Symmetric Hodge Feedback Selects a Color Channel and an Interaction Order

A stated Hodge-contact hypothesis supplies an exact conditional selection rule: among the six maintained depths, its instantaneous color-volume channel is active only at three and four. The same geometry permits a velocity interaction at two, five, six and ten. This connects the color-channel calculation and the slow-source response with one geometric decomposition.

The premise is specific: the relevant local contact starts from the raw real Hodge map on the complete middle-form channel, and its instantaneous same-field energy uses that map's selfadjoint part. The mathematics below follows from this premise. Geometry and reciprocity alone do not prove that the field selects this kernel. At seven/eight the rule also activates a larger channel, so it is not a global uniqueness theorem for three/four.

1. Two-Sided Cancellation in an Instantaneous Self-Contact

On an oriented ordinary Euclidean spatial tangent space of dimension $2m$, the middle-form Hodge star satisfies

$$*^2=(-1)^mI,\qquad *^\dagger=(-1)^m*.$$

Define its selfadjoint and skew-adjoint parts:

$$H=\frac12(*+*^\dagger),\qquad A=\frac12(*-*^\dagger).$$

Then

$$\boxed{H=\begin{cases}*,&m\text{ even},\\0,&m\text{ odd},\end{cases} \qquad A=\begin{cases}0,&m\text{ even},\\*,&m\text{ odd}. \end{cases}}$$

For a real middle form $F$, $F^T*F=F^THF$. An antisymmetric map cancels when its two slots contain the same field. For a complex mode amplitude, the corresponding real instantaneous source is

$$J_H(F)=\operatorname{Re}(F^\dagger*F)=F^\dagger HF.$$

Thus the cancellation holds for every complex amplitude and polarization when $m$ is odd. It does not require a specially balanced state, equal mean generators or a tensor singlet. It cancels this specified source operator.

The complete chiral channel has rank

$$k_m=\binom{2m-1}{m-1}.$$

On a Hodge eigenchannel with eigenvalue $\lambda$, the filter acts as $H=\operatorname{Re}(\lambda)I$. The isometric odd-shadow map transfers the same channel to depth $2m-1$. These are local ordinary spatial constructions, compatible for example with a nested flat spatial flag. The odd-depth interpretation uses the declared neighboring even-carrier map; it is not an intrinsic odd-dimensional middle-form star.

No coefficient is fitted separately at any depth. The filter is the same geometric expression everywhere. Its periodicity and higher active controls must be retained when assessing the conditional rule.

2. Filter the Existing Color-Volume Interaction

Let $a,b,c$ denote three distinguishable equivalent-mode amplitudes in a complete chiral channel. Use the same filtered interaction at every rank:

$$E_6=\gamma\|Ha\wedge_{\rm mode}Hb\wedge_{\rm mode}Hc\|^2, \qquad \gamma>0.$$

Here the exterior product is in the mode-amplitude carrier. It is distinct from wedging spatial middle forms. Its squared norm is the determinant of their three-by-three Hermitian Gram matrix. The amplitudes describe modes of the one field, not additional spatial dimensions or exchanged particles.

At three/four, $H=\pm I_3$. The interaction is the existing squared complex volume $\gamma|\epsilon(a,b,c)|^2$. Its Hermitian norm preserves full complex unitary mode mixing; a specified parallel complex volume separates $SU(3)$ from the common phase. The equivalent-mode carrier, independent spin factor and actual connection still require their existing physical construction.

Contract the third slot against isotropic covariance $\sigma^2I$. At general rank $k\ge3$, the pair operator is

$$\sigma^2|\operatorname{Re}\lambda|^6(k-2)(I-\mathsf S).$$

At three/four this becomes

$$\boxed{\sigma^2(I-\mathsf S) =\sigma^2\left(\frac23I-2\sum_AT_A\otimes T_A\right).}$$

The familiar $qq$ contractions $-2/3$ and $1/3$ survive unchanged. At five/six and nine/ten the entire filtered operator is zero. This cancels the higher-rank three-form norm which, without the filter, retained a factor $k-2$ in the earlier pair calculation.

