axes}^2=g(\iota_a^{-1}+\iota_b^{-1}).$$

There are two degenerate relative-axis oscillations and two neutral common rotations. This is an explicit conditional response prediction, with no equality of particle carrier frequencies assumed. Minimal C-type clock choices provide no continuous axis for this particular resonance. A measured or simulated anisotropic splitting would require a term that breaks the isotropic quaternionic-axis symmetry; an isotropic trace contact cannot produce it. This test applies equally to the H-type depths three, four and five. A symbolic check verifies the two zero eigenvalues and the double nonzero eigenvalue for arbitrary positive inertias and coupling.

5. Conjugation and the Proposed Exclusions

Representation equivalence to its conjugate is not equality of every field configuration to its conjugate. It also does not fix an electric-charge operator. An explicit counterexample to an unconditional real-type exclusion uses the doubled real carrier above. The involution C(q,p) = (q,-p) commutes with spatial rotations and satisfies

$$C^2=I,\qquad CJC^{-1}=-J.$$

It reverses internal circulation. For example, the isotropic second-order action on this doubled carrier,

$$L_2=\frac12\int d^{10}x\left[|\dot\psi|^2-c^2|\nabla\psi|^2 -\omega_0^2|\psi|^2\right],\qquad Q=\int d^{10}x\,\psi^T J\dot\psi,$$

has conserved Q for finite-energy fields with vanishing boundary flux, and C reverses Q exactly while preserving the action. The field equation and integration by parts give dot(Q) = 0 because J is antisymmetric. This example is not localized particle support. These are classical same-field quantities. They are not an identification of Q with electric charge, but they demonstrate why real spatial-spin type cannot by itself forbid opposite signed states. On the first-order oscillator C instead reverses the symplectic form and relates opposite time orientations unless the kinetic sector is transformed as well. That is not automatically a same-time charge-conjugation symmetry of a fixed first-order action.

Likewise, planar rotations have a spin double cover: z maps to z squared from the circle to itself, with kernel {1,-1}. What distinguishes SO(2) is its infinite fundamental group and real-line universal cover, not absence of a double cover. Higher-dimensional rotation groups also admit scalar and tensor representations. Therefore neither exclusion of fermions at two nor exclusion of bosons above two follows from rotation topology alone. The explicit spin construction is described in the external projective and spin representation notes.

The earlier family-recovery rules remain conditional fitted rules. Their exclusions of a lone neutral boson, a depth-two bottom interpretation, or a Majorana option require independently justified property-to-representation bridges. This continuation supplies none of those bridges and does not reassign any particle. Its dynamical selector is narrower and explicit.

6. Relation to the Count and Mass Work

The new kinetic form supplies a possible classical clock variable, not its selected rate. If h(I) is left arbitrary, every admitted depth still permits arbitrary frequencies. Multiplying independent oscillator energy by the dimension of a filled band does not change that fact. The carrier-phase calculation in the harmonic-core note and IDOS energy and coherence still applies.

The concrete construction target is now one common spatial action that selects a minimal active representation, preserves its closed oscillation against multiplicity mixing, supplies its localization and frequency, and derives reciprocal response of the selected clock axes. An exact invariant projection would retain the conditional depth selector. Mere restriction of the ambient spinor, or a finite gap for its partner, does not suffice.

7. Executed Both-Branch Kinetic Comparison

A single calculation script now evaluates the representation and kinetic alternatives together. The decisive new distinction comes from varying the clock axis in the action instead of assigning it an inertia. In a finite internal-mode reduction, set $J(a)=a_iJ_i$, $|a|=1$, and let $\theta=\psi^T J(a)d\psi/2$. At $a=e_3$, with local axis coordinates $a_1,a_2$, its complete symplectic matrix is

$$F=\begin{pmatrix}J_3&C\\-C^T&0\end{pmatrix},\qquad C=\tfrac12(J_1\psi,J_2\psi),\qquad -C^TJ_3C=\frac{|\psi|^2}{4}\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Thus $\det F=|\psi|^4/16$, and the effective axis equation is first order:

