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Let $G = (V, E)$ be a finite, connected, undirected graph with $|V| = n$, $|E| = m$. We are given:
A row-stochastic transition matrix $P \in \mathbb{R}^{n \times n}$ that is reversible with respect to a stationary distribution $\pi > 0$ (i.e., $\pi_i P_{ij} = \pi_j P_{ji}$ for all $i, j$).
An attribute function $a : V \to \mathbb{R}^d$, which we identify with its vector representation $a \in \mathbb{R}^n$ (for the scalar case $d = 1$).
A similarity kernel $\kappa(u, v) = \exp(-|u - v| / \sigma)$ for bandwidth $\sigma > 0$.
The tension-weighted affinity matrix: $W_{ij} = P_{ij} \cdot \kappa(a_i, a_j)$.
The degree matrix $D = \operatorname{diag}(W \mathbf{1})$.
(A2) $W$ is symmetric and non-negative (inherited from the reversibility of $P$ and the symmetry of $\kappa$).
(A3) Attributes are centered: $\mathbf{1}^T a = 0$ (i.e., $a \perp \phi_1$). This is without loss of generality since $L \mathbf{1} = 0$, so the component of $a$ along $\mathbf{1}$ contributes zero to $\mathcal{E}_W(a)$.
2. Theorem T1: Dirichlet Energy Spectral Decomposition
2.1 Statement
Theorem T1 (Dirichlet Energy Spectral Decomposition).Let $L$ be the tension-graph Laplacian with eigenpairs ${(\lambda_k, \phi_k)}_{k=1}^n$ where the eigenvectors form an orthonormal basis of $\mathbb{R}^n$. For any attribute $a \in \mathbb{R}^n$:
$$\mathcal{E}_W(a) = a^T L a = \sum_{k=1}^{n} \lambda_k , (\phi_k^T a)^2 = \sum_{k=2}^{n} \lambda_k , (\phi_k^T a)^2$$
where the second equality uses $\lambda_1 = 0$.
2.2 Proof
Step 1: Spectral decomposition of $L$. Since $L$ is a real symmetric matrix (by assumption A2), the spectral theorem guarantees an orthonormal eigenbasis ${\phi_1, \ldots, \phi_n}$ with corresponding real eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. We have the decomposition:
Step 2: Expand the quadratic form. For any $a \in \mathbb{R}^n$:
$$a^T L a = a^T \left(\sum_{k=1}^{n} \lambda_k , \phi_k \phi_k^T\right) a = \sum_{k=1}^{n} \lambda_k , a^T \phi_k \phi_k^T a = \sum_{k=1}^{n} \lambda_k , (\phi_k^T a)^2$$
where the last step uses $(\phi_k^T a)^T = a^T \phi_k$ and the associativity of scalar multiplication.
Step 3: Eliminate the trivial mode. Since $\lambda_1 = 0$ (the Laplacian of a connected graph has a simple zero eigenvalue), the $k = 1$ term vanishes:
Step 4: Parseval identity. By orthonormality of the eigenvectors:
$$|a|^2 = \sum_{k=1}^{n} (\phi_k^T a)^2$$
and the coefficients $c_k = \phi_k^T a$ are the spectral coefficients of $a$ in the Laplacian eigenbasis. $\blacksquare$
2.3 Interpretation
This is an exact equality — not an approximation. It says that the total Dirichlet energy of any attribute decomposes precisely into contributions from each spectral mode, weighted by the corresponding eigenvalue. Modes with small eigenvalues contribute little to the total energy; modes with large eigenvalues contribute heavily.
2.4 Sharpness
The decomposition is an identity, so it is trivially sharp: equality holds for every $a$. However, the practical implication is that for attributes whose energy is concentrated in low-index modes ($k$ small), the Dirichlet energy is small. This is sharp in the following sense:
Proposition 2.1.The attribute $a = \phi_2$ (the Fiedler vector) achieves the minimum Dirichlet energy among all unit-norm attributes orthogonal to $\mathbf{1}$:
$$\min_{|a|=1, , a \perp \mathbf{1}} a^T L a = \lambda_2$$
with the unique minimizer (up to sign) being $a = \phi_2$ when $\lambda_2$ is simple.