At seven/eight the same rule retains a rank-35 channel with pair multiplier $33\sigma^2$. The existence and occupation of such field states are open. This prediction of the candidate cannot be erased by restricting the comparison to assigned particles.

The energy is phase neutral in each slot. Cancelling this color-volume source therefore does not force electric charge to vanish. Conversely an inactive interaction operator is not a proof that an arbitrary state is an invariant color singlet. The two claims remain distinct.

An uncorrected complex self-wedge would give $|a^Ta|^2$ on a real chiral triple. That expression is not $SU(3)$ invariant. The Hermitian filter and volume norm above avoid importing that different interaction.

3. The Antisymmetric Part Permits a Velocity Interaction

For real coordinates of a field pattern,

$$L_B=\frac g2F^T*\dot F.$$

Its symmetric part is a total derivative. The skew part contributes a gyroscopic force proportional to $A\dot F$, with no instantaneous work because $\dot F^TA\dot F=0$. Thus the same raw Hodge map distinguishes a position contact from a velocity contact by the parity of its middle degree.

An explicit reciprocal two-coordinate reduction is

$$L=\frac12(|\dot q|^2+|\dot b|^2) -\frac12(q^TCq+b^TDb) +g_0q^THb +\frac{g_1}{2}(q^TA\dot b-\dot q^TAb).$$

The vectors $q,b$ describe retained response and source-pattern modes of the same field. They are not a particle and an independent mediator. With the common diagnostic $C=2I$, $D=3I$, $g_0=g_1=1$, the complete potential is positive and the gyro matrix is antisymmetric across both participants. All complete normal frequencies are real. The full conserved energy includes both coordinates. No source-only variation is being used to claim reciprocity.

When $b$ is prescribed at frequency $\Omega$, its response vertex is $g_0H-i\Omega g_1A$. For a scalar receiver susceptibility $\chi(\Omega)$ times the form identity, the induced source response is

$$\chi(\Omega)\left[g_0^2H^2+ g_1^2\Omega^2A^\dagger A\right].$$

The static and velocity terms occupy complementary Hodge parities. Their coefficients have different physical units before a reference clock is chosen. The common dimensionless comparison above is a diagnostic, not an identification of their physical constants.

A nonzero gyro term also requires appropriate time-reversal data. If both coordinates are time-reversal even, a fixed gyro coefficient is odd under that transformation; circulation of the same field or different coordinate parities must supply the physical structure. The spatial Hodge identity alone does not derive it.

4. Shared Geometry with the Sawtooth Source

Use the maintained finite sawtooth preparation, source coefficients $c_p(K)$, cutoff $K=S(n,d)$ and sector fundamental ratios $r_d$. The conditional source is $b=f_K^2u$ with fixed unit form direction $u$. It is not yet an autonomous derived particle solution or an identity between field density and canonical coordinate.

For the same receiver

$$\chi(x)=\frac15\left(\frac3{2-x^2}+\frac2{3-x^2}\right),$$

the Hodge rule selects direct density response at three/four and derivative response at two, five, six and ten. All shape copies use the identity on the form channel. The already specified source/receiver scales and separate neutrino splitting calibration are retained, without fitting individual branches. The scalar receiver comparison does not yet calculate finite orientation acceptance in a real encounter.

Representative signed response sums with common dimensionless $g_0=g_1=1$ are:

The muon and tau absolute omitted-tail bounds are $1.029\times10^{-5}$ and $6.120\times10^{-7}$; the other displayed sums are complete. The report includes all remaining assigned integer-cutoff branches. Photon and unresolved bottom cutoffs are not invented to extend this massive-waveform prescription.

Within this action, a common geometric rule now supplies the derivative choice responsible for the small depth-five source response. The effect still uses its calibrated slow fundamental; any sufficiently slow source would be suppressed in that channel. Similar charged-lepton values remain conditional spectral responses, not exact electric-charge universality. These signed conservative sums are not measured force strengths, probabilities or weak rates.

This bare canonical gyro action differs from the fast-band-induced velocity action, where induced inertia and finite gaps altered the complete response. Results cannot be transferred between those actions without retaining their terms and assumptions.