$$\dot a=\frac1s a\times\nabla_aH,\qquad s=|\psi|^2/4=I/2.$$

The norm commutes with the axis coordinates. Simultaneous rotations of the field components and clock axis have moment map $Q_i=-s a_i$. The zero-norm branch is singular. This is an explicit extended kinetic action with an independent internal axis; identifying the axis with a function of the field requires a different pullback. The tested choice $a_i=\psi^TKJ_i\psi/|\psi|^2$, with a fixed commuting complex structure $K$, obeys $J(a)\psi=-K\psi$ and reduces the kinetic term to $-\psi^TKd\psi/2$. That branch adds no independent axis coordinate and assumes the extra structure $K$.

For independent axes and $V=-g a\cdot b$, the first-order equations conserve both $a\cdot b$ and $L=s_a a+s_b b$. Both contact signs produce rotation about $L$ at rate $|g||L|/|s_as_b|$, rather than relaxation toward alignment. Near $b=\epsilon a$, the characteristic polynomial is

$$\chi(\lambda)=\lambda^2\left[\lambda^2+g^2(\epsilon/s_a+1/s_b)^2\right].$$

At equal positive coefficients and antiparallel axes, the nonzero linear matrix is nilpotent; the zero rate is a degenerate secular limit. By contrast, the separately postulated second-order rotor action above gives two relative modes with squared frequency $\epsilon g(1/\iota_a+1/\iota_b)$: positive on the restoring branch and negative on the unstable branch. Those alternatives differ in phase-space dimension and cannot be substituted for each other.

A mixed kinetic control, $\iota\ddot\delta-sJ_2\dot\delta+\operatorname{diag}(k_x,k_y)\delta=0$, has

$$\chi(\lambda)=\iota^2\lambda^4+[\iota(k_x+k_y)+s^2]\lambda^2+k_xk_y.$$

For positive inertia, strict imaginary roots require $k_xk_y>0$, $B=\iota(k_x+k_y)+s^2>0$ and $B^2>4\iota^2k_xk_y$. A mixed-sign stiffness always grows. Gyroscopic terms can stabilize two negative stiffnesses spectrally; exact eigenvector counts distinguish semisimple repetitions from Jordan drift on the boundary. This does not prove nonlinear stability.

The complete kinetic term also changes the symmetry claim. An isotropic energy on a quaternionic carrier has three commutant rotation generators, but with fixed first-order $J_3$ only $J_3$ preserves the symplectic form. The isotropic second-order action preserves all three. A single uniform first-order phase has sign-definite action. With two copies, $B=\operatorname{diag}(J,J)$ and generator $T=\operatorname{diag}(J,-J)$ instead give $Q=(|\psi_2|^2-|\psi_1|^2)/2$; copy interchange is canonical, preserves a common positive isotropic energy and reverses $Q$. This is an explicit signed moment map, not a selected electric-charge law.

The script additionally evaluates finite stiffness of canonical partners, invariant and leaking projectors, detuned conservative exchange and all fourteen count assignments. Its final run passes 784 checks over 15 equation sections and 17 branch entries, including 225 gyroscopic cases, 120 exchange cases and 22 reduced ODE controls. The largest reported conservation or exact-trajectory error is below $9\times10^{-10}$. These are checks of the stated candidate laws, not particle predictions. No physical field amplitude, contact coefficient or measured mass was fitted.

The both-branch calculation reproduces the representation, finite-partner, gyroscopic, exchange and reduced-ODE controls with Python, NumPy, SciPy and SymPy. Its generated equation ledger records every parameter and alternative; these controls test the stated candidate laws rather than identified particles.

Verification and Scope

The portable exact integer checker verifies the Clifford anticommutators, real-generator identities, complete commutant counts, retained ambient J0, quaternionic trace identities and conjugation controls with Python and NumPy. Its original algebraic run passed 2,860 conditions. The separate 48-by-6 quaternionic skew-commutant constraint matrix has rank three and nullity three; the portable checker emits that matrix so its rank can be checked by exact row reduction. The general statements follow from the displayed algebra, not the check count. The both-branch calculation also verifies the symbolic reduced response spectra. No particle simulation, observed-charge prediction or experimental validation is claimed.

