Exact Normal Response Kernels and Their Spectral Gaps

The useful positive result is an exact localized solution of a geometrically defined finite-width same-field response. It supplies a canonical normal operator, its complete reduced energy, actual bound levels, and responding translation and phase zero modes. Its curvature/depth ratio is sqrt(2/3), while its frozen bound-level spacing/depth is 1/2. A separate exact local-power family can attain the historical ratio 2, but selecting that ratio fixes an otherwise unexplained golden-ratio nonlinearity and produces only one frozen bound level. Thus achieving the curvature ratio does not supply the proposed normal gap. A third, ordinary warped spatial geometry gives exact normal localization at depths 4, 5, 6 and 10, but its tangential modes change the normal well: geometric localization alone is not band independence.

1. Grounded target and normalization

The hadronic-mass analysis, including its later Sections 64–76, defines the historical target

$$R_{\rm curv}=\frac{\hbar\Omega_{\rm curv}}{A}=2, \qquad \Omega_{\rm curv}^{2}=\frac{V''(0)}{m_\perp}, \qquad A=V(\infty)-V(0)>0.$$

The original normal Schrödinger convention is $H_\perp=-\hbar^2\partial_y^2/(2m_\perp)+V$. A potential tending to zero at infinity makes $A=-V(0)$. The depth must be a difference, because a constant shift of the energy origin cannot change a curvature or spacing.

This is a local curvature diagnostic. It is not automatically a bound-level difference, a frequency of the fully responding nonlinear field, or a whole baryon pole. Sections 64–76 withdraw the claim that the trial 154.808228 MeV is an established half-contribution. They also retain the separate unresolved status of the scalar scale identification, reciprocal factor 7/6 and frequency multiplier 4/3. None is derived here. An off-diagonal sector block alone has no standalone eigenfrequency spectrum; exact elimination retains its spectral parameter.

The historical Gaussian calculation imposes both $V=-A\exp(-y^2/\ell^2)$ and the harmonic ground-state width $\ell^2=\hbar/(m_\perp\Omega)$. It obtains ratio 2 from those two conditions. A Gaussian is not an exact stationary solution of that Gaussian-density nonlinear well. The exact cubic sech profile instead gives sqrt(2). The stated scalar-Soler diagnostic is $\hbar\Omega_{\rm loc}/A=\sqrt{2\omega/m}<\sqrt2$ in natural units, with its Schrödinger-type kinetic interpretation explicit; it is not the full Dirac fluctuation spectrum. Coherent projective overlap gives $V_N''(0)=2AN$ in its specified angular coordinate, including zero at $N=0$. A physical ray metric would additionally be required to convert that derivative into a proper-distance curvature.

The related scalar-contact, contact-audit and band-consistency analyses establish the remaining safeguards. The scalar-contact module requires an actual band-independent normalized coefficient, not merely the absence of a band label in a bare action. The contact audit distinguishes point, constant and smooth symbols and retains variation of the field-dependent ray. The band-consistency note requires common bare compressions or, for eliminated channels, consistency of their frequency-dependent Schur maps. Those distinctions are applied below.

Canonical action and dimensions

Use a classical complex normal envelope, not a position-probability amplitude,

$$S=\int dt\left[\frac{i\hbar}{2}\int(\psi^*\dot\psi-\dot\psi^*\psi)\,dy-E[\psi]\right], \quad Q=\int|\psi|^2dy,\quad a=\frac{\hbar^2}{2m}>0.$$

The canonical norm $Q$ remains explicit and is not calibrated to a particle amplitude. With dimensionless $Q$, $[\psi]=L^{-1/2}$, $[a]=EL^2$. The stationary condition for $\psi=e^{-i\mu t/\hbar}\phi$ is $\delta(E-\mu Q)=0$. Its carrier frequency is $\mu/\hbar$, whereas $\Omega_{\rm curv}$ and fluctuation frequencies have distinct definitions. The negative $\mu$ below is an envelope frequency relative to this action's zero, not an identified negative particle mass.