5. The Physical Choices Still to Derive

The kernel hypothesis is global and simple, but not forced by symmetry alone. The complementary Hermitian map

$$K=\frac{*-*^\dagger}{2i}$$

is nonzero precisely where $H$ vanishes, and $H^2+K^2=I$. Its current changes sign under complex conjugation; its squared energy is nevertheless time-reversal even. Generic norm overlaps also remain allowed. A field law must explain why the symmetric Hodge-volume channel is the relevant strong response and how other allowed terms behave. No absence of every higher-sector interaction has been proved.

Reciprocity constrains the full kernel. A real antisymmetric fibre map $A$ can occur in the symmetric two-slot block $\left(\begin{smallmatrix}0&A\\-A&0\end{smallmatrix}\right)$, or combine with a spatial derivative. Thus reciprocity alone does not require discarding every skew fibre contribution. The stated local instantaneous same-field Hodge self-contact is a narrower premise.

Source ordering also matters in nested geometry. A higher middle form can be contracted along selected ordinary normal directions and restricted to a lower plane. For the six/ten-to-four control maps $R$, the parent-filtered source is zero, whereas $H_4R$ is generally nonzero. The checker finds squared Hilbert–Schmidt norm $6$ for this difference. The candidate therefore requires the physical source to be formed from the complete pattern's active-sector geometry before projection to an observer or receiving pattern. Deriving that order from one reciprocal field action remains necessary; nesting alone does not provide it.

With those premises stated, the result is a genuine conditional sector discriminator: three/four are the only active complete color-volume channels among the maintained six depths. It does not distinguish up-type from down-type assignment between that pair, select the neutral depths among themselves, prove a global dimensional uniqueness, derive the observed strong dynamics, or replace the occupied-singlet question. Its value is a concrete common action whose selection and failure modes can now be calculated.

Verification

The single checker passes 4,021 conditions across 31 cases, recorded with exact source hashes in its report. It covers every middle-form basis block through depth twelve, all maintained sector and neighboring controls, filtered complex-volume covariance and pair-Casimir factors, complete reciprocal positive mode actions, position-versus-velocity variation, allowed complementary interactions, nested-source ordering, and the all-sector sawtooth comparison. No particle simulation is used.

Passive Finite-Delay Control

A positive local wave-network completion realizes the symmetric filter at zero frequency while conserving energy in a complementary output. Its finite-delay transfer is a different interaction: substituting it for every instantaneous color filter generally restores a nonzero higher-sector volume. Exact finite-cutoff sixth moments and rigorous muon/tau bounds verify this control. The instantaneous action above retains its stated exact zeros; the delayed alternative cannot inherit them without an additional law. The same comparison shows that source-sized delays can remove the slow-clock neutrino suppression obtained with a receiver-set time scale.

An Explicit Source from the Same Spinor Field

The spinor-bilinear construction supplies the complete middle-form source directly from classical spatial field components. Its Hodge phase gives the same all-sector instantaneous parity, with full Clifford kinetics retained. At three/four the form is a symmetric spinor pair; its determinant factors into three pair areas. Uniform unit-spinor polarization gives exactly isotropic triplet covariance and preserves the earlier pair-Casimir contraction. One pointwise spinor square remains a constrained source, so independent color mixing and physical occupation are still separate requirements.

A Separate Normal Color Index in the Same Current

An explicit adjoint-color spinor current now combines this Hodge filter with the independently acting ordinary normal triplet. Its spatial middle-form index and normal color index are distinct. The common positive quartic has the same maintained-sector zeros, full source variation and canonical conservation, while its three/four pair algebra retains scalar and spin-dependent terms alongside the standard color Casimir. This avoids identifying the spatial spin-one triplet itself with color. The kernel premise, active seven/eight control and other allowed geometric couplings remain explicit.

The Positive Current Has a Self-Wedge Action

A local chiral self-contact gives the same occupied-sector current energy and variation through graded exterior multiplication and the oriented Clifford-volume identity. It includes both spatial source chiralities and has an explicit depth-three normal contraction. This provides another concrete geometric formulation of the kernel premise, without establishing that this contact is the unique interaction or removing other allowed terms.