Research status

Particle interaction baseline

Particles interact through extended tails with changes in mutual influence propagating no faster than light over their separation. Keep both full spatial patterns, their multiple orientations, shape, phase and reciprocal response. Within the ten-dimensional matter domain, seven directions are unresolved by a three-dimensional observer; infinitely many additional directions are unbound for every particle. Neither a mirrored three-plus-three position lock nor an independent-offset encounter penalty is assumed. Electromagnetic dimensional reduction remains a derivation target. Conventional antisymmetric amplitudes remain external comparisons.

Research Update: 5 November 2026

The case for particle geometry and sector depths now connects the mass formula to several explicitly combined constructions. The central advance is the reuse of geometry across properties and spatial dynamics, rather than the existence of a field equation in ten dimensions alone.

The compact plane-parity charge prescription is a fitted hypothesis with geometric consequences, not a new independent prediction of the charge table. Known reference hypercharge branching, per-depth seed labels and conserved physical electric sources remain distinct. Positive quartic contact, focusing density dynamics, centered-only localization and the additional-window law have different assumptions; results are not transferred between them.

The spatial premise remains one field on infinitely many ordinary spatial directions plus time, with matter occupying up to ten and every remaining direction unbound for every particle. These constructions add no mediator or compactification and do not replace the electron's external deterministic trajectory with internal field circulation.

Follow count selection, joint property carriers, plane geometry and fitted charge, and localization through the six-depth comparison. The twenty-volume guide retains the broader programme. A complete physical mass operator selecting the observed spectrum and its interactions remains unfinished. The dated entries below record the preceding stages under their own hypotheses.

Research Update: 1 November 2026

The new case for particle geometry and sector depths now brings the mass formula, electron-based scale calculation, spin, color, charge and shared-assumption tests into one readable account with a glossary. The twenty-volume research map gives the broader theory and the distinct candidate actions.

The current physical premise is one field on infinitely many ordinary spatial dimensions plus time, with matter occupying up to ten and every remaining direction unbound for every particle. Particles have different bound dimensional involvement within that space. There are no compactified directions, additional physical manifolds or mediator particles. The electron retains its literal deterministic external trajectory. Uniform extension through a free direction is unchanged by translation along that direction.

The current clock picture has $C=S(n,d)$ portions at one fundamental per pattern kind. Additive event counting gives average rate $C$ times that fundamental even with uneven phase offsets. Electron, muon and tau share a fundamental; their registered mass frequencies differ by their counts. The physical registration law and the selection of sustained counts remain open, rather than being consequences of counting alone.

The joint mass-and-ratio appendix reports conditional coverage of about $2.84\times10^{-20}$ and $7.33\times10^{-18}$ for its two complete mass windows. These are retrospective numerical comparisons under fixed coefficients and observed precision bounds, not probabilities that IDWT is true. The electron is the sole measured anchor; W, down and up are scored targets. Correlated mass ratios and geometric results are not multiplied as independent probabilities.

The compact-shape and inertia account and clock-connectivity proposal state the new equations and their limitations. Count selection, sector-scale origin, bottom, the neutrino absolute scale and a single complete interacting mass operator remain unresolved. The newest proposals are not presented as independently accepted physical results.

The dated entries below preserve the research history and each calculation's own action. Statements of the physical premise in older entries describe that stage; the current premise and interpretation are those above.

Research Update: 17 October 2026

The substantially revised sector case connects the mass comparisons to the latest dimensional criteria, shared-property constructions and complete-inertia calculation. Its technical evidence map retains links to the earlier detailed arguments and their corrections. Results that work at both six and ten are included alongside results that favor particular depths.

The physical picture is one shared higher-dimensional space, with at least ten spatial dimensions. Particles occupy layers of different dimensionality, which can be described as nested submanifolds. The familiar three-dimensional observer view is not the full ambient space. The current working geometry is flat; its total dimension and the confinement mechanism remain open. Older projective and round-sphere results below keep their stated mathematical premises. Neither their curvature nor their finite-volume normalization automatically transfers to the flat-space construction.

Dimensional Criteria and Shared Properties

The minimal real-spinor clock criterion allows exactly depths $2,3,4,5,6,10$ within $1$ through $10$. Its premise is one irreducible real spin module with an invariant first-order kinetic structure. Common-carrier multiplicities restore clocks at other depths, so the result is a conditional discriminator. The finite energetic extensions supply additional model tests, not a completed physical state selector.

The middle-Hodge rank theorem singles out three complete channels at depths three and four over all positive dimensions. The selfadjoint Hodge contact is active at those two among the maintained assignments, with higher controls also active. Physical color, confinement and the absence of other strong responses still require their dynamical bridge.

The normal charge construction gives the relative weights $0,2/3,-1,-1/3$ on a six-real-direction normal carrier and keeps every assigned matter charge available under one occupation rule. It does not select the entire depth assignment. Actual count fields on round spatial spheres also have full observer half-spin transformation at both six and ten. Their minimal balanced carriers and closed-circle fields expose further higher-dimensional capabilities, including depths five, six and ten. Their internal field rotation remains distinct from literal electron motion.

Mixing, Orbits and Static Response

The count-ratio mixing study compares $|V_{us}|^2=1/20$ and $|V_{cb}|^2=3/1771$ with dated empirical benchmarks. It retains the searched candidate set, chance-control limits and failed third-generation and neutrino comparisons. An isotropic nested-band projection identity supplies a conditional mathematical bridge; it does not derive a complete physical mixing matrix.

The flat orbit calculation gives the exact deterministic equation

$$\ddot q=2\beta J\dot q-(\delta^2-\beta^2)q.$$

The higher-dimensional static-response calculation derives $1/(4\pi r)$ by integrating the higher-dimensional Green kernel over unused observer coordinates. A different confined-response operator gives that asymptotic falloff. Neither calculation assumes compactification or mediator particles. A finite physical registration rule, complete source normalization and reciprocal dynamics remain to be supplied.

Complete Reference Inertia in the Shared Space

An explicit local ambient action now retains all fourteen supplied massive-state count clocks in flat ten-dimensional physical space. The complete supporting structure contributes to the stress and boost response. In units with $c=1$,

$$E_0=M_{\rm boost}=A_0H_0, \qquad H_0=\mu_dS(n,d), \qquad E=\gamma E_0,\quad P=\gamma E_0v.$$

The action's finite field carrier, occupation tensors, sector units, indices and normalization are supplied. At each reference branch, the nonnegative quadratic form is the constrained co-rotating form of $E-eH_0Q$, not a claim about the raw energy Hessian at arbitrary charge. Reference characteristics are controlled, but this action does not inherit nonlinear persistence from earlier different actions. General exterior and interface evolution, physical charge, reciprocal pair forces and observed lifetimes require their own checks.

The physical mass operator is therefore still unfinished. The new result establishes a common complete-inertia realization of the supplied frequency family; it does not yet derive the family, the units or the occupied indices. The earlier dated updates below record their own action versions and should be read with those scopes intact.

Research Update: 13 October 2026

The sector-structure case now connects the geometric counts to actual spatial fields, their rotation forms and conditional persistence under explicit effective actions. Measured masses are known inputs to constrained spectral inference; the resulting depths and indices become inputs to these dynamical realization tests. This reverse construction is legitimate, but does not independently derive its constitutive law or turn a fit into an experimental prediction.