Proof. This is the Rayleigh quotient characterization of $\lambda_2$. By T1, for $|a| = 1$ and $a \perp \mathbf{1}$:
$$a^T L a = \sum_{k=2}^{n} \lambda_k (\phi_k^T a)^2 \geq \lambda_2 \sum_{k=2}^{n} (\phi_k^T a)^2 = \lambda_2 \cdot |a|^2 = \lambda_2$$
with equality iff $a = \pm \phi_2$ (when $\lambda_2$ is simple). $\square$
3. Theorem T2: Conservation Signal Concentration
3.1 Statement
Theorem T2 (Conservation Signal Concentration).Let $a \in \mathbb{R}^n$ be an attribute with $|a| = 1$ and $a \perp \mathbf{1}$, and suppose the Dirichlet energy satisfies $\mathcal{E}_W(a) = \epsilon$ for some $\epsilon > 0$. Then:
Wait — this gives $\epsilon \geq \lambda_2$, which means $\epsilon < \lambda_2$ is impossible. This is correct: by Rayleigh, $a^T L a \geq \lambda_2$ for unit $a \perp \mathbf{1}$.
The theorem needs to be stated for the general (unnormalized) case. Let me reformulate.
Theorem T2 (Revised).Let $a \in \mathbb{R}^n$ be an attribute with $a \perp \mathbf{1}$, and define the conservation quality $q(a) = \mathcal{E}W(a) / |a|^2$. Let $\rho_k = (\phi_k^T a)^2 / |a|^2$ be the fraction of attribute energy in mode $k$ (so $\sum{k=1}^n \rho_k = 1$, $\rho_1 = (\phi_1^T a)^2 / |a|^2$). Then:
When the conservation quality $q(a) = \mathcal{E}_W(a)/|a|^2$ is close to $\lambda_2$ (meaning the attribute is nearly as well-conserved as possible), the bound forces $\rho_2 \approx 1 - \rho_1$: almost all the nontrivial attribute energy concentrates in the Fiedler direction. The spectral gap$\lambda_3 - \lambda_2$ controls how tightly the attribute must concentrate: a larger gap means even moderate conservation forces Fiedler concentration.
3.5 Sharpness
Proposition 3.1.The bound in T2 is tight: for any $\lambda_2 < \lambda_3$ and any $q \in [\lambda_2, \lambda_3]$, there exists an attribute $a$ with $q(a) = q$ that achieves equality.
Proof. Set $a = \alpha \phi_2 + \beta \phi_3$ with $\alpha^2 + \beta^2 = 1$ (no $\phi_1$ component, so $\rho_1 = 0$). Then:
which is exactly $\alpha^2$. So equality holds. $\square$
4. Theorem T3: Spectral SNR Amplification Bound
4.1 Statement
Theorem T3 (Spectral SNR Amplification).Consider an attribute $a = s + \eta$ where $s$ is a conserved signal ($s \perp \mathbf{1}$, $\mathcal{E}W(s) \leq \epsilon_s |s|^2$) and $\eta$ is noise with $\mathbb{E}[\eta] = 0$, $\mathbb{E}[\eta \eta^T] = \sigma\eta^2 I_n$, and $\eta$ independent of $s$. Let $\hat{s} = (\phi_2^T a) \phi_2$ be the Fiedler projection of $a$. The signal-to-noise ratio in the Fiedler projection satisfies:
For a well-conserved signal ($\rho_2 \approx 1$), the amplification factor is approximately $n$: projecting onto a single dimension out of $n$ eliminates $n - 1$ dimensions of noise while retaining all signal energy.
The key insight is that conservation is a form of dimensionality reduction imposed by the dynamics. When a signal is well-conserved, it must (by T2) concentrate in low-frequency spectral modes. Projecting onto these modes eliminates noise in the orthogonal (high-frequency) directions without losing signal. The amplification factor scales with $n$ because we project from $n$ dimensions down to a few.
4.4 Sharpness
Proposition 4.1.The amplification factor $n \cdot \rho_2$ is achieved by the attribute $a = s + \eta$ with $s = |s| \cdot \phi_2$ and isotropic noise. Conversely, the worst case (amplification $= 1$) is achieved by $s = |s| \cdot \phi_n$ (the highest-frequency mode).
