We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy
M=UL−1⋯U0. Its spectrum is
λa,k=1−cos((2πk−θa)/L), where
eiθa∈spec(M). Thus the exact history sector is isomorphic to
Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step
U, the minimal number of clock events is the projective order of
U. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by
U(r)×ZL on a rank-
r history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
Computing quantum ground states is generically difficult, but additional structure can sometimes allow diagonalization to be recast as a more feasible problem. For example, when the desired ground state is sparse in a given basis, diagonalization can be facilitated via graph search. We make this reformulation precise by introducing the eigenwalk problem, which seeks the support of a sparse eigenvector of a Hermitian matrix by exploring the graph induced by its nonzero entries. However, it is not obvious whether the relevant support vertices must always be efficiently reachable by a search on the graph. To resolve this question, we prove that for every sparse eigenvector, there exists a (possibly different) sparse eigenvector with the same eigenvalue whose support is tightly localized in the graph, with diameter scaling only linearly in the sparsity and independently of the total number of vertices. As a consequence, if a
2n-dimensional,
poly(n)-sparse Hamiltonian has an
O(1)-sparse extremal eigenvector and one support element is known, then an exact eigenvector with the same eigenvalue can be computed classically in
poly(n) time. The same conclusion follows when the
O(1)-sparse eigenvector is non-extremal, provided that it is sparser than every eigenvector with a different eigenvalue. These results hold with no assumptions on the degeneracy, locality, spectral width, or spectral gap of the Hamiltonian, and the underlying support-localization principle also extends to problems beyond exact diagonalization, such as sparse principal component analysis.
A recurring proposal in emergent-geometry programmes is that a simple positive local rule acting on relational degrees of freedom will select a sparse, manifold-like graph. We show that one of the most natural such rules does the opposite.On the all-pairs relation-qubit substrate, let an edge ik toggle coherently whenever i and k share a neighbour, and let H_N = -lambda A_N with lambda > 0 be the resulting relation-only Hamiltonian. The operator contains no target dimension, valence, distance potential, connectivity reward or preferred graph. We prove that its unrestricted ground phase is asymptotically dense, with one-edge occupation 5/6.Three ingredients are independent and are stated separately. A generic spectral-observable lemma converts any pointwise row-degree bound R <= beta X into the ground-state estimate >= rho(A)/beta. A sharp local operator inequality T <= N_Delta/sqrt(2) on a single triangle gives, with a connected variational witness, the rigorous density bound liminf E[eta_N] >= 25 sqrt(10)/108, approximately 0.7320, which does not depend on the sharper result that follows. An exact all-cut PPT witness with constant c = 25 sqrt(5)/36, together with a symmetric (N-2)-extension and the Christandl-Koenig-Mitchison-Renner bound, upgrades this to triangle-marginal productization at p = 5/6. A separate argument removes the sector restriction: the component partition is dynamically invariant, and an exact compression identity shows the global Perron sector is the connected sector, so the statement concerns the ground phase of H_N itself rather than a chosen sector.We state the ceiling precisely. This establishes local triangle productization and one-edge density; it does not establish global product-state convergence, arbitrary propagation of chaos, the exact first 1/N density coefficient, or anything about dynamics from disconnected initial data. It kills a natural dense-phase architecture. It does not solve, or bear on the truth of, quantum gravity, and no novelty is claimed.
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph.
Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from
k-Local Hamiltonian to SIH
3k+3 up to a known global energy scale and inverse-polynomial approximation.
Sharper constructions give QMA-completeness of SIH
3 and of geometrically local SIH
8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH
3 and QMA-completeness of 1-Pinned Stoq-SIH
4. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields
O(Q3.15) Pauli string terms in the number of qubits
Q, compared to
O(Q4+) for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives
Q4.25 for LiH and
Q3.92 for H
2O, spanning
Q3.9–
Q4.3). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as
∼1/M (empirically
≈M−0.85–
M−1) with the number of spatial orbitals
M, while Gaussian ERI tensors remain near-dense at
O(1). Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as
O(Q3.36), and the Pauli 1-norm
λ=∑i∣ci∣—which controls both Trotter step counts and qubitization query complexity—scales as
O(Q1.69) with
R2=0.997. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H
2; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of
O(Q2.5), independent of molecular topology (exponent spread
0.02 across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref],
Q3.9–
Q4.3 span fitted by us)—an advantage that grows with system size due to the
∼1.5 exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH
2/H
2O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning
Z=1–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count:
NPauli=11.10×Q (exact, non-identity terms;
d-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (
ma=mc,
mb=md) adopted deliberately for sparsity; the exact global-
ML Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives
≈2.5× more terms with the same scaling, under which the
d-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.