If an unreduced action instead has symplectic coefficient $Z>0$, gradient coefficient $a_b$, and power energy $-g_b|\psi|^{2\sigma+2}/(\sigma+1)$, canonical rescaling $\psi_c=\sqrt Z\psi$ gives $a=a_b/Z$, $g=g_b/Z^{\sigma+1}$. Frequencies must use those canonical coefficients. For a Hartree quartic, its canonical coupling is $g_b/Z^2$.

Under a coordinate change $y=b x$, the unitary density convention is $\psi_x(x)=\sqrt b\psi_y(bx)$. Then $a_x=a/b^2$, $m_x=b^2m$, and $V_x''(0)=b^2V_y''(0)$. Thus $\Omega_{\rm curv}$, $A$, and their ratio are invariant. Changing coordinates while leaving the kinetic coefficient unchanged changes the model. More generally the proper kinetic metric and its induced measure must be transformed together.

2. Finite-width same-field normal response

Choose one positive inverse differential kernel on the ordinary normal line:

$$K_\ell=(1-\ell^2\partial_y^2)^{-1},\qquad K_\ell(y)=\frac{e^{-|y|/\ell}}{2\ell},\qquad \ell>0.$$

Its integral is one. It is geometrically defined by the normal metric, a positive length and decaying boundary conditions. This choice is a proposed effective same-field response, not an independent interaction field or a mediator particle. Its derivation by elimination of actual master-field modes remains open. The response is instantaneous in this envelope model; no relativistic propagation claim follows.

Take

$$E_H=a\int|\psi'|^2dy -\frac g2\iint\rho(y)K_\ell(y-y')\rho(y')\,dy\,dy', \quad \rho=|\psi|^2,\quad g>0,$$

where $[g]=EL$. Symmetry of $K$ removes the factor 1/2 in the variation exactly once:

$$\frac{\delta E_H}{\delta\psi^*} =-a\psi''-g(K_\ell*\rho)\psi.$$

Consequently

$$i\hbar\dot\psi=-a\psi''-W[\psi]\psi, \qquad W[\psi]=gK_\ell*|\psi|^2.$$

The convenient identity $(1-\ell^2\partial_y^2)W=g\rho$ only computes this functional; $W$ has no separately postulated physical degrees of freedom here. The boundary condition $W\to0$ fixes its inverse uniquely.

Exact stationary branch

Let

$$k=\frac1{2\ell},\quad \phi(y)=B\operatorname{sech}^{2}(ky),\quad W(y)=A\operatorname{sech}^{2}(ky).$$

Direct differentiation gives

$$\phi''=k^2(4-6\operatorname{sech}^2ky)\phi,$$ $$(1-\ell^2\partial_y^2)W =A(1-4\ell^2k^2)\operatorname{sech}^2ky +6A\ell^2k^2\operatorname{sech}^4ky.$$

Both field equations therefore hold exactly with

$$A=6ak^2,\quad B^2=\frac{9ak^2}{g},\quad \mu=-4ak^2.$$

There is no imposed harmonic approximation in this profile. Its natural amplitude width is $k^{-1}=2\ell$; the amplitude tail decays as $e^{-|y|/\ell}$. The branch norm is

$$Q=\frac{4B^2}{3k}=\frac{12ak}{g} =\frac{6a}{g\ell}=\frac{3\hbar^2}{mg\ell}.$$

Thus at fixed $a,g,\ell$ this explicit solution occupies a specific norm; arbitrary independently prescribed $Q$ need not have this sech-squared shape. It is a consistency condition, not an amplitude fit to a target mass. The action may have other stationary profiles, which were not scanned.

The exact reduced energy follows from $\int\operatorname{sech}^4t\,dt=4/3$ and $\int\operatorname{sech}^6t\,dt=16/15$:

$$T=\frac45ak^2Q,\qquad \int\rho W\,dy=\frac45AQ=\frac{24}{5}ak^2Q,$$ $$\boxed{E_H=T-\frac12\int\rho W\,dy=-\frac85ak^2Q.}$$

Meanwhile $\mu Q=T-\int\rho W=-4ak^2Q$. Summing two mean-field endpoint terms would therefore miscount the interaction. Neither $E_H/Q$ nor $\mu$ equals the local curvature energy.

