A uniform replacement for the forward spectral-gap argument

20 October 2026. New self-contained repair based on the user-supplied Web GPT Pro argument, with explicit constants and a coordinating mathematical check. This is not a claim that the printed §2 proof was complete. It replaces its global/fiberwise orthogonality inference, the omitted small-positive-remainder case and the tube argument. No later theorem is imported to prove the comparison below.

Statement and exact scope

Let $M=\Gamma\backslash\mathbb H$ be a fixed connected hyperbolic surface with an isolated simple lowest eigenvalue $\lambda_0$, a positive real ground state $\phi$ normalized by $\int_M\phi^2=1$, and a scalar gap $\gamma_0>0$:

Here $Q(h)=\int_M|\nabla h|^2$ is the closed Laplace quadratic form; $\gamma_0$ means the bottom of the spectrum on the ground-state orthogonal complement, even if there is no second discrete eigenvalue. For the BGS $\delta>1/2$ setting, positivity, simplicity and isolation of $\lambda_0$ are the required fixed-base spectral inputs. Their external derivation is not reproved in this note.

Fix a finite generating set $S$ of $\Gamma$. Let $\rho$ vary over the finite-quotient regular representations, with invariant unit vector $e$, and suppose expansion supplies one fixed $\kappa>0$ such that

For equivariant sections $F(sz)=\rho(s)F(z)$, we prove

where $c>0$ is independent of representation dimension and cover degree. The proof uses the actual quotient representation; identifying a congruence image with all of $\mathrm{SL}_2(\mathbb Z/q\mathbb Z)$ and establishing expansion for its full modulus range remain separate arithmetic inputs. The reverse implication from a surface gap to expansion is not proved here.

Separate the invariant component

Write $F=h e+F_\perp$, where $F_\perp(z)\perp e$ pointwise. The scalar $h$ is automorphic and satisfies $h\perp\phi$. Because these orthogonal fiber projections are constant and commute with differentiation, both the $L^2$ norm and the energy split orthogonally. The scalar part costs at least $\gamma_0\|h\|_2^2$. It remains to bound the pointwise perpendicular part.

Normalize $\|F_\perp\|_2=1$ and define

Integration by parts using $\Delta\phi=\lambda_0\phi$ gives the ground-state transform

For compactly supported smooth sections this is a direct identity. For the full quadratic-form domain, take the closure of that core: multiplication by $\phi$ identifies the weighted norm/form with the original shifted norm/form. The resulting identity does not require the pointwise norm of $F_\perp$ to equal $\phi$.

Small energy forces mass in a fixed ball

Put $f=\|F_\perp\|$. The weak Kato inequality gives $Q(f)\le Q(F_\perp)$, and $\|f\|_2=1$. Write

The scalar gap implies

Choose a fixed small compact ball $B$ inside a fundamental cell, and set $\beta=\int_Bw>0$. If $E<\gamma_0\beta/16$, then

Thus $\int_Bw\|V\|^2\ge\beta/4$. This estimate applies to every sufficiently small positive $E$, rather than only to an exact ground-state norm.

One compact weighted Poincaré estimate

Choose a fixed connected compact Lipschitz region $D\subset\mathbb H$ containing $B$ and all $sB$, $s\in S$. A locally finite fundamental-cell tiling covers $D$ using finitely many cells, say $N$. The integrand $w\|\nabla V\|^2$ is invariant, since $\rho$ is unitary. Therefore

On $D$, the positive smooth weight $w$ has positive minimum and finite maximum. The scalar weighted Poincaré inequality on this fixed region, summed over orthonormal coordinates, gives its weighted mean $v$ with

where $A=C_DN$ is independent of representation dimension. Since $V(z)\perp e$ everywhere, also $v\perp e$.

The mass estimate on $B$ yields

If also $E<\beta/(16A)$, then $\|v\|\ge1/4$.

For each $s\in S$, equivariance and change of variables from $B$ to $sB$ give

Expansion now forces

A cover-independent lower bound

Take

If $E<c_\perp$, all preceding inequalities hold and the last one implies $E\ge\kappa\beta/(64|S|A)\ge c_\perp$, a contradiction. Thus $E\ge c_\perp$. Restoring arbitrary norms and the invariant component gives the claimed gap with

Every constant comes from the fixed base surface, fixed generators or the uniform quotient-expansion constant. In particular, no constant depends on the number of components. The same lifted calculation works for the discrete-group orbifold formulation, choosing $B$ away from elliptic fixed points and using the scalar orbifold quadratic form; ineffective central elements simply restrict the allowed equivariant sections.

Checking conclusion: this supplies a complete replacement for the forward expansion-to-spectral-gap comparison under the stated fixed-base spectral and uniform-expansion inputs. It is not an audit of those imported inputs, the reverse comparison, the transfer-operator machinery, uniform lattice-point counting or the affine-sieve deductions.