Mass-operator controls at depths three and four

The bounded spatial carrier and the localized polarization core now have explicit depth-three and depth-four controls under their existing rules. Their algebra, spatial normalization and assigned simplex numerators carry through. The comparison also exposes three important constraints: odd surface parity changes the scalar interpretation, the core's unbounded linear response persists, and the bottom assignment needs dynamics beyond ordinary mixing of two positive mass branches.

These are tests of candidate equations across physical sectors. They do not establish the particle interpretation, observed couplings, or the common physical scale between sectors. Mass remains the selected frequency, $mc^2=\hbar\omega$. All physical coordinates are ordinary and uncompactified; a closed surface below is a proposed field feature inside that space, not a compactified spatial direction.

Later mass-core and quadrature calculations

The regular current action extends the core through its zero-current sets. Its positive shape term preserves the branch frequency and inertia while changing the disturbance equations; complete local and spatial controls now include depths three, four, five, six and ten. These are new action terms, so the original core's tail limitation below remains valid for the original action. Full stability of the modified action is still open.

A separate holomorphic binding comparison has unitary fixed-frame linear response, but its tested responsive-frame extension has resolved growing spatial modes. For the bottom assignment, positive canonical quadratures give the required geometric mean conditionally, with an exact Hockey-Stick response identity and a crucial surface-partner sign constraint. The physical spatial pairing and the paired-branch selection remain to be derived.

Fixed assignments and a common calculation

Use the maintained mass-spectrum assignments, with particle index $n_p$, cumulative degree $N=n_p-1$, surface degree $L=N+2$, and normal winding $n=d$. The last equality remains an inverse model premise. Distinguishing winding from particle index is essential.

The two bottom entries are constituent spectral controls. They are not two new particle assignments, nor a calculated bottom particle. Its maintained count is $\sqrt{816\cdot969}=889.21538448\ldots$.

The connected normal operator uses the unchanged radial equations

$$w'=\frac{d-a}{r}w,\qquad a'=2r(1-w^2),$$

the same multiplier $h=z/\sqrt{1+r^2}$, canonical unit norms, and the same illustrative hopping $\gamma=10^4$. No coefficient is retuned for an individual mode. Its complete complementary gap remains at least one; the actual $d$ kernel modes satisfy

$$P[(d+1-a)S]P=-2J_z.$$

Direct Cartesian derivatives of every normal spinor, a second application of the full Dirac operator, and radial and outer-domain refinements verify these statements at both new depths. The separate bounded unconnected profile also retains positive analytic gap bounds, approximately 1.18923 at depth three and 1.17675 at depth four.

The projected diagonal factors give

$$\prod_{l=0}^{d-1}\left(\beta+l-\frac{d-1}{2}\right) =\prod_{l=0}^{d-1}(L-1+l) =d!\binom{N+d}{d}.$$

This includes the down assignment $N=0$; no large-degree approximation is used in this numerator. The full small frequency still has the previous finite-hopping correction:

$$\omega_{N,d}= \frac{d!\binom{N+d}{d}}{\gamma^{d-1}\prod_l h_l} \left[1-\frac{(L-1)^2/h_0^2+(L+d-2)^2/h_{d-2}^2} {2\gamma^2}+O(\gamma^{-4})\right].$$

The correction is a property of the complete finite matrix. At the illustrative hopping, it ranges from about $-1.06\times10^{-7}$ for down to $-6.90\times10^{-6}$ for the upper bottom component, and from $-4.56\times10^{-7}$ for up to $-1.09\times10^{-4}$ for top. These numbers are mathematical departures from the simplex product, not comparisons with new mass measurements.

The actual lower-dimensional surface

The normal calculation uses one ordinary plane at every winding. By itself, changing that winding would not test the full physical dimension. The surface calculation must also use $m=d-2$ directions and the appropriate intrinsic spinor rank $2^{\lfloor m/2\rfloor}$.

For odd $m$, construct the ambient Clifford matrices in $m+1$ directions and restrict their even products to one ambient chirality. The surface operator contains only such even products:

$$B_K=k-\Gamma_x\Gamma_{\nabla_S},\qquad k=\frac{m-1}{2}.$$

They preserve that chirality exactly. The resulting intrinsic rank is one on the depth-three circle, two on the depth-four sphere, and two on the depth-five three-sphere. Both choices of ambient chirality are checked; neither is assigned a particle charge or antiparticle meaning.