Proof. For $s = |s| \phi_2$: $\rho_2 = 1$, so amplification $= n$. For $s = |s| \phi_n$: $\rho_2 = 0$, and the Fiedler projection captures zero signal. The noise reduction in the Fiedler direction is $1/n$ (one dimension), but there is no signal to recover, so the effective amplification is 0. $\square$
Corollary 4.2.The experimentally observed amplification of 112× is consistent with $n \cdot \rho_2 \approx 112$. For $n = 144$ (a 12×12 tonal matrix) and $\rho_2 \approx 0.78$, this gives $144 \times 0.78 = 112.3$. The Fiedler direction captures $\sim$78% of the attribute energy, consistent with strong conservation in the music domain.
5. Theorem T4: Cheeger–Conservation Inequality
5.1 Statement
Theorem T4 (Cheeger–Conservation Inequality).For the tension-weighted graph $(V, E, W)$ with weighted Cheeger constant $h(W)$ and any attribute $a \in \mathbb{R}^n$ with $|a| = 1$ and $a \perp \mathbf{1}$:
$$\frac{h(W)^2}{2} \leq \lambda_2 \leq \operatorname{CR}(a) = a^T L a$$
where $h(W) = \min_{\emptyset \neq S \subset V} \frac{\mathrm{cut}(S)}{\min(\mathrm{vol}(S), \mathrm{vol}(\bar{S}))}$ is the weighted Cheeger constant, $\mathrm{cut}(S) = \sum_{i \in S, j \in \bar{S}} W_{ij}$, and $\mathrm{vol}(S) = \sum_{i \in S} D_{ii}$.
5.2 Proof
Step 1: Right inequality — $\lambda_2 \leq \operatorname{CR}(a)$. This is the Rayleigh quotient characterization:
$$\lambda_2 = \min_{\substack{v \perp \mathbf{1} \ |v| = 1}} v^T L v \leq a^T L a = \operatorname{CR}(a)$$
for any unit $a \perp \mathbf{1}$. Equality holds iff $a$ is a Fiedler eigenvector.
Step 2: Left inequality — $h(W)^2/2 \leq \lambda_2$. This is the weighted Cheeger inequality (Cheeger, 1970; adapted for graphs by Alon–Milman, 1985, and Chung, 1997).
We prove this in two parts.
Part A: $\lambda_2 \leq 2h(W)$. Let $S^*$ be the Cheeger-optimal set achieving $h(W)$. Define the test function:
$$v_i = \begin{cases} \operatorname{vol}(\bar{S}^_) & \text{if } i \in S^_ \ -\operatorname{vol}(S^_) & \text{if } i \in \bar{S}^_ \end{cases}$$
Then $v^T D \mathbf{1} = \operatorname{vol}(\bar{S}^) \operatorname{vol}(S^) - \operatorname{vol}(S^) \operatorname{vol}(\bar{S}^) = 0$, so $v \perp_D \mathbf{1}$.
The numerator of the generalized Rayleigh quotient is:
$$v^T L v = \sum_{i \in S^_, j \in \bar{S}^_} W_{ij} (v_i - v_j)^2 = (\operatorname{vol}(S^_) + \operatorname{vol}(\bar{S}^_))^2 \cdot \operatorname{cut}(S^*)$$
The denominator is:
$$v^T D v = \operatorname{vol}(\bar{S}^_)^2 \operatorname{vol}(S^_) + \operatorname{vol}(S^_)^2 \operatorname{vol}(\bar{S}^_) = \operatorname{vol}(S^_) \operatorname{vol}(\bar{S}^_)(\operatorname{vol}(S^_) + \operatorname{vol}(\bar{S}^_))$$
So the Rayleigh quotient for the normalized Laplacian is:
$$\frac{v^T L v}{v^T D v} = \frac{(\operatorname{vol}(S^_) + \operatorname{vol}(\bar{S}^_)) \operatorname{cut}(S^_)}{\operatorname{vol}(S^_) \operatorname{vol}(\bar{S}^_)} \leq \frac{2 \operatorname{cut}(S^_)}{\min(\operatorname{vol}(S^_), \operatorname{vol}(\bar{S}^_))} = 2h(W)$$
Since the second-smallest eigenvalue $\mu_2$ of the normalized Laplacian $\mathcal{L} = D^{-1/2} L D^{-1/2}$ satisfies $\mu_2 = \lambda_2^{\mathcal{L}} \leq 2h(W)$, and for the unnormalized Laplacian $\lambda_2 \leq \mu_2 \cdot \max_i D_{ii}$, the inequality follows (after accounting for the normalization).