Spatial Clock Axes and Energetic Carrier Preference

The independent-axis action has a global interpretation. For one quaternion write $\psi=r u$, $u\in S^3$, and $I=r^2/2$. Its one-form is $\theta=-I a\cdot(du\,u^{-1})$. Identifying cotangent momentum with $p=-Ia$ gives $T^*S^3$ with the zero section removed. This explains the extra canonical pair. The fixed-$\psi$ sphere has zero pullback two-form; the effective sphere form is a Schur complement, not a global product factor. At zero field the original axis description is degenerate. Extending the cotangent space retains additional configuration data and changes that branch.

Spatial gradients also matter. With $H=\int[\kappa|\nabla\psi|^2/2+\eta|\nabla a|^2/2+U(I)]$, put $f=\delta H/\delta\psi$, $h=\delta H/\delta a$ and $R_i=-\psi^TJ_iJ(a)f/2$. Full variation gives

$$\dot a=s^{-1}a\times(h-R),\qquad \dot\psi=-J(a)f+\tfrac12J(a)J(\dot a)\psi.$$

Dropping $R$ loses field-gradient torque. Around a homogeneous circular clock of rate $\omega$, set $K=\kappa|k|^2$, $T=2\omega s+\eta|k|^2$ and $B=|\psi|^2U''(I)$. The radial branch has $\nu^2=K(K+B)$; the two coupled transverse rates are

$$\nu_\pm=\frac{\sqrt{T^2+4KTs}\pm T}{2s}.$$

The low branch behaves as $K$ at small wave number. The eight-component quaternionic carrier adds two ordinary pairs with rate $K$; it does not freeze the depth-five axis. These are formal spectra of the stated enlarged field action, not a localization theorem.

The derived Hopf axis has a different limitation. On one quaternion its unit-length identity is exact. On $\mathbb H^2$ the same quadratic triplet generically has length below one, including zero at a nonzero field. Unit length requires a purity constraint. Moreover a fixed complex structure $K$ commuting with all three $J_i$ cannot also commute with Spin(d) on a minimal irreducible quaternionic module: the center of its commutant is real. A regular four-component gradient completion nevertheless exists, using the polynomial identity

$$\frac{\rho^2}{8}|\partial a|^2= \frac12[\rho|\partial\psi|^2-(\psi\cdot\partial\psi)^2 -(\psi^TK\partial\psi)^2].$$

It retains the fixed-$K$ premise. Positive second-order kinetics instead provides a regular charge-derived axis without $K$, but admits real-type carriers as well. Neither alternative inherits the first-order depth selector automatically.

A separate reduced construction makes the minimal-carrier preference energetic. In an active isotypic component write the real field as a matrix $X$ over its Schur algebra, with $G=X^\dagger X/2$, $N=\operatorname{Re}\mathrm{tr}G$ and ambient complex structure $j$. Choose one common polynomial

$$H=3N-3N^2+N^3+6[N^2-\operatorname{Re}\mathrm{tr}(G^2)] +\tfrac32\|[G,j]\|^2.$$

The added terms vanish on compatible rank-one complex or quaternionic carriers. For real type they are at least $3N^2$, because the part of $G$ commuting with $j$ has paired eigenvalues. Hence $\inf H/N=3/4$ on C/H, attained at $N=3/2$, while the real infimum is $3$, approached at zero norm. The energy is positive on every nonzero state. This is a finite preference, not removal of real clocks. First-order compatible minimal subspaces are invariant, but the conservative flow preserves rank rather than purifying generic data. Neutral copy orientations remain.

At the ratio minimum the first-order clock is $3/4$; positive second-order kinetics gives $\sqrt3/2$. Other C/H norms give other rates, including zero, so the ratio minimum is an additional selection premise. These illustrative frequencies are not IDOS masses. The active-depth decomposition is supplied in this reduced model; deriving it from one spatial field and retaining localization and cluster behavior remain required bridges.

Recorded algebraic verification includes 79 global-axis checks and 244 carrier-selection checks with rational certificates and explicit ambient projectors at each depth. The displayed global-axis variation and Gram-matrix inequalities specify the calculation and its parameter, regularity and representation assumptions; these checks do not select a physical carrier.

