We exhibit an exact three-mode packet realization of a mediated three-sector system in which forward transport strictly dominates complementary reconstruction on a nontrivial coherence window. The coefficient hierarchy forces the forward amplitude to remain close to the interior coherence kernel while the return amplitude is uniformly suppressed at order 1/318. Whenever the coherence kernel exceeds 1/158, forward dominance holds exactly. Since the kernel equals 1 at t=0, a nontrivial forward-dominance interval is guaranteed to exist. The result is structural and occurs in a closed system without environmental coupling or irreversibility.
We exhibit an exact three-mode packet realization of a mediated three-sector system in which forward transport strictly dominates complementary reconstruction on a nontrivial coherence window. The coefficient hierarchy forces the forward amplitude to remain close to the interior coherence kernel while the return amplitude is uniformly suppressed at order 1/318. Whenever the coherence kernel exceeds 1/158, forward dominance holds exactly. Since the kernel equals 1 at t=0, a nontrivial forward-dominance interval is guaranteed to exist. The result is structural and occurs in a closed system without environmental coupling or irreversibility.
This paper establishes a structural and exact result for mediated three-sector systems, demonstrating asymmetry between transport and reconstruction channels in a closed finite-dimensional setting. A minimal three-sector model is introduced in which outer states are coupled only through a mediated intermediate state. Using second-order reduction, it is shown that outer-sector transport is governed by inverse detuning structure, while asymmetry between channel-resolved amplitudes is controlled by detuning imbalance. This produces suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary evolution. An exact realization of this mechanism is then provided through a three-mode packet model derived from the Q₅ framework. In this setting, forward and return amplitudes are computed explicitly, yielding a coefficient hierarchy in which forward contributions are order-one while return contributions are suppressed by approximately three orders of magnitude. This leads to a rigorously defined coherence window in which forward amplitude strictly dominates return amplitude. The results establish that forward dominance is not an isolated numerical artifact, but arises from a structural asymmetry inherent in mediated multi-channel systems. Together, the structural theorem and exact realization provide a unified account of reconstruction suppression and transport dominance in a closed-system setting. This work forms part of a broader program investigating operator structure, grading behavior, and sector-resolved dynamics on the five-dimensional hypercube (Q₅), with related results available in accompanying white papers and operator decomposition studies.
We study a minimal three-sector mediated system and derive the second-order effective dynamics on the outer subspace. We show that mediated transport is governed by an inverse detuning structure, while asymmetry between transport channels is controlled by detuning imbalance. This produces structural suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary inversion. The complete proof and unification with the forward dominance theorem is given in the companion paper (Theorem 11b (Unified), DOI: 10.5281/zenodo.19380323).
We isolate a self-contained three-frequency effective subsystem and prove a uniform upper bound relating the complementary return amplitude to the interior coherence factor. The suppression coefficients 1/320, 1/318, 1/316 form an arithmetic family with common difference -2. The central result is the bound |A_return(t)| ≤ (1/318)|R_int(t)| + 1/50244, holding for all t ∈ ℝ under unit normalization. A corollary establishes that when at least two fibres are occupied, and frequencies are distinct, dephasing forces the return amplitude to remain small at generic post-coherence times, up to isolated recurrences. An interpretive remark notes that any added decoherence mechanism would convert bounded return into sustained suppression. The theorem is stated as pure mathematics: no physical interpretation is assumed.
The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the r-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of r and k---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at r=3 (k=2,3,4), r=4 and r=5 (k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios qr, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.
What limits quantum coherence on a causal network: distance, noise, or the act of recording which structure was taken? This two-paper study develops a unified framework for answering that question. The first paper proves a general spacetime law for coherence transport in quantum networks. It separates three physically distinct effects: finite-speed causal propagation, contraction of pre-existing coherence by a quantum channel, and long-range coherence injected by non-local structural changes. The resulting bound reveals two unavoidable regimes: an early-time decay law for coherence prepared across large separations, followed by a spatial coherence envelope once the corresponding causal depth has been reached. The second paper turns to a different regime in which the network structure itself is coherently controlled. It proves that two alternative structures remain perfectly indistinguishable until quantum matter acquires amplitude in the region where they differ. More generally, it derives causal bounds on visibility loss, structure–matter entanglement, and the optimal probability of determining which structure was present. Numerical network-walk experiments illustrate exact causal protection, coherent visibility revivals, and the suppression of those revivals when environmental records are introduced. Together, the papers distinguish structural difference from irreversible decoherence: non-locality or alternative geometry alone need not destroy coherence. The decisive mechanism is whether information distinguishing the alternatives is physically acquired and irreversibly recorded. Included works:A General Causal Coherence Transport Law and Causal Protection of Coherently Superposed Network Structures.
A completely positive channel N that is covariant for a group G acting on a carrier W = ⊕_s V_s (pairwise inequivalent irreducibles) admits two natural bilinear read-outs of its sector structure: a "realign reading" A_phys[s,t] = Tr[P_s N(P_t)]/d_s, the channel's coefficient matrix on the sector algebra, and a "no-realign reading" A_ext[s,t] = (1/d_s) sum_{a,A} <a|P_s N(|a><A|) P_t|A>, a score. We prove three things. First, covariance forces N(P_t) to be block-scalar, so A_phys is the coefficient matrix c_{st} in N(P_t) = sum_s c_{st} P_s (Schur's lemma). Second, the score is the Gram matrix of the channel's sector-overlap vectors, A_ext[s,t] = (1/d_s) <w_s, w_t> with w_s = (Tr[P_s K_k])_k; hence its diagonal is a manifest sum of squares (nonnegative for any channel) and D·beta is Hermitian positive semidefinite in the sector-trace metric D = diag(d_s), where beta = A_ext. Third, the score equals the sector-return diagonal, A_ext = diag(A_phys), if and only if two independent conditions hold: (I) the overlap vectors of distinct sectors are orthogonal, and (II) within each sector a trace/Frobenius balance sum_k |Tr K_k^{(ss)}|^2 = sum_k ||K_k^{(ss)}||_F^2. Condition (II) has a Schumacher entanglement-fidelity reading: the maximally-mixed entanglement fidelity of the sector-s diagonal sub-channel equals A_phys[s,s]/d_s. We call channels meeting (I)+(II) sector-return. A specific Sp(3) = USp(6)-covariant channel on the 35-dimensional sl(6) adjoint (split 14 + 21) is sector-return exactly, while generic covariant channels — the identity among them — are not; in the correct metric its diagonal sub-channels are scalar-unital and sit exactly at the (II)-balance, and a numerical spectral pass from exact-rational operators resolves them as Hermitian positive semidefinite of Choi rank D^2 = 36 (D = 6 the memory dimension), with dimension-indexed spectra whose leading eigenvalue is the sector-return coefficient. We close with the open problem this sharpens: whether covariant compression through a finite memory (a D-dimensional auxiliary memory, not a minimal Stinespring environment) forces sector-return.