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

Anthony Jordan Blair

Published Oct 13, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.23333493

Related papers

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

Anthony Jordan Blair

Oct 13, 20260 citations

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

Anthony Jordan Blair

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

Oct 12, 20260 citations

Quantum Structural Theory of Harmony (QSTH 8.7) — Lambda_lock Technical Note: From the Schrödinger Ledger of Possibility to Objective Settlement of Reality

Rostislav Stepanik

Jun 6, 20260 citations

Boundary Compensation Layers VI: Irrecoverable Loss Classes and Sufficiency Thresholds Protocol Envelopes, Refinement Limits, Persistent Indistinguishability and the No-Resurrection Principle

Alex Malachevsky

Jul 13, 20260 citations

Operational Observability and Contract-Induced Resource Transitions in Quantum Realisability

Mark Whitlock

Aug 21, 20260 citations

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

Anthony Jordan Blair

Published Oct 13, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23333493

Related papers

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

Anthony Jordan Blair

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.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.

Oct 13, 20260 citations

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 -- 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

Quantum Structural Theory of Harmony (QSTH 8.7) — Lambda_lock Technical Note: From the Schrödinger Ledger of Possibility to Objective Settlement of Reality

Rostislav Stepanik

This QSTH 8.7 working publication develops Lambda_lock as a technical candidate condition for the transition from coherent possibility toward stable record formation. The publication follows the QSTH 8.x sequence, especially QSTH 8.6.XS, where the Schrödinger equation was interpreted as a ledger of coherent possibility, and QSTH 8.6.XP, where Lambda_lock was introduced as a candidate threshold for objective settlement. QSTH 8.7 then asks the next technical question: what minimal structure is required for possibility to stop remaining only possible and become a stable, directionally readable record? Within this framework, Lambda_lock is not presented as a finished physical constant or confirmed law. It is introduced as a candidate lock condition, threshold window, and structured functional boundary. The publication also develops Gamma_lock as a candidate functional describing accumulated locking contribution over a finite transition window. A central modeling route is expressed as: integral Gamma_lock(t) dt >= Lambda_lock -> R_stable The text introduces several operational layers: candidate admissibility amplitude A_rec, the Gamma_lock integral, the Lambda_lock threshold condition, Sigma-wall and E_wall interface contributions, spin-locking contribution, gate-off null test, and the bridge toward Hessian settlement stability. The publication also emphasizes computability. Several components can already be explored as toy-models: threshold integrals, exponential rise and damping regimes, candidate non-Hermitian locking terms, gate-off limits, and later Hessian stability tests once a lock-potential or settlement landscape is defined. This text belongs to the QSTH 8.x working sequence. It is not presented as a confirmed physical theory, but as a structured conceptual, mathematical, and methodological technical note intended for future formalization, numerical testing, falsification, and comparison with null models. Additional candidate modeling relations used in this technical note include: A_rec = candidate admissibility amplitude Gamma_lock = candidate locking functional H_eff = nabla^2 Phi_lock H_eff > 0 -> stable settlement Gate-off / null test: Gamma_lock -> 0 These expressions are not presented as confirmed physical laws. They are structured candidate relations intended for future toy-model construction, numerical testing, falsification, and comparison with null models. Short description QSTH 8.7 develops Lambda_lock as a candidate technical lock condition for the transition from coherent possibility to stable record formation. It introduces Gamma_lock, A_rec, Sigma-wall, E_wall, spin-locking contribution, gate-off null testing, and the bridge toward Hessian settlement stability. Methodological status This publication is part of the QSTH CORE/CAND/SUPPORT/FUTURE framework. It should be read as a structured working publication, not as a confirmed physical model. Lambda_lock and Gamma_lock are candidate constructs. Their role is to provide a disciplined modeling language for future toy-model construction, numerical testing, falsification, and comparison with standard null models such as decoherence, open quantum systems, and non-Hermitian effective dynamics. Computability note Several parts of the proposed framework are already computable as toy-models. A candidate Gamma_lock function can be selected and integrated over a finite transition window. Threshold conditions of the form integral Gamma_lock(t) dt >= Lambda_lock can be numerically tested. Gate-off limits can be imposed by setting Gamma_lock to zero or by removing the proposed locking contribution. Candidate exponential, damping, growth, and phase-sensitive regimes can be compared against null models. These are not yet confirmed physical laws. They are structured modeling entry points for future numerical testing, falsification, and scientific collaboration. This record belongs to the QSTH 8.x publication sequence. It follows QSTH 8.6.XS — Schrödinger Equation as a Ledger of Possibility and QSTH 8.6.XP — Lambda_lock and Objective Settlement. QSTH 8.7 provides the technical bridge from the concept of possibility and objective settlement toward a structured candidate locking