Curvature of the complete normal band

The finite normal band in A complete finite normal band is exact on a product of its surface and normal coordinates. An ordinary Euclidean tube is different: its tangential radius varies across the normal profile. Retaining that variation mixes normal bands but gives a small, explicit correction in the tested narrow-profile regime.

The calculations below concern the declared linear parent and background. They do not derive the background from the master field. A small residual in this equation is not yet a particle fixed point or an inertia proof.

Keep the actual radial metric

For a round S_R^m feature in ordinary R^(m+2), write r=R+y_1 and z=y_2. The containing metric is

$$ds^2=dr^2+dz^2+r^2d\Omega_m^2,\qquad r>0. \tag{1}$$

The unitary factor r^(-m/2) converts its radial measure to dr dz. The corresponding first radial derivative includes m/(2r) before that unitary conversion. The following normal fields and their norms use the converted representation. This keeps the physical volume factor rather than assuming an exact product metric.

Set x=y_1/a and epsilon=a/R. At a fixed angular spinor eigenvalue beta, the target frequency term in the parent becomes

$$\tau_x\frac{\beta}{1+\epsilon x}. \tag{2}$$

The normal coefficients, R, a, gamma and M are held common between the assigned modes. This is a specific extension of the declared parent into the tube, not a derivation of its normal potentials. It uses no compactification or additional physical dimensions.

Retain every forced normal band

In the fixed-number basis, multiplication by x changes the target normal degree by one. It retains both chirality components. With w_0 the complete parent frequency and psi_0 its full normalized field, expand

$$\begin{aligned} H(\epsilon)&=H_0+\sum_{j\ge1}\epsilon^jV_j, &V_j&=(-1)^j\beta\tau_x x^j,\\ w(\epsilon)&=w_0+\sum_{j\ge1}\epsilon^j w_j, &\psi(\epsilon)&=\psi_0+\sum_{j\ge1}\epsilon^j\psi_j. \end{aligned} \tag{3}$$

V_j acts on the target components; the additional components still enter the complete resolvent and canonical norm. Use intermediate normalization <psi_0,psi_j>=0 for j>0. The coefficient equations are

$$\begin{aligned} w_k&=\left\langle\psi_0, \sum_{j=1}^{k}V_j\psi_{k-j}\right\rangle,\\ (w_0-H_0)\psi_k &=\sum_{j=1}^{k}(V_j-w_j)\psi_{k-j}. \end{aligned} \tag{4}$$

The inverse in the base band is restricted to the phase-orthogonal complement. In every other forced band it includes all parent branches and both signs. At order k, every possible normal degree lies between q-k and q+k. Thus a coefficient calculation has a finite complete set of paths, without dropping an infinite tail at that order.

The second coefficient provides a separate useful check:

$$w_2=\langle\psi_0,V_2\psi_0\rangle \,+\sum_{s=\pm1} \left\langle V_1\psi_0, (w_0-H_{q+s})^{-1}V_1\psi_0\right\rangle. \tag{5}$$

Normal parity makes every odd frequency coefficient vanish. It does not make the first field correction vanish: that correction occupies both adjacent bands.

The assigned modes have small relative corrections

At gamma=10000 and M=10^18, write w/w_0=1+c_2 epsilon^2+c_4 epsilon^4+c_6 epsilon^6+... . The complete coefficient calculation gives

The first field-correction norms are 5.49399714, 5.45149742 and 11.57966292, respectively. These multiply epsilon. Their values come from the complete spatial response, not from dividing a large source by a guessed single gap.

The near equality of the first two c_2 values makes their leading ratio correction only about -0.03944564 epsilon^2. This is a conditional geometric comparison at fixed coefficients. It does not select epsilon, explain why a physical particle chooses the band, or make the finite-gamma frequency law exact.

schwinger-tube-20260914T034426_921830Z.json passes 54 checks. Independent 100- and 120-digit calculations agree, the full second-order resolvent agrees with the recurrence, and every generated component satisfies its coefficient equation. A separate finite-curvature control diagonalizes the complete canonical parent through different normal cutoffs. Its error relative to the sixth-order series falls from about 1.32e-15 at epsilon=.004 to 5.16e-18 at epsilon=.002. This is the expected eighth-order scaling. Normal cutoff refinement and reversal of epsilon agree independently.

Bound the full spatial residual

A Taylor series by itself does not prove convergence of the complete curved problem. Instead, take its explicit field through order K=6, psi_app, and its frequency w_app, and bound their residual directly. The field is a finite sum of actual Hermite functions, including every generated normal component.

Keep a smooth radial cutoff chi equal to one for x>=-1/(4 epsilon) and zero for x<=-1/(2 epsilon). Its transition can be chosen with |chi'|<=2 pi epsilon. The resulting spatial field vanishes before the radial origin and has r>=R/2 wherever it is nonzero. No Gaussian is silently extended through r=0.

On this retained region, the exact rational remainder obeys

$$\left|\frac1{1+\epsilon x} -\sum_{j=0}^{K}(-\epsilon x)^j\right| \le2\epsilon^{K+1}|x|^{K+1}. \tag{6}$$

Let r_poly be the complete residual after replacing (2) by the polynomial through degree K, with all higher powers generated by psi_app and w_app still retained. Let P_t denote the target components. The normal derivative commutator with chi then gives

$$\begin{aligned} \|(H-w_{\rm app})\chi\psi_{\rm app}\| \le{}&\|r_{\rm poly}\| +2|\beta|\epsilon^{K+1}\|x^{K+1}P_t\psi_{\rm app}\|\\ &+2\pi\epsilon\sqrt{M/2}\, \|\psi_{\rm app}\|_{x<-1/(4\epsilon)}. \end{aligned} \tag{7}$$

All norms are complete spatial norms. Hermite polynomial coefficient bounds and an upper incomplete-gamma bound control the final tail. The same estimate bounds the small norm removed by chi. Thus (7) can be divided by the retained full norm without losing normalization.

Under the illustrative hierarchy epsilon=10^(-9), gamma=10000 and M=10^18, the computed right sides of (7), divided by the frequency and full norm, are approximately 9.39e-43, 1.09e-44 and 5.04e-29 for the three assigned controls. These numerical evaluations need their high-precision and spatial checks; they are not interval-arithmetic certificates. The tail is retained analytically even though ordinary floating-point arithmetic would underflow it to zero.

For a self-adjoint realization of this declared parent, any normalized field with residual delta satisfies

$$\|e^{-itH}\psi-e^{-itw_{\rm app}}\psi\|\le |t|\delta. \tag{8}$$

This follows directly by integrating its forced evolution. It provides a finite-time approximate-mode statement and a nearby spectral value. It does not by itself prove an isolated exact eigenstate, nonlinear stability, or a self-supported particle.

The stated width hierarchy is compatible with the parent's finite-speed bound when c_t<=sqrt(2/3)c. Its width, high-frequency normal coefficients, background energy and selection still require a same-field derivation. Those physical questions remain separate from the small residual in (7).

schwinger-quasimode-20260914T035406_949092Z.json passes 67 checks. The analytic Hermite tail bounds agree with independent integrals, and direct spatial quadrature of the rational radial term lies below (7). The full residual is stable between 120 and 140 decimal digits. The initial checker used too little precision when subtracting unit-size terms to verify a remainder near 10^(-67); its failed report and exact source are preserved. Increasing that check's precision resolves the cancellation without changing the mode construction or the residual bound.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.