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Generic Schmidt-Rank Width of Fixed Weighted Phase Networks: Maximum-Matching Width, Template Diversity, and Single-Parameter Global Saturation

Byoungwoo Lee

Published Aug 9, 2026• 0 citations
Publisher

Abstract

This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph $G=(V,E)$ with a fixed real edge-weight template $W=(W_e)_{e\in E}$, we consider the commuting weighted-phase evolution $$U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),$$ with a single scalar evolution parameter $t$. For every bipartition $A\vert{}A^c$, let $\nu_G(A)$ be the maximum matching number of the crossing graph $G[A,A^c]$. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates $W$ such that, outside one locally finite exceptional set of evolution times, the same scalar $t$ simultaneously saturates the matching capacity of every nontrivial bipartition: $$\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)$$ for all nontrivial cuts $A\vert{}A^c$. Consequently, $$\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)$$ for generic fixed weighted-phase networks, where $\chi_{\mathrm{wd}}$ is Schmidt-rank width and $\operatorname{mmw}(G)$ is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support $K_n$, $$\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,$$ so almost every nonuniform fixed template attains $$\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,$$ which is the largest Schmidt-rank width permitted for any $n$-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is $$\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},$$ with $\operatorname{ord}(e^{i\theta})=\infty$ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: $$\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,$$ $$\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),$$ $$\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).$$ This separation is not restricted to dense complete graphs. Since $$\operatorname{rw}(G) \le \operatorname{mmw}(G),$$ graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse $O(n)$-coupler networks with generic Schmidt-rank width $\Theta(n)$. The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension $$D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),$$ whereas generic nonuniform complete templates require $$D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.$$ Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.

DOI
10.5281/zenodo.21861894

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quant-pharXiv:2609.21209•Sep 21, 2026•0 citations
This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph G=(V,E)G=(V,E)G=(V,E) with a fixed real edge-weight template W=(We)e∈EW=(W_e)_{e\in E}W=(We​)e∈E​, we consider the commuting weighted-phase evolution UW(t)=exp⁡(it∑e={u,v}∈EWenunv),U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),UW​(t)=exp(it∑e={u,v}∈E​We​nu​nv​), with a single scalar evolution parameter ttt. For every bipartition A∣AcA\vert{}A^cA∣Ac, let νG(A)\nu_G(A)νG​(A) be the maximum matching number of the crossing graph G[A,Ac]G[A,A^c]G[A,Ac]. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates WWW such that, outside one locally finite exceptional set of evolution times, the same scalar ttt simultaneously saturates the matching capacity of every nontrivial bipartition: log⁡2SR⁡A∣Ac(∣ΨW(t)⟩)=νG(A)\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)log2​SRA∣Ac​(∣ΨW​(t)⟩)=νG​(A) for all nontrivial cuts A∣AcA\vert{}A^cA∣Ac. Consequently, χwd(∣ΨW(t)⟩)=mmw⁡(G)\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)χwd​(∣ΨW​(t)⟩)=mmw(G) for generic fixed weighted-phase networks, where χwd\chi_{\mathrm{wd}}χwd​ is Schmidt-rank width and mmw⁡(G)\operatorname{mmw}(G)mmw(G) is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support KnK_nKn​, mmw⁡(Kn)=⌈n3⌉,\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,mmw(Kn​)=⌈3n​⌉, so almost every nonuniform fixed template attains χwd=⌈n3⌉,\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,χwd​=⌈3n​⌉, which is the largest Schmidt-rank width permitted for any nnn-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is χwdunif=log⁡2min⁡{⌈n3⌉+1, ord⁡(eiθ)},\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},χwdunif​=log2​min{⌈3n​⌉+1,ord(eiθ)}, with ord⁡(eiθ)=∞\operatorname{ord}(e^{i\theta})=\inftyord(eiθ)=∞ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: Clifford⟶χwd=1,\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,Clifford⟶χwd​=1, generic uniform⟶χwd=Θ(log⁡n),\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),generic uniform⟶χwd​=Θ(logn), generic nonuniform⟶χwd=Θ(n).\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).generic nonuniform⟶χwd​=Θ(n). This separation is not restricted to dense complete graphs. Since rw⁡(G)≤mmw⁡(G),\operatorname{rw}(G) \le \operatorname{mmw}(G),rw(G)≤mmw(G), graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse O(n)O(n)O(n)-coupler networks with generic Schmidt-rank width Θ(n)\Theta(n)Θ(n). The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension Dunif=⌈n3⌉+1=Θ(n),D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),Dunif​=⌈3n​⌉+1=Θ(n), whereas generic nonuniform complete templates require Dgen=2⌈n/3⌉=2Θ(n).D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.Dgen​=2⌈n/3⌉=2Θ(n). Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.