model. This publication prepares later QSTH branches on Spin-Locking and Structural Orientation, Schrödinger Equation with QSTH Locking Term, Hessian Geometry of Record Settlement, Entropic Genesis of Photon, and the later M-independent horizon framework. The publication includes a computability section and a gate-off / null-test discipline intended as an open invitation for future toy-model construction, numerical testing, falsification, and scientific collaboration. One-line public summary QSTH 8.7 introduces Lambda_lock as a candidate technical lock condition through which coherent possibility may become a stable, directionally readable record. Diamond sentence QSTH 8.7 does not claim that Lambda_lock is already a confirmed law. It defines the gate where such a law could begin to be modeled, tested, and possibly falsified.This QSTH 8.7 working publication develops Lambda_lock as a technical candidate condition for the transition from coherent possibility toward stable record formation. The publication follows the QSTH 8.x sequence, especially QSTH 8.6.XS, where the Schrödinger equation was interpreted as a ledger of coherent possibility, and QSTH 8.6.XP, where Lambda_lock was introduced as a candidate threshold for objective settlement. QSTH 8.7 then asks the next technical question: what minimal structure is required for possibility to stop remaining only possible and become a stable, directionally readable record? Within this framework, Lambda_lock is not presented as a finished physical constant or confirmed law. It is introduced as a candidate lock condition, threshold window, and structured functional boundary. The publication also develops Gamma_lock as a candidate functional describing accumulated locking contribution over a finite transition window. A central modeling route is expressed as: integral Gamma_lock(t) dt >= Lambda_lock -> R_stable The text introduces several operational layers: candidate admissibility amplitude A_rec, the Gamma_lock integral, the Lambda_lock threshold condition, Sigma-wall and E_wall interface contributions, spin-locking contribution, gate-off null test, and the bridge toward Hessian settlement stability. The publication also emphasizes computability. Several components can already be explored as toy-models: threshold integrals, exponential rise and damping regimes, candidate non-Hermitian locking terms, gate-off limits, and later Hessian stability tests once a lock-potential or settlement landscape is defined. This text belongs to the QSTH 8.x working sequence. It is not presented as a confirmed physical theory, but as a structured conceptual, mathematical, and methodological technical note intended for future formalization, numerical testing, falsification, and comparison with null models. Additional candidate modeling relations used in this technical note include: A_rec = candidate admissibility amplitude Gamma_lock = candidate locking functional H_eff = nabla^2 Phi_lock H_eff > 0 -> stable settlement Gate-off / null test: Gamma_lock -> 0 These expressions are not presented as confirmed physical laws. They are structured candidate relations intended for future toy-model construction, numerical testing, falsification, and comparison with null models. Short description QSTH 8.7 develops Lambda_lock as a candidate technical lock condition for the transition from coherent possibility to stable record formation. It introduces Gamma_lock, A_rec, Sigma-wall, E_wall, spin-locking contribution, gate-off null testing, and the bridge toward Hessian settlement stability. Methodological status This publication is part of the QSTH CORE/CAND/SUPPORT/FUTURE framework. It should be read as a structured working publication, not as a confirmed physical model. Lambda_lock and Gamma_lock are candidate constructs. Their role is to provide a disciplined modeling language for future toy-model construction, numerical testing, falsification, and comparison with standard null models such as decoherence, open quantum systems, and non-Hermitian effective dynamics. Computability note Several parts of the proposed framework are already computable as toy-models. A candidate Gamma_lock function can be selected and integrated over a finite transition window. Threshold conditions of the form integral Gamma_lock(t) dt >= Lambda_lock can be numerically tested. Gate-off limits can be imposed by setting Gamma_lock to zero or by removing the proposed locking contribution. Candidate exponential, damping, growth, and phase-sensitive regimes can be compared against null models. These are not yet confirmed physical laws. They are structured modeling entry points for future numerical testing, falsification, and scientific collaboration. This record belongs to the QSTH 8.x publication sequence. It follows QSTH 8.6.XS — Schrödinger Equation as a Ledger of Possibility and QSTH 8.6.XP — Lambda_lock and Objective Settlement. QSTH 8.7 provides the technical bridge from the concept of possibility and objective settlement toward a structured candidate locking model. This publication prepares later QSTH branches on Spin-Locking and Structural Orientation, Schrödinger Equation with QSTH Locking Term, Hessian Geometry of Record Settlement, Entropic Genesis of Photon, and the later M-independent horizon framework. The publication includes a computability section and a gate-off / null-test discipline intended as an open invitation for future toy-model construction, numerical testing, falsification, and scientific collaboration. One-line public summary QSTH 8.7 introduces Lambda_lock as a candidate technical lock condition through which coherent possibility may become a stable, directionally readable record. Diamond sentence QSTH 8.7 does not claim that Lambda_lock is already a confirmed law. It defines the gate where such a law could begin to be modeled, tested, and possibly falsified.

