Observational comparisons and remaining proofs

IDWT proposes that particle patterns of one field have different dimensional involvement in one spacetime with infinitely many ordinary spatial directions plus time, with matter occupying up to ten. Its maintained depths are two, three, four, five, six and ten. No additional physical manifold, compactification or particle-exchange force is used. A physical identification must connect a proposed field pattern to measured frequencies, inertia, charge, spin, response and detection, with the same action and normalization throughout.

The present evidence combines a constrained numerical mass structure, conditional dimensional discriminators and explicit higher-dimensional field constructions. These have different evidential roles. A result working at six and ten can support a shared lepton property without choosing either depth. A selector must exclude alternatives under the same stated rule. A fitted formula can be useful without being a forward derivation. This volume collects the current comparisons and the remaining mathematical bridges.

See the research map for the whole reading set, the new particle-geometry case for the full connected argument and reproduction instructions, and mass candidates for the explicit actions. The earlier sector case remains available with its historical scope.

Reviewed Connections as of 5 November 2026

There is a meaningful intermediate milestone short of observed-state selection. The normalized cone law selects completed binomial rungs at supplied coefficients. The prepared-field realization carries spin and normal character data, and the centered-window law joins controlled first-simplex weights, localization and weak translating fields. These are established conditional results.

The open target is selection and persistence of the observed large rungs, active carriers and occupations, with coefficient/scale origin, physical charge normalization, causal separated-particle response and appropriate lifetimes. The six-depth fixed-exponent profiles add numerical spatial compatibility, not those missing physical identifications.

Current advances and their next mathematical requirements

The current portion proposal is $C=S(n,d)$ contributions at one kind-dependent fundamental $\Omega$, giving average mass-frequency $C\Omega$. Even phase spacing is a special case; arbitrary fixed offsets retain additive event multiplicity. A physical registration law remains to be derived, and Fourier harmonics are not separate physical resonators.

These results belong to explicitly identified actions and domains. The mass operator remains unfinished: no current theorem chooses the observed $n$ or $C$. At fixed depth selecting either is the same problem because $S(n,d)$ increases strictly with $n$. A reference configuration, a stable fixed-count shape or a connectivity plateau is not that selection.

1. Evidence must retain its actual premises

Four distinctions determine what a calculation establishes:

These distinctions require keeping assumptions visible and testing a coherent realization. A failed trial excludes its stated action or domain, not every realization of the intended mechanism. Conversely, success in one frozen component calculation does not transfer automatically to a changed action, spatial geometry or kinetic normalization.

A useful common law must accommodate several properties together. Reusing the same geometry across mass, spin and coupling is explanatory structure, but the resulting checks are not statistically independent. Their likelihoods cannot simply be multiplied.

2. The simultaneous mass and depth comparison

The maintained count law is

$$S(n,d)=\binom{n+d-1}{d},\qquad m_{n,d}=\mu_dS(n,d),\qquad \omega_{n,d}=m_{n,d}c^2/\hbar.$$

Here $n$ is a particle index, $N=n-1$ a cumulative degree, $d$ a physical-depth assignment, and $\mu_d$ a mass unit. A representation rank, winding, spatial harmonic degree or field amplitude is none of these. Mass-frequency conversion is ordinary physics; the unknown is the field mechanism sustaining the selected frequency.

The massless photon convention $n=0,d=2$ gives zero rest count, not zero physical photon frequency. Bottom's count is the mean of two neighbouring rungs, 892.5, as volume 5 sets out; which description of its level is the physical one is open.

The non-neutrino numerical recipe uses sector coefficients

$$g_{22}=1445/2,\quad g_{33}=8\sqrt7,\quad g_{44}=12/\sqrt7,\quad g_{66}=g_{10,10}=1/4,$$

with fixed reference counts $C_2^\star=C_3^\star=1$, $C_4^\star=15$, $C_6^\star=18564$, and

$$\mu_d=\frac{m_e}{C_d^\star}\sqrt{g_{dd}/g_{66}}, \qquad \mu_{10}=\mu_6.$$

The last equality is now the common charged-lepton-kind fundamental premise; an earlier overlap construction required an additional equal-overlap identification. These are sector coefficients, not already derived physical force constants. The electron is the sole measured dimensional reference when evaluating this recipe. W, down and up remain scored targets, not extra anchors.