This work develops a finite-resolution counterpart of Schmidt-rank width for weighted phase networks and uses it to distinguish algebraic Schmidt-rank activation from spectrally resolved entanglement complexity. For a bipartition A∣AcA|A^cA∣Ac with Schmidt coefficients s1≥s2≥⋯s_1 \ge s_2 \ge \cdotss1​≥s2​≥⋯, define the discarded Schmidt-tail mass τrA(ψ)=(∑j>rsj2)1/2,\tau_r^A(\psi)=\left(\sum_{j>r}s_j^2\right)^{1/2},τrA​(ψ)=(∑j>r​sj2​)1/2, and the corresponding ϵ\epsilonϵ-resolved Schmidt rank RϵA(ψ)=min⁡{r:τrA(ψ)≤ϵ}.R_\epsilon^A(\psi)=\min\left\{r:\tau_r^A(\psi)\le\epsilon\right\}.RϵA​(ψ)=min{r:τrA​(ψ)≤ϵ}. The global tail-resolved Schmidt-rank width is then χres(ϵ)(ψ)=min⁡Tmax⁡e∈E(T)log⁡2RϵAe(ψ),\chi_{\mathrm{res}}^{(\epsilon)}(\psi)=\min_T\max_{e\in E(T)}\log_2R_\epsilon^{A_e}(\psi),χres(ϵ)​(ψ)=minT​maxe∈E(T)​log2​RϵAe​​(ψ), where TTT ranges over subcubic decomposition trees and Ae∣AecA_e|A_e^cAe​∣Aec​ is the bipartition induced by deleting the tree edge eee. At zero resolution, ϵ=0\epsilon=0ϵ=0, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: χres(ϵ)(∣G⟩)=log⁡2⌈(1−ϵ2) 2rw⁡(G)⌉,0≤ϵ<1,\chi_{\mathrm{res}}^{(\epsilon)}(|G\rangle)=\log_2\left\lceil(1-\epsilon^2)\,2^{\operatorname{rw}(G)}\right\rceil,\qquad0\le\epsilon<1,χres(ϵ)​(∣G⟩)=log2​⌈(1−ϵ2)2rw(G)⌉,0≤ϵ<1, where rw⁡(G)\operatorname{rw}(G)rw(G) is the graph rank-width. Consequently, graph families with linear rank-width retain linear tail-resolved Schmidt-rank width at every fixed finite resolution ϵ<1\epsilon<1ϵ<1. The paper next establishes a gauge-stable perturbation theory. After optimization over cut-local diagonal phases, Schmidt truncation tails satisfy the Lipschitz estimate $$\left|\tau_r^A(\psi)-\tau_r^A(\phi)\right|\led_{\mathrm{diag}}^A(\psi,\phi),$$ which yields a global resolution-shift sandwich for χres(ϵ)\chi_{\mathrm{res}}^{(\epsilon)}χres(ϵ)​. For balanced pairwise weighted phase networks with crossing phase errors ηe\eta_eηe​, ddiagA≤14∥ηE(A,Ac)∥2.d_{\mathrm{diag}}^A\le\frac14\left\|\eta_{E(A,A^c)}\right\|_2.ddiagA​≤41​​ηE(A,Ac)​​2​. If a fixed coupling template WWW is implemented with coupling error ΔW\Delta WΔW and timing error Δt\Delta tΔt, then η=t ΔW+Δt W+Δt ΔW,\eta=t\,\Delta W+\Delta t\,W+\Delta t\,\Delta W,η=tΔW+ΔtW+ΔtΔW, giving an explicit certificate that converts finite calibration and timing uncertainty into a guaranteed resolved-width bound. The tail-resolved width also controls approximate subcubic tree-tensor-network complexity. If DTTN(ϵ)D_{\mathrm{TTN}}^{(\epsilon)}DTTN(ϵ)​ denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error ϵ\epsilonϵ, then $$2^{\chi_{\mathrm{res}}^{(\epsilon)}}\leD_{\mathrm{TTN}}^{(\epsilon)}\le2^{\chi_{\mathrm{res}}^{\left(\epsilon/\sqrt{2n-3}\right)}}.$$ The central application concerns the Clifford point of the complete weighted phase network. Consider ∣ΨV(λ)⟩=exp⁡[i∑e(π+λVe)nunv]∣+⟩⊗n.|\Psi_V(\lambda)\rangle=\exp\left[i\sum_e\left(\pi+\lambda V_e\right)n_u n_v\right]|+\rangle^{\otimes n}.