Non-unitary lattice fingerprints of the Yang–Lee minimal model M(2,5): effective central charge and lowest scaling dimension from energy-level scaling, with a negative result for a channel-layer observable
We test whether the infrared fixed point of a non-unitary “parent” model — the object whose categorical data we certify — is the Yang–Lee minimal model M(2,5), by direct lattice measurement on the A₄ RSOS(4,3) chain. Three results are reported. (1) The effective central charge is measured by ground-state energy-level scaling, c_eff = 0.40 ± 0.02, with the error bar certified by the deviations of two known baselines (unitary M(4,5) → 0.70, non-unitary M(3,5) → 0.60). (2) The lowest scaling dimension is measured from the excited-gap scaling, h_φ = −1/5, with the topological spin θ_φ = e^{−2πi/5} following as a one-step corollary of the standard spin–weight relation (cross-checked against the Itzykson–Saleur–Zuber relation); this is done with no unitary oracle. (3) A channel-layer observable λ* proposed as a second carrier of the categorical data is shown, robustly, to sit ≈ 0 — at least 20× below the CP-lock target 1/φ² ≈ 0.382 across the raw, filtered, and far-edge extractions — a negative result classified as an observable degeneracy: the carrier dies, while the categorical identity is carried independently by (1) and (2). Honest-history sections (a withdrawn spectral-matching claim; a discarded unitary-oracle route; the data-cut dependence of the λ* decay law) are retained on purpose. Posture: realize, not decide — the parent IR being M(2,5) is measured; whether the cosmic charge sector equals (2,5) is out of scope and belongs to the companion framework/holography line. Grading convention used throughout. We keep four evidence levels distinct and never weld them into one: measured (extracted from lattice data), analytic (closed-form / CAS), structural self-check (algebraic consistency of the categorical data), and one-step corollary via the standard spin–weight relation.
We test whether the infrared fixed point of a non-unitary “parent” model — the object whose categorical data we certify — is the Yang–Lee minimal model M(2,5), by direct lattice measurement on the A₄ RSOS(4,3) chain. Three results are reported. (1) The effective central charge is measured by ground-state energy-level scaling, c_eff = 0.40 ± 0.02, with the error bar certified by the deviations of two known baselines (unitary M(4,5) → 0.70, non-unitary M(3,5) → 0.60). (2) The lowest scaling dimension is measured from the excited-gap scaling, h_φ = −1/5, with the topological spin θ_φ = e^{−2πi/5} following as a one-step corollary of the standard spin–weight relation (cross-checked against the Itzykson–Saleur–Zuber relation); this is done with no unitary oracle. (3) A channel-layer observable λ* proposed as a second carrier of the categorical data is shown, robustly, to sit ≈ 0 — at least 20× below the CP-lock target 1/φ² ≈ 0.382 across the raw, filtered, and far-edge extractions — a negative result classified as an observable degeneracy: the carrier dies, while the categorical identity is carried independently by (1) and (2). Honest-history sections (a withdrawn spectral-matching claim; a discarded unitary-oracle route; the data-cut dependence of the λ* decay law) are retained on purpose. Posture: realize, not decide — the parent IR being M(2,5) is measured; whether the cosmic charge sector equals (2,5) is out of scope and belongs to the companion framework/holography line. Grading convention used throughout. We keep four evidence levels distinct and never weld them into one: measured (extracted from lattice data), analytic (closed-form / CAS), structural self-check (algebraic consistency of the categorical data), and one-step corollary via the standard spin–weight relation.
The Rényi entropies of a non-unitary critical system are not fixed by a single power of the central charge but by the full four-point function of twist fields in a cyclic orbifold, constrained by null-vector equations. For the Yang–Lee universality class—the minimal model with —Dupic, Estienne and Ikhlef (DEI) computed several of these twist correlators in closed form. We provide an independent lattice realization and calibration of those exact correlators on two distinct critical lattice models: the critical Restricted Solid-On-Solid (RSOS) model with and the quantum Ising chain in an imaginary longitudinal field. Both are diagonalised exactly, and the lattice-to-continuum correspondence is certified on three independent objects: the low-lying spectrum reproduces the tower with effective central charge and no exceptional-point degeneracy on the RSOS side; the single-interval vacuum and ground-state twist correlators reproduce the exact two- and four-point functions, including a quantitative determination of the sub-leading fusion channel; and the vacuum two-interval four-twist function is reproduced with a crossing self-consistency of . Along the way we correct a typographical error in the DEI Temperley–Lieb Hamiltonian. We define precisely the boundary-height-sector-resolved trace functional that the lattice certifies, and we do not address the microscopic-realization dependence of the physical two-interval mutual information, which involves a choice of observable subalgebra and is left to a separate study.