Curvature, bound levels and responding frequencies

For the frozen self-well $V=-W$,

$$V''(0)=2Ak^2,\qquad \hbar\Omega_{\rm curv}=\sqrt{4aAk^2},\qquad \boxed{\frac{\hbar\Omega_{\rm curv}}A=\sqrt{\frac23}.}$$

The actual frozen normal Hamiltonian is

$$H_f=-a\partial_y^2-6ak^2\operatorname{sech}^2(ky).$$

Its two square-integrable bound modes have energies $-4ak^2$ and $-ak^2$, with respective profiles $\operatorname{sech}^2ky$ and $\operatorname{sech}ky\tanh ky$. The next threshold solution is not square-integrable. Hence

$$\boxed{\Delta E_f=3ak^2=A/2,}$$

which differs from the curvature diagnostic. These levels can be obtained by factorizing with $D_\lambda=\partial_y+\lambda k\tanh ky$, since $D_\lambda^\dagger D_\lambda=-\partial_y^2+\lambda^2k^2-\lambda(\lambda+1)k^2\operatorname{sech}^2ky$, and iterating $\lambda\to\lambda-1$. No probability interpretation is used.

For the full responding field, write $\psi=e^{-i\mu t/\hbar}(\phi+u+iv)$. Its linear equations are

$$\hbar\dot u=L_-v,\qquad -\hbar\dot v=L_+u,$$ $$L_-=-a\partial_y^2-W-\mu, \qquad L_+u=L_-u-2g\phi K_\ell*(\phi u).$$

They imply $L_-\phi=0$ and $L_+\phi'=0$: phase and translation zero modes. Thus translating the complete self-well has zero restoring frequency, despite its positive local curvature for a probe in the frozen well. The full nonzero scalar fluctuation spectrum and nonlinear stability are not calculated here. For a vector envelope with at least two components and interaction depending on total density, an orthogonal polarization disturbance has zero first-order density response. In that specified complementary channel the frozen first excited state does give the full linear frequency $3ak^2/\hbar$; this is conditional on the vector carrier and is still not a baryon pole.

Galilean translation of this reduced action gives momentum $P=mQv$ and energy $E_H(v)=E_H(0)+mQv^2/2$. It has positive translational coefficient $mQ$, but this is a nonrelativistic envelope result. It does not establish equality to complete relativistic rest energy divided by $c^2$.

3. Exact ratio-two construction and why it is insufficient

A bounded one-parameter comparison family is

$$E_\sigma=a\int|\psi'|^2dy-\frac g{\sigma+1}\int|\psi|^{2\sigma+2}dy, \qquad \sigma>0,$$

with $[g]=EL^\sigma$. Full variation yields $-a\psi''-g|\psi|^{2\sigma}\psi$. For every $\varepsilon>0$ it has the exact stationary solution

$$\phi=B\operatorname{sech}^{1/\sigma}(ky),\quad k=\sigma\sqrt{\varepsilon/a},\quad B^{2\sigma}=\frac{(\sigma+1)\varepsilon}{g},\quad \mu=-\varepsilon.$$

The self-well is $-A\operatorname{sech}^2ky$, with $A=(\sigma+1)\varepsilon$. Its curvature ratio is

$$\boxed{R_{\rm curv}(\sigma)=\frac{2\sigma}{\sqrt{\sigma+1}}.}$$

The cubic comparison is $\sigma=1$, giving sqrt(2). There is no universal sqrt(2) bound for all local nonlinearities. Exactly ratio 2 occurs at

$$\boxed{\sigma=\frac{1+\sqrt5}{2}.}$$

This is a correct changed-action construction. It is not a geometric derivation of that exponent: it was solved from the target ratio. Choosing it solely to obtain 2 is one explicit constitutive choice, not new evidence for the historical hadronic chain.

Its norm and complete energy are

$$Q=\left(\frac{(\sigma+1)\varepsilon}{g}\right)^{1/\sigma} \frac{\sqrt a}{\sigma\sqrt\varepsilon} \frac{\sqrt\pi\,\Gamma(1/\sigma)}{\Gamma(1/2+1/\sigma)},$$ $$T=\frac{\sigma}{\sigma+2}\varepsilon Q, \qquad E_\sigma=\frac{\sigma-2}{\sigma+2}\varepsilon Q.$$

For fixed common norm and coefficients the formula selects $\varepsilon$ when $\sigma\ne2$; it does not require a separate norm for every mode. The stationary branch changes under changes of physical norm, even though rescaling an eigenvector of a fixed linear Hamiltonian does not change its eigenvalue.