∣ΨV​(λ)⟩=exp[i∑e​(π+λVe​)nu​nv​]∣+⟩⊗n. For every fixed resolution 0<ϵ<12,0<\epsilon<\frac{1}{\sqrt{2}},0<ϵ<2​1​, the tail-resolved width remains equal to 111 for all sufficiently small ∣λ∣|\lambda|∣λ∣, independently of the perturbation direction VVV. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width ⌈n3⌉,\left\lceil\frac{n}{3}\right\rceil,⌈3n​⌉, whereas a uniform perturbation activates only log⁡2(⌈n3⌉+1).\log_2\left(\left\lceil\frac{n}{3}\right\rceil+1\right).log2​(⌈3n​⌉+1). Thus the zero-resolution and zero-perturbation limits do not commute. For an open, dense, full-measure class of nonuniform perturbation directions, lim⁡ϵ↓0lim⁡λ→0χres(ϵ)=1,\lim_{\epsilon\downarrow0}\lim_{\lambda\to0}\chi_{\mathrm{res}}^{(\epsilon)}=1,limϵ↓0​limλ→0​χres(ϵ)​=1, while lim⁡λ→0, λ≠0lim⁡ϵ↓0χres(ϵ)=⌈n3⌉.\lim_{\lambda\to0,\ \lambda\neq0}\lim_{\epsilon\downarrow0}\chi_{\mathrm{res}}^{(\epsilon)}=\left\lceil\frac{n}{3}\right\rceil.limλ→0, λ=0​limϵ↓0​χres(ϵ)​=⌈3n​⌉. This establishes a sharp distinction between algebraic Schmidt activation and finite-resolution activation: infinitesimal symmetry breaking can create maximal exact Schmidt-rank width while remaining invisible at any fixed finite spectral resolution sufficiently close to the Clifford point. The results connect weighted phase and fixed ZZZZZZ/Ising networks, graph width parameters, Schmidt-spectrum stability, and approximate tensor-network complexity. The global phase-error certificate is intentionally conservative: it does not by itself establish thermodynamic robustness against a size-independent coherent error on every edge.
We investigate bipartite entanglement in qubit hypergraph states across an arbitrary fixed bipartition. Using the real equally weighted (REW) representation, the Schmidt rank across the cut can be computed as the real rank of a phase-cleaned cross-cut sign matrix. Whereas graph states admit an exact cut-rank rule, because the cross-cut phase is purely bilinear, hypergraph states typically contain higher-degree cross-cut interactions, for which the cut-rank rule fails. Our approach certifies entanglement by fixing a single computational-basis assignment on a subset of qubits, thereby selecting a submatrix on an active slice. When this restriction removes all higher-degree cross-cut residues, the remaining cross-cut phase becomes bilinear up to cut-local terms. We call the resulting submatrices residual-free bilinear cores and show that they yield an exponential Schmidt-rank lower bound in terms of the F2\mathbb{F}_2F2​-rank of an exposed core matrix. We further give a combinatorial sufficient condition, phrased as a disjoint bridge matching, that guarantees the existence of large full-rank cores for broad families of CCZ-type bridge patterns, and we present a search-and-verify procedure that constructs and certifies such cores directly from the hyperedge description.