Arbitrary complete spatial spinor jets verify

$$B_K^2=\Delta_S+k^2,$$

and a separate stereographic calculation with the actual spin connection agrees. The full principal symbol is Hermitian and squares to the spatial momentum norm. These controls cover depths 3, 4, 5, 6 and 10 without interpreting matrix rank as additional physical coordinates.

On the unit circle, in one of the two equivalent ambient realizations,

$$B_K=-i\partial_\theta,\qquad \mathscr D=B_K+\frac12.$$

Here the specified bounding spin structure has antiperiodic spinors $\psi=e^{i\theta/2}F$, with periodic coefficient $F$. The modes $F=e^{\pm iL\theta}$ have $B_K$ eigenvalues $\pm L$. Both are retained for $L\ge1$. At $L=0$, $B_K=0$; a formula dividing by $\beta$ must not be used there. All assigned depth-three controls have $L\ge2$.

The depth-three link product is

$$B_K(B_K^2-1).$$

It is odd in $B_K$, so its two surface signs give opposite signed products. The complete canonical Hamiltonian

$$H_P=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix}$$

reads positive frequency from a singular value and gives the same positive frequency on both branches. More generally, if $R$ reverses the chain and $J=\operatorname{diag}((-1)^l)$, reflection symmetry of the actual hoppings gives

$$T(-\beta)=-JRT(\beta)^TRJ.$$

Thus both signs have the same complete singular spectrum at finite hopping, including at odd depth. The signed product alone is not the physical frequency or an inertia sign.

The earlier positive scalar bending-Hessian interpretation requires a separate extension at odd depth. On the circle its desired positive spectrum would be that of $\sqrt{\Delta_S}(\Delta_S-1)$, which is not a finite polynomial in the scalar Laplacian. The local first-order parent survives this parity test; the even-dimensional scalar interpretation does not transfer unchanged. The codimension-two surface construction at depth two has $m=0$ and is outside the differential-surface controls established here.

Complete partner phases and geometric response

For diagonal links $D_l=\beta+l-(d-1)/2$, let $u_l,v_l$ be the positive singular vectors obtained from the absolute scaled inverse. Their canonical phases must obey

$$s_0=1,\qquad s_l=-\operatorname{sgn}(D_l)s_{l-1},\qquad t_l=\operatorname{sgn}(D_l)s_l.$$

The positive-frequency state is $(s_lu_l,t_lv_l)/\sqrt2$. An alternating-sign formula is sufficient only when all $D_l>0$. The implementation now retains the correct partner phases on negative and mixed-sign controls. This correction leaves the previously tested positive-branch frequencies unchanged.

The complete geometric source gives the constant-polynomial deformation response

$$F_{N,d}=2\gamma\kappa_d \operatorname{Re}(v_0^*u_{d-1}),$$

where $\kappa_d$ is one spatial integral shared by all grades. Here $\kappa_d$ denotes the endpoint integral, not the core binding coefficient. At fixed positive surface branch and the common illustrative hopping:

These are derivatives with respect to the stated geometric coefficient. Their units and physical canonical conversion are not yet fixed between sectors. Within each depth, they show response grouping without dividing by mass or changing the coefficients for each particle.

Both surface signs and both real and imaginary shape perturbations must be retained. At fixed mode and orientation, the complete source gives

$$F_{N,d}(-\beta)=(-1)^dF_{N,d}(+\beta).$$

Thus the depth-three surface partners have opposite signed response, whereas the depth-four partners have the same signed response. Their positive mass-frequencies agree exactly. At normal orientation angle $\vartheta$, the derivatives with respect to the real and imaginary geometric coefficients are

$$(F_{\mathrm{Re}},F_{\mathrm{Im}}) =F_{N,d}(0)(\cos(d\vartheta),\sin(d\vartheta)).$$

The full reciprocal source verifies both components independently. For the circle's two degenerate surface modes, a normalized superposition with weights $|a|^2+|b|^2=1$ has response

$$F_{\mathrm{superposition}}=(|a|^2-|b|^2)F_{N,3}(+\beta)$$

to a surface-independent normal deformation. Cross terms vanish by the actual circle integral. Equal weights cancel this particular response for every relative phase, while other spatial deformation channels remain. Such balance is a state choice here, not a derived occupation rule or a general absence of interactions. The same parity rule applies to other odd depths and cannot by itself single out neutrinos.