The frozen potential has parameter $\lambda=1/\sigma$: its bound energies are

$$E_j=-ak^2(\lambda-j)^2,\qquad j=0,1,\ldots<\lambda.$$

At the ratio-two exponent, $\lambda=(\sqrt5-1)/2<1$. It has only one bound level. There is no bound-to-bound normal gap equal to the curvature energy. Its continuum begins a distance $\varepsilon=A/(\sigma+1)$ above the stationary level, not a distance $2A$.

The full local linearization is $L_-=-a\partial_y^2+\varepsilon-g\phi^{2\sigma}$, $L_+=-a\partial_y^2+\varepsilon-(2\sigma+1)g\phi^{2\sigma}$, again with phase and translation zero modes. Its fluctuation poles must be solved using those operators, rather than the frozen curvature.

4. Band independence: an exact sufficient construction and its cost

The strongest exact positive statement uses an orthogonal product geometry and an interaction depending on the full tangential marginal. Let the ordinary spatial patch be $Q_{d-1}\times\mathbb R_y$, with product volume, and define

$$\rho_\perp(y)=\int_{Q_{d-1}}|\Psi(x,y)|^2d\mu(x).$$

Take one action on the full field with kinetic energy

$$E_{\rm kin}=\int dy\,[a\|\partial_y\Psi\|_x^2+\langle\Psi,H_\parallel\Psi\rangle_x]$$

and either $-g\iint\rho_\perp K_\ell\rho_\perp/2$ or $-g\int\rho_\perp^{\sigma+1}/(\sigma+1)$. Both are explicit same-field functionals. The first is finite-width in the normal direction but nonlocal in the tangential directions. It is an effective proposal, with this nonlocality stated rather than concealed.

For any normalized tangential eigenmode $H_\parallel u_N=E_Nu_N$, the field $\Psi=u_N(x)\phi(y)e^{-i(E_N+\mu)t/\hbar}$ solves the complete product-model equation whenever $\phi$ solves the normal equation above. Its marginal is exactly $|\phi|^2$. Thus the normal width, well and normal frequency increments are independent of $N$, without dividing by a contact multiplicity. The total frequency retains the tangential $E_N$. No claimed simplex law for $E_N$ is supplied by this calculation.

If $P_N$ are actual spectral-band projectors of this same $H_\parallel$, the normal operator is $I_\parallel\otimes H_f$. For the inclusion $J_N:\mathcal B_N\to\mathcal B_{N+1}$, $$(J_N^\dagger\otimes1)(1_{\mathcal B_{N+1}}\otimes H_f)(J_N\otimes1) =1_{\mathcal B_N}\otimes H_f.$$

The nonlinear density also agrees on the embedded field. This is a genuine compatible bare family under canonical inclusions. It does not replace the changing projective coherent evaluation maps in the maintained point-contact construction. Nor does it prove that the full master action factorizes this way.

By contrast, a local tangential interaction generally produces $g_N=g\int|u_N|^{2\sigma+2}d\mu$, or its quartic analogue, after projection. Those moments are band dependent unless independently constrained. A non-product metric or a moving tangential projector can also couple normal and tangential modes. The band-blind construction therefore requires an actual product kinetic structure, reducing tangential bands, common physical normal norm/branch, and the marginal interaction. Calling the coefficient band blind does not establish these conditions.

