Second adjacency eigenvalue of the inner-face Apollonian stages

Rad Sarar Anando's MathOverflow question 516405, revision 1, GUID 8C9D6CD5-E3A3-4B25-BDB6-FF5A9A3D3A2B, asks whether the second largest ordinary adjacency eigenvalue, counted with multiplicity, satisfies

$$\lim_{m\to\infty}\theta_2(G_m)/2^{m/2}=\sqrt2.$$

Here $G_0$ is plane $K_4$ with outer triangle $abc$ and centre $o$. Every step inserts one vertex in each inner triangular face and joins it exactly to the three corners; the outer triangle is never subdivided. The source note attributes the quotation and adapted exposition to the author under CC BY-SA 4.0.

The dominant old vertices have exponentially growing degree, while their common late neighbours grow only linearly. A fixed old cutoff exposes this separation through the Gram matrix of the old-to-late adjacency block.

The exact all-stage spectral limit is Proved by D5.S3.Combinatorics.Apollonian.SecondEigenvalueLimit.result : InnerFaceStages.limitClaim. It quantifies along every natural stage, including odd stages, without an additional degree or spectral hypothesis. This is the formal completion of the accepted ordinary proof attributed in the source note, rather than a claim to a new ordinary solution or worldwide priority. The forward preregistration is issue #24989, comment 6550398124. It retains unit 24989, cutoff 2026-11-01T11:00:00Z and predictions: []; the earlier unit 24905 settlement is unchanged.

Fix $k$ and partition stage $m>k$ into $H=V(G_k)$ and later vertices $L$. The old-to-late block $B$ has Gram diagonal $\deg_{G_m}(v)-\deg_{G_k}(v)$ and off-diagonal entries at most $2(m-k)$. An inner vertex born at $j$ has degree $3,2^{m-j}$; the centre has degree $3,2^m$ and each outer seed vertex has degree $2^{m+1}+1$. Thus $2^{-m}BB^\top$ converges on fixed $H$ to a diagonal matrix whose second largest entry is $2$.

Edges induced on $L$ orient from newer to older endpoints, with outdegree at most $3$ and indegree at most $3,2^{m-k-1}$. The directed-edge Cauchy--Schwarz bound makes the normalized remainder at most $3\sqrt2,2^{-k/2}$ asymptotically. Ordered eigenvalue perturbation, positive Gram/dilation spectrum with multiplicities, and then $k\to\infty$ give the accepted ordinary proof. This route does not assume its graph bounds.

Changing the outer-face convention, inserting one random face at a time, using weighted or normalized adjacency, replacing the ordered second eigenvalue by the spectral radius, or proving only a finite-stage or subsequence statement changes the question. The posted numerical values and preliminary bounds are not assumptions.

The finite source construction and its graph laws belong to InnerFaceStages, with their canonical exposition. The exact proposition proved by SecondEigenvalueLimit.result is InnerFaceStages.limitClaim; secondEigenvalue uses the decreasing real ordinary-adjacency spectrum, with index one and multiplicities, and the normalization is a real power. cutoffEquiv is the exact old/late vertex bijection, and cutoff_adjacency identifies every entry with the corresponding ordinary adjacency block. The final result combines these combinatorial statements with the spectral and analytic interfaces below. The actual late adjacency norm interface belongs to CutoffSpectralBounds. Its bound uses the proved birth orientation and the finite Schur operator bound. For every cutoff $k$ and later increment $t\ge1$, it gives $|C|/2^{(k+t)/2}\le3\sqrt2/2^{k/2}$ with real exponents, including odd stages. The fixed-cutoff matrix convergence belongs to CutoffGramLimit: $2^{-(k+t)}BB^\top$ tends to the diagonal with weights $\operatorname{faceCount}(k,u)/2^k$ on the fixed actual vertex set. The real unequal-block identity belongs to CutoffOrderedSpectrum. It identifies the second decreasing eigenvalue of $[[0,B],[B^\top,0]]$ with the square root of the second Gram eigenvalue when the latter is positive. The positive prefix retains signs, zero padding and algebraic multiplicities; the old and late vertex sets need not have equal sizes. The final proof identifies the limiting second Gram eigenvalue as two, proves eventual positivity, transports the cutoff spectrum with its multiplicities, and controls the actual full adjacency remainder. Its canonical exposition records the Proved source-resolution claim.

The unfinished registration audit remains separate from mathematical proof under CLAUDE.md section 3.9. There are 27 public targets: 21 retained records, comprising eight validated and thirteen unfinished records, plus six faithful failed attempts with no retained source. No target lacks an attempt; nineteen lack a completed registration. The thirteen retained gaps are nine InnerFaceStages records (three source.observation_data, six source.params), CutoffGramLimit.normalized_gram_tendsto, both public CutoffSpectralBounds bounds and SecondEigenvalueLimit.result (four source.selection_missing in total). These retain IE-C050 and no certificate. The three source.observation_data targets are InnerFaceStages.adjacent_embed, inner_faces_distinct and late_edge_oriented; the six source.params targets are cutoff_adjacency, cutoff_gram_diag, cutoff_gram_offdiag, degreeCount_late_le, late_oriented_indegree_le and old_vertices_iff. The six failed targets are InnerFaceStages.faceCount_new, the B-hosted Schur norm theorem, RectangularDilation.eigenvalues_ofReal, both unequal-dilation identities and the cutoff ordered-spectrum identity. Their respective readings are failure of observational dependence for the constant-three count, absence of the RCLike (ULift Real) negative witness, contract.discovery:rigid_universes, lifted finite-matrix transparency failure for both unequal identities, and absence of a compiled finite second-Gram positivity witness. Existing valid sources and binding evidence remain in place; the resolution claim does not assert complete registration.

• Proved: the centre has limiting Gram weight three, the three outer seeds have weight two, and every other weight is at most two. Thus the second decreasing limiting eigenvalue is exactly two, with multiplicity.
• Proved: fixed-cutoff Gram convergence gives convergence of its ordered second eigenvalue and eventual positivity. The unequal-block dilation identity then yields the cutoff second-eigenvalue limit $\sqrt2$.
• Proved: exact vertex transport decomposes the adjacency error into the fixed old block and the actual late induced block. After normalization, the former tends to zero with the stage and the latter is bounded by $3\sqrt2/2^{k/2}$. Choosing the cutoff first and then the stage proves the full natural-stage limit, including odd stages. The graph bounds are consequences of the construction, not supplied spectral assumptions.
• Unresolved adjacent questions: limits for further leading eigenvalues, rates of convergence, and variants using other face conventions or operators are not settled by this theorem. The fixed-cutoff method gives a route to examine them, but the necessary limiting weights, ordered positive-prefix conditions and remainder estimates must be checked for each target. No new such theorem is asserted here.
• Source scope: this resolves the stated second-eigenvalue question at the mathematical level by formalizing the attributed ordinary argument. The source's broader question about other leading eigenvalues remains outside this result. The nineteen unfinished registration targets remain audit obligations under the linked issue.

The bounded documentary source and literature screen does not establish historical priority or exhaustive worldwide openness. The explicit unread literature limits are retained in the source note.