Jun 6, 20260 citations

Boundary Compensation Layers VI: Irrecoverable Loss Classes and Sufficiency Thresholds Protocol Envelopes, Refinement Limits, Persistent Indistinguishability and the No-Resurrection Principle

Alex Malachevsky

his paper develops the sixth and final foundational module of the BCLayers branch of the Boundary Compensation programme. BC-Layers Iintroduced finite-resolution readout channels. BC-Layers II formalized layermorphisms, preserved invariants, loss ledgers and evolution drift. BC-LayersIII developed task-relative multi-channel sufficiency. BC-Layers IV treatedhidden-sector elimination as a certified Schur–Feshbach response morphism.BC-Layers V organized parameter-dependent reductions into local responseatlases with pole walls, recalibration maps, transition budgets and mandatorycertificate reset.The present paper asks the limiting question: which losses can be repairedby admissible refinement, and which distinctions remain inaccessible under every protocol allowed by a declared observation class? Let Xb be an upper realization class and let P be a registered family of admissible passive observationprotocolsΠ : X −→ D b Π.The protocol envelope isΠP(X) =Π(X)Π∈P,and it induces the persistent indistinguishability relationX ∼P Y ⇐⇒ Π(X) = Π(Y ) for every Π ∈ P.A distinction is irrecoverable relative to P when it survives this complete admissible envelope.The central No-Resurrection Principle states that no deterministic postprocessing, finite channel aggregation or passive adaptive decision treeconstructed solely from protocols in P can distinguish a persistently indistinguishable pair. The result also extends to randomized post-processing:identical descriptor inputs produce identical output probability laws when theauxiliary randomness is independent of the tested realization. State-updatingor destructive protocols require a stronger history-indistinguishability semantics and are not covered by initial-state equality alone.For a task Q : X → Y b , the paper distinguishes single-protocol, finite-family,envelope, approximate and asymptotic recoverability. Along a refinement family Πr, the exact task oscillationEr(Q) = sup{dY (Q(X), Q(Y )) : Πr(X) = Πr(Y )}is monotone nonincreasing. For finite-resolution data, the robust two-scale oscillationEr,δ(Q) = sup{dY (Q(X), Q(Y )) : dr(Πr(X), Πr(Y )) ≤ δ}separates descriptor tolerance δ from admissible task uncertainty ε. Its monotonicity requires calibrated transport of metrics and tolerances across refinement maps.The paper classifies recoverable, resolution-induced, task-irrelevant,equivalence-calibrated, protocol-structural and probe-class-relative losses.It separates information-theoretic recoverability from computational tractability, and it treats the full protocol envelope as a set-theoretic information upperbound whose topological, measurable, computable and operational structuresrequire separate declarations.The principal statuses include RECOVERABLE_BY_REFINEMENT, FINITE_SUFFICIENCY_THRESHOLD, ROBUST_SUFFICIENCY_THRESHOLD, PROTOCOL_STRUCTURAL_LOSS, NO_RESURRECTION_CERTIFIED, HISTORY_SEMANTICS_REQUIRED, andREADY_FOR_BCM.The paper does not claim an ontologically absolute information limit, doesnot identify deterministic