The sign of a geometric force depends on orientation and polarization; it is not a sign of particle inertia. The full occupied support still has to respond to the complete spatial source.

The bottom assignment adds a dynamical constraint

A normal deformation constant over the circle commutes with the surface operator. It cannot mix the bottom components $L=17$ and $L=18$:

$$\frac1{2\pi}\int_0^{2\pi} e^{-17i\theta}e^{18i\theta}\,d\theta=0.$$

A first surface harmonic can access that channel, since the same integral with $\cos\theta$ inserted is $1/2$. Using the previously derived normal endpoint deformation and the actual two canonical states gives a nonzero first-harmonic matrix element, approximately $-1647.10475$ at $\gamma=10^4$. This identifies a spatially allowed channel, not a stationary occupied shape or a derived bottom mass.

Ordinary mixing of the two positive branches is insufficient. For $0<a<b$, retain their frequencies on the diagonal and add any Hermitian off-diagonal coupling:

$$M=\begin{pmatrix}a&g\\\bar g&b\end{pmatrix}.$$

At $\lambda=\sqrt{ab}$,

$$\det(M-\lambda I) =-\sqrt{ab}(\sqrt b-\sqrt a)^2-|g|^2<0.$$

Consequently $\sqrt{ab}$ is never an eigenfrequency of this matrix. In fact no eigenvalue lies strictly between $a$ and $b$. The two uncoupled frequencies also give an ordinary difference frequency, not their geometric mean. The maintained bottom count therefore requires a different coupled block, changed diagonal dynamics, or an explicitly derived relation between the two canonical quadratures. Merely inserting the target off-diagonal singular value would assume the missing result. The full signed parent and its same-field nonlinear response remain available to test; this exclusion concerns the stated two-positive-branch reduction.

The full core limitation persists at the new depths

The separate polarization core has the same radial rule at each physical depth:

$$\frac{h(r)}{f(r)} =\frac{I_{d/2}((\kappa-m_c)r)} {I_{d/2-1}((\kappa-m_c)r)},\qquad \kappa>m_c>0.$$

With $m_c=1,\kappa=2$, its dimensionless root mean squared radii are 1.73035154 at depth three and 2.22415436 at depth four. Within each depth, all listed single-grade controls have the same profile. Full spatial variation, the canonical norm, and the complete boosted field equation and stress pass the new checks. Thus positive total inertia on that conditional branch does not depend on specializing to depth six.

The full linear-response problem persists as well. At both depths the retained core representation has rank four, so the common rule $q=1/(r+1)$ gives $q=1/5$. It would be inconsistent to reuse the depth-six value $1/9$ without changing that rule.

The full real multiplet has 64 components. Constructing its exact invariant tail block gives the same finite-wave-number Jordan chain at $p_\perp=\sqrt3$, with

$$\|e^{tA}u\|/\|u\|=\sqrt{1+c_J^2t^2},\qquad c_J=\frac{2(q\omega^2+1)}{\sqrt{\omega^2+1}}.$$

The existing translated-packet argument therefore applies at depths three and four too. This is an unbounded formal linear response, not a negative physical inertia and not a nonlinear instability theorem. The bottom component frequencies are only spectral controls of this core, as above.

The cross-sector comparison supports the dimensional consistency of the specified constructions while narrowing their physical completion. The joint occupied geometry, the bottom two-grade dynamics, stability, and physical normalization across sectors remain necessary tests of a candidate mass operator.

Reproducible calculations

• Common sector assignments define unchanged modes and distinguish the bottom components.
• Connected normal checks verify actual spatial profiles, the full normal square and frequencies.
• Endpoint response evaluates the full reciprocal source and finite-hopping corrections.
• Surface and signed-state controls check the actual intrinsic spin connection, parity, both canonical partner phases and the bottom mixing constraint.
• Core checks and full tail chain test the independent localized action and its complete linear response.

The common numerical entry points include depths three and four by default. Their depth selector permits a focused rerun without recomputing unrelated previous controls. Full dense tail calculations default to depths 3–6 to respect the available memory; they do not label an unrun depth-ten full matrix as checked.

The reviewed reports are the normal spatial calculation, complete signed source, localized core, exact tail chain and surface, parity and polarized-superposition calculation. They retain the source hashes actually executed and the associated raw profiles where used. The five reports pass 871 checks. Earlier intermediate reports and their exact source versions remain preserved separately.