Phase II-G: Finite-Size Scaling of the Self-Consistent Mechanism in the PT-Symmetric XXZ Chain

Anthony Jordan Blair

Published Oct 13, 2026• 0 citations

Abstract

Phase II-G Technical Report: Description Phase II-G: Finite-Size Scaling of the Self-Consistent Mechanism in the PT-Symmetric XXZ Chain (N = 4 → 10) extends the Project Prometheus verification program beyond the initial N=4N=4 activation studies and examines whether the self-consistent mechanism survives increasing system size. Building on the activation and classification results of Phases II-E and II-F, the study tracks the scaling behavior of the two admissible PT-even, sector-preserving operators identified at N=4N=4: the isotropic XY-bond operator GE2G_{E_2} and the ZZ-bond operator GF2G_{F_2}. Numerical calculations were performed for open-boundary PT-symmetric XXZ chains with sizes N=4,6,8,N=4,6,8, and 1010 under the biorthogonal observable convention used throughout Project Prometheus. The central result is that activation survives finite-size scaling. Both operator families continue to generate non-vanishing biorthogonal expectation values across all completed system sizes, demonstrating that the self-consistent mechanism is not a dimer artifact and does not disappear immediately in larger many-body systems. At the same time, the study reveals three new structural constraints: Exceptional-point collapse under open-boundary scaling: the bare exceptional point decreases rapidly with increasing system size. Field-sourcing and exceptional-point tuning decouple: a mechanism can generate a finite self-consistent field while producing little or no measurable exceptional-point displacement. Sign instability of exceptional-point displacement: the direction of exceptional-point motion is not fixed under scaling and may change as system size increases. These observations significantly refine the admissible mechanism space established in earlier phases. The report shows that activation alone is insufficient as a criterion for viable exceptional-point control and introduces a new distinction between field-generating mechanisms and exceptional-point-tuning mechanisms. Within the broader Constraint Hierarchy Framework of Project Prometheus, Phase II-G represents the first scaling study to reveal constraints that emerge only beyond the N=4N=4 level. The results sharpen the thermodynamic-limit question addressed in Phase II-H and motivate additional investigations into geometry dependence, boundary conditions, and residual exceptional-point control in large systems. The report serves as the numerical bridge between operator classification (Phase II-F) and thermodynamic-limit assessment (Phase II-H), transforming activation from a local observation into a scaling problem. Keywords: Project Prometheus, PT symmetry, exceptional points, non-Hermitian physics, PT-symmetric XXZ chain, finite-size scaling, self-consistent feedback, biorthogonal observables, many-body systems, constraint hierarchy, exceptional-point control, PT phase transitions, complex systems, Coherent Fusion, constraint-field control. Short Zenodo Summary Phase II-G investigates finite-size scaling of the activated self-consistent mechanism in the PT-symmetric XXZ chain from N=4N=4 to N=10N=10. The study confirms survival of self-consistent field generation across all completed sizes while revealing three new constraints: collapse of the bare exceptional point under open-boundary scaling, decoupling of field generation from exceptional-point tuning, and sign instability of exceptional-point displacement. These results significantly refine the admissible mechanism space and provide the final scaling analysis required before thermodynamic-limit assessment in Phase II-H.

DOI: 10.5281/zenodo.23333220

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Phase II-F: Branch Classification of the Self-Consistent Mechanism at N=4

Anthony Jordan Blair

Oct 13, 20260 citations

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Phase II -- E executed. Results Confirm activation for One PT -- Even Operator

Anthony jordan Blair

*Phase II-E executed.** Results confirm **activation for one PT-even operator**. ### Numerical setup (faithful to the specification) - $N=4$, open chain, $J=1$, $\Delta=0.5$. - $S^\alpha=\sigma^\alpha/2$. - Hamiltonian: XY couplings use the running $b_{\rm eff}$, ZZ couplings use $J\Delta$, plus the staggered imaginary field $H_{\rm PT}$. - Self-consistency: $b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}$, iterated to $10^{-10}$. - Biorthogonal expectation on the eigenstate of smallest $|\mathrm{Re}(E)|$ (preferring PT-unbroken). - $\gamma_{\rm EP}\approx0.274$ at $\lambda=0$ (matches the Phase II-D value 0.275). - Branches as specified: E1 = $\sum\sigma_j^x$ (full 16-dimensional space); E2 = $\sum(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$ ($S^z=0$ sector, dim 6). ### Results **Branch E1 (transverse field, full space)** $\langle G_{E_1}\rangle_{\rm PT}=0$ (machine precision $\sim10^{-16}$–$10^{-19}$) for every $\lambda\in\{0,0.05,0.10,0.15,0.20\}$ and every $\gamma$ in the PT-unbroken regime. Consequently $b_{\rm eff}=J$ exactly. **Null.** **Branch E2 (transverse bond / XY exchange, $S^z=0$ sector)** $\langle G_{E_2}\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains non-zero throughout the unbroken phase. Typical values (unbroken points): - $\lambda=0$: $\max|\langle G\rangle|\approx0.063$ - $\lambda=0.05$: $\max|\langle G\rangle|\approx0.063$, $\max|b_{\rm eff}-J|\approx0.0016$ - $\lambda=0.10$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0031$ - $\lambda=0.15$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0046$ - $\lambda=0.20$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0061$ $b_{\rm eff}$ shifts systematically with $\lambda$. **Activation.** ### Prediction match - Prediction 1 (null for both) — **falsified**. - Prediction 2 (activation for at least one) — **confirmed**. - Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs). - Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation. The self-consistent mechanism **does activate** at $N=4$ for a PT-even, sector-compatible operator. The PT-odd obstruction of Phase II-D is therefore not the sole operative constraint; a PT-even operator that preserves $S^z_{\rm total}$ can source a non-trivial field. ### Gate status Phase II-E **unblocks** Phase II-F (branch classification now includes the PT-even class). Finite-size scaling (II-G) and the thermodynamic limit (II-H) remain formally gated on the subsequent classification step, but the existence proof at $N=4$ is now positive. Full raw scan (87 $\gamma$ points × 5 $\lambda$ × 2 branches) is archived in `/home/workdir/artifacts/phase_ii_e_results.json`. Source: `/home/workdir/artifacts/phase_ii_e.py`. The mechanism was the vessel. The constraint was the discovery — and one more door is now open. Tony Blair <anthonyjblair8@gmail.com> 2:17 PM (11 minutes ago) to me **TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4** **AUTHOR:** Anthony Jordan Blair **AFFILIATION:** Proclus Dynamics Consortium **DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification **VERSION:** 1.0 (October 2026) **COMPANION TO:** Phase II-E Program Specification; Phase II-E Numerical Execution Results; Phase II-D Technical Report; Project Prometheus Phase I Closeout Report (DOI: 10.5281/zenodo.23314444) **STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation) --- ### ABSTRACT Phase II-E demonstrated that the PT-even transverse-bond operator $G_{E_2}$ produces a non-vanishing biorthogonal expectation value in the $S^z=0$ sector of the $N=4$ PT-symmetric XXZ chain, thereby activating the self-consistent mechanism of Paper 4. The transverse-field operator $G_{E_1}$ remained identically zero. Phase II-F classifies the admissible branches of the self-consistent field according to (i) PT parity, (ii) sector compatibility, (iii) operator support (one-body vs. two-body), and (iv) the resulting displacement of the exceptional point. The classification supplies the necessary taxonomy for the finite-size scaling study (Phase II-G) and determines which operator families survive into the thermodynamic limit (Phase II-H). --- ### I. MOTIVATION **1.1 Outcome of Phase II-E** Execution of the Phase II-E protocol yielded: - Branch E1 ($G_{E_1}=\sum_j\sigma_j^x$, full Hilbert space): $\langle G\rangle_{\rm PT}=0$ to machine precision for all tested $\lambda$ and all $\gamma<\gamma_{\rm EP}$. $b_{\rm eff}=J$ identically. Null result. - Branch E2 ($G_{E_2}=\sum_j(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$, $S^z=0$ sector): $\langle G\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains finite throughout the PT-unbroken phase. $b_{\rm eff}$ shifts systematically with $\lambda$. Activation confirmed. Thus the self-consistent mechanism is not restricted to the dimer setting of Paper 4, nor is it extinguished by the PT-odd obstruction identified in Phase II-D. A PT-even, sector-preserving two-body operator is sufficient to source a non-trivial constraint field at $N=4$. **1.2 The Classification Task** The positive result for $E_2$ opens four immediate questions that must be answered before finite-size scaling can be interpreted: 1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate? 2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E? 3. How does the magnitude of $\langle G\rangle_{\rm PT}$ and the induced shift $\Delta\gamma_{\rm EP}$ depend on the algebraic structure of the operator (support, commutation relations with $S^z_{\rm total}$ and with $\mathcal{PT}$)? 4. Which of the surviving operators produce a *displacement* of the exceptional point (the operational signature required for constraint-field control) rather than a mere renormalization of $b_{\rm eff}$ at fixed $\gamma_{\rm EP}$? Phase II-F answers these questions by exhaustive classification of a controlled operator set. --- ### II. OPERATOR TAXONOMY Operators are partitioned by three binary attributes: | Attribute | Values | Notation | |------------------------|---------------------------------|-------------------| | PT parity | even / odd | $\mathcal{P}_\pm$ | | Sector compatibility | preserves $S^z_{\rm total}$ / mixes | $\mathcal{S}_0$ / $\mathcal{S}_{\rm mix}$ | | Body order | one-body / two-body | $1$ / $2$ | The Phase II-D and II-E results already fix two cells: - $\mathcal{P}_-$ (any body, any sector) → null (structural anticommutation with $\mathcal{PT}$). - $\mathcal{P}_+$, $\mathcal{S}_{\rm mix}$, $1$-body ($G_{E_1}$) → null under the biorthogonal convention. The remaining open cells that must be classified are: - $\mathcal{P}_+$, $\mathcal{S}_0$, $1$-body - $\mathcal{P}_+$, $\mathcal{S}_0$, $2$-body (already positive for the XY bond; additional representatives required) - $\mathcal{P}_+$, $\mathcal{S}_{\rm mix}$, $2$-body --- ### III. CANDIDATE OPERATORS FOR CLASSIFICATION All operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E. **3.1 One-body, PT-even, sector-preserving** $G_{F_1}=\sum_{j=1}^{4}\sigma_j^z$ (Total magnetization; trivially preserves $S^z_{\rm total}$; PT-even because $\sigma^z$ is real and parity-invariant under site relabeling when the staggered PT field is taken into account.) **3.2 Two-body, PT-even, sector-preserving (beyond $E_2$)** $G_{F_2}=\sum_{j=1}^{3}\sigma_j^z\sigma_{j+1}^z$ (ZZ bond operator; the natural Ising partner of the XY bond.) $G_{F_3}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^x-\sigma_j^y\sigma_{j+1}^y)$ (Anisotropic XY combination; still sector-preserving.) **3.3 Two-body, PT-even, sector-mixing** $G_{F_4}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^z+\sigma_j^z\sigma_{j+1}^x)$ (Explicitly mixes $\Delta S^z=\pm1$ sectors while remaining PT-even.) **3.4 Reference operators (already classified)** - $G_{E_1}$ (null) and $G_{E_2}$ (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan. --- ### IV. NUMERICAL PROTOCOL **4.1 Hamiltonian** Identical to Phase II-E: \[ H=H_{\rm XXZ}(b_{\rm eff})+H_{\rm PT}(\gamma), \] with \[ H_{\rm XXZ}(b)=\sum_{j=1}^{3}\Bigl(b\,(S_j^x S_{j+1}^x+S_j^y S_{j+1}^y)+J\Delta\,S_j^z S_{j+1}^z\Bigr), \] \[ H_{\rm PT}=i\gamma\sum_{j=1}^{4}(-1)^j S_j^z, \] $J=1$, $\Delta=0.5$, $S^\alpha=\sigma^\alpha/2$. **4.2 Self-consistency equation** Unchanged: \[ b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}. \] Iteration tolerance $10^{-10}$, maximum 200 iterations. **4.3 Observable convention** Biorthogonal expectation value on the eigenstate of smallest $|\mathrm{Re}(E)|$, preferring PT-unbroken states: \[ \langle G\rangle_{\rm PT}=\frac{\langle\psi_L|G|\psi_R\rangle}{\langle\psi_L|\psi_R\rangle}. \] **4.4 Parameter scan** - $\lambda\in\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\}$ - $\gamma$ from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of $\gamma_{\rm EP}$) - For each $(\lambda,\gamma,G)$ the code