task factorization with the Fisher–Neyman factorization thehis paper develops the sixth and final foundational module of the BCLayers branch of the Boundary Compensation programme. BC-Layers Iintroduced finite-resolution readout channels. BC-Layers II formalized layermorphisms, preserved invariants, loss ledgers and evolution drift. BC-LayersIII developed task-relative multi-channel sufficiency. BC-Layers IV treatedhidden-sector elimination as a certified Schur–Feshbach response morphism.BC-Layers V organized parameter-dependent reductions into local responseatlases with pole walls, recalibration maps, transition budgets and mandatorycertificate reset.The present paper asks the limiting question: which losses can be repairedby admissible refinement, and which distinctions remain inaccessible under every protocol allowed by a declared observation class? Let Xb be an upper realization class and let P be a registered family of admissible passive observationprotocolsΠ : X −→ D b Π.The protocol envelope isΠP(X) =Π(X)Π∈P,and it induces the persistent indistinguishability relationX ∼P Y ⇐⇒ Π(X) = Π(Y ) for every Π ∈ P.A distinction is irrecoverable relative to P when it survives this complete admissible envelope.The central No-Resurrection Principle states that no deterministic postprocessing, finite channel aggregation or passive adaptive decision treeconstructed solely from protocols in P can distinguish a persistently indistinguishable pair. The result also extends to randomized post-processing:identical descriptor inputs produce identical output probability laws when theauxiliary randomness is independent of the tested realization. State-updatingor destructive protocols require a stronger history-indistinguishability semantics and are not covered by initial-state equality alone.For a task Q : X → Y b , the paper distinguishes single-protocol, finite-family,envelope, approximate and asymptotic recoverability. Along a refinement family Πr, the exact task oscillationEr(Q) = sup{dY (Q(X), Q(Y )) : Πr(X) = Πr(Y )}is monotone nonincreasing. For finite-resolution data, the robust two-scale oscillationEr,δ(Q) = sup{dY (Q(X), Q(Y )) : dr(Πr(X), Πr(Y )) ≤ δ}separates descriptor tolerance δ from admissible task uncertainty ε. Its monotonicity requires calibrated transport of metrics and tolerances across refinement maps.The paper classifies recoverable, resolution-induced, task-irrelevant,equivalence-calibrated, protocol-structural and probe-class-relative losses.It separates information-theoretic recoverability from computational tractability, and it treats the full protocol envelope as a set-theoretic information upperbound whose topological, measurable, computable and operational structuresrequire separate declarations.The principal statuses include RECOVERABLE_BY_REFINEMENT, FINITE_SUFFICIENCY_THRESHOLD, ROBUST_SUFFICIENCY_THRESHOLD, PROTOCOL_STRUCTURAL_LOSS, NO_RESURRECTION_CERTIFIED, HISTORY_SEMANTICS_REQUIRED, andREADY_FOR_BCM.The paper does not claim an ontologically absolute information limit, doesnot identify deterministic task factorization with the Fisher–Neyman factorization the

Jul 13, 20260 citations

Operational Observability and Contract-Induced Resource Transitions in Quantum Realisability