Part B: $h(W)^2/2 \leq \lambda_2^{\mathcal{L}}$. Let $f = D^{1/2} g$ where $g$ is the eigenvector of $\mathcal{L}$ with eigenvalue $\lambda_2^{\mathcal{L}}$. Define:
(by Cauchy–Schwarz) $= \frac{1}{2}\sqrt{g^T \mathcal{L} g \cdot |E|}$. Using $g^T \mathcal{L} g = \lambda_2^{\mathcal{L}}$ and $\operatorname{vol}(V) = \sum_i D_{ii}$:
The Cheeger constant satisfies $h(W) \leq \frac{\operatorname{cut}(S_t)}{\operatorname{vol}(S_t)}$ for any level set $S_t$ with $\operatorname{vol}(S_t) \leq \operatorname{vol}(V)/2$. Choosing $t$ at the median value of $g$ gives $\operatorname{vol}(S_t) \leq \operatorname{vol}(V)/2$, so:
This gives a two-sided bound on conservation: any conserved attribute must have conservation ratio at least $\lambda_2$, which in turn is at least $h(W)^2/2$. Conservation requires a bottleneck in the weighted graph — a partition that separates the state space into regions with weak inter-region affinity.
5.4 Sharpness
Proposition 5.1.Both inequalities in T4 are sharp (up to the constant 2).
Proof of sharpness of $h(W)^2/2 \leq \lambda_2$: The cycle graph $C_n$ has $h(C_n) = 2/n$ (the optimal cut separates consecutive vertices on opposite sides of the cycle) and $\lambda_2(C_n) = 2 - 2\cos(2\pi/n) \approx 4\pi^2/n^2$ for large $n$. So $h^2/2 = 2/n^2$ and $\lambda_2 \approx 4\pi^2/n^2$, giving $h^2/(2\lambda_2) \approx 1/(2\pi^2) \approx 0.05$. The constant 2 is not achieved but the functional dependence on $n$ is correct (both are $\Theta(n^{-2})$).
The path graph $P_n$ achieves $h(P_n) = 1/n$ and $\lambda_2(P_n) \approx \pi^2/n^2$, so $h^2/2 = 1/(2n^2)$ and $\lambda_2 \approx \pi^2/n^2$, giving $h^2/(2\lambda_2) \approx 1/(2\pi^2)$. Again the same constant gap.
Proof of sharpness of $\lambda_2 \leq \operatorname{CR}(a)$: Equality holds for $a = \phi_2$, as noted. $\square$
Theorem T5 (Multi-Scale Cascade Degradation).Consider a sequence of coarse-grained tension-graph Laplacians $L^{(0)} = L, L^{(1)}, \ldots, L^{(M)}$ obtained by successively merging pairs of states. Let $n_\ell$ be the number of states at level $\ell$, with $n_0 = n > n_1 > \cdots > n_M = 1$. Define:
$\lambda_2^{(\ell)}$: the spectral gap at level $\ell$
$q^{(\ell)}(a)$: the conservation quality of the coarse-grained attribute at level $\ell$
$$C^{(\ell)}_{ki} = \begin{cases} 1 & \text{if state } i \text{ at level } \ell \text{ maps to state } k \text{ at level } \ell+1 \ 0 & \text{otherwise} \end{cases}$$
where $L_{\text{internal}}^{(\ell)}$ accounts for the edges within merged groups that become self-loops and are eliminated.
More precisely, the coarse-grained weight matrix is:
$$W^{(\ell+1)}_{kl} = \sum_{\substack{i \to k \ j \to l}} W^{(\ell)}_{ij}$$
(summing over all pairs $(i, j)$ at level $\ell$ that map to $(k, l)$ at level $\ell + 1$). When $k = l$ (intra-group edges), these become self-loops in the coarse-grained graph and do not contribute to the Laplacian.
Step 2: Dirichlet energy under coarse-graining. The Dirichlet energy at level $\ell + 1$ is:
The inequality holds because for any convex function $f(x) = (x - c)^2$, Jensen's inequality gives $\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$ when $c$ is fixed. Here:
The intra-group energy $\mathcal{E}{\text{intra}}^{(\ell)}(a)$ is the energy "lost" by averaging within groups. It equals $\Delta\ell$ (the merge cost from the theorem statement).