The forward count action has two stationary equations: the auxiliary label equals the spatial Dirac shell, and its count-energy derivative fixes the polarization. Their common value gives the simplex frequency. The convex dispersion continuation preserves every eligible count node and open affine source neighborhoods. Under the fully varied positive-wave action, all fourteen maintained assignments retain their exact clocks, nonnegative complete energy and conditional nonlinear persistence of sufficiently regular small spatial disturbances. Its finite constitutive window, ordinary sphere and preferred-time density are explicit premises. Physical localization, complete translational inertia and common normalization remain open.

The same fields have minimal balanced carriers with one commuting spinor column at depths five, six and ten. Their local rotation forms differ by dimension. The complete observer-rotation calculation includes the spatial derivative and connection, giving half-spin transformation under the declared observer subgroup. A balanced classical field's mean rotation charge is distinct from its representation weight. These internal fields do not replace literal deterministic electron motion.

The accompanying constraints are part of the result. The continuous source's full spatial index test establishes growing modes for two assigned full-matrix zonal fields. The curvature continuation keeps every clock fixed while certifying growth over explicit intervals; restricted-pair persistence alone is insufficient. The positive action's prepared pair-contact bound is nonnegative in the tested orthogonal-column preparation, so isolated persistence does not yet supply physical binding. Coherent phase response is treated separately in the actual coherent-field calculation.

The finite-guide shape response derives tension and shorter-range mismatch repulsion from one declared guide action. Its orientation instability and endpoint assumptions remain explicit. Distinct endpoint clocks require the dynamic boundary response and correct stress; rapid direct driving does not inherit the static force. Allowing an actual frame twist to relax keeps its bulk inward force positive and ties the force derivative to longitudinal stiffness. Internal rotation gives positive bulk momentum response, while finite end cores and complete particle inertia remain open. The three/four guide restriction is inherited from the existing Hodge rule.

The deterministic orbit projection also connects cylindrical, annular-breathing and petal appearances of one existing autonomous trajectory. Its full radius stays fixed while the projected radius varies. The maintained example uses four active directions inside its six-dimensional ambient patch; it does not independently select depth six or a detector law.

The release includes the associated research notes, checkers and recorded results, with later corrections linked to their earlier actions. These are conditional mathematical advances. A complete same-field particle and a derived physical color or charge law remain research targets.

Research Update: 11 October 2026

Electron-orbit interpretation: the baseline is literal deterministic motion in six ordinary physical spatial dimensions, as stated in the actual-orbit correction. The observer map uses coordinate shares and trajectory dwell times, not a Born position law. Conventional hybrid orbitals and Born--Oppenheimer energies are external comparisons; they do not close IDWT molecular binding. Classical same-field mode calculations remain separately conditional and do not replace the electron trajectory. The existing autonomous Coulomb-petal solutions are actual-orbit calculations, with their source-selection and multi-electron limitations retained.

The updated case for investigating the sector structure now connects polarization, normal curvature and phase through one explicit geometry. In the selected ten-dimensional spatial parent, the depth-four carrier has six ordinary perpendicular directions. Its rank-three normal geometry supplies singlet/triplet branching and a complete chiral polarization metric, with the exact identity $\|dJ\|_F^2=16ds_{\rm FS}^2$. The geometric color synthesis sets out the derivations and their physical scope.

The canonical normal connection has full $U(3)$ holonomy at the declared three/four carrier, only central $U(1)$ at five/six, and no normal carrier at ten. Depth two can support full $U(4)$, even on a two-dimensional base; excluding a separate strong response there still requires a dynamical selection rule. Its determinant phase line is calculated explicitly, but does not select observed electric charge. Compatible actual embeddings realize the full three-component transport, while a parallel curved comparison has $SO(3)$ transport. This makes the actual connection, rather than a component count alone, part of the test.

The common orientation action combines this metric and curvature with a neutral deformation mode. Its principal speeds are real and at most one throughout a proved sufficient moving-background regime, which includes every static orientation background. An exact smooth finite-energy solution outside that regime loses hyperbolicity. A complete causal model therefore needs control or completion of its large-derivative terms. Constraining the connection to geometry also changes the source equation: the old independent-connection Gauss constraint cannot be transferred to this action.

With an independently varied connection, the mixed quadratic/quartic dielectric instead gives exact commuting flux-guide thresholds: continuous onset with diverging width at depth three, and finite-width binding at four. The full stress and transverse inertia are included. These are controlled conditional results. Non-Abelian comparison fields prevent extending the commuting separation bound to a full confinement claim. The publication includes the exact checkers and frozen reports for these results, together with the growth, source and connection-choice corrections needed to assess them.

Research Update: 9 October 2026

The shared-mathematics constructions now connect actual spatial-band evaluation to the simplex contact eigenvalue, kinetic dressing, geometric transport and overlap defects. The same integration derives a volume-to-color-pair operator, conditional charge fractions and a complete-channel weak-anomaly parity test. Directed angular kernels give exact simplex and cross-polytope energy minima, together with their small-oscillation spectra. One script checks all of these across the maintained depths and higher controls. The physical carrier, interaction coefficients and occupied branches remain explicit assumptions; the results have been integrated into the mass, mixing, coupling, color and bond-geometry documents.

The expanded case for investigating the sector structure brings the mass-frequency comparisons together with spatial spectra, observer spin, handedness, charge geometry and reciprocal contact. In the maintained law the simplex count is exactly the frequency multiplier; the open spectral question is which spatial operator and same-field dynamics produce that spectrum and select its occupied states. The introduction gives an accessible route into these connections.

The quark-depth color criterion proves that complete middle-Hodge channels have rank three only at depths three and four, with an explicit chiral restriction between them. A minimal-spinor rotation-Casimir criterion reaches the same pair under an additional carrier premise; the full paired carrier fails that selector. The common interaction and occupation rule, including the absence of higher-sector strong forces, remain unresolved. The source-pinned checker supplies 36,489 exact comparisons, rather than particle simulations.

The shared normal geometry connects a conditional six-normal frame discriminator to charge weights, complete compatible sector flags, spin sections and geometric contact tensors. Its rotation-alignment and character continuation links that same carrier to oriented volume and higher-rank controls. The charge occupation comparison leaves the maintained matter assignments allowed; it does not establish a four-of-six-sector preference. These normal-gap results are distinct from the tests at the actual quark depths.

The synthesis also explains existing action-level bridges: a normalized local self-current constrains count-scaled standing-wave frequencies and supplies conserved energy and a flux bound; canonical kinetic metrics and spatial marginals give reciprocal cross-depth vertices. The constructions retain their separate action and topology assumptions. Their value is the increasingly precise set of constraints on a common physical realization, not a claim that one completed field law already produces all particle properties.

The finite sawtooth extension now takes the maintained mass-frequency as an upper harmonic endpoint. Its exact fundamental density coefficient is $1-1/K$, and its reciprocal temporal overlap supplies phase weights for the spatial contact, color and angular operators. Positive weights preserve angular minima and color eigenspaces; specified relative phases split the angular response spectrum. One common canonical receiver gives electron, muon and tau direct dynamic responses agreeing within $0.009\%$ under the maintained scales. Direct depth-five density response remains appreciable; a velocity-source action instead produces small depth-five response under a separately labelled splitting calibration. The latter is an action-dependent spectral result, not derived neutrality or a weak rate. Harmonic occupation, spatial readout, receiver scale and moving source orientation remain physical questions.