Reciprocal pairs, orientation and relative phase

The same marginal action also supplies a precise two-pattern diagnostic. In a common orthogonal carrier let $$\Psi(x,y,t)=u_1(x)\phi_1(y-Y_1)e^{-i\omega_1t+i\theta_1} +u_2(x)\phi_2(y-Y_2)e^{-i\omega_2t+i\theta_2}.$$ Here each $\omega_i$ is its own isolated carrier frequency; synchronization is not imposed. For unit tangential norms, put $c_{12}=\langle u_1,u_2\rangle$ and $\vartheta(t)=\theta_2-\theta_1-(\omega_2-\omega_1)t$. The exact marginal is $$\rho_\perp=|\phi_1|^2+|\phi_2|^2 +2\operatorname{Re}\left(c_{12}e^{i\vartheta}\phi_1^*\phi_2\right).$$ Inserting this into the one Hartree functional retains all phase and polarization interference terms. When $c_{12}=0$, the pair interaction contribution reduces to $$E_{12}=-g\iint\rho_1(y-Y_1)K_\ell(y-y')\rho_2(y'-Y_2)\,dy\,dy'.$$ It depends only on $Y_1-Y_2$, so $F_1=-\partial_{Y_1}E_{12}=-F_2$. Each endpoint mean field is the functional derivative of this one pair energy. For nonzero $c_{12}$, phase-dependent interference remains and cannot be replaced by a density-only collision count. The two-center preparation need not be stationary; the nonlinear equation determines its subsequent response.

For a conditional two-component transverse control, write a unit polarization as $e=(\cos\chi,e^{i\delta}\sin\chi)$. Then $$c_{12}=\cos\chi_1\cos\chi_2 +e^{i(\delta_2-\delta_1)}\sin\chi_1\sin\chi_2.$$ This retains linear, circular and elliptical preparations and is invariant when both are rotated by the same physical transverse rotation. It is a classical polarization control, not a derived photon solution. Higher-dimensional unit polarizations use the same inner product, without imposing photon transversality universally. The normal direction and all geometric data must rotate together under a common spatial rotation; changing only one pattern changes its overlap. The scalar normal law has no special depth-five coefficient or acceptance rule. It supplies no fixed-axis or rare-alignment effect at five. A full cross-depth encounter additionally needs the actual common-field embeddings; independently chosen component spaces do not define those embeddings.

5. Ordinary warped geometry: exact profile and explicit band dependence

A geometric control uses the complete ordinary spatial metric

$$ds_d^2=dy^2+R^2\cosh^2(ky)\,h_{d-1},\qquad p=d-1,$$

where $h_{d-1}$ is a dimensionless metric on a closed ordinary spatial manifold with a normalized constant mode, for example a sphere. A closed spatial sphere here is a stated global physical geometry, not compactified extra dimensions. The metric is a prescribed candidate neck; no equation deriving it from $\Psi_\infty$ is claimed. $R$ and $k^{-1}$ are common geometric inputs.

For $-\Delta_hu_j=\lambda_ju_j$, the scalar kinetic energy $a\int|\nabla\Psi|^2d\mu_g$, with the same canonical symplectic measure, gives normal operator

$$-a[\partial_y^2+pk\tanh(ky)\partial_y] +\frac{a\lambda_j}{R^2}\operatorname{sech}^2ky.$$

The exact unitary measure transformation $\chi=\cosh^{p/2}(ky)f$ yields

$$H_j=-a\partial_y^2+\frac{ap^2k^2}{4} -a\left[\frac{p(p-2)k^2}{4}-\frac{\lambda_j}{R^2}\right]\operatorname{sech}^2ky.$$

Thus geometry creates a normal well for the constant tangential mode only when $p>2$, or $d>3$. The transformed ground mode is $\chi_0\propto\operatorname{sech}^{(d-3)/2}(ky)$; the physical field profile is

$$f_0\propto\operatorname{sech}^{d-2}(ky),\qquad E_0=a k^2(d-2).$$

It is finite norm for $d>3$, since its physical weighted density is proportional to $\operatorname{sech}^{d-3}(ky)$. This is an actual stationary scalar solution in the prescribed full metric, with total energy $E_0Q$ and frequency $E_0/\hbar$. It is not a spinor identification or a self-generated geometry.

The asymptotic threshold is $ap^2k^2/4$, so the depth relative to threshold, not the absolute value at the origin, is $A_g=ak^2(d-1)(d-3)/4$. Canonical curvature gives

$$\boxed{R_g(d)=\frac4{\sqrt{(d-1)(d-3)}}\quad(d>3).}$$

No maintained depth gives ratio 2 under this law. Depth five is not uniquely selected: it belongs to the same threshold sequence and has the same cubic-well ratio as the flat cubic control. Its marginally allowed next level is a non-normalizable threshold state, not a discrete oscillator spacing.