On a constrained lattice realization of the non-unitary Yang–Lee / minimal model, the two-interval entanglement of a region is not fixed until one specifies the algebra of local observables retained inside that region. We show that on the restricted solid-on-solid (RSOS) chain the constraint-preserving local algebra of a region carries a non-trivial center: the height variable on the region’s boundary commutes with every interior generator, and this is forced by locality together with the adjacency constraint rather than chosen. Fixing the boundary height selects the electric-center algebra; absorbing it selects the trivial-center algebra. The two choices are the lattice counterparts of the center choices of Casini, Huerta and Rosabal for gauge theories, and the electric-center block decomposition is the constrained-lattice analog of the anyonic charge-sector decomposition standard in anyonic entanglement. Because the biorthogonal reduced density matrix of a non-unitary model is not positive, the weights of the center sectors form a signed normalized distribution: individual sector weights can be negative while their sum is exactly one. We report this distribution as a reproducible trace-level fact and trace each negative weight to a negative Boltzmann weight of the non-unitary theory. A single model then furnishes distinct two-interval Rényi mutual informations for its two center functionals, and, at fixed center, for two different twist operators. We read these numbers as the values of distinct well-posed quantities rather than as competing estimates of one quantity. A positive-definite entropy for the non-unitary center decomposition is left open.
We identify and separate a preliminary gauge obstruction from three logically distinct source requirements for a counting field to activate a Yang–Lee-compatible charge-odd direction in the class of local charge processes studied here. Gate 0 asks whether the counted jump observable is only a boundary term: if its weights are exact, the tilted generator is related to the untilted one by a similarity transformation and the long-time spectrum is unchanged. Once a non-boundary count exists, three further conditions concern the source itself: a charge-odd source must be symmetry-allowed, real analyticity together with charge conjugation must pin it to the imaginary axis on the counting line, and the zero mode to which it couples must be dynamically live rather than a c-number in a fixed charge sector. The last statement is a state-source inertness condition and is algebraically distinct from Gate 0. A 3×3 factorial test of source permission against sector openness contains nine conditions, exactly four of which are live; they are precisely the four carrying both ingredients. A symbolic elimination independently reproduces the source structure and its phase. We then construct a tilted three-state chain in which these requirements are realised. Its tilted generator has an exact antiunitary symmetry, from which we derive and verify that the real part of the charge expectation vanishes on the unbroken branch and splits into an equal-and-opposite pair after the finite-size branch point. Finite-time Lee–Yang zeros of the moment generating function approach that certified branch point as 1/t²; a spectral-side prediction of the convergence amplitude agrees with the zero-side measurement to 2.7–5.2%, improving with size, and the second-order Jordan secular term is observed directly. Two independent finite-size procedures return a common critical-field estimate within 0.02%. We do not promote that agreement to proof of thermodynamic branch-point accumulation, which remains unresolved in the available size window. The same calculations establish, equally sharply, the boundary of what these signatures can decide. The 1/t² law and the associated fixed-size zero-density exponent −1/2 are generic kinematics of a second-order branch point and therefore cannot identify a Yang–Lee universality class. On a published benchmark with a known answer, the dynamical-exponent estimator remains 7.7–11.8% from the truth at the largest comparable sizes and a linear 1/L extrapolation overshoots by 17%. An observable-specific correction-amplitude zero is stable across sizes, explaining why the best-looking finite-size working point can be the worst point from which to extrapolate; a separate rank-one residual test is reserved for deciding whether it is an effective-action property. We therefore keep two inference levels separate. Source-level/static Yang–Lee compatibility does not by itself fix the stochastic-time dynamical exponent. The stronger spectral-CFT route used for a Virasoro-tower, Casimir-energy and central-charge reconstruction additionally requires z=1. Conditional on that gate, three spectral anchors would overdetermine the low-energy tower through target-value-free closure identities; only if those identities close are the lowest weight, effective central charge and central charge to be read out blind, with the velocities and tilt-induced drift cancelling.