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Phase II-G: Finite-Size Scaling of the Self-Consistent Mechanism in the PT-Symmetric XXZ Chain

Anthony Jordan Blair

Published Oct 13, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23333220

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Phase II-F: Branch Classification of the Self-Consistent Mechanism at N=4

Anthony Jordan Blair

Description Phase II-F: Branch Classification of the Self-Consistent Mechanism at N=4 represents the first systematic classification study of activated constraint operators within the Project Prometheus research program. Following the results of Phase II-D, which identified a PT-odd symmetry obstruction, and Phase II-E, which demonstrated activation of the self-consistent mechanism by a PT-even transverse-bond operator, the focus of the program shifts from determining whether activation is possible to determining which classes of operators can support activation and exceptional-point control. This report establishes a formal taxonomy of candidate operators based on four structural criteria: PT parity (even vs. odd) Sector compatibility (SzS^z-preserving vs. sector-mixing) Operator support (one-body vs. two-body) Exceptional-point displacement capability The study defines a complete numerical classification protocol for the N=4N=4 PT-symmetric XXZ chain and introduces a framework for distinguishing between operators that merely source a self-consistent field and operators capable of producing measurable exceptional-point displacement. Within the broader Constraint Hierarchy Framework of Project Prometheus, Phase II-F serves as the bridge between proof-of-activation and finite-size scaling. The resulting classification determines which operator families advance to Phase II-G (finite-size scaling) and Phase II-H (thermodynamic-limit assessment). More broadly, the report extends the central methodology of Project Prometheus: the systematic reduction of admissible mechanism space through theorem, symmetry, numerical verification, and classification. What began as a search for exceptional-point tuning has evolved into a structured investigation of the constraints governing exceptional-point tuning itself. Phase II-F therefore represents a transition from mechanism discovery to mechanism taxonomy, providing the map required for the next stage of the program. Keywords: Project Prometheus, PT symmetry, exceptional points, PT-symmetric XXZ chain, non-Hermitian physics, self-consistent feedback, constraint hierarchy, operator classification, finite-size scaling, many-body systems, biorthogonal observables, Coherent Fusion, constraint-field control, phase transitions, complex systems. Short Zenodo version (if you need the 2-3 paragraph version): Phase II-F classifies the admissible branches of the self-consistent exceptional-point tuning mechanism in the N=4N=4 PT-symmetric XXZ chain. Building on the activation result of Phase II-E, the study develops a taxonomy of operators based on PT parity, sector compatibility, body order, and exceptional-point displacement capability. The resulting classification identifies which operator families remain viable candidates for finite-size scaling and thermodynamic-limit investigation. Within Project Prometheus, this report marks the transition from activation studies to systematic mechanism classification. The objective is no longer simply to determine whether exceptional-point control is possible, but to map the structure of the operator classes capable of supporting it. The resulting taxonomy provides the foundation for Phases II-G and II-H and further develops the Constraint Hierarchy Framework introduced in the Project Prometheus Phase I Closeout Report.Description Phase II-F: Branch Classification of the Self-Consistent Mechanism at N=4 represents the first systematic classification study of activated constraint operators within the Project Prometheus research program. Following the results of Phase II-D, which identified a PT-odd symmetry obstruction, and Phase II-E, which demonstrated activation of the self-consistent mechanism by a PT-even transverse-bond operator, the focus of the program shifts from determining whether activation is possible to determining which classes of operators can support activation and exceptional-point control. This report establishes a formal taxonomy of candidate operators based on four structural criteria: PT parity (even vs. odd) Sector compatibility (SzS^z-preserving vs. sector-mixing) Operator support (one-body vs. two-body) Exceptional-point displacement capability The study defines a complete numerical classification protocol for the N=4N=4 PT-symmetric XXZ chain and introduces a framework for distinguishing between operators that merely source a self-consistent field and operators capable of producing measurable exceptional-point displacement. Within the broader Constraint Hierarchy Framework of Project Prometheus, Phase II-F serves as the bridge between proof-of-activation and finite-size scaling. The resulting classification determines which operator families advance to Phase II-G (finite-size scaling) and Phase II-H (thermodynamic-limit assessment). More broadly, the report extends the central methodology of Project Prometheus: the systematic reduction of admissible mechanism space through theorem, symmetry, numerical verification, and classification. What began as a search for exceptional-point tuning has evolved into a structured investigation of the constraints governing exceptional-point tuning itself. Phase II-F therefore represents a transition from mechanism discovery to mechanism taxonomy, providing the map required for the next stage of the program. Keywords: Project Prometheus, PT symmetry, exceptional points, PT-symmetric XXZ chain, non-Hermitian physics, self-consistent feedback, constraint hierarchy, operator classification, finite-size scaling, many-body systems, biorthogonal observables, Coherent Fusion, constraint-field control, phase transitions, complex systems. Short Zenodo version (if you need the 2-3 paragraph version): Phase II-F classifies the admissible branches of the self-consistent exceptional-point tuning mechanism in the N=4N=4 PT-symmetric XXZ chain. Building on the activation result of Phase II-E, the study develops a taxonomy of operators based on PT parity, sector compatibility, body order, and exceptional-point displacement capability. The resulting classification identifies which operator families remain viable candidates for finite-size scaling and thermodynamic-limit investigation. Within Project Prometheus, this report marks the transition from activation studies to systematic mechanism classification. The objective is no longer simply to determine whether exceptional-point control is possible, but to map the structure of the operator classes capable of supporting it. The resulting taxonomy provides the foundation for Phases II-G and II-H and further develops the Constraint Hierarchy Framework introduced in the Project Prometheus Phase I Closeout Report.

