A finite-scale bound for the complete spatial mass branch

The bounded completion permits more than a formal leading-order statement. Its angular structure and complementary gap give an explicit bound on the remaining mass-frequency correction, including the infinite radial complement.

The theorem below is an operator estimate. Its numerical evaluation uses independently refined radial coefficients, rather than rigorous interval enclosures of those coefficients. The large scale separations used as examples establish a sufficient mathematical regime; they are not selected physical lengths or particle-specific adjustments.

The full complement here is the infinite normal plane at a fixed chosen surface eigenvalue $\beta$, within the maintained separated carrier. The theorem does not establish that separation in a complete curved physical sector or control interactions with every other surface harmonic.

A gap above the two dressed endpoints

Write the scaled completed operator as

$$\mathcal H(\epsilon)=A_g+\epsilon B+\epsilon^2 C_2, \qquad \epsilon=1/\Lambda. \tag{1}$$

Here $B=\widetilde B_-+\widetilde B_-^\dagger$, and $C_2=C_{--}+C_{--}^\dagger$. Let $P$ project onto the two normalized zero modes of $A_g$, and $Q=I-P$.

Using the unperturbed geometric gap $\delta=2/\sqrt{d+1}$, set

$$v=g\sqrt d,\qquad \mu=g\min_l h_l-\frac{v^3}{\delta(\delta-v)},\qquad \boxed{\eta=\frac{\min(\mu,\delta-v)} {[1+v/(\delta-v)]^2}>0.} \tag{2}$$

The endpoint-isolation conditions of the completion note are assumed. For an interior angular sector, eliminate the complement of the original normal kernel at zero energy. Its inverse norm is at most $1/(\delta-v)$; its chiral Schur correction is bounded by the cubic term in (2). The remaining two-state Schur block therefore has smallest singular value at least $\mu$.

The triangular factors in this elimination and their inverses have norm at most $1+v/(\delta-v)$. Applying them to the inverse gives (2). In an endpoint sector the Schur block is zero by its unpaired chirality; the same argument applies to the inverse on the orthogonal complement of its dressed zero mode. Outside these angular sectors the original complementary gap applies. Hence

$$\boxed{\|(QA_gQ)^{-1}\|\le1/\eta.} \tag{3}$$

This estimate concerns the full spatial Hilbert space. It does not replace that space by a finite radial matrix.

All added coefficients have global bounds

The neighboring term satisfies

$$\|B\|\le b:=|\beta|+\frac{d+1}{2}+gd. \tag{4}$$

The last contribution uses $r^2(1-w^2)\le a\le d$. The two-step term requires a bound on $r^3(1-w^2)$. Let $R_0=\sqrt{2d}$. For $r\ge R_0$, the same flux inequality gives $w^2\ge1/2$. With $q=-\log w$,

$$\Delta q=2(1-e^{-2q})\ge2q, \qquad 0\le q\le\tfrac12\log2. \tag{5}$$

The maximum principle, applied to the decaying exponential supersolution, gives

$$1-w^2\le2q\le(\log2)e^{-\sqrt2(r-R_0)}. \tag{6}$$

Thus, with $r_* =\max(R_0,3/\sqrt2)$, a global coefficient bound is

$$M_d=\max\left(R_0^3,(\log2)r_*^3e^{-\sqrt2(r_*-R_0)}\right), \qquad \boxed{\|C_2\|\le c_2:=gM_d/4.} \tag{7}$$

No numerical far-field cutoff is used to assert boundedness.

Bound the zero-energy Schur series

For complex $\epsilon$, let $\nu(\epsilon)=|\epsilon|b+|\epsilon|^2c_2$. The complementary inverse exists whenever $\nu<\eta$. Choose the analytic radius

$$\boxed{\rho=\frac{\eta}{b+\sqrt{b^2+2c_2\eta}}, \qquad \rho b+\rho^2c_2=\eta/2.} \tag{8}$$

At zero spectral parameter the exact Schur operator is

$$F(0,\epsilon)=P\mathcal HP -P\mathcal HQ(Q\mathcal HQ)^{-1}Q\mathcal HP. \tag{9}$$

It is analytic inside this circle, and its norm on the circle is at most $\nu+\nu^2/(\eta-\nu)=\eta$. Chiral symmetry makes its two diagonal entries zero. Its off-diagonal entry $t(\epsilon)$ has the angular expansion

$$t(\epsilon)=a_d\epsilon^d+a_{d+2}\epsilon^{d+2}+\cdots, \qquad a_d=\frac{F_d(g)}{g^{d-1}}\prod_{l=0}^{d-1}(n+l). \tag{10}$$

The two endpoints differ by angular weight $d$. A neighboring step costs one power of $\epsilon$, and a double step costs two. No smaller power connects them. Reversing a step adds an even number of powers, which supplies the parity of the series.