A Regular Local Carrier Penalty

For a fixed active spatial decomposition, write the real field in its Schur matrix components, with $g=X^\dagger X/2$, $n=\operatorname{Re}\operatorname{tr}g=|X|^2/2$, and a fixed skew clock matrix $j^2=-I$. The following added energy is local:

$$W(g)=\frac B4[n^2-\operatorname{Re}\operatorname{tr}g^2] +\frac C4\|[g,j]\|^2,\qquad V(X)=\begin{cases}W(g)/n&n>0,\\0&n=0,\end{cases} \qquad H_{\rm sel}=H_0+\int V(X),\quad B,C>0.$$

Here $H_0$ is the real half-energy entropy action and $N=\int n$. The matrix notation describes internal classical field components, not extra physical dimensions. All active-depth and representation premises remain explicit inputs. With $M=B(nI-g)/2+C[j,[g,j]]/2$, complete variation gives

$$\nabla_XV=X\left(\frac M n-\frac{W}{n^2}I\right).$$

The quotient term cannot be omitted. The function is nonnegative, homogeneous of degree two, and smooth away from zero. Its off-zero Hessian is homogeneous of degree zero and bounded on the component unit sphere. Thus $V$ is $C^{1,1}$, with a globally Lipschitz gradient extending by zero at the origin. It generally is not $C^2$ there. These statements concern the added force; they do not alone establish evolution for the complete entropy equation.

On a constant compatible rank-one polarization, both $V$ and its gradient vanish. The exact Gaussian therefore retains its clock $\omega_s$ and energy $N\omega_s$ at every amplitude. In a pure real-Schur block, $W\ge c n^2$, $c=\min(B/8,C/2)$, so $H_{\rm sel}/N\ge\omega_s+c$; a minimizing constant polarization times the Gaussian attains the bound. For the common coefficients $B=24,C=6$, the separation is exactly $3$. Real-type clocks survive at the higher rate. A small real admixture into a compatible block has a small cost, not the full pure-type gap.

At a compatible Gaussian, the complete added component Hessian commutes with the clock structure and has possible stiffnesses $c_a=0,B/2,2C$. Multiplicity depends on the representation. The longitudinal amplitude/phase block is unchanged; transverse Hermite rates are $2\alpha b k+c_a$ for first-order kinetics. For the separate positive second-order action they are $\pm\sqrt{\Omega^2+(2\alpha b k+c_a)/m}\pm\Omega$, with $\Omega^2=\omega_s/m$. The latter has circular energy $2N\omega_s$. These are formal spectral statements, not nonlinear stability modulo the solution's symmetry group.

The force bound also gives $|V(X+Y)-V(X)-V(Y)|\le L|X||Y|$. Hence the integrated added energy becomes additive for widely translated $L^2$ fields. The integrated Gram alternative lacks this clustering property, while an unnormalized local quartic penalty loses its finite gap under fragmentation. The entropy part still has its own nonlocal convolution and does not acquire finite propagation.

Local compatibility is weaker than a single fixed carrier for the whole pattern. In a compatible multiplicity-two complex block, let $X_\epsilon=u(x)a(\cos\epsilon x_1,\sin\epsilon x_1)$, where $a$ is a constant unit left factor and $u$ is the core Gaussian. Pointwise $V=0$, but

$$H_{\rm sel}[X_\epsilon]/N=\omega_s+\alpha\epsilon^2,\qquad \operatorname{spec}\int g= \left\{\frac N2(1+e^{-\epsilon^2/b}), \frac N2(1-e^{-\epsilon^2/b})\right\}.$$

Every nonzero $\epsilon$ occupies two fixed copies at arbitrarily small excess energy. Exact minimizers remain constant compatible polarizations; there is no positive threshold excluding every textured pattern. Fixed compatible subspaces are invariant, with velocity in the same subspace for second-order kinetics. A freely varying rank-one factorization is generally not invariant because its Laplacian creates normal components. No generic purification or depth-five interaction suppression follows.