The expanded sector case now states these connections quantitatively, including the color-rank charge fractions and higher-depth parity controls. The spatial integration results contain 2,529 checks; the sawtooth results contain 2,023. Their source-pinned calculations and five supporting derivations are available in the reading edition. Verification of these equations is distinct from observational validation of their physical premises.

Research Update: 4 October 2026

The complete sector-assignment test now covers all 120 exclusive block permutations and every positive integer index under the unchanged numerical scales, fixed electron reference and shared charged-lepton unit. Only the maintained arrangement reaches every existing individual mass accuracy. The closest admissible alternative has 8.88588 times the RMS logarithmic error. The proof and downloadable code retain the unresolved bottom discrepancy, input scale assumptions and separate finite reference-null scope.

The rewritten paper, The Case for Investigating IDWT's Sector Structure, presents a reproducible case for independent investigation: rigid combinatorial mass relationships followed by the reuse of spatial geometry across orbit, spin, helicity and charge constructions. It explains how shared premises can constrain several properties without separate adjustments, while stating the additional choices. The depth-six and depth-ten count polynomials are an inverse construction on specified spatial spectra; the available half-spin states are not selected over the higher-spin states by the quadratic law. Supporting calculations, scale provenance and five concrete research questions make the proposal open to verification and challenge.

Recent spatial work has also produced full-field solutions in a common saturated-current action at depths two, four, six and ten. Complete conserved stress gives the isolated boost inertia $E/c^2$. These are particle-construction results within that action, rather than independent evidence selecting those depths. The tested stability windows do not establish unrestricted stability, and no observed particle or complete mass-frequency selection law has been identified. Independent-clock pair preparations retain the two isolated resonance frequencies; short projected evolution does not yet establish a persistent interaction or a measured coupling.

The dated entries below describe their own actions and assumptions. Their favorable results require a checked common-action construction before they can be combined. The physical mass operator, occupation rule and common polarization-sensitive interaction remain open.

Research Update: 28 September 2026

The latest work develops localized wave support, complete inertia and reciprocal contact alongside the earlier spatial simplex operator. These are conditional candidate actions. A physical mass operator, a selected particle spectrum and a common cross-depth coupling law remain open.

1. Counting, frequency selection and internal-motion inertia derives Pascal phase weights, supported profile identities and the complete boost response. Conditional phase-action quantization supplies discrete roots, with restrictions from spatial depth and stability.
2. A scale-covariant action for count scaling and shared contact gives exact complete-energy scaling under one coefficient rule. A projected source supplies an analytic exterior stability bound; a shared density square supplies both self-contact and reciprocal pair contact.
3. Reciprocal contact with deforming spatial patterns evolves the two-grade field equations at depths three and five. Contact changes through deformation while momentum is exchanged reciprocally, with mesh, timestep and domain controls.

The count family in those graph calculations is $3,10,35,126,\ldots$; it is not the full maintained particle assignment spectrum. The graph's physical spatial origin, photon polarization, cross-depth contact and absolute frequency selection remain unresolved. Its amended action has promising isolated-state controls, but finite-volume occupied spectra are not a nonlinear stability theorem.

The recent standing spatial support, current support ledger and canonical contact bounds qualify earlier surface and current constructions. Results below retain their stated action and date; a later repair does not retroactively validate a different earlier action.

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. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Spatial mass operators, polarized reciprocal coupling and the sustaining-source and inertia requirements, with correction routes through the earlier localized actions.

Spatial mass operator and source: 14 September 2026

The current spatial investigation has a complete finite normal Dirac kernel and a positive gap to every other normal mode. Under its stated winding and surface assumptions, the projected frequency approaches the assigned simplex product with one coefficient per depth. The carrier calculation retains the full normal spectrum; its coefficient texture and physical source remain inverse inputs.

The complete polarized pair keeps mixed currents, actual surface overlaps and reciprocal separation forces. A defined common probe gives a muon/electron response ratio about 1.11 while their frequency ratio is about 207. This is a conditional example of different frequencies with comparable responses. It is not an observed coupling, a photon identification or a neutrino suppression mechanism.

The normal stress audit gives, at unit canonical norm,

$$M^{\mathrm{wave}}_{ij}=\omega\delta_{ij}+T_{ij}.$$

One directional response is negative in the tested simplex regime. Independent spatial-force integrals and the exact contracted boost confirm the stress contribution. This is the partial wave response in a prescribed background. The supporting field must contribute its own energy, momentum and stress before this is a complete particle calculation. Mass remains the selected frequency in equivalent units.

The shape-source constraints show that a common restoring energy and scalar norm adjustment alone cannot support both assigned branches at the same reference texture in the tested separable completion. A joint same-field source changes those equations and remains the next physical requirement. The scale audit also keeps the normal hopping and ordinary cavity radius in the frequency conversion.

The earlier localized-current construction below has a different conditional inertia result. Its retained spatial frame now repairs a particular polarized principal defect, but this extension has not been combined with the new normal carrier into one physical mass operator.

Localized mass and coupling: 13 September 2026

The earlier reconstructed current action has actual localized spatial solutions under its stated inverse premises. Its frequency and inertia calculation gives

$$\omega_N=\omega_{0,d}\sum_{k=0}^{N}\binom{k+d-1}{d-1} =\omega_{0,d}\binom{N+d}{d},\qquad E_0=\hbar\omega_NQ,\qquad M_{ab}=\frac{\hbar\omega_NQ}{c^2}\delta_{ab}.$$

Here $N$ is the occupied coefficient grade and $Q$ the canonical field norm. The depth-six electron and muon retain $N=12$ and $N=34$. Their continuously oscillating solutions share one spatial core and, under the common $Q=1$ condition, have the corresponding mass-frequency inertia. Field magnitude enters the spatial fixed-point equation; it is not used as a separate mass variable.

These are exact results of an inverse action. The shell weights, its physical spatial mode carrier, and the common norm have not been selected by the master field. The common-gap core calculation also gives an explicit radius law $L_{\mathrm{rms},d}=\gamma_d c/m_c$, with a dimension-dependent factor $\gamma_d$ determined by that binding law. Equating its core frequency $m_c$ with $\omega_{0,d}$ remains an additional inverse identification.

Different masses with a common initial coupling response

The same action gives identical lower-grade probe energies and complete probe sources for different larger target grades at the same depth and common profile. The calculation retains both polarizations and reciprocal initial forces. Two independently solved cores realize that identity, while the target's full deformation can still depend on its grade. Their binding currents can cancel even when each isolated core is regular; a smooth binding comparison removes this specific fractional-current singularity without changing the frequency hierarchy.

These are initial same-depth pair results. A complete encounter, cross-depth vertices and the observed photon and neutrino coupling pattern have not yet been derived.

What the stability calculations establish

A trace contribution to the mass current preserves the same core, frequency and inertia:

$$F_q(R)^2=qr\operatorname{Tr}(R^2) +(1-q)[\operatorname{Tr}R]^2,\qquad 0<q\leq1.$$

Here $r$ is a component rank, not a spatial dimension. The original norm has $q=1$. It has a nilpotent high-momentum compression with secular amplification despite real limiting frequencies. For $0<q<1$ and a nonzero gap, the complete constant far-field linearization instead admits a positive conserved perturbation norm and a bound uniform in time and momentum. Equal trace and exchange contractions give the explicit comparison $q=1/(r+1)$, hence $1/9$ at depth six; this contraction is an inverse choice rather than a uniquely derived interaction.