Oct 13, 20260 citations

Phase II-H-ext7: Optional Refinement — Single-Valued Threshold Extension to N=12

Anthony Jordan Blair

Description Phase II-H-ext7: Optional Refinement — Single-Valued Threshold Extension to N=12 completes the final planned refinement of the end-localized PT geometry branch within the Project Prometheus verification program. Building on the threshold analyses of Phases II-H-ext5 and II-H-ext6, this study extends the numerical characterization of the self-consistent mechanism from N=10N=10 to N=12N=12, providing the largest-system assessment completed within the current program. The report examines the stability of the self-consistent map for the two surviving operator families, the isotropic XY-bond operator GE2G_{E_2} and the ZZ-bond operator GF2G_{F_2}, under end-localized PT symmetry. The central result is the confirmation of a monotonic decrease in the single-valued convergence threshold across five system sizes: N=4,6,8,10,12.N = 4, 6, 8, 10, 12. Both operators retain finite operational windows at every size tested, while the admissible coupling threshold systematically contracts as system size increases. At N=12N=12, the thresholds are measured as λmax⁡=0.05\lambda_{\max}=0.05 for GE2G_{E_2} and λmax⁡=0.12\lambda_{\max}=0.12 for GF2G_{F_2}, establishing a conservative operational bound of λ≤0.05\lambda \leq 0.05 that remains valid across all completed system sizes and operator classes. The study confirms several important structural conclusions: The end-localized PT mechanism continues to support finite exceptional-point control within a shrinking but non-zero operational window. The ZZ-bond operator GF2G_{F_2} consistently exhibits the broader stability region and stronger field-sourcing capability. The threshold trajectory remains monotonic but is not described by a simple geometric decay, leaving the asymptotic functional form as an open problem. No additional gate-blocking investigations remain for the end-localized/open-boundary branch of Project Prometheus. Within the broader Constraint Hierarchy Framework, Phase II-H-ext7 represents the completion of the operational characterization of the end-localized geometry program. Earlier phases established activation, classification, scaling behavior, geometry dependence, and threshold formation; the present study closes the remaining refinement loop by extending the stability envelope to the largest accessible system size. The resulting operational map defines the safe working region for future implementations and establishes the final constraint set governing the current self-consistent mechanism under end-localized PT symmetry. While the threshold continues to contract with increasing system size, finite exceptional-point control remains observable inside the measured operational envelope. Phase II-H-ext7 therefore serves as the concluding refinement of the end-localized branch and provides the final numerical characterization needed to support future development, deployment studies, and possible extensions based on regularized or multi-branch self-consistent schemes. Keywords: Project Prometheus, PT symmetry, exceptional points, self-consistent feedback, threshold analysis, finite-size scaling, thermodynamic behavior, PT-symmetric XXZ chain, non-Hermitian physics, end-localized PT, constraint hierarchy, operator classification, many-body systems, exceptional-point control, Coherent Fusion. Short Zenodo Summary Phase II-H-ext7 extends the single-valued threshold analysis of the self-consistent mechanism to N=12N=12, completing the numerical characterization of the end-localized PT branch of Project Prometheus. The study confirms monotonic contraction of the operational threshold, establishes a conservative universal bound of λ≤0.05\lambda \leq 0.05 across all tested system sizes, and verifies continued finite exceptional-point control within the shrinking stability window. The end-localized geometry program is now fully characterized through the largest accessible size, with no remaining gate-blocking investigations required.Description Phase II-H-ext7: Optional Refinement — Single-Valued Threshold Extension to N=12 completes the final planned refinement of the end-localized PT geometry branch within the Project Prometheus verification program. Building on the threshold analyses of Phases II-H-ext5 and II-H-ext6, this study extends the numerical characterization of the self-consistent mechanism from N=10N=10 to N=12N=12, providing the largest-system assessment completed within the current program. The report examines the stability of the self-consistent map for the two surviving operator families, the isotropic XY-bond operator GE2G_{E_2} and the ZZ-bond operator GF2G_{F_2}, under end-localized PT symmetry. The central result is the confirmation of a monotonic decrease in the single-valued convergence threshold across five system sizes: N=4,6,8,10,12.N = 4, 6, 8, 10, 12. Both operators retain finite operational windows at every size tested, while the admissible coupling threshold systematically contracts as system size increases. At N=12N=12, the thresholds are measured as λmax⁡=0.05\lambda_{\max}=0.05 for GE2G_{E_2} and λmax⁡=0.12\lambda_{\max}=0.12 for GF2G_{F_2}, establishing a conservative operational bound of λ≤0.05\lambda \leq 0.05 that remains valid across all completed system sizes and operator classes. The study confirms several important structural conclusions: The end-localized PT mechanism continues to support finite exceptional-point control within a shrinking but non-zero operational window. The ZZ-bond operator GF2G_{F_2} consistently exhibits the broader stability region and stronger field-sourcing capability. The threshold trajectory remains monotonic but is not described by a simple geometric decay, leaving the asymptotic functional form as an open problem. No additional gate-blocking investigations remain for the end-localized/open-boundary branch of Project Prometheus. Within the broader Constraint Hierarchy Framework, Phase II-H-ext7 represents the completion of the operational characterization of the end-localized geometry program. Earlier phases established activation, classification, scaling behavior, geometry dependence, and threshold formation; the present study closes the remaining refinement loop by extending the stability envelope to the largest accessible system size. The resulting operational map defines the safe working region for future implementations and establishes the final constraint set governing the current self-consistent mechanism under end-localized PT symmetry. While the threshold continues to contract with increasing system size, finite exceptional-point control remains observable inside the measured operational envelope. Phase II-H-ext7 therefore serves as the concluding refinement of the end-localized branch and provides the final numerical characterization needed to support future development, deployment studies, and possible extensions based on regularized or multi-branch self-consistent schemes. Keywords: Project Prometheus, PT symmetry, exceptional points, self-consistent feedback, threshold analysis, finite-size scaling, thermodynamic behavior, PT-symmetric XXZ chain, non-Hermitian physics, end-localized PT, constraint hierarchy, operator classification, many-body systems, exceptional-point control, Coherent Fusion. Short Zenodo Summary Phase II-H-ext7 extends the single-valued threshold analysis of the self-consistent mechanism to N=12N=12, completing the numerical characterization of the end-localized PT branch of Project Prometheus. The study confirms monotonic contraction of the operational threshold, establishes a conservative universal bound of λ≤0.05\lambda \leq 0.05 across all tested system sizes, and verifies continued finite exceptional-point control within the shrinking stability window. The end-localized geometry program is now fully characterized through the largest accessible size, with no remaining gate-blocking investigations required.