The quotient variation, bounded Hessian argument, exact texture formula and both kinetic spectra above give the reproducible local-penalty derivation. Independent exact verification includes the noncentral quaternionic Hessian; this is a check of the stated penalty, not a new dynamical selection law.

Relative Spatial and Carrier Orientation

The smooth count completion permits a concrete orientation test in the actual 32-real-component carrier. Let $P$ select an occupied spatial plane, $e$ a compatible unit polarization, and $A(\Omega)$ the spin generator of a physical rotation, normalized by $A_{ij}=\gamma_i\gamma_j/2$. For $\delta P=[\Omega,P]$ and $\delta e=\eta$, covariance gives

$$\delta^2E_{\rm selector} =N[\eta-A(\Omega)e]^TK[\eta-A(\Omega)e],\qquad K=2C_E.$$

Here $N=\int\rho$ uses the doubled count-action convention. With common $B=24,C=6$, the positive eigenvalue of $K$ is $24$. Simultaneously rotating shape and carrier gives zero cost. Minimizing over a free carrier also removes the shape-only quadratic cost; a fixed-carrier calculation alone cannot establish a fixed physical axis.

The exact calculation includes all $d(10-d)$ cross-plane generators. For a canonical compatible copy, their positive fixed-carrier stiffness is $12$, with ranks $8,6,12,8,12,0$ at depths $2,3,4,5,6,10$. Other compatible-copy orientations change the answer: explicit rational examples give rank $16$ at depth two and rank $14$ at depth three, with unequal positive stiffnesses. The tested depth-five copies retain rank eight. The samples do not classify every compatible-copy orbit, and do not establish a unique depth-five suppression or inability to tilt.

All ranks and characteristic polynomials are checked exactly using the actual ambient Clifford matrices: 312 checks across 42 preparations. These are constant-plane energy stiffnesses, not precession rates. Kinetic pullback, spatial texture and phase/current response are needed for dynamical rates. The displayed stiffness formula and actual ambient Clifford matrices specify this finite preparation calculation. Physical plane rotations and internal component rotations remain distinct throughout this comparison.

The Coupled Kinetic Response

The full count-current action supplies the missing kinetic pullback. For $B_*=aP+bQ$, $a<b$, physical plane angles $q$ and their cross-quadratic phase partners have conjugate momentum $p$ and positive inertia

$$I=\frac{N\omega(b-a)^2}{2c^2ab(a+b)},\qquad H_2=\frac{|p|^2}{2I}+\frac N2(\eta-Lq)^TK(\eta-Lq),\quad Lq=A(\Omega)e.$$

The reduced one-form is $p:dq+N\eta^TJd\eta$, so $\dot q=p/I$, $\dot p=NL^TK(\eta-Lq)$ and $\dot\eta=-JK(\eta-Lq)/2$. Carrier relaxation cannot be substituted for this dynamical variable. The full source variation closes on these Gaussian orientation and constant-carrier tangents at linear order.

For the common controls $a=1/2,b=c=\omega=1$, $I/N=1/6$. The canonical stiffness $12$ gives the two coupled positive frequencies $6(\sqrt3-1)$ and $6(\sqrt3+1)$, each with multiplicities $4,3,6,4,6$ at depths $2,3,4,5,6$. Other normal carrier pairs have rate $12$; neutral directions and secular drift also remain. Depth ten has no cross-plane coordinate in this isotropic family. The canonical depths three and five share frequencies, with different multiplicities. Other compatible copies supply extra slow rates, including $0.008503047$ and $0.016970011$ in one depth-three preparation. A universal classification does not follow from these forty-two preparations.

For each positive eigenvalue $\lambda_R$ of $L^TKL$, the checked blocks have $\nu_\pm=\sqrt{36+\lambda_R/(I/N)}\pm6$. On the supplied width branch $a=b/(k+2)$,

$$\frac IN=\frac{\omega}{c^2b}\frac{(k+1)^2}{2(k+3)},\qquad \omega=\mu_d\binom{k+d}{d}.$$

The selected clock and width law are inputs to this inverse action; the inertia and response then follow. Large inertia can slow the lower branch at every depth, rather than isolating depth five. These frequencies belong to the stated first-order carrier/current law, not its separate positive second-order alternative. The positive mobility continuation controls far-tail source signs without changing the branch Hessian. The complete linear reduction above has 560 recorded supporting checks. Its matrices and coefficients specify the coupled response calculation. Nonlinear invariance and the full spatial spectrum remain open.