The higher angular response classifies the even-depth matrix angular blocks and checks 200 finite electron and muon spectra across two radial refinements and both current norms. No exponentially growing mode is resolved within those ranges. This is not a theorem for all radial wavelengths, all angular labels, or nonlinear disturbances of the spatially varying core.

Two limitations remain explicit. A regular fixed-unit-norm family makes the physical Hamiltonian unbounded below, ruling out a global energy-minimum proof without itself establishing instability. Separately, the field-carried spatial frame has defective principal blocks for complex polarized rays. Changing the mass potential does not repair that kinetic issue.

How this connects to the preceding calculations

The moving-cavity inertia control and common-size balance isolate the kinetic and sustaining-stress requirements. The separate surface-bending construction has the exact simplex frequency polynomial under a declared geometric time law; its canonical field projection exposes the required boundary contribution. These are distinct conditional routes, not interchangeable derivations of one selected master action.

The localized current calculation now supplies a concrete same-field frequency, sustaining profile and canonical inertia example. Its physical selection, regular evolution at entirely zero current and complete particle stability remain open. The sections below preserve earlier calculations and their individual scopes; read their mass and coupling claims with the later results and corrections above.

Working backward from the mass operator

A direct curved-space causality test excludes finite propagation for the fixed polynomial inverse actions when applied to every spatial mode. It retains the full field and both frequency signs. The inverse frequency match therefore needs a controlled effective interpretation or different causal nonlinear dynamics that generate the selected particle states.

The inverse local scalar-law test keeps the original Dirac kinetics. It reconstructs a real interaction only when the complete spatial source has scalar form and equal-current points require the same law. Unchanged paired Jacobi profiles fail the assigned simplex frequency comparison; independently differentiated nonlinear dressed controls recover a common local law and its coupling derivative. The desired self-consistent particle fields remain unselected.

The dressed excited-branch calculation now connects the scalar response to spatial stability at the unchanged assigned indices. A derived near-resonance survives full nonlinear profile readjustment and converges under independent spatial differentiation and larger cutoffs. The tested local law still supplies neither the required mass ratio nor an observed decay rate; its value and coupling derivative must be constrained together in the inverse construction.

Localized scalar interaction: current result

The local scalar calculation now includes spatially localized encounters. The exact axial family retains full complex polarization vectors and transverse localization; an independent Cartesian encounter calculation checks its three-dimensional response and extends the resonant quartet beyond four modes. Opposite incoming phases reverse approximately $1.77\%$ chiral-norm transfer in the stated benchmark. This is classical field norm, with conserved total norm.

In five-dimensional textured preparations, the same local dynamics changes a prescribed spatial compatibility measure in either direction. Both signs survive rescaling each outgoing pattern to unit norm. The time-ordered source calculation derives the leading quadratic change and computes it through independent full-field coupling variations. The response follows from the equation; a tendency toward alignment is not imposed.

Frequency accounting distinguishes an absent resonant negative-output Born channel for positive incoming scattering data from a permitted finite-time response. The spectral signs remain without a particle-antiparticle assignment.

The field preparations, flat charts and scalar coefficient are specified controls. Stable occupied particle patterns, physical normalization across distinct depths, and observed electric and weak coupling rates remain unidentified. These results do not select the global geometry of ordinary three-dimensional space.

Reciprocal evolution of spatial profiles

Spatially invariant scalar-null fields now have a complete minimal-module classification: constant linear zero-source classes exist at every odd spatial depth, not uniquely five. The full null-plane response has an explicit density-dependent stability threshold, including all spatial directions and complex polarizations. Moving nonnull planes connect to that threshold and give an exact covariant response from the same local term. These unbound controls select neither physical particles nor observed charges; their static directional limits must be retained before identifying a scalar coefficient with an interaction strength.

Complete scalar response now fixes the pointwise source rank and its exact weakest response. A vanishing background scalar retains a strong responsive direction; general spatial kinetics can activate the scalar, while an actual aligned eigenplane keeps its zero. These results distinguish full-source cancellation, a projected channel zero and a dynamically protected spatial preparation.

The local polarized pulse equations now supply an autonomous causal comparison using the written scalar cross term of the same field. Exact mismatches depend on the full polarizations and phases, and the nonlinear collision can increase or decrease normalized outgoing pattern compatibility. Independent full spatial evolution, refined grids and the conserved combined polarization matrix check both responses. The fields remain flat, transversely unbound controls; no physical particle or observed channel is identified by those results.

A common spatial kernel and its reciprocal projection

A common contact kernel across bands

Band consistency makes the operator bridge testable: standard coherent contacts do not obey fixed bare compression under the stated cumulative identification, while eliminating omitted modes requires an explicit Schur-compatible family. The finite-resolution spectrum quantifies how a specified smooth symbol approaches point contact and how fixed-width and shrinking-width limits differ.

Spatial counting and current projection

Oscillator source marginals turn the weighted Hockey-Stick identity into an exact reciprocal spatial map. Constructing the full bilinear current first instead produces a normal mode contraction, so the source count is not a universal particle coupling. Spatial point contact has a position-dependent trace density with a different central count from projective ray contact. Both use fixed, explicitly stated spatial measures.

Spatial scalar contact

The inverse simplex construction classifies a local polynomial family with the exact simplex values as full paired frequencies. It gives nine choices up to simultaneous sign at depth six and eighty-one at depth ten. These reconstructed actions change both kinetic and scalar terms; they preserve the other spinor families and use the complete canonical norm. Their particle selection, nonlinear interaction and causal interpretation remain additional tests.

Its complete zero-form spectrum and local Dirac-pair extension are explicit. The relative polynomial coefficients follow from geometry; selecting this derivative energy in the master action, its connection, occupied field and physical normalization remains unresolved.

The full spinor bundle retains two additional positive spectral families and its zero modes. The required connection and sixth-order polynomial are explicit inputs; their master-action origin, occupied states and physical frequency relation remain open.

Direct inheritance of one particle's mass family from another is an additional ansatz in that test. The sectors are nested; individual particles need not be contained in one another or be restrictions of one another's field solutions.

The full normal Dirac extension obtains the absolute tangential Dirac operator from a specified spatial half-cylinder. Canonical normalization cancels its factor in a constant bulk scalar mass. Local quartic compression gives an inverse-width coefficient but also a nonzero complementary normal source. The complete simplex operator and physical normal binding remain unresolved. The full-line local equation has no nonzero finite-norm stationary normal field in its fixed spatial Killing channel, even with all four complex normal amplitudes and the nonlinear profile retained.

An exact local scalar-gap branch now solves the written interaction on the ordinary spinor bundle of a specified round six-dimensional geometry. Opposite Killing modes give a nonzero constant current, and a full harmonic decomposition proves complete spatial spectral stability. The coefficient depends on norm, volume and radius; it has no derived simplex enhancement. Adding the conformal derivative-current term destabilizes high harmonics even on this nonzero branch. Nonlinear orbital stability remains open.