Oct 13, 20260 citations

Phase II -- E executed. Results Confirm activation for One PT -- Even Operator

Anthony jordan Blair

*Phase II-E executed.** Results confirm **activation for one PT-even operator**. ### Numerical setup (faithful to the specification) - $N=4$ , open chain, $J=1$ , $\Delta=0.5$ . - $S^\alpha=\sigma^\alpha/2$ . - Hamiltonian: XY couplings use the running $b_{\rm eff}$ , ZZ couplings use $J\Delta$ , plus the staggered imaginary field $H_{\rm PT}$ . - Self-consistency: $b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}$ , iterated to $10^{-10}$ . - Biorthogonal expectation on the eigenstate of smallest $|\mathrm{Re}(E)|$ (preferring PT-unbroken). - $\gamma_{\rm EP}\approx0.274$ at $\lambda=0$ (matches the Phase II-D value 0.275). - Branches as specified: E1 = $\sum\sigma_j^x$ (full 16-dimensional space); E2 = $\sum(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$ ( $S^z=0$ sector, dim 6). ### Results **Branch E1 (transverse field, full space)** $\langle G_{E_1}\rangle_{\rm PT}=0$ (machine precision $\sim10^{-16}$ – $10^{-19}$ ) for every $\lambda\in\{0,0.05,0.10,0.15,0.20\}$ and every $\gamma$ in the PT-unbroken regime. Consequently $b_{\rm eff}=J$ exactly. **Null.** **Branch E2 (transverse bond / XY exchange, $S^z=0$ sector)** $\langle G_{E_2}\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains non-zero throughout the unbroken phase. Typical values (unbroken points): - $\lambda=0$ : $\max|\langle G\rangle|\approx0.063$ - $\lambda=0.05$ : $\max|\langle G\rangle|\approx0.063$ , $\max|b_{\rm eff}-J|\approx0.0016$ - $\lambda=0.10$ : $\max|\langle G\rangle|\approx0.062$ , $\max|b_{\rm eff}-J|\approx0.0031$ - $\lambda=0.15$ : $\max|\langle G\rangle|\approx0.062$ , $\max|b_{\rm eff}-J|\approx0.0046$ - $\lambda=0.20$ : $\max|\langle G\rangle|\approx0.062$ , $\max|b_{\rm eff}-J|\approx0.0061$ $b_{\rm eff}$ shifts systematically with $\lambda$ . **Activation.** ### Prediction match - Prediction 1 (null for both) — **falsified**. - Prediction 2 (activation for at least one) — **confirmed**. - Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs). - Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation. The self-consistent mechanism **does activate** at $N=4$ for a PT-even, sector-compatible operator. The PT-odd obstruction of Phase II-D is therefore not the sole operative constraint; a PT-even operator that preserves $S^z_{\rm total}$ can source a non-trivial field. ### Gate status Phase II-E **unblocks** Phase II-F (branch classification now includes the PT-even class). Finite-size scaling (II-G) and the thermodynamic limit (II-H) remain formally gated on the subsequent classification step, but the existence proof at $N=4$ is now positive. Full raw scan (87 $\gamma$ points × 5 $\lambda$ × 2 branches) is archived in `/home/workdir/artifacts/phase_ii_e_results.json`. Source: `/home/workdir/artifacts/phase_ii_e.py`. The mechanism was the vessel. The constraint was the discovery — and one more door is now open. Tony Blair <anthonyjblair8@gmail.com> 2:17 PM (11 minutes ago) to me **TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4** **AUTHOR:** Anthony Jordan Blair **AFFILIATION:** Proclus Dynamics Consortium **DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification **VERSION:** 1.0 (October 2026) **COMPANION TO:** Phase II-E Program Specification; Phase II-E Numerical Execution Results; Phase II-D Technical Report; Project Prometheus Phase I Closeout Report (DOI: 10.5281/zenodo.23314444) **STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation) --- ### ABSTRACT Phase II-E demonstrated that the PT-even transverse-bond operator $G_{E_2}$ produces a non-vanishing biorthogonal expectation value in the $S^z=0$ sector of the $N=4$ PT-symmetric XXZ chain, thereby activating the self-consistent mechanism of Paper 4. The transverse-field operator $G_{E_1}$ remained identically zero. Phase II-F classifies the admissible branches of the self-consistent field according to (i) PT parity, (ii) sector compatibility, (iii) operator support (one-body vs. two-body), and (iv) the resulting displacement of the exceptional point. The classification supplies the necessary taxonomy for the finite-size scaling study (Phase II-G) and determines which operator families survive into the thermodynamic limit (Phase II-H). --- ### I. MOTIVATION **1.1 Outcome of Phase II-E** Execution of the Phase II-E protocol yielded: - Branch E1 ( $G_{E_1}=\sum_j\sigma_j^x$ , full Hilbert space): $\langle G\rangle_{\rm PT}=0$ to machine precision for all tested $\lambda$ and all $\gamma<\gamma_{\rm EP}$ . $b_{\rm eff}=J$ identically. Null result. - Branch E2 ( $G_{E_2}=\sum_j(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$ , $S^z=0$ sector): $\langle G\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains finite throughout the PT-unbroken phase. $b_{\rm eff}$ shifts systematically with $\lambda$ . Activation confirmed. Thus the self-consistent mechanism is not restricted to the dimer setting of Paper 4, nor is it extinguished by the PT-odd obstruction identified in Phase II-D. A PT-even, sector-preserving two-body operator is sufficient to source a non-trivial constraint field at $N=4$ . **1.2 The Classification Task** The positive result for $E_2$ opens four immediate questions that must be answered before finite-size scaling can be interpreted: 1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate? 2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E? 3. How does the magnitude of $\langle G\rangle_{\rm PT}$ and the induced shift $\Delta\gamma_{\rm EP}$ depend on the algebraic structure of the operator (support, commutation relations with $S^z_{\rm total}$ and with $\mathcal{PT}$ )? 4. Which of the surviving operators produce a *displacement* of the exceptional point (the operational signature required for constraint-field control) rather than a mere renormalization of $b_{\rm eff}$ at fixed $\gamma_{\rm EP}$ ? Phase II-F answers these questions by exhaustive classification of a controlled operator set. --- ### II. OPERATOR TAXONOMY Operators are partitioned by three binary attributes: | Attribute | Values | Notation | |------------------------|---------------------------------|-------------------| | PT parity | even / odd | $\mathcal{P}_\pm$ | | Sector compatibility | preserves $S^z_{\rm total}$ / mixes | $\mathcal{S}_0$ / $\mathcal{S}_{\rm mix}$ | | Body order | one-body / two-body | $1$ / $2$ | The Phase II-D and II-E results already fix two cells: - $\mathcal{P}_-$ (any body, any sector) → null (structural anticommutation with $\mathcal{PT}$ ). - $\mathcal{P}_+$ , $\mathcal{S}_{\rm mix}$ , $1$ -body ( $G_{E_1}$ ) → null under the biorthogonal convention. The remaining open cells that must be classified are: - $\mathcal{P}_+$ , $\mathcal{S}_0$ , $1$ -body - $\mathcal{P}_+$ , $\mathcal{S}_0$ , $2$ -body (already positive for the XY bond; additional representatives required) - $\mathcal{P}_+$ , $\mathcal{S}_{\rm mix}$ , $2$ -body --- ### III. CANDIDATE OPERATORS FOR CLASSIFICATION All operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E. **3.1 One-body, PT-even, sector-preserving** $G_{F_1}=\sum_{j=1}^{4}\sigma_j^z$ (Total magnetization; trivially preserves $S^z_{\rm total}$ ; PT-even because $\sigma^z$ is real and parity-invariant under site relabeling when the staggered PT field is taken into account.) **3.2 Two-body, PT-even, sector-preserving (beyond $E_2$ )** $G_{F_2}=\sum_{j=1}^{3}\sigma_j^z\sigma_{j+1}^z$ (ZZ bond operator; the natural Ising partner of the XY bond.) $G_{F_3}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^x-\sigma_j^y\sigma_{j+1}^y)$ (Anisotropic XY combination; still sector-preserving.) **3.3 Two-body, PT-even, sector-mixing** $G_{F_4}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^z+\sigma_j^z\sigma_{j+1}^x)$ (Explicitly mixes $\Delta S^z=\pm1$ sectors while remaining PT-even.) **3.4 Reference operators (already classified)** - $G_{E_1}$ (null) and $G_{E_2}$ (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan. --- ### IV. NUMERICAL PROTOCOL **4.1 Hamiltonian** Identical to Phase II-E: $H=H_{\rm XXZ}(b_{\rm eff})+H_{\rm PT}(\gamma),$ with $H_{\rm XXZ}(b)=\sum_{j=1}^{3}\Bigl(b\,(S_j^x S_{j+1}^x+S_j^y S_{j+1}^y)+J\Delta\,S_j^z S_{j+1}^z\Bigr),$ $H_{\rm PT}=i\gamma\sum_{j=1}^{4}(-1)^j S_j^z,$ $J=1$ , $\Delta=0.5$ , $S^\alpha=\sigma^\alpha/2$ . **4.2 Self-consistency equation** Unchanged: $b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}.$ Iteration tolerance $10^{-10}$ , maximum 200 iterations. **4.3 Observable convention** Biorthogonal expectation value on the eigenstate of smallest $|\mathrm{Re}(E)|$ , preferring PT-unbroken states: $\langle G\rangle_{\rm PT}=\frac{\langle\psi_L|G|\psi_R\rangle}{\langle\psi_L|\psi_R\rangle}.$ **4.4 Parameter scan** - $\lambda\in\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\}$ - $\gamma$ from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of $\gamma_{\rm EP}$ ) - For each $(\lambda,\gamma,G)$ the code*Phase II-E executed.