Gauge geometry

In the four-plus-six spatial reference split, the base has four ordinary spatial dimensions and its perpendicular complement has six. Time is a separate coordinate in the infinite-dimensional physical spacetime. This four-plus-six split concerns ten spatial directions in the matter domain.

• Relaxed Frame Twists Link Guide Force and Longitudinal Stiffness — Full frame rotation, relaxed force, phase stiffness, an exact tilt channel and a complete rotating energetic bound.

• Distinct Endpoint Clocks and the Dynamic Shape Response — Frequency-dependent boundary stress, source work, exact count-harmonic averages and the finite-segment radiation threshold.

• Competing Ranges from the Shape Energy of a Geometric Guide — Actual neutral-frame shape channels produce a short-range outward correction to tension; reciprocal torque and guide-width bounds delimit the proposed balance.

• Finite-Amplitude Localized Coefficients and Their Clock Window — Dynamical amplitudes replace the norm constraint, with complete stress and polarization checks; an exact clock bound exposes the missing mass-spectrum mechanism.

• Phase-Complete Coefficient Rotation and Polarization Positivity — A complete fixed-norm wave comparison includes phase and nonpure directions, with one analytic polarization positivity identity and a scoped radial stability comparison.

• Localized Rotation and Constrained Stability in the Orientation Law — A common causal action with a declared normal-axis potential supplies radially localized profiles, complete inertia and full target-polarization checks at depths two through six.

• Sources, Energy Exchange and Current Response Through Canonical Orientation — A constrained source variation has reciprocal complete stress and causal wave principal blocks, with exact current-driven stability controls.

• Completing an Orientation Defect with Its Phase and Amplitude — Complete spinor energy exposes a phase obstruction and a smooth amplitude-zero lift, with its required supporting force and full-ray metric limits.

• A Causal Determinant Law and Exact Normal-Curvature Defects — One timelike constitutive law retains the ray metric, supplies complete causal principal cones and exact curvature-carrying defects, and exposes static and width-inertia limits.

• What the Geometric Color Construction Currently Establishes — A common geometric synthesis separates the rank-three carrier, complete polarization metric, gated guides and three distinct connection actions.

• Chiral Ray Geometry and Normal Rotation Have the Same Pure-State Metric — An exact factor sixteen connects the complete rank-three projective polarization metric to ordinary normal rotation.

• Curvature Generated by Normal Rotation Structure — Explicit pullbacks certify U3/U4 holonomy while rank two generates only its central circle; determinant transport and changed Gauss variation are derived.

• A Common First-Derivative Normal Orientation Action — A combined action has exact static principal cones, a sharp electric orientation cost and an explicit rapid-motion hyperbolicity boundary.

• Parallel Quadratic Normal Geometry and a Distinct Triplet Algebra — A complete curved-parent flag distinguishes parallel SO3 triplet transport from the full SU3 graph connection.

• Retained Normal Rotation Structure and Compatible Transport — A positive normal orientation term, exact compatible connection, electric continuation obstruction and unchanged commuting vacuum history.

• Dielectric Growth, Critical Concentration and the Three/Four Guide Window — Exact critical escape, supercritical concentration and bounded nonlinear completions delimit flux-guide robustness.

• Geometric Flux Guides Survive a Quadratic Dielectric Correction — Sharp arbitrary-shape minimization distinguishes continuous and finite-width binding at three/four while retaining the occupied-sector gate.

• Induced Normal Curvature, Spatial Inertia and Retained Orientation — Squared-linear-curvature bounds are distinguished from the complete background Hessian, carrier preservation and a finite-propagation two-mode comparison.

• Induced Normal Color Geometry on One Nested Spatial Flag — One actual nested spatial graph generates U3 and exposes the quadratic and odd-pullback assumptions needed for sector selection.

• Exact-Zero Dielectric Requires a Rule for Vacuum Connection Histories — Full local field equations, complete probe work and neutral-clock controls expose the zero-carrier connection ambiguity.

• A Dynamical Color-Neutral Pair Phase and Its Central Character — Explicit canonical pair reduction with the actual spin shift, determinant characters and all-sector spatial-current controls.

• Compact Non-Abelian Gauss Data Delimit the Flux-Confinement Result — Full classical Gauss trials retain the sector gate but give a separation-independent energy upper bound for opposite fixed sources.

• Classical Color-Current Cancellation and Complete Spatial Dynamics — Canceled positive modes, a free-wave direct-sum branch and the residual SU2 packet and symplectic constraints.