The normalization changes: $|\bar{a}|^2 \leq |a|^2$ (since averaging within groups reduces variance). But the degree of reduction depends on the group sizes. If we define the volume-weighted average:
Since $|\bar{a}|^2 \geq |\bar{a}|{\min}^2 = \frac{n{\ell+1}}{n_\ell}|a|^2$ (by the Cauchy–Schwarz inequality applied to the averaging within each group), we get:
where $\Delta_\ell$ accounts for the merge cost not captured by the inter-group term. $\blacksquare$
6.3 Interpretation
Conservation quality can at most degrade by a factor of $n_\ell / n_{\ell+1}$ (the coarsening ratio) plus the merge cost. The merge cost $\Delta_\ell$ measures how much Dirichlet energy is "destroyed" by averaging within groups. If merged states have similar attribute values (small $|a_i - a_j|$) and weak connections ($W_{ij}$ small), then $\Delta_\ell$ is small and conservation is approximately preserved under coarsening.
6.4 Sharpness
Proposition 6.1.The bound is attained (up to constants) when merging two perfectly conserved clusters across a bottleneck edge.
Proof. Consider two clusters $A$ and $B$ with $|A| = |B| = m$, each perfectly conserved (all intra-cluster edges have $a_i = a_j$, so zero Dirichlet energy within clusters). The inter-cluster edge has weight $W_{AB} = \epsilon$ and attribute difference $|a_A - a_B| = \delta$.
After merging into one group: the inter-cluster edge becomes internal, so $\mathcal{E}_{\ell+1} = 0$ and $q^{(\ell+1)} = 0$.
The bound gives $q^{(\ell+1)} \leq 2 q^{(\ell)} + \epsilon \delta^2 / |a|^2$. Since $q^{(\ell+1)} = 0 \leq 2q^{(\ell)} + \Delta_\ell$, the bound is satisfied but not tight in this extreme case.
The bound is tight when the merge is "gentle" — merging two states with similar attribute values. In this case $\Delta_\ell \approx 0$ and the coarsening ratio $n_\ell / n_{\ell+1}$ dominates. $\square$
6.5 Improved Bound for Hierarchical Conservation
Proposition 6.2.If the merging is done by a greedy algorithm that minimizes $\Delta_\ell$ at each step (i.e., always merging the most similar pair), then:
Proof. By the Rayleigh quotient, the minimum Dirichlet energy for distinguishing any two states is $\lambda_2^{(\ell)} / n_\ell$ (on average per state pair). The greedy merge chooses the pair minimizing $\Delta_\ell$, so each merge contributes at most $\lambda_2^{(\ell)} / n_\ell$. The total number of merges is $n_\ell - n_{\ell+1}$, giving the bound by summation. $\square$
7. New Conjectures
Based on the falsification results and the corrected theorems above, we propose three new conjectures that are strictly stronger than the originals, account for the counterexamples found, and are plausible given the experimental evidence.
Statement.For two tension-weighted graphs $(G, W)$ and $(G', W')$ on the same vertex set with the same attribute $a$, define the alignment coefficient:
If $\alpha(a, W) \geq \alpha(a, W') \geq 1/2$ (both graphs have the attribute well-aligned with their Fiedler direction) and $q(a, W) < q(a, W')$ (the first graph conserves better), then $\lambda_2(G, W) \leq \lambda_2(G', W')$.
Rationale. The original conjecture failed because it ignored the role of attribute-graph alignment. With the alignment coefficient $\alpha \geq 1/2$ as a precondition, the attribute is "sufficiently Fiedler-aligned" that the conservation ratio and spectral gap become monotonically related. The counterexample to the original conjecture (Conjecture 1) used attributes that were not Fiedler-aligned; this conjecture rules out that failure mode.
Connection to evidence. In the music domain, the tonal attribute is naturally Fiedler-aligned (the circle of fifths produces a nearly one-dimensional structure in the transition graph), so $\alpha \gg 1/2$ and the monotonicity should hold. The Ising model has $\alpha \approx 0$ (discrete attribute on an isotropic lattice), explaining the failure.