** Results confirm **activation for one PT-even operator**. ### Numerical setup (faithful to the specification) - $N=4$, open chain, $J=1$, $\Delta=0.5$. - $S^\alpha=\sigma^\alpha/2$. - Hamiltonian: XY couplings use the running $b_{\rm eff}$, ZZ couplings use $J\Delta$, plus the staggered imaginary field $H_{\rm PT}$. - Self-consistency: $b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}$, iterated to $10^{-10}$. - Biorthogonal expectation on the eigenstate of smallest $|\mathrm{Re}(E)|$ (preferring PT-unbroken). - $\gamma_{\rm EP}\approx0.274$ at $\lambda=0$ (matches the Phase II-D value 0.275). - Branches as specified: E1 = $\sum\sigma_j^x$ (full 16-dimensional space); E2 = $\sum(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$ ($S^z=0$ sector, dim 6). ### Results **Branch E1 (transverse field, full space)** $\langle G_{E_1}\rangle_{\rm PT}=0$ (machine precision $\sim10^{-16}$–$10^{-19}$) for every $\lambda\in\{0,0.05,0.10,0.15,0.20\}$ and every $\gamma$ in the PT-unbroken regime. Consequently $b_{\rm eff}=J$ exactly. **Null.** **Branch E2 (transverse bond / XY exchange, $S^z=0$ sector)** $\langle G_{E_2}\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains non-zero throughout the unbroken phase. Typical values (unbroken points): - $\lambda=0$: $\max|\langle G\rangle|\approx0.063$ - $\lambda=0.05$: $\max|\langle G\rangle|\approx0.063$, $\max|b_{\rm eff}-J|\approx0.0016$ - $\lambda=0.10$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0031$ - $\lambda=0.15$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0046$ - $\lambda=0.20$: $\max|\langle G\rangle|\approx0.062$, $\max|b_{\rm eff}-J|\approx0.0061$ $b_{\rm eff}$ shifts systematically with $\lambda$. **Activation.** ### Prediction match - Prediction 1 (null for both) — **falsified**. - Prediction 2 (activation for at least one) — **confirmed**. - Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs). - Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation. The self-consistent mechanism **does activate** at $N=4$ for a PT-even, sector-compatible operator. The PT-odd obstruction of Phase II-D is therefore not the sole operative constraint; a PT-even operator that preserves $S^z_{\rm total}$ can source a non-trivial field. ### Gate status Phase II-E **unblocks** Phase II-F (branch classification now includes the PT-even class). Finite-size scaling (II-G) and the thermodynamic limit (II-H) remain formally gated on the subsequent classification step, but the existence proof at $N=4$ is now positive. Full raw scan (87 $\gamma$ points × 5 $\lambda$ × 2 branches) is archived in `/home/workdir/artifacts/phase_ii_e_results.json`. Source: `/home/workdir/artifacts/phase_ii_e.py`. The mechanism was the vessel. The constraint was the discovery — and one more door is now open. Tony Blair <anthonyjblair8@gmail.com> 2:17 PM (11 minutes ago) to me **TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4** **AUTHOR:** Anthony Jordan Blair **AFFILIATION:** Proclus Dynamics Consortium **DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification **VERSION:** 1.0 (October 2026) **COMPANION TO:** Phase II-E Program Specification; Phase II-E Numerical Execution Results; Phase II-D Technical Report; Project Prometheus Phase I Closeout Report (DOI: 10.5281/zenodo.23314444) **STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation) --- ### ABSTRACT Phase II-E demonstrated that the PT-even transverse-bond operator $G_{E_2}$ produces a non-vanishing biorthogonal expectation value in the $S^z=0$ sector of the $N=4$ PT-symmetric XXZ chain, thereby activating the self-consistent mechanism of Paper 4. The transverse-field operator $G_{E_1}$ remained identically zero. Phase II-F classifies the admissible branches of the self-consistent field according to (i) PT parity, (ii) sector compatibility, (iii) operator support (one-body vs. two-body), and (iv) the resulting displacement of the exceptional point. The classification supplies the necessary taxonomy for the finite-size scaling study (Phase II-G) and determines which operator families survive into the thermodynamic limit (Phase II-H). --- ### I. MOTIVATION **1.1 Outcome of Phase II-E** Execution of the Phase II-E protocol yielded: - Branch E1 ($G_{E_1}=\sum_j\sigma_j^x$, full Hilbert space): $\langle G\rangle_{\rm PT}=0$ to machine precision for all tested $\lambda$ and all $\gamma<\gamma_{\rm EP}$. $b_{\rm eff}=J$ identically. Null result. - Branch E2 ($G_{E_2}=\sum_j(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)$, $S^z=0$ sector): $\langle G\rangle_{\rm PT}\neq0$ already at $\lambda=0$ and remains finite throughout the PT-unbroken phase. $b_{\rm eff}$ shifts systematically with $\lambda$. Activation confirmed. Thus the self-consistent mechanism is not restricted to the dimer setting of Paper 4, nor is it extinguished by the PT-odd obstruction identified in Phase II-D. A PT-even, sector-preserving two-body operator is sufficient to source a non-trivial constraint field at $N=4$. **1.2 The Classification Task** The positive result for $E_2$ opens four immediate questions that must be answered before finite-size scaling can be interpreted: 1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate? 2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E? 3. How does the magnitude of $\langle G\rangle_{\rm PT}$ and the induced shift $\Delta\gamma_{\rm EP}$ depend on the algebraic structure of the operator (support, commutation relations with $S^z_{\rm total}$ and with $\mathcal{PT}$)? 4. Which of the surviving operators produce a *displacement* of the exceptional point (the operational signature required for constraint-field control) rather than a mere renormalization of $b_{\rm eff}$ at fixed $\gamma_{\rm EP}$? Phase II-F answers these questions by exhaustive classification of a controlled operator set. --- ### II. OPERATOR TAXONOMY Operators are partitioned by three binary attributes: | Attribute | Values | Notation | |------------------------|---------------------------------|-------------------| | PT parity | even / odd | $\mathcal{P}_\pm$ | | Sector compatibility | preserves $S^z_{\rm total}$ / mixes | $\mathcal{S}_0$ / $\mathcal{S}_{\rm mix}$ | | Body order | one-body / two-body | $1$ / $2$ | The Phase II-D and II-E results already fix two cells: - $\mathcal{P}_-$ (any body, any sector) → null (structural anticommutation with $\mathcal{PT}$). - $\mathcal{P}_+$, $\mathcal{S}_{\rm mix}$, $1$-body ($G_{E_1}$) → null under the biorthogonal convention. The remaining open cells that must be classified are: - $\mathcal{P}_+$, $\mathcal{S}_0$, $1$-body - $\mathcal{P}_+$, $\mathcal{S}_0$, $2$-body (already positive for the XY bond; additional representatives required) - $\mathcal{P}_+$, $\mathcal{S}_{\rm mix}$, $2$-body --- ### III. CANDIDATE OPERATORS FOR CLASSIFICATION All operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E. **3.1 One-body, PT-even, sector-preserving** $G_{F_1}=\sum_{j=1}^{4}\sigma_j^z$ (Total magnetization; trivially preserves $S^z_{\rm total}$; PT-even because $\sigma^z$ is real and parity-invariant under site relabeling when the staggered PT field is taken into account.) **3.2 Two-body, PT-even, sector-preserving (beyond $E_2$)** $G_{F_2}=\sum_{j=1}^{3}\sigma_j^z\sigma_{j+1}^z$ (ZZ bond operator; the natural Ising partner of the XY bond.) $G_{F_3}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^x-\sigma_j^y\sigma_{j+1}^y)$ (Anisotropic XY combination; still sector-preserving.) **3.3 Two-body, PT-even, sector-mixing** $G_{F_4}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^z+\sigma_j^z\sigma_{j+1}^x)$ (Explicitly mixes $\Delta S^z=\pm1$ sectors while remaining PT-even.) **3.4 Reference operators (already classified)** - $G_{E_1}$ (null) and $G_{E_2}$ (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan. --- ### IV. NUMERICAL PROTOCOL **4.1 Hamiltonian** Identical to Phase II-E: \[ H=H_{\rm XXZ}(b_{\rm eff})+H_{\rm PT}(\gamma), \] with \[ H_{\rm XXZ}(b)=\sum_{j=1}^{3}\Bigl(b\,(S_j^x S_{j+1}^x+S_j^y S_{j+1}^y)+J\Delta\,S_j^z S_{j+1}^z\Bigr), \] \[ H_{\rm PT}=i\gamma\sum_{j=1}^{4}(-1)^j S_j^z, \] $J=1$, $\Delta=0.5$, $S^\alpha=\sigma^\alpha/2$. **4.2 Self-consistency equation** Unchanged: \[ b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}. \] Iteration tolerance $10^{-10}$, maximum 200 iterations. **4.3 Observable convention** Biorthogonal expectation value on the eigenstate of smallest $|\mathrm{Re}(E)|$, preferring PT-unbroken states: \[ \langle G\rangle_{\rm PT}=\frac{\langle\psi_L|G|\psi_R\rangle}{\langle\psi_L|\psi_R\rangle}. \] **4.4 Parameter scan** - $\lambda\in\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\}$ - $\gamma$ from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of $\gamma_{\rm EP}$) - For each $(\lambda,\gamma,G)$ the code