Using the archived comparison inputs, the units in MeV/$c^2$ are approximately 27.47064959, 4.701857978, 0.1451024874 and 0.00002752633861 at depths two, three, four and six. Representative signed residuals are:

These reproduce a dated benchmark, not current world averages or an uncertainty-weighted common-scheme fit. Quark mass conventions matter. A small percentage discrepancy is not automatically agreement within experimental uncertainty; the muon residual is much larger than its measurement uncertainty.

The charged-lepton ratio is exactly scale-free within the prescription:

$$\frac{m_\mu}{m_e}=\frac{3838380}{18564}=\frac{3515}{17} \simeq206.76470588.$$

It is about 17.3 parts per million below the archived benchmark ratio. The neighboring count steps obey

$$\frac{S(n+1,d)-S(n,d)}{S(n,d)}=d/n,\qquad \frac{S(n,d)-S(n-1,d)}{S(n,d)}=d/(n+d-1).$$

At the muon rung the downward step is 15% and the upward step about 17.14%. This is genuine numerical rigidity: shifting one integer cannot repair a tiny residual, and changing the shared unit changes the electron too. It is not by itself a coincidence probability.

The complete assignment theorem compares all 120 permutations of intact family blocks among $\{2,3,4,6,10\}$, retaining all benchmark masses, distinct depths, the electron reference equation and the common lepton unit. It covers every positive integer index rather than a finite cutoff. Only the maintained assignment reaches every existing individual accuracy under that prescription. The electron factor already contains its reference count; recovering it is input consistency. The substantive comparison is simultaneous compatibility of the remaining masses. This theorem neither derives the scale factors nor excludes all alternative scale laws.

The newer joint absolute-mass and ratio assessment retains the electron-only anchor and the actual shared prescription. Absolute placement and ratios use overlapping mass data, so their probabilities cannot be multiplied as independent events. Its observed-precision coverage is conditional on the stated benchmark, candidate rules and reference distribution; it is not automatically a calibrated global discovery probability. The deliberately relaxed scale-fitting diagnostic answers a separate question and remains a valid labelled control. Neither replaces the uncapped assignment theorem, which is a deterministic uniqueness result under its own prescription.

3. Neutrino, bottom and tau are essential controls

For the maintained depth-five neutrino counts, the scale-free splitting ratio is

$$\frac{11628^2-2002^2}{65780^2-2002^2}=0.0303498422349.$$

The archived solar/atmospheric input set in the sector-evidence case gives 0.0297593170770, a +1.98% comparison residual. A different dated set retained in the neutrino module gives a different residual. This ratio is a fixed prediction, and JUNO's sub-percent measurement of the two splittings decides it (volume 16). Solar calibration supplies a conditional absolute unit and normal-ordering mass-state values, not three directly measured absolute masses. Flavor states and mass eigenstates remain different. The physical law must preserve zero ordinary electric charge alongside nonzero weak response; inserting a neutral projector or a tiny coefficient to meet this target would be an input.

Bottom lies midway between levels 16 and 17. The arithmetic mean of 816 and 969 is 892.5, which at the down-type unit of 4.67 MeV gives 4,168 MeV against $4{,}180\pm30$ MeV. A physical mechanism for a two-rung level is open. The geometric mean $\sqrt{816\cdot969}$ is not that mechanism: a two-positive-branch Hermitian matrix with fixed diagonal frequencies $a<b$ cannot generate $\sqrt{ab}$ by an ordinary off-diagonal coupling:

$$\det\!\left[\begin{pmatrix}a&g\\\bar g&b\end{pmatrix} -\sqrt{ab}\,I\right] =-\sqrt{ab}(\sqrt b-\sqrt a)^2-|g|^2<0.$$

Changing the signed block, diagonals or nonlinear action is a new hypothesis requiring its full equations. Bottom's own count does not enter the quark mixing matrix, which is built from the corner overlaps of the other five quarks (volume 19).