• A Neutral Spatial Frame Realizes the Quartic Color Dielectric — Exact area and gradient inequalities recover the scalar tube while retaining neutral amplitude, angle and orientation modes.

• Sharp Plane Bounds Give Linear Energy for Finite Commuting Sources — Finite fixed-source energy is bounded between tension times separation and that same linear term plus a fixed cap cost.

• A Positive Hodge-Gated Dielectric Supports Finite-Width Color Flux at Three and Four — Exact support, positive tension, complete transverse inertia and scalar fixed-flux stability follow from a declared neutral-mode constitutive action.

• A Hodge-Gated Gauss Constraint Preserves Complete Spatial Singlet Branches — The actual Gauss source is constrained in inactive sectors while the correct residual-singlet projectors retain all active spatial directions.

• A Positive Gradient Gauss Law Separates Static Confinement from Signal Propagation — A conserved finite-energy quadrupole distinguishes positive static response, ordinary mobility and the physical signal cone; positive ordinary spectral channels give a scoped complementary bound.

• Geometric Color Channels, Pair Response and Charge Consistency — A volume contraction supplies the pair-Casimir operator; the same channel rank enters charge fractions and an all-depth weak-anomaly parity test.

Topic navigation; Research index lists all notes.

• A Fourth-Order Static Response Has a Three-and-Four Neutral-Pair Window — A common conditional static pole distinguishes divergent isolated cost from finite neutral pairs only at three/four; source, propagation and stiffness controls preserve its actual scope.

• Real Normal Curvature Cancels Odd Characters and Permits Color Pair Response — Reality cancels odd trace characters, while a positive connection response supplies a distinct pair sign; maximal-degree selection and inactive-source constraints remain physical questions.

• Positive Spatial Kinetics Constrain the Color-Current Polarization — A full tangent-symbol control constrains both chiral sources, preserves color/spin independence, and checks pair response at independent momentum directions.

• A Parity-Even Self-Contact Produces the Occupied-Sector Hodge Rule — Graded self-contact supplies the same positive current action, while spatial chirality and normal contraction keep both parity and the three-shadow geometry explicit.

• Independent Normal Color and Two-Sided Hodge Contact Share One Current Action — A full spinor adjoint current separates the spatial form index from normal color, preserves the all-sector parity and yields exact spin/Casimir contractions.

• Normal Spatial Rotations Give Independent Color on a Four-Dimensional Spatial Base — Full tangent symbols and the actual metric connection distinguish normal color from the base Hodge triplet, with complete depth-two and higher-sector controls.

• A Pure Spatial Spinor Pencil Links the Four-Plane, Normal Frame and Charge Carrier — A derived four-plane combines with the universal six-normal criterion and the charge stabilizer; a positive potential and complete higher-rank controls retain the physical assumptions.

• The Spinor Field Generates the Middle-Hodge Source — Clifford bilinears supply the middle-form carrier and exact pair-area volume; constrained source states keep independent color as a separate physical question.

• Symmetric Hodge Feedback Selects a Color Channel and an Interaction Order — Selfadjoint and skew Hodge parts distinguish instantaneous color-volume contact from velocity response across every maintained depth, including higher controls and alternative kernels.

• Auxiliary gauge carrier — A determinant-trivial auxiliary bundle realizes the charge branching globally but is not the tangent spinor bundle or a derived family of modes.

• Shared Normal Geometry for Charge and Sector Frames — One normal-spinor construction links a conditional six-direction criterion, the charge spectrum, polarization frames, compatible spatial embeddings and curvature-dependent contact, with explicit competing realizations.

• A Three-Channel Hodge Criterion for the Quark Sectors — Middle-Hodge and lowest polarization-Casimir channels select three/four; the interaction and higher-sector force absence remain to derive.

• 20261009-normal-charge-character-and-oriented-volume — Normal Charge, Rotation Alignment and Oriented Volume — normal-charge-character-and-oriented-volume.md — The same exterior carrier links rank-three charge asymmetry to a cubic normal-volume polynomial, and its charge weights to relative rotation alignment. Exact higher-rank and correlated-frame controls delimit the conditional six-normal preference.

• Paired Spatial Modes, Color Neutrality and the Mass Count — Conjugate pairing supplies a trace singlet at six and ten; common controls identify the occupation, electric-character and mass-normalization conditions still needed.