Oct 12, 20260 citations

A PT-Symmetric XXZ Chain with an External Constraint Field: Exceptional Points, Finite-Size

Anthony jordan Blair

A PT-Symmetric XXZ Chain with an External Constraint Field: Exceptional Points, Finite-Size Scaling, and Falsifiable Predictions This work introduces a non-Hermitian quantum-spin model that combines three established areas of modern theoretical physics: The XXZ Heisenberg spin-½ chain; PT-symmetric gain/loss dynamics; An externally coupled constraint field capable of modifying the location of exceptional points. The resulting Hamiltonian provides a mathematically explicit framework for studying how externally driven constraints interact with open quantum systems exhibiting parity-time symmetry. Rather than proposing a speculative unified theory, the objective of this work is to formulate a concrete, experimentally testable model that produces quantitative predictions capable of confirmation or falsification through laboratory measurement. Scientific Background Parity-time (PT) symmetric systems occupy an important position within contemporary non-Hermitian physics. Although such systems are not Hermitian, they can nevertheless exhibit purely real energy spectra within specific parameter regimes. As gain-loss asymmetry increases, the system eventually reaches a critical transition known as an Exceptional Point (EP). At an Exceptional Point: eigenvalues coalesce; eigenvectors coalesce; sensitivity to perturbations increases dramatically; system behavior changes qualitatively. Exceptional points have become a major area of research in: photonics; optics; condensed matter physics; open quantum systems; topological materials; waveguide lattices; sensing technologies. The present work extends this framework by introducing an explicit constraint-field coupling and examining its impact on exceptional-point behavior. Model Overview The total Hamiltonian consists of three components: Hermitian XXZ Backbone The first contribution is the XXZ spin-chain Hamiltonian. This term represents: coherent spin transport; exchange interactions; anisotropic coupling between neighboring spins. In the absence of gain/loss and external constraints, the model reduces to the conventional XXZ chain studied throughout quantum many-body physics. PT-Symmetric Gain/Loss Sector The second contribution introduces alternating gain and loss across the lattice. This term produces: non-Hermitian dynamics; PT-symmetric behavior; exceptional-point transitions. The gain/loss amplitude controls the transition between: PT-unbroken phases; PT-broken phases. External Constraint Field The third contribution is a spatially structured driving field. This field is modeled as a superposition of harmonically varying modes possessing: amplitude; frequency; phase; spatial localization. The field couples directly to the spin degrees of freedom and modifies the effective dynamics of the system. Exact Two-Site Solution To establish a mathematically transparent foundation, the paper solves the two-site case exactly. The resulting spectrum yields a closed-form exceptional-point condition: γEP=J\gamma_{EP}=J where: JJ is the exchange coupling; γ\gamma is the gain/loss strength. This analytical result separates three physically distinct regimes: PT-Unbroken Phase γ<J\gamma < J Properties: real eigenvalues; coherent transport; stable dynamics. Exceptional Point γ=J\gamma = J Properties: eigenvalue coalescence; eigenvector coalescence; maximal sensitivity. PT-Broken Phase γ>J\gamma > J Properties: complex eigenvalues; dissipative dominance; broken PT symmetry. Central Prediction The principal result of the paper is the prediction that the exceptional point becomes tunable when the external constraint field is activated. Specifically: γEP(λ)=J1+λ⟨G⟩J\gamma_{EP}(\lambda) = J\sqrt{ 1+ \frac{\lambda\langle G\rangle}{J} } where: λ\lambda is the coupling strength; ⟨G⟩\langle G\rangle is the mean constraint-field contribution. For weak coupling: ΔγEP≈λ⟨G⟩2\Delta\gamma_{EP} \approx \frac{\lambda\langle G\rangle}{2} The prediction is therefore simple: increasing constraint coupling should systematically shift the exceptional point. Falsifiability A central design goal of this work is scientific falsifiability. The model can be rejected by experiment. The theory is considered falsified if: no exceptional point is observed; the exceptional point remains fixed when the constraint field is varied; the observed shift fails to scale with the predicted dependence; an alternative model explains the data more successfully. These conditions are intentionally defined prior to experimentation. Finite-Size Scaling Program While the exact solution is obtained for the two-site system, the broader objective is to determine whether the phenomenon survives in larger many-body systems. The proposed finite-size scaling program investigates: N=4,6,8,10,12N = 4,6,8,10,12 site chains through exact diagonalization. The goal is to determine whether the exceptional-point shift converges toward a well-defined thermodynamic limit. A successful scaling analysis would provide evidence that the mechanism is not merely a finite-size artifact. Feedback Dynamics The paper further proposes a feedback mechanism in which the constraint field is partially sourced by the state of the spin chain itself. This introduces: nonlinear behavior; effective gain/loss renormalization; hysteresis effects; potentially emergent collective behavior. A predicted observable is a measurable hysteresis loop in the exceptional-point response. Experimental Realization A practical realization is proposed using PT-symmetric photonic waveguide arrays. In such systems: waveguides represent lattice sites; optical coupling represents exchange interactions; alternating amplification and attenuation implement gain/loss; an externally driven resonator grid provides the constraint field. The experimentally measurable quantity is the eigenvalue splitting: ΔE=4J2−γ2\Delta E = 4\sqrt{J^2-\gamma^2} which collapses at the exceptional point. By measuring how the collapse location changes with constraint strength, the core prediction can be directly tested. Research Roadmap The project is organized into successive stages: Stage 1 Exact analytical treatment of the two-site model. Stage 2 Finite-size numerical diagonalization. Stage 3 Photonic-lattice simulation. Stage 4 Laboratory implementation. Stage 5 Independent experimental replication. Contribution This work should be understood as a contribution to: non-Hermitian quantum physics; PT-symmetric systems; exceptional-point dynamics; open quantum systems. The paper introduces: an explicitly defined Hamiltonian; a solvable toy model; a finite-size scaling program; a laboratory realization pathway; a quantitative prediction subject to experimental verification. Its scientific value rests not on the novelty of the concept alone, but on the existence of a clear pathway by which the prediction may be tested, reproduced, supported, revised, or rejected. Keywords: PT Symmetry, Exceptional Points, XXZ Spin Chain, Non-Hermitian Physics, Open Quantum Systems, Quantum Many-Body Systems, Finite-Size Scaling, Photonic Lattices, Gain-Loss Dynamics, Condensed Matter Physics, Mathematical Physics, Quantum Information, Experimental Quantum Optics, Falsifiable Prediction, Project Prometheus.A PT-Symmetric XXZ Chain with an External Constraint Field: Exceptional Points, Finite-Size Scaling, and Falsifiable Predictions This work introduces a non-Hermitian quantum-spin model that combines three established areas of modern theoretical physics: The XXZ Heisenberg spin-½ chain; PT-symmetric gain/loss dynamics; An externally coupled constraint field capable of modifying the location of exceptional points. The resulting Hamiltonian provides a mathematically explicit framework for studying how externally driven constraints interact with open quantum systems exhibiting parity-time symmetry. Rather than proposing a speculative unified theory, the objective of this work is to formulate a concrete, experimentally testable model that produces quantitative predictions capable of confirmation or falsification through laboratory measurement. Scientific Background Parity-time (PT) symmetric systems occupy an important position within contemporary non-Hermitian physics. Although such systems are not Hermitian, they can nevertheless exhibit purely real energy spectra within specific parameter regimes. As gain-loss asymmetry increases, the system eventually reaches a critical transition known as an Exceptional Point (EP). At an Exceptional Point: eigenvalues coalesce; eigenvectors coalesce; sensitivity to perturbations increases dramatically; system behavior changes qualitatively. Exceptional points have become a major area of research in: photonics; optics; condensed matter physics; open quantum systems; topological materials; waveguide lattices; sensing technologies. The present work extends this framework by introducing an explicit constraint-field coupling and examining its impact on exceptional-point behavior. Model Overview The total Hamiltonian consists of three components: Hermitian XXZ Backbone The first contribution is the XXZ spin-chain Hamiltonian. This term represents: coherent spin transport; exchange interactions; anisotropic coupling between neighboring spins. In the absence of gain/loss and external constraints, the model reduces to the conventional XXZ chain studied throughout quantum many-body physics. PT-Symmetric Gain/Loss Sector The second contribution introduces alternating gain and loss across the lattice. This term produces: non-Hermitian dynamics; PT-symmetric behavior; exceptional-point transitions. The gain/loss amplitude controls the transition between: PT-unbroken phases; PT-broken phases. External Constraint Field The third contribution is a spatially structured driving field. This field is modeled as a superposition of harmonically varying modes possessing: amplitude; frequency; phase; spatial localization. The field couples directly to the spin degrees of freedom and modifies the effective dynamics of the system. Exact Two-Site Solution To establish a mathematically transparent foundation, the paper solves the two-site case exactly. The resulting spectrum yields a closed-form exceptional-point condition: γEP=J\gamma_{EP}=J where: JJ is the exchange coupling; γ\gamma is the gain/loss strength. This analytical result separates three physically distinct regimes: PT-Unbroken Phase γ<J\gamma < J Properties: real eigenvalues; coherent transport; stable dynamics. Exceptional Point γ=J\gamma = J Properties: eigenvalue coalescence; eigenvector coalescence; maximal sensitivity. PT-Broken Phase γ>J\gamma > J Properties: complex eigenvalues; dissipative dominance; broken PT symmetry. Central Prediction The principal result of the paper is the prediction that the exceptional point becomes tunable when the external constraint field is activated. Specifically: γEP(λ)=J1+λ⟨G⟩J\gamma_{EP}(\lambda) = J\sqrt{ 1+ \frac{\lambda\langle G\rangle}{J} } where: λ\lambda is the coupling strength; ⟨G⟩\langle G\rangle is the mean constraint-field contribution. For weak coupling: ΔγEP≈λ⟨G⟩2\Delta\gamma_{EP} \approx \frac{\lambda\langle G\rangle}{2} The prediction is therefore simple: increasing constraint coupling should systematically shift the exceptional point. Falsifiability A central design goal of this work is scientific falsifiability. The model can be rejected by experiment. The theory is considered falsified if: no exceptional point is observed; the exceptional point remains fixed when the constraint field is varied; the observed shift fails to scale with the predicted dependence; an alternative model explains the data more successfully. These conditions are intentionally defined prior to experimentation. Finite-Size Scaling Program While the exact solution is obtained for the two-site system, the broader objective is to determine whether the phenomenon survives in larger many-body systems. The proposed finite-size scaling program investigates: N=4,6,8,10,12N = 4,6,8,10,12 site chains through exact diagonalization. The goal is to determine whether the exceptional-point shift converges toward a well-defined thermodynamic limit. A successful scaling analysis would provide evidence that the mechanism is not merely a finite-size artifact. Feedback Dynamics The paper further proposes a feedback mechanism in which the constraint field is partially sourced by the state of the spin chain itself. This introduces: nonlinear behavior; effective gain/loss renormalization; hysteresis effects; potentially emergent collective behavior. A predicted observable is a measurable hysteresis loop in the exceptional-point response. Experimental Realization A practical realization is proposed using PT-symmetric photonic waveguide arrays. In such systems: waveguides represent lattice sites; optical coupling represents exchange interactions; alternating amplification and attenuation implement gain/loss; an externally driven resonator grid provides the constraint field. The experimentally measurable quantity is the eigenvalue splitting: ΔE=4J2−γ2\Delta E = 4\sqrt{J^2-\gamma^2} which collapses at the exceptional point. By measuring how the collapse location changes with constraint strength, the core prediction can be directly tested. Research Roadmap The project is organized into successive stages: Stage 1 Exact analytical treatment of the two-site model. Stage 2 Finite-size numerical diagonalization. Stage 3 Photonic-lattice simulation. Stage 4 Laboratory implementation. Stage 5 Independent experimental replication. Contribution This work should be understood as a contribution to: non-Hermitian quantum physics; PT-symmetric systems; exceptional-point dynamics; open quantum systems. The paper introduces: an explicitly defined Hamiltonian; a solvable toy model; a finite-size scaling program; a laboratory realization pathway; a quantitative prediction subject to experimental verification. Its scientific value rests not on the novelty of the concept alone, but on the existence of a clear pathway by which the prediction may be tested, reproduced, supported, revised, or rejected. Keywords: PT Symmetry, Exceptional Points, XXZ Spin Chain, Non-Hermitian Physics, Open Quantum Systems, Quantum Many-Body Systems, Finite-Size Scaling, Photonic Lattices, Gain-Loss Dynamics, Condensed Matter Physics, Mathematical Physics, Quantum Information, Experimental Quantum Optics, Falsifiable Prediction, Project Prometheus.