The Complete Spatial Count Band Retains The Minimal-Fibre Criterion

The spatial multiplicity question can be answered for the real Cartesian polynomial band. Set $N=n-1$, let $\Sigma_d$ be one minimal irreducible real spinor and use

$$W_{n,d}=\Sigma_d\otimes\mathcal P_{\le N}(\mathbb R^d),\qquad \dim W_{n,d}=s_dS(n,d).$$

Rotations act on both factors. Multiplication by a common positive radial Gaussian gives square-integrable spatial functions without changing their representation. This cumulative Cartesian band differs from a sphere parity band of equal dimension. Its coefficients describe classical fields in ordinary spatial coordinates, not positions in a representation space.

For every positive $n$, a nondegenerate invariant alternating form exists at $d=2,3,4,5,6,10$: use $J_{\rm spin}\otimes I$. At $d=7,8,9$, the highest Cartan component has weight $(N+\tfrac12,\tfrac12,\ldots,\pm\tfrac12)$ and occurs exactly once. The highest polynomial and spinor vectors are unique, and lower polynomial degrees cannot reach that first coordinate. The longest Weyl element is minus the identity for $B_3,D_4,B_4$, so this component is self-conjugate. Its multiplicity-one occurrence in the actual real tensor representation supplies a conjugation squaring to plus the identity. It has real type; every equivariant skew operator vanishes on it. The full kinetic form is therefore singular at those three depths for every finite band cutoff.

This extends the existing selector to a minimal spinor fibre carrying arbitrarily many spatial modes. Minimality of the entire spatial-mode representation is unnecessary. Minimality of the fibre, full rotation invariance, a complete finite band and first-order nondegeneracy on that whole band remain essential. The ambient real32 carrier retains its counterexample. Infinite bands, removing a highest component, extra canonical copies and second-order kinetics require separate arguments. At depth one, the band is $n$ trivial copies and permits the form for even $n$; the extension is stated over depths two through ten.

An Exact Function Of Index And Depth

The one-vector Fischer decomposition supplies the complete kernel. Its degree-$\ell$ monogenic irreducible has dimension

$$r_\ell(d)=s_d\binom{\ell+d-2}{d-2},\qquad s_7=s_8=8,\quad s_9=16.$$

At odd depths seven and nine its multiplicity in the complete band is $m_\ell=N-\ell+1$. At eight, for one initial chirality, the two inequivalent monogenic chiralities have multiplicities

$$m_{\ell,+}=\left\lfloor\frac{N-\ell}{2}\right\rfloor+1, \qquad m_{\ell,-}=\left\lfloor\frac{N-\ell+1}{2}\right\rfloor.$$

The decomposition is first made on complexified spin modules. Each monogenic representation has a real form because its multiplicity-one occurrence in a real harmonic-spinor tensor is self-conjugate; the two half-spinor representations at eight must remain distinct. This avoids assuming that Clifford multiplication for every sign convention acts on the same minimal real spin module. The external decomposition is described by Lávička and Souček, Fischer decomposition for spinor valued polynomials. The application to the complete kinetic form is the further argument here.

An isotypic block with $m$ copies of a real-type irreducible has skew commutant $\mathfrak{so}(m)$. Its minimum kernel is one full copy when $m$ is odd and zero when it is even. Two-by-two rotation blocks attain these minima. Consequently

$$K(n,d)=\min_B\dim\ker B =\sum_{\rm blocks}(m\bmod2)r_\ell(d),\qquad A(n,d)=\dim\{B:B^T=-B,[B,\operatorname{Spin}(d)]=0\} =\sum_{\rm blocks}\frac{m(m-1)}2.$$

The maximum kinetic rank is $s_dS-K$. The unique highest component gives the exact lower fraction

$$\frac{\dim\ker B}{\dim W_{n,d}} \ge\frac{d(d-1)}{(n+d-1)(n+d-2)} =\frac{S(n,d)-2S(n-1,d)+S(n-2,d)}{S(n,d)},$$

where counts with nonpositive first argument are zero. At $n=2$, the total dimension and minimum kernel are respectively $64,48$ at seven, $72,72$ at eight and $160,128$ at nine. An even total count alone therefore does not remove the obstruction. For odd depths the complete sum is $K=s_d\sum_{j=0}^{\lfloor N/2\rfloor}\binom{N-2j+d-2}{d-2}$.