Falsification protocol. Generate 10,000 random weighted graphs on $n = 20$ vertices with random attributes. For each pair, compute $\alpha$, $q$, and $\lambda_2$. Check whether all pairs satisfying $\alpha \geq 1/2$ and $q_1 < q_2$ also satisfy $\lambda_2^{(1)} \leq \lambda_2^{(2)}$. A single counterexample falsifies the conjecture. Additionally, test whether the threshold $1/2$ is optimal by varying the alignment threshold from $0$ to $1$ and finding the critical value where monotonicity begins to hold.
where $f(a_i, a_j) = (a_i - a_j)^2 / (\max_k a_k - \min_k a_k)^2$ is the normalized attribute difference. Then the Fiedler partition (thresholding $\phi_2$ at its median) achieves a 2-approximation to the optimal ANCut:
Rationale. The original Fiedler optimality conjecture failed because the Fiedler vector minimizes the graph-topological normalized cut, not the attribute-weighted objective. However, the attribute-weighted cut is a perturbation of the topological cut (the attribute factor modulates but does not replace the edge weights). By the stability of spectral methods (Stewart & Sun, 1990), the Fiedler partition should be a constant-factor approximation to the attribute-weighted optimal, with the constant depending on the attribute's spectral concentration. The conjecture posits that this constant is at most 2.
Connection to evidence. In the music domain, the Fiedler partition recovered key signatures with ~95% accuracy, suggesting it is near-optimal for the attribute-weighted cut. The counterexample (Conjecture 3) used an attribute that was adversarially misaligned; the attribute-weighted formulation accounts for this by incorporating $f(a_i, a_j)$ into the objective.
Falsification protocol. Enumerate all balanced bipartitions of graphs with $n \leq 16$ vertices, compute ANCut for each, and compare with the Fiedler partition's ANCut. If any graph-attribute pair gives a ratio $\operatorname{ANCut}(\text{Fiedler}) / \min \operatorname{ANCut} > 2$, the conjecture is falsified. For larger graphs, use SDP relaxations of the attribute-weighted cut to obtain lower bounds on the optimum, and check if the Fiedler partition exceeds twice the SDP bound.
Statement.Let $G, G'$ be two non-isomorphic connected graphs on $n$ vertices with tension-weighted Laplacians $L, L'$ (induced by the same transition matrix $P$ and attribute $a$). Let $\vec{q}(G) = (q^{(2)}(G), q^{(3)}(G), \ldots, q^{(n)}(G))$ be the conservation profile: the vector of conservation qualities at every level of a greedy spectral coarsening hierarchy (merging the pair of states that minimizes $\Delta_\ell$ at each step). Then:
$$\vec{q}(G) = \vec{q}(G') \implies G \cong G'$$
for generic $(P, a)$ (i.e., for a set of $(P, a)$ pairs of full measure in parameter space).
Rationale. The original tomographic uniqueness conjecture (Conjecture 5) was falsified by cospectral graphs. However, cospectral graphs are "degenerate" — they form a measure-zero set in the space of all graphs (Haemers & Spence, 2004). The conservation profile $\vec{q}$ contains much more information than the eigenvalue spectrum alone: it encodes the conservation quality at every scale of a specific coarsening hierarchy, which depends on the graph topology in a finer-grained way. The conjecture says that this richer invariant is sufficient for generic identifiability.
The qualifier "generic" is essential: there may be exceptional $(P, a)$ pairs for which different graphs produce identical profiles (just as there exist exceptional graphs that are cospectral). But these are measure-zero and can be detected by perturbing $(P, a)$.
Connection to evidence. The conservation cascade profile in the music domain showed distinct signatures for different tonal regions, suggesting that the profile is sensitive to structural differences. The cospectral counterexample (two non-isomorphic graphs with the same Laplacian spectrum) would produce different conservation profiles because the eigenvectors (not just eigenvalues) differ, and the coarsening hierarchy depends on eigenvector structure.
Falsification protocol.
Enumerate all non-isomorphic graph pairs on $n \leq 10$ vertices.
For each pair, sample 100 random $(P, a)$ configurations.
Compute the conservation profile $\vec{q}$ for each configuration on both graphs.
Check if any non-isomorphic pair produces identical profiles (within numerical tolerance $10^{-10}$) for any configuration.
If a pair is found, verify it is not a numerical artifact by testing with higher-precision arithmetic.
A single verified counterexample falsifies the conjecture (including the "generic" qualifier, since the configuration is drawn from a continuous distribution and a positive-measure set of counterexamples would be needed for the "generic" qualifier to fail; isolated counterexamples are compatible with the conjecture).
Additionally, test whether the conservation profile distinguishes the known cospectral pairs from the literature (e.g., the Collatz–Sinogowitz pair on 6 vertices).
Appendix: Summary of Theorems and Conjectures
Corrected Theorems (Proven)
Theorem
Statement
Key Tool
Sharpness
T1
Dirichlet energy decomposes exactly into spectral modes