Tau's depth-ten mass uses the common charged-lepton unit and comes out 0.064% below the benchmark. Three results now bear on it. The tau is the only complete simplex of its count at any depth from 2 to 21; each charged lepton is a quark plus a neutrino, with $23=1+22$; and the conserved charge integral at depth ten is exact, with spin one half per unit norm (volume 10, volume 19). A universal adjacent-depth transition rule has no depth-nine or depth-eleven partner for tau, so the rule cannot be extended from the two complete doublets. The tau's decays are edge splits instead.

4. Geometry may select, permit or merely describe

The new particle-geometry case and the earlier sector case contain results whose scopes must remain distinct. The older comparison table below retains its stated carriers; the newer flat-space and differential-field results are detailed in volume 19:

A higher-dimensional theory must construct the confinement, boundary structure, source and registration that reproduce established observations. Merely substituting ten for three in an unchanged three-dimensional equation and declaring a mismatch is not a test of every ten-dimensional realization. Conversely, integrating away directions to obtain a desired static power does not yet derive the physical source or its moving, polarized response.

The real-space clarification is equally important. A real amplitude sphere, a symmetric coefficient tensor or paired-real complex notation is a mathematical description, not automatically a physical spatial manifold. Round and projective spatial models retain their additional topology and metric hypotheses. Their spectra and normalizations cannot be transferred to flat ordinary space by renaming the coordinates. No compactified length or tower is available to hide that bridge.

5. One complete action must pass the frequency, energy and charge tests

An inverse construction is legitimate: the observed frequencies are known targets, and common functional forms may be fitted before their physical source is understood. The test is what the fitted law constrains and predicts beyond its inputs. A mass mechanism needs one explicit action, its kinetic normalization and all retained/eliminated field modes, not just a polynomial of an operator with the desired eigenvalues.

The exact count actions described in volume 08 demonstrate how stationary feedback equations can give

$$\omega_{N,d}=u_d\binom{N+d}{d},\qquad E=Q\omega$$

on their stated spatial branches. The count spectrum, coherent-ray operator, occupied-band preparation and sector unit remain different premises in different candidates. A positive-wave persistence result does not transfer unchanged to a first-order proper-current action. A finite chain's singular frequency and a signed numerator product are not interchangeable either. Both signs of actual roots, both relevant coefficient branches and all occupied depths must be retained with the same common coefficients.

Canonical quantities must be derived from the same full action. For real fields $y^I$,

$$\Pi_I^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu y^I)},\quad h=\sum_I\Pi_I^0\partial_ty^I-\mathcal L,\quad \sigma^i{}_j=\delta^i{}_j\mathcal L-\Pi_I^i\partial_jy^I.$$

For a derivative-dependent participation gate, $\Pi$ includes its gate-load derivatives. A positive algebraic vacuum potential does not establish a positive full raw Hamiltonian, because the gate also contributes to momenta and Hessians. The algebraic density constraint must be eliminated from its true equation, including horizontal phase terms where present, before identifying physical perturbation directions.

The older two-clock material/vacuum candidate has two conserved charges. Its stationary identity is

$$E-\omega Q_\theta-\Omega Q_\psi =-\langle\mathcal L_{\rm stat}\rangle.$$

The rotating generator on the left is not the raw energy. The material response contribution can be signed and must not be replaced by a positive residual square. In that candidate, the proposed energy-clock normalization requires both

$$Q_\theta=A_0,\qquad E=A_0\omega,$$

while retaining $Q_\psi$, complete spatial storage and supporting stress. The empirical conversion is matched only with the supplied or independently derived $A_0=\hbar$ bridge. Normalizing an eigenfunction to one does not derive that physical action unit or a field amplitude.

For a smooth radial state in ten spatial dimensions the measure is $(\pi^5/12)r^9dr$. If $H_r$ is radial physical stress and $I$ its trace, the full Euler equations give

$$(r^{10}H_r)'=r^9I.$$

Global virial follows only with the actual origin and infinity stress limits. On a finite exterior cut the accepted normalization identity has the form

$$E_{\rm tail}-\omega Q_{\theta,\rm tail} =\langle\mathcal B\rangle_{\rm tail} +\frac{\pi^5}{120}b^{10}H_r(b).$$

The two tail quantities are separately positive for a nonzero massive exterior, but differ by boundary stress. Full smooth matching cancels the internal boundary terms. A reference core's pressureless identity is therefore a useful exact test, not proof of global mass normalization.