Cauchy's coefficient bound now gives an explicit tail:

$$\boxed{ |t(\epsilon)-a_d\epsilon^d| \le\Delta:=\eta\frac{x^{d+2}}{1-x^2}, \qquad x=|\epsilon|/\rho<1.} \tag{11}$$

For a maintained nonzero index, division by $\ell=|a_d\epsilon^d|$ gives a relative error proportional to $\epsilon^2$, with all coefficients specified. The bound is conservative at high depth, but it resolves the small mass branch without using an absolute operator-norm error as though it were its frequency.

The physical eigenfrequency also retains a positive metric

For real $\epsilon$, put $a=\eta-\nu$. If $\ell+\Delta<a$, the exact energy-dependent elimination has

$$F(E)=F(0)-E[I+K(E)],\qquad K(E)=V^\dagger A_Q^{-1}(A_Q-E)^{-1}V\ge0, \tag{12}$$

where $A_Q=Q\mathcal HQ$ and $V=Q\mathcal HP$. Positivity follows because $A_Q$ and $A_Q-E$ have the same spectral sign below the complementary gap. The norm is bounded by

$$\|K(E)\|\le\kappa:= \frac{\nu^2}{a[a-(\ell+\Delta)]}. \tag{13}$$

Monotonicity of the two Schur eigenvalues then bounds the positive small eigenfrequency of $\mathcal H$:

$$\boxed{\max\left(0,\frac{\ell-\Delta}{1+\kappa}\right) \le E\le\ell+\Delta.} \tag{14}$$

The actual dimensionless parent frequency is $\Lambda E$, and the physical frequency is $c\Lambda E/R$. The relative bound is unchanged by these conversions. The metric term is positive and accounts for the norm of the eliminated spatial field; it is not an additional mass parameter.

Numerical scale of this sufficient regime

With one common trial $g=0.02$ and $\gamma=10^7$ for the comparison, the refined radial inputs give these approximate worst relative Schur-tail bounds among the maintained controls at each depth:

The positive metric bound is included separately in the complete report. Depth ten is the demanding control; the estimate at substantially smaller separation need not be informative. These are sufficient error estimates, not a physical determination of $R/L$ or evidence that such large separation is realized. Equal scales across depths are not asserted.

The exact theorem uses the true hopping integrals and endpoint coefficient. The displayed numbers use converged approximations to them. Their refinement error is much smaller than the displayed margins, but a machine-certified numerical theorem would require interval enclosures.

Frequency calibration exposes the length hierarchy

The large-separation example also has a physical cost that the error bound does not resolve. Let $\ell_{\rm ref}=c/\omega_{\rm ref}$ be the frequency-equivalent length of one reference particle at a depth. Fixing its mass-frequency in the leading operator fixes

$$\boxed{\frac{R_d}{\ell_{\rm ref}} =\frac{|F_d(g)|}{\gamma^{d-1}} \prod_{l=0}^{d-1}(n_{\rm ref}+l),\qquad \frac{L_d}{\ell_{\rm ref}}=\frac g\gamma \frac{R_d}{\ell_{\rm ref}}.} \tag{15}$$

For the same illustrative $g=0.02,\gamma=10^7$, the implied leading ratios are:

These are profile-length ratios under the specified calibration, not additional compactified spatial directions. They do not establish the validity of the effective action at those lengths. A smaller separation would make the hierarchy less severe, but needs a sharper finite-scale calculation where the conservative high-depth bound is uninformative. The complete same-field supporting energy and length selection must therefore be solved together. Spectral error control alone does not provide a physically selected cavity scale.

Verification

angular_mass_remainder_checks.py passes 588 checks. It verifies the global coefficient comparisons, evaluates all maintained controls at two physical couplings, and retains both informative and uninformative finite-scale bounds. An independent exact finite carrier checks (14) against its small singular value and its zero-energy Schur product. The input report and raw radial arrays are identified by their complete hashes. The 588-check remainder report retains every evaluated bound, including regimes where it does not resolve the small frequency.