We compute the entanglement entropy of a closed conjugate Weil pair $\{c,q-c\}$ along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly ${S_{\mathrm{ent}}}(n)=\log {r_{\mathrm{pair}}}(n)$, the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions, ${r_{\mathrm{pair}}}(n)$ is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of $10^{-6}$. The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is $\log 2$ conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-$q$ task is to measure the contraction law of ${r_{\mathrm{pair}}}(n)$ and compare its exponent with the pair-capacity exponent $\delta_{\mathrm{pair}}$.
We compute the entanglement entropy of a closed conjugate Weil pair {c,q−c} along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly Sent(n)=logrpair(n), the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions, rpair(n) is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of 10−6. The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is log2 conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-q task is to measure the contraction law of rpair(n) and compare its exponent with the pair-capacity exponent δpair.
The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair {c,q−c}, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection Πq is spectrally atomic, equal to span{e0,eξc,eq−ξc}; removing the zero mode leaves the two-dimensional Born–Infeld parity pair {eξc,eq−ξc}, identified with the spin-21 representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so Sentadm=log2 for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank rpair(n)=R∞−R(n) defined on the full Gram–Schmidt basis (rank q at saturation, not the rank-3 admissible projection). E1 proves Sent(n)=log(rpair(n)) unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis [Hres]. The metaplectic dilation ϕω is excluded as a bridge for [Hres] because it acts between blocks, not within the residual support of one block.
The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies wj=1 for every admitted Weil fingerprint direction, across q∈{29,61,101,151} and multiple generic blocks. The equality was left as an open analytic strengthening. We close it here by explicit computation. The k=3 Weil fingerprint of Heis3(Z/qZ) for any block (c1,c2,c3) is, up to a global phase, the discrete Fourier mode u^A=(e2πiAt/q)t=0q−1 at frequency A=c1b1+c2b2+c3b3modq, where (b1,b2,b3) are the b-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving wj=∥wjGS∥/∥vj∥=1 exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified; conditional on the companion paper's supplied purity of the joint state, this fixes the diagonal Schmidt-partner pairing. For general canonical blocks, the equal-count identity is recovered, while the literal Schmidt pairing remains conditional on a canonical block identification. Combined, these results prove analytically that, conditional on the companion paper's supplied purity of the joint state, the Schmidt spectrum of the conjugate pair is flat on the residual support for matched blocks, and the reduced entropy is Sent(n)=logrpair(n), where rpair(n)=R∞−R(n) is the residual rank.
A recent inelastic-neutron-scattering experiment on the heavy-fermion strange metal Ce3Pd20Si6 reports a scale-free increase of the quantum Fisher information (QFI) density as the strange-metal state forms, witnessing high multipartite entanglement. We ask whether this signature is reproduced by the non-injective projection framework of Cosmochrony, and we isolate the structural condition under which it is. We first reduce the QFI density to the projective spectral data: using linear response, the QFI density is a weighted integral of the imaginary part of the dynamical susceptibility, whose infrared behaviour is controlled by the projective stability gap ΔΠ and the edge spectral weight of the projective stability operator LΠ. This yields a clean dichotomy. A finite gap forces the QFI density to saturate; a gap that closes with an anomalous edge density of states of exponent a forces fQ(T)∼T1−a, so that scale-free growth requires a>1. We then evaluate the two ingredients by exact diagonalisation. The single winding sector w=2 reduces to the critical almost-Mathieu (Harper) operator at flux 2/q; its gap closes as ΔΠ∼q−1 but its edge spectral dimension is ds→2, giving a→1, the marginal boundary, hence only logarithmic growth. The full Cayley substrate of Heis3(Z/qZ), whose homogeneous growth dimension is D=4, instead has an edge spectral dimension ds≈4, giving a≈2 and fQ(T)∼T−1, a genuine power law. The conclusion is structural: in this framework the scale-free QFI growth is a property of the full nilpotent substrate and not of any isolated composite, consistent with the collective, multipartite character of the measured entanglement. The required thermal closure ΔΠ(T)∼T is obtained at the level of the effective heat-kernel response: the framework identifies the diffusion scale with inverse temperature, and the induced linear-response flow gives ΔΠ(T)=2dsT, so that the chain from the nilpotent substrate to the linear prediction fQ(T)∼T−1 closes within the effective response regime. The predicted exponent is of the right order but steeper than observed. Numerically, the Born–Infeld bounded-response correction can soften the effective exponent from the linear value αth=1 towards the observed αexp≈0.7, with a size-stable crossing at a partial substrate saturation. The asymptotic exponent and the saturation level realised by the material remain open.
Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most 4, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.
The quantum structure sub-programme comprises three companion papers on the binary icosahedral group 2I, plus one independently published result. Q1 proves an exact, unconditional Fourier-support rigidity theorem in the finite Weil representation of the Heisenberg group, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics; it establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule. Q2 proves an exact finite-group fact: the spin-21 and spin-23 representations of 2I induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or SU(2) fixed-point/sector-selection argument. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis: given that a bipartite state on a supplied composition is invariant under the diagonal action of 2I — not derived from admissibility or from Born–Infeld indiscernibility — the state is uniquely the SU(2) singlet, with an explicit Casimir correlator, for all five admissible sectors of 2I; it does not derive that invariance hypothesis, phase coherence, or the Born rule. No chain currently connects these three results to each other or to the structures of quantum mechanics. The Bell paper, published independently in Quantum Reports, establishes the structural non-applicability of Bell-type factorizability in non-injective effective descriptions; the proof of that central theorem uses only non-injectivity and informational completeness. This note records the current, narrow scope of each constituent and the open problems that remain.
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∣Ac with Schmidt coefficients s1≥s2≥⋯, define the discarded Schmidt-tail mass τrA(ψ)=(∑j>rsj2)1/2, and the corresponding ϵ-resolved Schmidt rank RϵA(ψ)=min{r:τrA(ψ)≤ϵ}. The global tail-resolved Schmidt-rank width is then χres(ϵ)(ψ)=minTmaxe∈E(T)log2RϵAe(ψ), where T ranges over subcubic decomposition trees and Ae∣Aec is the bipartition induced by deleting the tree edge e. At zero resolution, ϵ=0, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: χres(ϵ)(∣G⟩)=log2⌈(1−ϵ2)2rw(G)⌉,0≤ϵ<1, where 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. 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). For balanced pairwise weighted phase networks with crossing phase errors ηe, ddiagA≤41ηE(A,Ac)2. If a fixed coupling template W is implemented with coupling error ΔW and timing error Δt, then 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) denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error ϵ, 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. For every fixed resolution 0<ϵ<21, the tail-resolved width remains equal to 1 for all sufficiently small ∣λ∣, independently of the perturbation direction V. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width ⌈3n⌉, whereas a uniform perturbation activates only 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, limϵ↓0limλ→0χres(ϵ)=1, while limλ→0,λ=0limϵ↓0χres(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 ZZ/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.