The conformal scalar-current response is now solved on an exact zero-current Killing background in six dimensions. Its full complex response includes the spatial derivative partner. The simplex value enters a collective-frequency polynomial, and any nonzero coefficient of the added derivative-current term gives unstable high harmonics on this branch. This tests the specified action and background.

An exact Killing-spinor transfer now carries the scalar simplex operator to the complete spinor bundle of the specified round geometry. The local self-adjoint operator has adjacent simplex values on opposite Dirac signs. Its direct scalar mass insertion fails Hermiticity, while the real action takes their average. This narrows the mathematical bridge without deriving a master mass term.

A conformal scalar differential operator on a specified round even-dimensional spatial sphere has the exact normalized simplex spectrum. Conformal covariance fixes the recurrence, and independent spatial differentiation verifies it. Its master origin and physical mass interpretation remain open.

The direct local-current derivative calculation derives the written scalar source on the finite jets of one physical field. Its full variation repeats the local response blocks without multiplying their eigenvalues by the coefficient count. Curved connection splittings preserve this result. A finite derivative identification therefore needs an actual physical projection and its kinetic metric before it can supply the proposed mass contact.

Their nonconstant densities have isotropic first moments, and an exact source closure proves the absence of exponential modes in the complete spatial linearization, including angular and form polarizations. This establishes a candidate spatial branch, while the master kernel, occupied field, nonlinear stability and physical mass identification remain open.

Under spatial covariance, that condition makes the whole holomorphic band degenerate. The mean kernel has exact nonlinear orbital stability; the previously derived static response approaches its constant-source limit on fixed bands with a quantified remaining correction. Shape selection and a full nonlinear, canonically normalized master reduction remain separate requirements.

The scalar gap calculation derives a self-consistent branch and an exact scalar response frequency for a specified global kernel, including the full complex linearization. Its count enhancement requires an extensive canonical field norm. At fixed total norm, admitting higher kinetic shells lowers or removes the positive gap. The kernel, occupation and physical rest-mass identification remain to be derived; this does not reassign particle dimensionalities.

The nonlinear mass reduction derives a stationary effective action retaining complementary spatial Dirac modes and their induced normalization. Its complete complex variation gives different cubic corrections to normalized frequency and total energy. With a positive scalar potential, a nonzero complementary source makes that correction strictly nonzero; a source preserving the selected zero-mode line gives an exact linear branch. This is a precise test for a proposed common contact scale, not a derived master mass operator or a stability theorem for the occupied particles.

The local scalar normalization calculation separates the nonlinear coefficient, stationary frequency and total energy. An exact massive one-dimensional control gives a nonlinear fixed-norm frequency. In the flat massless reduction, an analytic radial tail argument excludes nonzero finite-norm stationary waves of the standard regular radial form.

Variational coherent contact

The full two-family action drives generic coherent preparations into shape modes, with changing contact fixed by conserved energy and common rotation moments. Its aligned coherent set has an explicit ambient Hessian and finite-mode nonlinear orbital stability. These are positive-density component-action results; the physical spatial Dirac reduction and observed particle coupling are still open.

The graph pairing variation gives a well-defined candidate action retaining the simplex coefficient. Its complete self-variation cancels the apparent ray stiffness; it does not select a coherent spatial pattern. A two-ray first-order reduction has reciprocal precession and a conserved relative overlap.

Coherent quartic variation derives the complete smooth quartic coefficient dynamics. Coherent tensors are exactly its sharp extrema; the full mode Hessian and conserved energy give finite-mode orbital stability. The frequency multiplier is $D_N^2/D_{2N}$, so coherent-state selection alone does not produce the point-contact mass law. Physical spatial kinetics, occupied particle modes and the graph-contact action still require derivation.

Spatial current and propagation benchmarks

The complete translation response fixes inertia and normal stiffness separately. A bound-depth-two scalar Dirac displacement has two polarizations but speed one half on the zero-bare-mass branch. The conditional spatial electric current, forward polarized scattering and canonical spin response now share one magnetic moment and satisfy the corresponding low-frequency coefficient relations. The six-dimensional scalar ground profile's $g_{\rm mag}=2.07691916$ differs from the measured electron value. The full charge-continuity calculation also distinguishes a rotation-neutral preparation from a conserved local electric charge.

A specified stationary component action supplies exact finite null pulses, an interaction from full spatial deformation and its retarded frequency response. The same field's continuum supplies the outgoing radiation; its coefficient and low-frequency logarithm are derived. These are conditional mathematical results. The action's identification with the master field, stable occupied particle modes, electric charges and neutrino weak interaction remain open.

The local pulse collision law fixes reciprocal delays from the other pulse's energy. Its spatial radiation calculation retains both complex polarizations and derives a channel-specific circular cancellation and a positive low-frequency emitted-energy coefficient.

Complete spatial polarization

The full polarization descriptors retain all constant amplitudes and phases in a specified spatial multiplet. Within one six-dimensional chiral family the circulation two-form suffices; at ten dimensions it can miss four-form polarization information. Complete equal-current fields demonstrate the consequence in a reciprocal six-inside-ten rotation-moment model: identical densities and current tensors can accompany different full polarization responses to the same lower partner.

Bound-coordinate textures add spatial phase and orientation variation, together with its exact kinetic remainder and complementary equation. These results expand the individual-particle description. The selected multiplets and declared rotation interaction are conditional; stable occupied particle modes, the physical interaction operator and measured rates remain to be derived.

Generalized coupling across particle sectors

The generalized calculation retains both objects' spatial polarization and allows the higher-dimensional geometry to determine the lower response. Complete nonlocal variations give normalized quadrupole vertices with exact relative-orientation zeros. Conservative scattering and direct wavepacket propagation turn those vertices into reflected and transmitted flux, retaining the distinction between a small overlap and a narrow resonance.

A six-dimensional complex structure compresses to two equal rotation planes in five dimensions. Their spin action has a neutral doublet, while the three-plane six-dimensional action has no neutral weight. Varying the physical null axis induces nonzero traceless geometric curvature on this doublet. This supplies a concrete example of neutrality under one spatial operation with a surviving geometric response under another.

Physical identification remains open: the full higher source has a mixed term acting on the doublet, scalar self-binding does not select its occupied polarizations, and the photon current must be derived. The actual nested geometry and its motion must enter that projection. The simple photon Pauli probe, including all target polarization transitions, has a positive total response bound at both five and six dimensions. These conditional results therefore provide a specific research route without assigning neutrino charge or predicting its interaction rate.

The current generalized sector coupling calculation derives reciprocal marginal maps, canonical vertices, and the static Gram response correction from eliminating stable modes of the same field for arbitrary nested sectors. IDWT has no mediator particles. The calculation identifies proportional response profiles as the condition for rank one and separates charge universality from profile-dependent coupling strengths. Neutrinos provide a channel-specific control: electric neutrality with surviving weak interactions. These are verified conditional operator results; physical source maps, field dynamics, and particle coupling constants remain to be derived. The earlier electron investigations below remain separate applications within the wider theory.

The full spatial Dirac extension now solves self-consistent scalar bound profiles and a nested pair in which both partners deform under one source. Complete angular response retains both spatial polarizations and both ordinary and conjugate fluctuations. A statically inaccessible higher source pattern can acquire a lower-sector response linear or quadratic in small frequency. An orthogonal higher circulation gives an exact three-map classification of persistent barriers; the actual incident mode still has to be derived. Consistent reconstruction also restores source terms hidden by a bare projection, and the centered source recovers the translation mode and its inertia.