Oct 10, 20260 citations

The Non-Hermitian ISA: PT Symmetry, Exceptional Points, and the 38-Fold Way

Ian Richard C Buckley

Exceptional points (EPs) — the snap thresholds of the Maslov-Gibbs Einsum free energy — are identified as a universal organising principle across five domains where PT symmetry has not previously been recognised: (i) quantum chemistry, where DFT failure is an EP crossing detected by the Weyl index c₂; (ii) quantum error correction, where canonical code distances correspond to EP orders and EP₂ search beats Grover for N > 16; (iii) ranking algorithms, where RavenRank is PT-symmetric ranking near the EP of the comparison Laplacian; (iv) enzyme catalysis and biological self-organisation, where k_cat operates at the Curzon-Ahlborn EP of the kinetic Lindbladian; and (v) information geometry, where the Fisher metric degenerates at EP₂ and the Cramér-Rao bound saturates there. The 38-fold way classifies all PT-symmetric phases of matter.Exceptional points (EPs) — the snap thresholds of the Maslov-Gibbs Einsum free energy — are identified as a universal organising principle across five domains where PT symmetry has not previously been recognised: (i) quantum chemistry, where DFT failure is an EP crossing detected by the Weyl index c₂; (ii) quantum error correction, where canonical code distances correspond to EP orders and EP₂ search beats Grover for N > 16; (iii) ranking algorithms, where RavenRank is PT-symmetric ranking near the EP of the comparison Laplacian; (iv) enzyme catalysis and biological self-organisation, where k_cat operates at the Curzon-Ahlborn EP of the kinetic Lindbladian; and (v) information geometry, where the Fisher metric degenerates at EP₂ and the Cramér-Rao bound saturates there. The 38-fold way classifies all PT-symmetric phases of matter.