Universality classes of non-unitary critical theories in two-dimensions are characterized by the central charge. However, experimental determination of the central charge of a non-unitary critical theory has not been done before because of the intrinsic difficulty that complex parameters usually occur in non-unitary theory, which is not physical. Here we propose to extract the effective central charge of the non-unitary critical point of a two-dimensional lattice model from the quantum coherence measurement of a probe spin which is coupled to the lattice model. A recent discovery shows that quantum coherence of a probe spin which is coupled to a bath is proportional to the partition function of the bath with a complex parameter. Thus the effective central charge of a non-unitary conformal field theory may be extracted from quantum coherence measurement of a probe spin which is coupled to a bath. We have applied the method to the Yang-Lee edge singularity of the two-dimensional Ising model and extracted the effective central charge of the Yang-Lee edge singularity with good precision and tested other predictions of non-unitary conformal field theory. This work paves the way for the first experimental observation of the effective central charge of non-unitary conformal field theory.
In a compression-based account of charge, five distinct objects share the single name “charge”: a global conserved charge , an einselected reduced record , a -odd Yang–Lee ordering operator , a decaying Gaussian occupation , and a trajectory charge-transfer count . We separate them inside an instrumented, charge-conserving open quantum process and ask which relations are invariants of the full process and which are only diagnostics of a monitoring scheme or of the no-jump conditional slice. In minimal completely-positive models we establish: (i) [THM] if every jump commutes with and , charge-diagonal monitoring makes the reduced record an exact fixed point, and [CALC] its survivor -algebra is the two-dimensional diagonal (charge-sector) algebra; a charge-violating perturbation makes it a finite-lifetime metastable record whose lifetime rises with the monitoring rate in the strong-monitoring (Zeno) branch. (ii) [THM] -covariance does not imply charge conservation — amplitude damping is covariant yet lowers the occupation. (iii) [CALC] the asymptotic () stationary fixed-point -algebra obeys a three-way classification: for a bare record; with unchanged charge centre when a non-abelian seed survives (intra-sector protected structure); and a refined centre when a second abelian label survives. Charge labels all asymptotic survivors iff the stationary and there are no nonzero-frequency peripheral modes; the weaker excludes neither intra-sector survivors nor rotating (nonzero-frequency) peripheral modes, which are counted separately from . (iv) [CALC] a genuine finite-time trajectory Lee–Yang zero of a two-site transport counting statistics exists (complex Newton root, residual , winding number one); we do not call it an edge. The two strongest results are negative: [NEG] the no-jump effective Hamiltonian is not a representation invariant of the physical generator — at an exceptional point the eigenvalue splitting opens from to under a redundant Lindblad representation that leaves fixed to , so the exceptional point itself is not a process invariant; and [NEG] in the stated model, at a no-jump exceptional point no non-analyticity of the physical scaled cumulant generating function is detected in the scanned range. One minimal model simultaneously instantiates three charge objects () and two derived dynamical objects (); and remain comparison entries supplied by other minimal models / companions. In that model the conserved record and the zero-generating transferred charge are different objects: the ledger and language are unified, but a direct, representation-invariant bridge from the record to the dynamical Lee–Yang zeros is not established. We do not derive , a dynamic exponent , or any universality class.