Readjusting both profiles under scalar amplitude changes gives a symmetric coupling matrix with a common scaling identity across several nested depths. Normal propagation and uniform boosts follow from the full Dirac partner. These are conditional calculations in the specified scalar moment model. Several bound radial branches are unstable, their rest polarizations remain degenerate, and reflection symmetries can preserve that degeneracy even with unequal scalar confinement axes. Polarization waves can change the spatial current while leaving the scalar source fixed. The envelope reduction also has an explicit omitted spatial residual for varying density. Its reduced spectral criterion is separate from full particle stability. The positive scalar source has no localized zero bound frequency, leaving the protected photon mode as a separate dynamical requirement. The linked generalized coupling note provides the individual derivations and executable checks.

The source-frequency calculation now separates the bound carrier from the actual spatial clock sampled in an interaction. Exact two-frequency fields produce currents at twice the bound frequency with fixed canonical norm. Their slower members grow wider and retain finite current amplitudes, so slow frequency alone does not establish weak coupling. The incoming-mode calculation also separates a free spatial shape from an applied force: low-frequency barriers can depend on cancellation between individually visible helicities. The occupied five-dimensional mode and physical photon source remain the requirements for applying those results to neutrinos.

The complete periodic pair response now retains both clocks, both scalar centers and reciprocal source feedback. The polarization selection calculation gives exact cancelling subspaces for a specified source sideband, including its incoming spatial pattern and common-rotation covariance. A checked lower traceless pattern has a complete finite spectral cancellation at rest and an amplitude linear in small normal momentum. This is a channel-specific geometric barrier; its physical photon projection and particle preparation remain to be identified.

The readjusted two-frequency pair can retain a finite lower frequency when the lower mean scalar source tends to zero, because the containing field still confines it. The scalar center is singular at exactly zero signed normalization. The slow collective shape poles retain their canonical residue signs and the separate instability of the full field. Neither a small mean source nor a small pole residue supplies a stable neutrino identification by itself.

Scope

The research notes connect derivations, conditional models, corrections and numerical evidence. Linked modules and historical excerpts provide mathematical context and provenance; their claims retain the assumptions and limitations identified by subsequent audits.

Reading the record

The long interaction notebook is now 99 section-sized notes. Original section numbers and mathematical text are preserved. A section title can report an intermediate claim later narrowed or corrected; read the later audit before treating a construction as established. Unqualified references to a numbered Section within an interaction note refer to the original interaction sequence.

Current boundaries

September 8 spatial-meaning correction: all sectors are nested submanifolds of the one infinite-dimensional physical manifold. Three dimensions are shared with us; this makes none of the six different in kind. The electron's six are also six of the tau's ten, making the electron lower-dimensional relative to the tau without dividing the tau into different kinds of space. Earlier wording such as “intrinsic sector” or “internal projective orbit” must not be read as an alternative ontology.

The attribution of three-dimensional experience to nuclear 3+3 locking is withdrawn. The general premise concerns the three-dimensional nuclear sector. A 3+3 grouping in a particular electron calculation is relative to the chosen observer; it is not an inherent split in the electron. Existing frame-lock calculations remain conditional models, with their physical application and dynamics to be justified.

The geometry premise audit now separates spatial submanifolds from coordinate and field representations. It also records why the old CP2/S5 candidate pair cannot supply the required smooth nesting. Candidate geometries and reduced models must be corrected to meet the fixed nesting premise.

Frozen response-kernel test: Frozen magnetic texture test allows spatial polarization variations while retaining the existing kernel and adding its stated positive bare gradient cost. An analytic supplement proves positive second variation for all nonconstant polarization textures at $\lambda<1/48$, modulo global ray rotations; the illustrative weak benchmark lies inside that sufficient regime. Nevertheless, 23 of 50 predeclared cases have negative curvature with direct nonlinear energy-lowering witnesses at stronger coupling. Localized polarization stationarity then compares the same preparation in five and six dimensions: five has a nonzero spatial first variation; six is stationary. Neither result rules out textured five-dimensional equilibria or proves full six-dimensional field, orbital, or molecular stability.

Two-path follow-up, September 7: Accurate cylindrical correction repairs nearby fixed-Casimir states and continues them to $h=0.100$, with fresh cross-method closure checks. Its physical application requires independent justification of the spinor, operator class, and target response.

September 7 independent audit: the saved nonzero cylindrical continuation candidates fail their stated closure tolerance under integration refinement; their earlier numerical certification is withdrawn. The fixed-Casimir chart and tested conservation identities remain valid. Projection does not turn a point into a cloud, and a frozen-route ligand minimum is not full molecular stability. Read the Independent correctness audit before using earlier success statements. It records corrected code, explicit witnesses, surviving results, and the limits of the audit itself.

Completed-coupling correction: solving the relative-equilibrium equations, rather than reusing the linear model's initial data, gives exact autonomous solutions of the quadratically completed Hamiltonian. The one-electron L=2 and L=4 branches produce four- and eight-petal linear projections and have no resolved exponential mode; ten-turn finite perturbations remain radially bounded. L=1 and all tested antipodal two-electron branches are unstable. The source magnitude and integer ratio remain prepared conserved data. See Completed Coulomb petals.

Latest mechanical reconnection: Coulomb angular response adds six-coordinate Kepler motion and a new classical angular-momentum source. The autonomous linear-coupling Hamiltonian has exact signed-rose projections, including even-index four/eight-petal planar images and a repelling two-electron family. The pair has a checked exponential instability; the prepared integer ratio is not dynamically selected; and the sizable quadratic magnetic completion changes the trajectories.

Latest response-extension result: the autonomous full 3+3+9 response model is implemented alongside the mixed-channel baseline. Equal gaps and couplings give a stationary relaxed family; physical-source norm bounds exclude the prescribed petal lifts in fourteen tested one- and two-electron cases. Adding within-plane channels therefore does not, by itself, close the petal dynamics. The original two-electron shared-source precession remains established. See Full response extension for the action, perturbations, inverse witnesses, and exact scope.

Physical scope: the working mass-formula electron assignment remains depth six. Orbit existence, stability and dimensional necessity are distinct mathematical questions. Counterfactual lower-dimensional models test which constructions require six dimensions; they do not change the working particle assignments. Named candidate spaces do not establish the physical geometry.

They already occur in V1; V2's inverse reconstruction is conditional on amplitude, normalization, symmetry, and phase assumptions not derived from the master interaction. The recent orbit results remain valid within their chosen models, but neither a carrier-dimension obstruction nor an S5 reproduction establishes physical particle depth. See the sector-geometry audit linked below.