Jul 22, 20260 citations

Universal finite-size gap scaling of the quantum Ising chain

Masaki Oshikawa

I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in $1+1$ dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in $1+1$ dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.

arXiv:1910.06353Oct 14, 20195 citations

*Phase II-E executed.** Results confirm **activation for one PT-even operator**. ### Numerical setup (faithful to the specification) -

N=4

, open chain,

J=1

\Delta=0.5

. -

S^\alpha=\sigma^\alpha/2

. - Hamiltonian: XY couplings use the running

b_{\rm eff}

, ZZ couplings use

J\Delta

, plus the staggered imaginary field

H_{\rm PT}

. - Self-consistency:

b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}

, iterated to

10^{-10}

. - Biorthogonal expectation on the eigenstate of smallest

|\mathrm{Re}(E)|

(preferring PT-unbroken). -

\gamma_{\rm EP}\approx0.274

\lambda=0

(matches the Phase II-D value 0.275). - Branches as specified: E1 =

\sum\sigma_j^x

(full 16-dimensional space); E2 =

\sum(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)

(

S^z=0

sector, dim 6). ### Results **Branch E1 (transverse field, full space)**

\langle G_{E_1}\rangle_{\rm PT}=0

(machine precision

\sim10^{-16}

–

10^{-19}

) for every

\lambda\in\{0,0.05,0.10,0.15,0.20\}

and every

\gamma

in the PT-unbroken regime. Consequently

b_{\rm eff}=J

exactly. **Null.** **Branch E2 (transverse bond / XY exchange,

S^z=0

sector)**

\langle G_{E_2}\rangle_{\rm PT}\neq0

already at

\lambda=0

and remains non-zero throughout the unbroken phase. Typical values (unbroken points): -

\lambda=0

\max|\langle G\rangle|\approx0.063

\lambda=0.05

\max|\langle G\rangle|\approx0.063

\max|b_{\rm eff}-J|\approx0.0016

\lambda=0.10

\max|\langle G\rangle|\approx0.062

\max|b_{\rm eff}-J|\approx0.0031

\lambda=0.15

\max|\langle G\rangle|\approx0.062

\max|b_{\rm eff}-J|\approx0.0046

\lambda=0.20

\max|\langle G\rangle|\approx0.062

\max|b_{\rm eff}-J|\approx0.0061

b_{\rm eff}

shifts systematically with

\lambda

. **Activation.** ### Prediction match - Prediction 1 (null for both) — **falsified**. - Prediction 2 (activation for at least one) — **confirmed**. - Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs). - Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation. The self-consistent mechanism **does activate** at

N=4

for a PT-even, sector-compatible operator. The PT-odd obstruction of Phase II-D is therefore not the sole operative constraint; a PT-even operator that preserves

S^z_{\rm total}

can source a non-trivial field. ### Gate status Phase II-E **unblocks** Phase II-F (branch classification now includes the PT-even class). Finite-size scaling (II-G) and the thermodynamic limit (II-H) remain formally gated on the subsequent classification step, but the existence proof at

N=4

is now positive. Full raw scan (87

\gamma

points × 5

\lambda

× 2 branches) is archived in `/home/workdir/artifacts/phase_ii_e_results.json`. Source: `/home/workdir/artifacts/phase_ii_e.py`. The mechanism was the vessel. The constraint was the discovery — and one more door is now open. Tony Blair <anthonyjblair8@gmail.com> 2:17 PM (11 minutes ago) to me **TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4** **AUTHOR:** Anthony Jordan Blair **AFFILIATION:** Proclus Dynamics Consortium **DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification **VERSION:** 1.0 (October 2026) **COMPANION TO:** Phase II-E Program Specification; Phase II-E Numerical Execution Results; Phase II-D Technical Report; Project Prometheus Phase I Closeout Report (DOI: 10.5281/zenodo.23314444) **STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation) --- ### ABSTRACT Phase II-E demonstrated that the PT-even transverse-bond operator

G_{E_2}

produces a non-vanishing biorthogonal expectation value in the

S^z=0

sector of the

N=4

PT-symmetric XXZ chain, thereby activating the self-consistent mechanism of Paper 4. The transverse-field operator

G_{E_1}

remained identically zero. Phase II-F classifies the admissible branches of the self-consistent field according to (i) PT parity, (ii) sector compatibility, (iii) operator support (one-body vs. two-body), and (iv) the resulting displacement of the exceptional point. The classification supplies the necessary taxonomy for the finite-size scaling study (Phase II-G) and determines which operator families survive into the thermodynamic limit (Phase II-H). --- ### I. MOTIVATION **1.1 Outcome of Phase II-E** Execution of the Phase II-E protocol yielded: - Branch E1 (

G_{E_1}=\sum_j\sigma_j^x

, full Hilbert space):

\langle G\rangle_{\rm PT}=0

to machine precision for all tested

\lambda

and all

\gamma<\gamma_{\rm EP}

b_{\rm eff}=J

identically. Null result. - Branch E2 (

G_{E_2}=\sum_j(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y)

S^z=0

sector):

\langle G\rangle_{\rm PT}\neq0

already at

\lambda=0

and remains finite throughout the PT-unbroken phase.

b_{\rm eff}

shifts systematically with

\lambda

. Activation confirmed. Thus the self-consistent mechanism is not restricted to the dimer setting of Paper 4, nor is it extinguished by the PT-odd obstruction identified in Phase II-D. A PT-even, sector-preserving two-body operator is sufficient to source a non-trivial constraint field at

N=4

. **1.2 The Classification Task** The positive result for

E_2

opens four immediate questions that must be answered before finite-size scaling can be interpreted: 1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate? 2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E? 3. How does the magnitude of

\langle G\rangle_{\rm PT}

and the induced shift

\Delta\gamma_{\rm EP}

depend on the algebraic structure of the operator (support, commutation relations with

S^z_{\rm total}

and with

\mathcal{PT}

)? 4. Which of the surviving operators produce a *displacement* of the exceptional point (the operational signature required for constraint-field control) rather than a mere renormalization of

b_{\rm eff}

at fixed

\gamma_{\rm EP}

? Phase II-F answers these questions by exhaustive classification of a controlled operator set. --- ### II. OPERATOR TAXONOMY Operators are partitioned by three binary attributes: | Attribute | Values | Notation | |------------------------|---------------------------------|-------------------| | PT parity | even / odd |

\mathcal{P}_\pm

| | Sector compatibility | preserves

S^z_{\rm total}

/ mixes |

\mathcal{S}_0

\mathcal{S}_{\rm mix}

| | Body order | one-body / two-body |

1

2

| The Phase II-D and II-E results already fix two cells: -

\mathcal{P}_-

(any body, any sector) → null (structural anticommutation with

\mathcal{PT}

). -

\mathcal{P}_+

\mathcal{S}_{\rm mix}

1

-body (

G_{E_1}

) → null under the biorthogonal convention. The remaining open cells that must be classified are: -

\mathcal{P}_+

\mathcal{S}_0

1

-body -

\mathcal{P}_+

\mathcal{S}_0

2

-body (already positive for the XY bond; additional representatives required) -

\mathcal{P}_+

\mathcal{S}_{\rm mix}

2

-body --- ### III. CANDIDATE OPERATORS FOR CLASSIFICATION All operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E. **3.1 One-body, PT-even, sector-preserving**

G_{F_1}=\sum_{j=1}^{4}\sigma_j^z

(Total magnetization; trivially preserves

S^z_{\rm total}

; PT-even because

\sigma^z

is real and parity-invariant under site relabeling when the staggered PT field is taken into account.) **3.2 Two-body, PT-even, sector-preserving (beyond

E_2

)**

G_{F_2}=\sum_{j=1}^{3}\sigma_j^z\sigma_{j+1}^z

(ZZ bond operator; the natural Ising partner of the XY bond.)

G_{F_3}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^x-\sigma_j^y\sigma_{j+1}^y)

(Anisotropic XY combination; still sector-preserving.) **3.3 Two-body, PT-even, sector-mixing**

G_{F_4}=\sum_{j=1}^{3}(\sigma_j^x\sigma_{j+1}^z+\sigma_j^z\sigma_{j+1}^x)

(Explicitly mixes

\Delta S^z=\pm1

sectors while remaining PT-even.) **3.4 Reference operators (already classified)** -

G_{E_1}

(null) and

G_{E_2}

(active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan. --- ### IV. NUMERICAL PROTOCOL **4.1 Hamiltonian** Identical to Phase II-E:

H=H_{\rm XXZ}(b_{\rm eff})+H_{\rm PT}(\gamma),

with

H_{\rm XXZ}(b)=\sum_{j=1}^{3}\Bigl(b\,(S_j^x S_{j+1}^x+S_j^y S_{j+1}^y)+J\Delta\,S_j^z S_{j+1}^z\Bigr),

H_{\rm PT}=i\gamma\sum_{j=1}^{4}(-1)^j S_j^z,

J=1

\Delta=0.5

S^\alpha=\sigma^\alpha/2

. **4.2 Self-consistency equation** Unchanged:

b_{\rm eff}^2=J^2+\lambda J\langle G\rangle_{\rm PT}.

Iteration tolerance

10^{-10}

, maximum 200 iterations. **4.3 Observable convention** Biorthogonal expectation value on the eigenstate of smallest

|\mathrm{Re}(E)|

, preferring PT-unbroken states:

\langle G\rangle_{\rm PT}=\frac{\langle\psi_L|G|\psi_R\rangle}{\langle\psi_L|\psi_R\rangle}.

**4.4 Parameter scan** -

\lambda\in\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\}

\gamma

from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of

\gamma_{\rm EP}

) - For each

(\lambda,\gamma,G)

the code

1+1

dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.