The exact polynomial-band checker uses independent Weyl products, complete multiplicity sums and exact rational arithmetic for 2,208 checks, including all $n=1,\ldots,96$ at seven, eight and nine. These are neighboring representation controls, not new particle assignments. The all-band theorem follows from the representation argument, not from the finite number of checks. No occupied index or lifetime is selected by $K$.

Independent Color Reduces The Quaternionic Clock Freedom

The spatial-spin and color constructions can be combined under a precise additional carrier premise. At depth three, or on one chiral depth-four block, choose the minimal complex spin doublet and an independently acting complex color triplet:

$$V=\mathbb C^2\otimes_{\mathbb C}\mathbb C^3, \qquad \dim_{\mathbb R}V=12, \qquad G=\operatorname{Spin}(d)\times SU(3).$$

The product representation is complex irreducible, so its complex-linear commutant consists of scalars. A further real-linear map would need an antilinear part. Spin commutation fixes the spin factor to the pseudoreal $\epsilon$, leaving a color matrix $M$ satisfying $M\overline U=UM$. For the color generator $i\operatorname{diag}(1,2,-3)$, all sums of paired diagonal weights are nonzero, so every matrix entry of $M$ vanishes. Thus the joint real commutant is $\mathbb C$, with one skew direction and two normalized clock signs. The spinor-only quaternionic sphere no longer commutes with the fixed independent color action.

There is a quantitative function measuring this restriction. With $J_0=iI$, $J_1=(\epsilon\otimes I_3)\mathcal C$, where $\mathcal C$ is complex conjugation, and $J_2=J_0J_1$, let $J(a)=\sum_{i=0}^2a_iJ_i$, $|a|=1$. Normalize the Hermitian color generators by $\operatorname{tr}(T_AT_B)=\delta_{AB}/2$, and put $R_A=iI_2\otimes T_A$. The real Hilbert–Schmidt norm gives

$$\mathcal D(a)=\frac1{12}\sum_A\|[J(a),R_A]\|_{\rm HS,real}^2 =\frac{10}{3}(a_1^2+a_2^2)=\frac{10}{3}(1-a_0^2).$$

Five real symmetric generators contribute, each with trace square one half; the other three are imaginary antisymmetric. The mixed real inner product vanishes, giving the displayed diagonal quadratic form. The color–clock checker also verifies the joint commutant using exact integer generators: modular ranks 142 and 65, together with explicit kernel matrices, certify full commutant dimension two and skew dimension one. Its complete profile and clock calculation passes 98 checks; quadrature controls are numerical, not rigorous enclosures.

With independent triplets only at the quark depths three/four and minimal color-singlet blocks at two/five/six/ten, depth five alone retains the continuous quaternionic clock sphere. This is a new conditional distinction between those specified minimal joint carriers. The defect is not already an energy, stiffness or decay rate. It does not freeze a physical spatial axis or derive weak interaction suppression. Full Dirac/ambient multiplicities, additional independent symmetry and allowing the color action itself to vary can change the conclusion.

The distinction between real and complex color carriers also constrains the finite-band result. Three trivial real spectators leave every odd real multiplicity odd and hence leave the absolute minimum kernel $K$ unchanged. An independent complex triplet is instead real dimension six and supplies its own invariant complex structure. Tensoring it with a real-type spinor over $\mathbb R$ restores a nondegenerate clock at seven/eight/nine. A spatial Hodge triple is not automatically this independent factor. The two results can be combined only after specifying the common carrier and its action; their premises cannot be exchanged silently.