• The positive rank-one scalar moment functional with a fixed nested 3-in-6 embedding now has an exact ground-state covariance solution. Its mixed block vanishes and its full covariance Hessian is positive. This excludes spontaneous oriented locking in that restricted ground-state model, not in the full V2 dynamics. A supplied mixed tensor has a calculable linear susceptibility.
• Point-contact and smooth-density operators differ. Their identification with the actual nonlinear dynamics remains an assumption to derive.
• Fixed reference-count normalization is compatible with band blindness, but the action must still select that normalization.
• A total-action constraint alone does not yield a 4/3 frequency multiplier or a baryon pole.
• CKM needs derived kinetic metrics, mass bases, and a charged-current vertex. A polar factor is not physical normalization.
• The generic free-pair geometric split is 2+4. Simultaneous-commutant restrictions are not a classification of all covariant operators.
• A rank-two orbital mixed tensor can select a unique SO(3) Procrustes frame. A full-rank source is not necessary; source generation remains open.
• A fixed, already selected oriented lock can constrain the orbital charge q. This does not prove that the full coupled Hamiltonian selects that lock or its frequency.
• The ordinary spatial Levi-Civita mixed contraction gives mean curvature, not the entire nine-component singlet-lock operator.
• A factorized cross-depth contraction with a complete unobserved doublet gives a scalar shift. A suitable nonfactorizing transition and its orientation source are still required.
• The CP5 obstruction is stronger than its nontrivial determinant: the complex tangent bundle admits no global rank-three plus rank-two split. A full-rank linear bridge is excluded; a selected line, nonlinear map, or field-induced base potential is still open.

Geometry selection at fixed electron depth six

The CP3-selection conclusions below are established mathematical results for their explicitly stated effective interactions. The master-connection work traces their inputs; it does not repeatedly reopen those conclusions.

Two forward effective-model calculations now let an unconstrained skew bivector vary its three rotation strengths before minimization:

• A filled bounded-response band plus positive quadratic cost selects three equal nonzero strengths in its condensed regime. The source occupation is essential: one occupied level prefers one plane. The filled real band has zero determinant Berry curvature and does not supply the earlier first-order electron dynamics.
• A normalized chiral Spin(6) polarization with a linear bivector coupling gives an exact completed-square minimum B=(g/(2mu))J(z).

The latter is the stronger current route. It still assumes the relevant pure chiral polarization, bivector coupling and cost, and amplitude kinetics. Fibre polarization is not automatically the spatial base. Neither model reopens the mass-formula depth assignment.

Master connection: explicit Dirac projection

The massive six-spatial-dimensional Dirac symbol now supplies an explicit four-component positive-energy polarization. Its scalar current projects to (m/E) times the density for fixed momentum, and its fixed-frame kinetic term supplies the projective Berry term. The nonforward current contains a calculated bivector term proportional to the incoming/outgoing momentum wedge. A scalar mean field therefore supplies an orientation operator at gradient order; the leading local source is a simple bivector, not yet the complete equal-strength self-response or nuclear orbit solution.

A separate normal-ordered fermionic Gaussian reduction matches the chiral auxiliary coefficient through g²/mu=kappa_eff/2, retaining the positive direct-density term. This is not the unmodified classical scalar equation: an exact mixed one-particle example has zero contact energy where the Gaussian closure gives a positive result. Field statistics, self-contact prescription, and closure must be specified before promoting that exchange route to a physical master reduction. Both results are recorded in the master-current and scalar-exchange notes linked below.

Magnetic implementation of the collective CP3 family

The minimally gauged Dirac–Maxwell candidate now supplies a Pauli coefficient g=e/(2m) and positive field cost Z_B||F||²/2. Enforcing the magnetic Bianchi constraint gives the nonlocal response F_*(k)=(g/Z_B) P_closed(k) M(k), not an unrestricted matrix proportional to M. The closed-two-form projector has five field amplitudes per nonzero wavevector and isotropic average I/3; neither count changes the six spatial dimensions.

This gives a concrete magnetic-field implementation at that level of reduction. Spatial textures, profile stability and orbital/electric terms remain the next spatial task. The magnetic-bivector bridge note records the full kernel and constraints.

Actual-orbit target and correction

The research target is the actual projected electron orbit in the 6D demo, not a conventional stationary orbital. The demo already constructs the ring/lobed trajectories and their observer-broadened time measures. The complete trajectories contain information beyond their second moments. The original proposal is Coulomb binding plus higher-dimensional precession. Revision ce71f75 explicitly implements assigned precession and frequency locking; 1f8a54a replaces that recipe with Hamiltonian evolution, and c6d6bcf–4701c3a construct its orbital frequency ratio within one representation action. The remaining task is to connect that existing mechanism to the physical electron–nucleus interaction, including its selection, physical clock/rate, and observer response—not to invent a mechanism for a previously unexplained petal shape. See the correction linked below for provenance and the distinction between successive models.

Kernel-source results

The four associated numerical diagnostics pass under both normal and optimized Python execution. An in-memory mutation audit reversed the electron response, doubled a collective rate, reversed the geometric lift, and reversed the finite-source electron Hamiltonian. Each incorrect variant was rejected in both normal and optimized execution by an explicit runtime condition. These checks validate the stated model calculations, not their physical identification.

• A positive-cost (3,3) cross-plane response generates the inverse kernel through static elimination: kappa=chi g², with no sign fitted to the precession. The source channel remains a new effective-model premise.
• The reciprocal dynamical tensor model has exact collective rotating solutions. Their common rate obeys a cubic. One low-rate example has no exponential linear growth by an exact full-tangent polynomial test, but retains neutral drift; a counter-rotating branch is unstable.
• The geometric Pauli-tensor map gives the explicit response operator and fixes I_e=hbar/2 when the projective action is normalized with hbar. The readout and preparation are chosen, and omitted coherent modes remain present. This is a coordinate-to-coefficient bridge, not a derivation of the apparatus coupling or of G from the response model.
• A finite singlet-plus-nine-excitation contact model derives a positive, saturating orientation kernel, susceptibility, and response frequency from an assumed gap and coupling. Its full reciprocal equations have a checked common rotating solution with rate .40069857 in a stated dimensionless example, compared with .5 in the harmonic approximation.

The instantaneous induced source can have a 1:1:3 spectral ratio at s=8/9, but freezing it gives the wrong trajectory. This is recorded as a counterexample, not advertised as closure of the petal generator. No microscopic nuclear current has yet been shown to supply the assumed (3,3) contact channel, its gap, or coupling. These remain the main physical source questions. The full spatial binding and observer/current maps are also not supplied by these finite effective models.

Dimensional discrimination: 2026-09-06

Correction after the depth-five reminder: V2's Q4 is CP2 and Q5 is S5. The earlier SO(4)/U(2) test used CP1, not Q4. Its carrier-family theorem below is not a particle-depth exclusion. An exact three-amplitude restriction of the current degenerate generator reproduces the prepared demo flow on both actual Q4 and Q5, with the original readout composed with a fixed linear embedding. A proposed source perturbation can leave that embedded copy; deriving its physical control and measured response is the next discrimination task. The full reciprocal source has not been reduced to Q5.

The follow-up lower-dimensional test gives a qualified positive result: six is the minimum vector-carrier dimension in the orthogonal-complex family for nontrivial static precession induced by an otherwise isotropic nuclear three-plane. In four dimensions the orientation invariant is fixed, the plane stabilizer acts transitively, and both the harmonic and finite contact-source equilibrium energies are constant.

This is not an "only 6D electrons" result. Retaining reciprocal source dynamics produces explicit four-dimensional collective rotations, checked by direct evolution and energy/rotation-charge conservation. Unlike the six-dimensional low-rate branch, those nonzero harmonic branches track the response resonance and have no regular bounded-response static limit. That contrast is a conditional discrimination target, not yet an atomic measurement proposal with a derived current, clock, or detector coupling.