As the final paper of the series, this paper constructs the minimal system in which two composites coexist — the twin minimal space model — and examines how much of the behavior that looks quantum mechanical is derived as a reading of hierarchy interfaces in a discrete relational model that assumes neither amplitudes nor continuous time. External face of the stable species: The handles (hairs) by which a composite is observed from outside divide into three ledgers: the face fundamental wave ν0 = 1/(2√s) (the dominant line of the self-container spectrum), the parent-record line νobs = |k|, and the energy-like norm R' = √s. The exact identity ν0R' = 1/2 (the zero point) links the first two, and R' never appears as a line — it is readable from outside as the spacing 1/R' of the odd comb (the particle-level realization of the record theorem). The external face is a minimal-uncertainty quasi-rectangular wave packet that carries a built-in odd comb of spacing 1/R' and is localized within an extent 2R' (half-width × fundamental wave = 1/2). Construction of the twin system: The initial value Ω0² = 2ν1² = 18 is an even label and, by the classification theorem (Paper 5), cannot exist as a single state — being born as a pair is forced by parity (the first event requires no dynamics). The allowed lattice separations of the twins (each s = 9) are quantized to |Δ|² ∈ {2, …, 20}, and |Δ|² ≡ 1 (mod 4) is entirely forbidden. The relational classes (21 under B4 × swap) are completely separated once the branch (cos/sin) channels are read in addition to the fringe intensity data (intensity-only fringe data degenerates to 18 kinds as ordered data, and to 11 kinds under the operationally available unordered reading) — the scale marks of space are born as chord sets of equal-frequency shells. The exact form of measurement: Inspecting the two conditions for measurement (affecting the system; detecting the change), the detection of change requires knowledge of the prior state and regresses infinitely; in this model, however, the record of the prior state is absent by censorship, and the λ ledger is append-only (the monotonic increase theorem, Paper 6), so the regress is structurally cut. Measurement = an append to the record (absence → presence); "collapse" is the appearance of a record, and its irreversibility reduces to the same single bit as the arrow of time. Computing the first decay of the twins explicitly, the appearance of the reference wave (s = 1) promotes the information class of the record from interferometric to holographic — the measuring apparatus is generated by the event itself. We also show that the final-state statistics depend on the choice of measure (one-shot equal weight over configurations 60/61 versus sequential conditional counting 396/403) — measure bifurcation, with an exact difference of +0.00098. Layer resolution of entanglement: The self-spectrum of a single fragment lies on all-components-even vectors and is therefore identically invisible in the logical-wave (odd) sector — only counts and relations can be read (the relational-only readout theorem). The decay of one twin instantaneously rewrites the branching statistics of the other (total variation 0.774), but this shift does not depend on the twin separation at all, and the influence does not propagate through the space of the child layer — instantaneity is not a speed but a mismatch of levels. Since order and attribution do not exist in the final record, operational no-signalling follows. An entangled pair, being an even label, cannot become a single particle at the layer above (parity protection). The same map applies to two-path interference of a single wave packet. Vertex deficits and the two hidden axes: An exhaustive inspection of the record continuity rule across an event (that the parent's identity line persists in the cross spectrum of the daughter system) derives, as the persistence condition, a conservation law of position vectors v5 ± v3 = ±v9 (a momentum analogue), which holds for parents in the (2,1,0,0) orbit but fails for parents in the (1,1,1,1) orbit by exactly one unit (one axis / one quantum). Furthermore, in the bookkeeping of the excitation degree ε = (−1)Σ|k| there exists a second one-bit deficit independent of this one (specific orbits at s = 7, 9). The visible four-axis ledgers demand two invisible directions — R on the curvature side and Q on the bare-parity side — and intermediate displacements along the invisible directions, being unrecorded, are permitted (the licensing principle for virtual displacements). This licence virtually unlocks two-body splittings that are forbidden in the visible ledgers, and decomposes every decay into sequences of cubic vertices (multiple paths to the same final state). Finally, we organize the correspondence between the obtained structure and standard theory as a transport principle (a theorem is transported whenever the counterparts of its premises are verified), and state explicitly the candidate answers this model gives to questions that remain postulates on the standard-theory side (the measurement problem, the location of the cut, the problem of time, dimensionality, the measure), together with falsifiability (gravitationally mediated entanglement experiments, etc.). Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus four figures (PNG). v0.3 (2026-06-12): Corrects Sec. 3.2 and the abstract: the twin pairs decompose into 21 relational classes under B4 × swap (not 18); intensity-only fringe data degenerates to 18 kinds (ordered) / 11 kinds (unordered, operational), and complete separation is recovered by the branch-channel readout. All other results are unchanged.