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×),r_{\mathrm{gen}} = 2^{\nu(G_\times)},rgen​=2ν(G×​), where ν(G×)\nu(G_\times)ν(G×​) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,qK_{p,q}Kp,q​, the exact rank is rank⁡Kθ=min⁡{p+1, q+1, ord⁡(eiθ)},\operatorname{rank} K_\theta = \min\left\{ p+1,\, q+1,\, \operatorname{ord}(e^{i\theta}) \right\},rankKθ​=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2K_{2,2}K2,2​ example therefore exhibits a rank stratification (4→3→24 \to 3 \to 24→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3.r_{\mathrm{str}}=4, \qquad r_{\mathrm{sym}}=3.rstr​=4,rsym​=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3
Strongly simulating a quantum circuit, that is, computing an output amplitude, can be done by summing the circuit's Feynman paths: a weighted count over assignments to Boolean path variables. The circuit's gates induce correlations among these variables, forming a graph whose structure controls several exact simulation routes. This sum-of-powers (SOP) viewpoint underlies recent simulators built on binary decision diagrams and weighted model counting. For a quadratic SOP with nnn variables, even modulus rrr, and a rank-decomposition of its variable graph of width kkk, our dynamic program (DP) computes an amplitude using only O(4kpoly(n))O(4^kpoly(n))O(4kpoly(n)) arithmetic operations. For Clifford+T+T+T circuits, the amplitude is given by an SOP with modulus 888. Rank-width never exceeds linear rank-width, which governs some decision-diagram approaches, and is at most one greater than the Markov--Shi contraction complexity of the circuit tensor network. Moreover, there are non-Clifford families of bounded rank-width where both competing parameters diverge. We also present a stabilizer-rank optimization, exploiting that the DP tables are stabilizer-type Gauss sums. Each subtree runs at the width price of its cut-ranks or at a magic price that discharges the non-Clifford phases below it. The resulting best total cost never exceeds O(4kpoly(n))O(4^kpoly(n))O(4kpoly(n)), yet is polynomial on mixed families where the pure rank-width and pure TTT-count guarantees are both exponential. Clifford amplitudes take polynomial time on any graph, the exact-amplitude consequence of Gottesman--Knill. A prototype evaluation on standard circuit benchmarks finds treewidth bucket elimination the strongest baseline, with the new rank-width DP complementary: it wins on structured families where treewidth blows up.
Graph states provide a bridge between graph theory and multipartite quantumentanglement, with cut-rank, rank-width, and related parameters playing a centralrole in both structural graph theory and the characterization of stabilizer states.Building on this interplay, we introduce a new scalar invariant—the S M NazmuzSakib entanglement rank entropy—defined as the Shannon entropy of the cut-rankdistribution over vertex subsets of fixed cardinality. For a simple graph G = (V, E)of order n, and k ∈ {1, . . . , n − 1}, we consider the random variable ρG(X) whereX ⊆ V is a uniformly random k-subset, and we define the Sakib entanglementrank entropy H(k)ρ (G) as the Shannon entropy (in bits) of its distribution.Forgraph states, where ρG(X) coincides with the Schmidt rank exponent across thebipartition (X, V \ X), H(k)ρ (G) is precisely the Shannon entropy of the bipartiteentanglement (in ebits) seen by random size-k bipartitions.We formally define the S M Nazmuz Sakib entanglement rank entropy, provethat it is invariant under local complementation and hence a local-Clifford invariantfor graph states, and study a balanced version Hbalρ (G) = H(n/2)ρ(G) (for even n).We then perform a data-based analysis on explicit graph families (paths, cycles,stars, complete graphs, and 2 × m grid graphs) of size up to n = 12, computingHbalρ (G), the average cut-rank, and the full cut-rank distributions. The resultingdataset shows that, while complete and star graphs have vanishing Sakib entangle-ment rank entropy, more “distributed” structures such as paths, cycles, and gridsexhibit increasing entropy with system size, despite similar ranges of cut-rank.Motivated by these observations, we formulate the S M Nazmuz Sakib entan-glement heterogeneity hypothesis, proposing that, at fixed rank-width and averagecut-rank, a larger Sakib entanglement rank entropy correlates with higher classicalsimulation complexity of the corresponding graph states.We outline how pub-licly available graph-state orbit datasets and average cut-rank techniques can becombined with the proposed invariant to further test this hypothesis.
High-dimensional bipartite entanglement depends on how probability is distributed across the Schmidt spectrum, whereas a single reference-state fidelity resolves only one spectral scale. We develop multiscale Schmidt-spectrum bounds that connect geometric measures with quantitative Schmidt-number witnesses. For any partition of a pure-state Schmidt spectrum, several nested Vidal tails determine block masses. We derive the sharp upper boundary of the associated normalized nuclear-norm coordinate and show that equality holds if and only if the spectrum is uniform within each block. Refining the partition gives a monotone hierarchy of tighter bounds whenever the added tail data distinguish unequal block means. We also solve a relaxed weighted multiscale optimization globally: one scalar parameter specifies its unique full-support optimizer and explicit value on the nontrivial branch. Using the established single-tail fidelity--resource curve as a baseline, we obtain exact-fidelity equality refinements, convex-roof lower bounds for mixed states, and quantitative calibrations of Schmidt-number witnesses. A higher-tail relation further bounds convex-roof extended negativity in terms of Vidal tails and identifies the pure-state equality spectra. Leakage-aware and joint-confidence formulations state how these bounds can be used with incomplete data. Multistep tails require block-resolved or independently certified spectral information; they are not determined by one projector expectation.