6. A physical spatial state must connect its core, transition and exterior

A particle-like classical candidate needs a finite complete field structure sustained by its own equations. Support size, translation, internal flow, phase dynamics and reciprocal response must coexist. A prescribed wall, background coefficient or external frame is an explicit conditional model, not an already derived supporting part of the particle. The field may flow internally in its zero-momentum frame; zero total momentum is not zero internal motion.

The earlier reviewed material constructions put actual response-charge saturation and common-volume shape into one action. Their selected reference bodies have exact count clocks, uniform participating density, free-traction balance and positive whole-reference boost inertia. The physical support remains ten-dimensional even when a carrier prefix is labelled by a lower depth. A finite coefficient rank or anisotropic ellipsoid does not prove a lower-dimensional layer, and an unbounded thin limit at fixed volume does not produce a bounded localized sheet.

A regular material-plus-Cartesian-vacuum continuation supplies an actual local smooth five-field exit on the origin subclass, with a forced negative amplitude slope and fixed readout traces below one. It also supplies constructed permanently inactive exteriors for boundary traces

$$(g_b/R)^2\le3/4,\qquad a_b^2\le1/2,$$

with arbitrary finite current and readout traces. Joint clipping of the actual mass-plus-mixed potential proves nonreactivation. Weighted norm comparison gives exponential value, derivative and stress decay, and identifies the remainder with a unique small tail at a finite radius. Uniqueness of the larger exterior/conormal relation is not established.

These are substantive local and exterior theorems, but they do not connect the active inner exit at $a^2=3/4$ to a compatible exterior trace at $a_b^2\le1/2$. A full connection must solve the intermediate Euler equations and match all field values, full conormals and stress. Inner existence plus exterior existence, a boundary-parameter count or a named matching map is insufficient. A stationary conormal degeneracy requires a null-force compatibility condition; it is not alone proof of nonexistence. Restricted slope/gate-work obstructions exclude stated nodeless permanently inactive classes, not all possible actual inner continuations.

An exact origin core plus a future-oriented nonzero exit has positive additional theta charge, so normalized completion requires a core radius smaller than its supplied reference scale. This necessary inequality does not select the radius or cancel the energy defect. On any reactivated component, $n_m=\sqrt{\det B}>0$ and future orientation must be used; a signed radial formula cannot become negative density to repair charge normalization.

The physical proof target is a solved complete state, not merely a trial profile with small residuals at one resolution. Evidence should retain actual equation residuals, domain, boundary conditions, field and conormal convergence, energy/charge convergence and the effect of increasing angular/radial resolution. A full-space obstruction must state its actual class. A failure of one restricted ansatz should not be generalized to the field theory or dimensional hypothesis.

7. Inertia, causality and persistence are separate requirements

Positive total inertia concerns the complete particle and supporting field, not the signed momentum response of a wave in prescribed coefficients. For a genuinely isolated positive-energy solution with the appropriate full stress balance and covariant dynamics, its boost response should reproduce the corresponding complete energy–momentum relation. A phase explanation of that response also needs phase-resolved complete dynamics.

The relevant kinetic form is the constrained one after actual algebraic equations and absent coordinates are treated. A rank-deficient unreduced matrix at an inactive edge is not automatically a ghost. Conversely, a normalization that divides by a vanishing gate coefficient can make a frozen matrix look regular without producing a bounded physical perturbation norm through the edge. The necessary checks include:

• Positive time inertia on retained physical dynamical directions, complete spatial stiffness where required, and real complete characteristic roots.
• Propagation in every physical direction and angle, with both signed cone comparisons and actual carrier polarizations; radial and transverse endpoint checks alone can miss mixed directions.
• The actual solved transition rather than only a reference core, plus the changing-rank constraints and transmission conditions.
• Complete raw-energy and charge-preserving comparisons rather than an unjustified identification of the principal action Hamiltonian with raw energy.
• Lower-order spectrum and nonlinear evolution on a stated norm/domain, separately from frozen principal health.