The chosen nonlinear readout can also map a circle in the lower-dimensional sector to the displayed petal formula. This does not make that circle a petalled spatial orbit; it shows that an unrestricted readout cannot by itself identify particle depth. See the dimensional-precession note linked below for the exact assumptions, counterexample, and diagnostic.

Auxiliary field-model results, not an orbit derivation

The scalar ground-state no-lock result now has a precise excited-state complement. Pure circulating oscillator states carry mixed angular momentum even when their position moments are isotropic. In the isolated scalar model, they attain an exact fixed-circulation energy bound and have no exponential linear instability. The bound controls covariance and opposite-circulation occupation, not the distance to a unique wavefunction.

More importantly, a coupled 3-in-6 stationary family now includes the nuclear response. Unequal observed/unobserved widths compensate the nuclear contribution and restore the oscillator degeneracy supporting circulation. The complete 99-dimensional moment linearization has an exactly oscillatory nonzero spectrum along the equal-coupling family for every integer degree N >= 2. This is a stationary solution of the imported scalar model. No map from it to the demo's actual orbit, generator, or clock has been established; its linear spectrum is not the stability spectrum of those electron orbits. Nonlinear coupled stability, unequal-coupling spectra, radiative selection, and physical band assignments remain open.

The local Dirac vector-potential reduction supplies a conditional spin-orbit vertex. A frozen simple bivector leaves a twofold spin degeneracy; orbital projection can remove it. An exactly solved supplied orbital-gap model quantifies the resulting entanglement and the large-gap approximation. Neither that gap nor the vector-potential identification has been derived from V2. No numerical particle mass, gauge coupling, or CKM fit is claimed.

Open physical source

The physical source of the existing Coulomb-plus-precession construction remains unresolved: the electron–nucleus interaction must supply the precession generator and its rate relative to the orbit clock. Producing a similar petal curve alone does not establish that physical identification. The scalar-potential and nuclear-response models below are diagnostics along possible routes, not established required ingredients.

For a fixed nuclear three-plane and invariant binding energy V(s), the cross-plane block precesses while s is conserved. Thus a nonzero binding slope can supply precession away from exact alignment, but a positive energy slope alone is not dynamical capture into the lock. This result does not yet include a separate relative-orbit clock or derive the physical petal readout; it leaves the action scale and kernel slope explicit.

An inverse nuclear orientation kernel now realizes that binding slope: K(u,J)=gamma(u)||P(u)JP(u)||²/2. With a fixed common nuclear plane, its normalized marginal is mean(gamma) s. The same energy supplies both electron and nuclear orientation variations. An inverse formula reconstructs V(s) from a specified precession-rate function, but a single orbit leaves infinitely many kernels compatible. The full fixed petal generator instead requires an oriented linear source; a bare nuclear plane does not select that source under its unbroken SO(3) times SO(3) symmetry. These are inverse compatibility results, not a first-principles nuclear current or binding law.

The electromagnetic first pass now covers the circle, two/four/six petals, and three smooth detuned alignments. It establishes conditional source constraints, not an atomic spectrum: the literal point-charge readout has cusps at ideal petal crossings, while a fixed bilinear current in the finite orbit model has only eigenvalue-difference frequencies. A conserved current pushforward and a checked two-time weak-field response specify the next calculation. The source bridge and physical clock remain unclosed.

A subsequent action/current pass supplies a concrete conditional bridge: twisting the spatial spinor bundle by O(L+2) places the degree-L orbit sections in its degree-zero line. The free Dirac operator annihilates that holomorphic component, so the orbit still needs a separately supplied Killing drive. Gauging that drive yields the matching transport current. Exact section integrals then give a geometric probe candidate. For L = 2 and 3, an explicit probe has zero four-mode compression but a nonzero response through other section modes. The physical twist, drive, observer map, ring bridge, and photon/nuclear normalization remain to be derived.

The drive-origin follow-up gives an exact scalar-potential compression to the Killing generator and an exact nonzero leakage norm. A positive-gap surrogate shows how omitted-mode corrections can spoil integer locking; its assumed gap is not the physical Dirac spectrum. Passive frame changes are distinguished from active motion by transforming the probe as well as the state. The ring's three-state generator now embeds in every degree k ≥ 2 section space, but its physical family/degree identification is open.

Nuclear electric polarization is now a dynamical variable in a reciprocal mean-field extension. Checked circle and two/four/six-petal preparations induce a nuclear dipole whose return field changes the electron state, with convergent energy balance. Feedback shifts and dipole-radiation interference are distinct; the illustrative runs do not establish radiative stabilization. Recoil alone does not cancel the neutral pair's relative dipole. A versioned ordinary proton benchmark gives only about 7.6×10⁻¹⁸ induced-to-electron dipole ratio at a fixed Bohr radius; this is not a bound on unspecified V2 source maps or on all atomic nuclear corrections. The microscopic response, local-field matrices, magnetic channels, and coherent radiation reaction remain open. Induced magnetic feedback has not been calculated; these electric-polarization results must not be reported as answering that question.

1. Derive the nuclear-projected potential and omitted-mode resolvent for the candidate line, testing whether they generate and protect the required Killing flow. Carry the electromagnetic probe through mode elimination; four-state unforced closure is not enough for response. Select the physical observer map and ring/degree assignment, and include the nuclear current.
2. Test whether the proposed observer-adapted frame model has a physical role, and derive any resulting coefficient and generator selection from the projected interaction without treating its coordinate grouping as an inherent electron split. Do not infer these from stationary field circulation or import a vector-potential spin-orbit vertex by analogy.
3. Preserve the distinction between the orbit-state coefficient ray, sector coordinate, and Dirac field on physical space when specifying the global physical map. Derive the observer response and physical rate for the existing orbit.
4. For the auxiliary gauge candidate, derive its connection and mode operator from the master projection. Its correct charge branching is not evidence of the required occupied-mode multiplicities or coupling normalization.

Status vocabulary

Checked algebra is exact within its specified model. A conditional mechanism still needs physical premises. A source audit reports what a passage says. A withdrawn inference exceeded its premises. Open marks a remaining proof obligation. Finite diagnostic success does not validate the master dynamics.

Related

Canonical scalar vertices for localized incoming fields

Positive-frequency scalar vertices give the exact massless singular strength $|\sin(\theta/2)|$ between complete positive-frequency bands. A fixed normalized input has inclusive squared strength $\sin^2(\theta/2)$ without a component-count multiplier. Spatial packets retain off-diagonal momentum interference, both complex polarizations and reciprocal local response. Odd-dimensional kinetic chirality can protect a scalar zero, but this is neither a unique depth-five selection nor an electric-charge or weak-rate derivation.

Resonant scalar wave quartets show that positive-frequency preparation does not generally forbid reciprocal chiral norm transfer. The full scalar source has a coherent four-wave contraction and sharp normalized bounds. Complete polarization vectors can remove a restricted three-dimensional channel cancellation; this supplies no depth-five suppression law or observed particle rate.

A radial control in six and ten dimensions

Complete radial polarization fields retains independent complex vector profiles and tangential current in an exact spherical family, including the even depths six and ten. Full Cartesian derivatives verify its local scalar evolution. The family has one radial center and does not identify bound particles or replace the missing general encounter calculation at those depths.