More decisively, changing the tangential mode changes $A_g$ by $-a\lambda_j/R^2$. At $\lambda_j\ge R^2k^2p(p-2)/4$ the attractive well disappears. The normal profile and normal eigenvalues are therefore not band blind. Restoring the measure correctly exposes this dependence even though the geometric action has no explicit band label.

6. Cross-depth scope and particle limitations

The product normal construction uses precisely one ordinary normal direction at every tested physical depth $d=2,3,4,5,6,10$. The same $a,g,\ell$, marginal law and branch norm give exactly the same one-dimensional normal result at all six depths. This is a controlled normal-channel comparison; it neither identifies that direction with all physical dimensions nor supplies tangential particle localization. Depth five has no special effect in this family. The scalar normal calculation also does not supply photon transverse polarization, handedness, spin, independent pattern rotation or a neutrino suppression rule. A factorized transverse vector control retains its two components without changing the normal result, but physical photon propagation and polarization dynamics remain unsolved.

Extending a normal construction to an isotropic full-space field is a changed problem. Fixed-norm dilation $\psi_L(x)=L^{-d/2}\psi(x/L)$ gives $T\sim L^{-2}$ and the local-power attraction $\sim-L^{-d\sigma}$. At the ratio-two exponent every maintained $d\ge2$ has $d\sigma>2$, so the corresponding unrestricted flat full-space energy is unbounded below under concentration. This does not disprove a normal envelope supported by different tangential dynamics, and it does not establish an observed decay process; it blocks calling that full-space extension a stable particle solely from its normal solution.

For the rotationally invariant full-space extension $(1-\ell^2\Delta_d)^{-1}$, Fourier scaling gives attraction $\sim-L^{-(d-2)}$ at small $L$ for $d>2$, with a logarithmic leading behavior at $d=2$. Its spatial Green kernel is singular for $d\ge2$, unlike the finite central value of the one-dimensional kernel. Thus the concentration comparison is subcritical at depths 2 and 3, coefficient-critical at 4, and attraction-dominated at 5, 6 and 10. The exact one-dimensional sech-squared profile is not automatically a solution of that full radial extension. These are scope checks with one shared law, not per-depth retuning or complete stability classifications.

The linear warped-geometry solution has a selected metric and a spatial neck, hence an external location in its current form. Deriving an inertially translating self-supported particle would require the same field to generate and move that geometry. The product Hartree solution has reciprocal self-response and Galilean motion, but its negative binding energy is only the reduced envelope energy and cannot serve as complete relativistic rest energy. These constructions remain useful kernel tests with explicit missing physical bridges.

7. Reusable conclusion and verification

The exact Hartree normal kernel is the principal reusable result. It supplies full variation, a finite-width stationary branch, canonical energy, an actual frozen excitation spacing, and responding zero modes. Its normal-only form can be made exactly consistent across cumulative bands with a specified marginal action. That action's master-field origin, full geometry and relativistic normalization remain premises to derive.

The exact power family supplies a second reusable identity: ratio 2 can be engineered without a Gaussian, but it neither uniquely determines the nonlinear law nor produces the requested oscillator gap. The warped-neck calculation gives a useful independent obstruction: a natural geometric normal well generally inherits tangential band dependence through the kinetic operator.

Accordingly, this bounded family does not close the hadronic mass calculation. It does establish why a future candidate must state which frequency it predicts and must check its kinetic measure and surviving band dependence before comparing a curvature number with a mass residual. The electron-anchored cross-sector normalization is a separate issue analyzed in the canonical count-response calculation.

The portable exact checker verifies the profile and Hartree response equations, energy and contraction identities, bound excited profile, full responding translation/phase zero modes, generic power family, unitary warped-metric transform, all six depth controls and canonical coordinate invariance. Run it with Python and SymPy; it writes a dated JSON report beside the script without a database or private inputs. Symbolic identities establish the stated exact branches; no numerical convergence or full stability claim is inferred from them.