The accepted local origin health theorem covers a short actual inner interval, not an entire determinant transition or endpoint-uniform nonlinear evolution. Local positive margins and exactly luminal Cartesian blocks are not a strictly positive all-block cone margin. A causal principal symbol also does not by itself establish nonlinear persistence or a measured lifetime.

Persistence must match the particle. The electron calls for comparatively persistent localized motion; a tau or muon may be a finite-lived resonance. A negative energy direction or growing linear mode is neither a universal rejection of unstable-particle candidates nor a derivation of their observed decay. Decay requires a timescale, available outgoing channels and complete conserved energy, momentum and charges. The known particle frequency remains the mass-frequency target; a slower shape disturbance must be named separately.

8. Pair interactions must use both actual patterns

The intended interaction mechanism is compatibility between real spatial oscillation patterns of the same field. Both objects have polarization, orientation, relative phase and response. Each pattern retains its actual isolated clock. In the portion model, same-kind fundamentals can agree while their count-multiplied mass-frequencies differ. Unlike fundamentals are not synchronized by assumption; evenly spaced and uneven internal phases both require testing.

A physical pair calculation must derive the force from position dependence of the complete common action, retain response and recoil of both objects, and transform covariantly under a common rotation of all physical data. Deformation toward compatibility or incompatibility is a hypothesis to calculate. A zero overlap alone does not give repulsion; repulsion needs an actual energy gradient or dynamical stress. Eliminated response modes and inverse operators must be identified as components of the same field, not independent mediators.

Bare coupling coefficients, overlap integrals, accelerations, scattering vertices and encounter rates are different quantities. Canonical energy/norm conversion and any required physical amplitude dependence must be explicit before an overlap is called a measured force. The absolute field amplitude remains uncalibrated; an absolute force claim must derive or independently calibrate its required scale. Amplitude-free geometric comparisons and ratios remain useful without such an absolute calibration. Unit-normalized trial shapes compare geometry; they do not calibrate a physical electron amplitude. The joint target is different particle mass-frequencies with groups of similar response associated with depth, without dividing by mass simply to collapse curves.

All six occupied depths are controls. In particular a proposed depth-five suppression must be compared with neighboring and other assigned depths under one law and full polarization space. An effect shared by generic odd depths, one chosen mode or a special inserted coefficient is not a neutrino explanation. Depth two must retain actual transverse linear, circular and elliptical polarization, not only an abstract two-state degeneracy. Massive higher-depth patterns need not inherit the photon's transversality. Depth-ten tau branches must be tested alongside six.

Charge requires its own normalization and conjugate. A geometric spectrum containing $0,2/3,-1,-1/3$ is useful compatibility under its occupation/representation premises, but does not uniquely assign the physical states or measure electric response. The same charge rule should carry the exact opposite conjugate on otherwise corresponding patterns. Magnetic moment, spin and response should be computed from that same normalized current; fitting the mass alone cannot determine them. Neutrino zero ordinary charge must coexist with nonzero weak channels. See same-field coupling, pair encounters and flavor, gauge and topology.

9. Literal electron motion and photon timing remain physical tests

The electron orbits the nucleus in four directions, the up quark's two planes. A nucleus is uniform across the pairs, so its remaining two coordinates do not change in an atom. Its external position $q(t)$, internal classical field, coefficient representation and detector registration are distinct.

The orbit is now tied to the spectra. The up quarks act through all four directions and the down quark through three, so $\omega_w^2/\Omega^2=4/3$ for every atom, and the light of a line has the frequency of this fourth-direction motion. With a nucleus of finite width the line at one Bohr radius comes within 0.13% of Lyman-γ (volume 18).

An exact flat-coordinate family has

$$q(t)=\rho e^{\beta Jt}(\cos\delta t\,e_1+\sin\delta t\,e_3), \qquad q''=2\beta Jq'-(\delta^2-\beta^2)q.$$

It is a capability result for the specified isoclinic force; the physical attractive force, selected frequency lock, reciprocal nuclear response and persistence must be supplied by the common dynamics. Older projective orbit models retain their additional spatial/representation premises.

A detector's accumulated registration may be described conditionally by a broadened time measure,

$$\rho_{\rm reg}(x)=\frac1T\int_0^T G_\sigma(x-\Gamma(t))\,dt.$$

It is the apparatus response along an actual deterministic curve, not a Born position law or probability cloud. Squared normalized coordinates, stationary field harmonics and geometric lobe counts do not by themselves establish this detector coupling. Molecular/crystal binding requires reciprocal moving nuclei and electron trajectories; sharing a four-component algebra with conventional hybridization is not that derivation. See orbits and binding and the actual-orbit correction.

The photon is an extended connection already spread from its source to where it makes contact, with an advancing phase. Nothing travels along it. Where it ends is fixed by the field's actual configuration, and the equations for that remain to be written. The target is to derive distance-dependent timing from the phase along the span, not assume that one phase difference at a single frequency equals a delay. Full energy, stress, recoil, polarization, interference, detector response and no-signalling consequences must remain consistent with observations. The zero-duration hypothesis concerns processes occurring entirely through the relevant free directions. It does not make bound-profile deformation or bound-direction transport instantaneous, and geometric adjacency alone does not establish conservation or no-signalling. See photon polarization.

10. Current measurements and open calculations

These are the sharpest present tests and the calculations most needed next.

Open within the geometry:

• Bottom's physical level description, and why the rung mean is its count.
• A rate rule for decays, and the channels the two operations do not yet reach: the lepton pair in beta and semileptonic decays, the muon and tau channels of charmless bottom decay, leptonic meson decays, the tau's muon channel and the strange tau decays.
• The lepton mixing magnitudes, which do not follow from the corner rule.
• The down quark's charge error $w/3$ at depth three.

Open within the mass and scale structure:

• The universal norm value.
• The 1.14% gap between the count and $1/\alpha^2$.
• The 0.26% residual of the boson unit.

Open within the atom and molecule:

• Hydrogen's binding energy on the same orbit as its line, and eccentric orbits.
• The conflict between the width that fits Lyman-γ and the muonic bound on that width.
• The doubling of shell capacity.
• The halogen factor 1.75, the polyene path and bond order as a count of phase slots.

Open within the field dynamics:

• The second-order energy of the uniform background and what selects a common pairing of directions.
• The stability of a contact-ended free extension.
• Observational refuters for the neutrino assignment, including what contact range a neutrino's cross-section requires (volume 15).

11. What would count as completion or further evidence

A physical mass-operator claim should provide one explicit action and a traceable chain from its geometry and retained modes to the following results:

1. A spatial mechanism selecting the common-frequency portions and their maintained counts, with all sector units and action conversions identified as supplied, fitted or derived.
2. An actual complete spatial state for each claimed branch, including sustaining stress, exterior and source-free matching; physical depth must follow from the state rather than coefficient rank alone.
3. Complete energy and momentum for the actual action, all its relevant charges and positive total inertia, with normalization and finite-cut boundary work retained.
4. Evolution and persistence appropriate to the branch, including causal/transmission domains and, for unstable particles, actual outgoing decay channels and timescales.
5. Reciprocal unlike-particle encounters using each object's clock, polarization and response, with force/current/rate normalization and depth controls.
6. The shared empirical properties of the identified particle: charge and conjugate, spin and detector interpretation, response strengths and any lifetime or branching comparison actually claimed.
7. Further consequences that distinguish the law from other actions fitting the same masses, with a declared benchmark, candidate family, uncertainty and null procedure where statistical claims are made.

A full mass family need not precede every useful result. A rigorously constructed particle-like state, a selective geometric constraint or a new measured consequence under one coherent action can advance the theory without completing it. Reviewed core, exterior, finite shape-energy and probe-response results sharpen these targets. The new cluster law supplies a scoped mechanical construction with explicit premises; the covariant rigid embedding and spectral connectivity proposal retain their pending independent-review status. No unresolved result becomes accepted because it would close the chain.

The next explanatory success must preserve the useful common structure already obtained while supplying a missing physical bridge. Agreement with selected numbers, exact algebra and conditional spatial capability each contribute evidence in their own scope. None alone identifies the whole proposed theory with nature.