Mark Whitlock

Paper 7 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a structural theory for classifying how representation-resource requirements change when the operational obligation imposed on an independently fixed native system is lawfully strengthened. Building on the contract-relative resource law of Paper 6, it asks which source-level features of a strengthening determine or constrain the resulting resource transition. The theory fixes the native system, weak and strong contracts, realisation category, tolerance and charged resource before optimisation. A finite-baseline gate requires the weak contract to possess a finite-resource realisation, preventing pre-existing category nonrealisability from being misclassified as a strengthening-induced transition. The source-level object is a Contract Distinction System (CDS), which records the native configurations challenged by the stronger obligation, required distinctions or recovery targets, success rules, inspection boundaries and interface/commonality structure. Semantic challenge refinement and CRC cofinal/refinement equivalence control dependence on alternative finite challenge presentations. A Certified Observability Transport Theory (COTT) then maps licensed source-level facts into necessary resource lower bounds inside a fixed representation category and tolerance regime. The resulting directed finitary demand profile supports two distinct mechanisms. Internal divergence occurs when a single strengthened contract contains finite restrictions of arbitrarily large certified demand and can therefore exclude every finite-resource strong realisation in the declared category. Family divergence occurs when every member of a parametrised family remains finitely realisable while the additional resource induced by strengthening grows without bound across the family. Lower recognition is kept logically separate from upper-side existence: bounded certified demand is not treated as an existence theorem, and finite regimes require independent construction, compactness/gluing or another valid upper certificate. Conditional on finite baseline feasibility, the resulting pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unbounded finite separation carried separately as a family-scaling modifier. Several exact witnesses separate these regimes. With three native labels, the same coarse distinction graph K3, representation category, Hilbert-dimension resource and discrimination threshold fixed, changing only interface commonality produces an exact minimum-dimension transition from 2 to 3 throughout 2/3 < τ ≤ 1/2 + √3/4. Thus interface structure can itself be representation-resource-bearing. For the canonical TUS OS target, the complete static Atlas has exact minimum Hilbert dimension 4, whereas recovery-complete selected-line instrument access has exact minimum 64, giving an exact strict-finite transition 4→64. The weaker state-only common-CPTP problem remains unresolved within 32≤d≤64, demonstrating that the stronger interface theorem cannot be transferred backward through a forgetful map. For periodically modulated decay processes, every finite member remains finitely realisable while the contract-induced capacity gap diverges with family size. For the fixed dual-Poisson process, every positive worst-case transparency strengthening eliminates all finite-resource strong realisations in the declared autonomous one-copy classical–quantum category. A synthetic Countable Pairwise Identity Ladder supplies a cross-interface internal-divergence test, while the Fanizza–Lumbreras–Winter result marks the separate outer boundary of baseline category nonrealisability. Together, the results establish that operational observability is a resource-bearing input to quantum realisability without being invariably resource-increasing. Resource transition depends on the typed structure of the operational obligation—not merely native-state count, predictive-state cardinality, distinction count or coarse distinction graph. Supporting General Theory Programme This paper is Paper 7 in the TUS OS quantum-operational and Answerability research programme. Building on Paper 6’s contract-relative resource theory, it develops operational observability and a structural classification of contract-induced resource transitions, distinguishing resource-neutral, strict finite, finite-resource breakdown and unresolved pointwise outcomes, with family divergence carried separately as a family-scaling modifier while separating lower-bound certification from constructive realisability. Papers 8–9 then move to Answerability as a generic mathematical object, developing the resource geometry of answerable inference and the Answerability Calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops the structural classification theory for contract-induced representation-resource transitions. It introduces the Contract Distinction System (CDS) as a representation-independent source description of the additional distinctions, recovery targets, challenge structure and interface/commonality obligations introduced by lawful contract strengthening; semantic challenge refinement and CRC cofinal/refinement equivalence to control presentation dependence; and a Certified Observability Transport Theory (COTT) that maps licensed source-level structure into necessary category- and tolerance-specific resource lower bounds. The resulting directed finitary-demand theory separates internal divergence within one fixed strengthened contract from family divergence across finitely realisable members, while keeping lower recognition logically independent from upper-side construction or compactness/gluing. Conditional on finite baseline feasibility, the pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unboundePaper 7 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a structural theory for classifying how representation-resource requirements change when the operational obligation imposed on an independently fixed native system is lawfully strengthened. Building on the contract-relative resource law of Paper 6, it asks which source-level features of a strengthening determine or constrain the resulting resource transition. The theory fixes the native system, weak and strong contracts, realisation category, tolerance and charged resource before optimisation. A finite-baseline gate requires the weak contract to possess a finite-resource realisation, preventing pre-existing category nonrealisability from being misclassified as a strengthening-induced transition. The source-level object is a Contract Distinction System (CDS), which records the native configurations challenged by the stronger obligation, required distinctions or recovery targets, success rules, inspection boundaries and interface/commonality structure. Semantic challenge refinement and CRC cofinal/refinement equivalence control dependence on alternative finite challenge presentations. A Certified Observability Transport Theory (COTT) then maps licensed source-level facts into necessary resource lower bounds inside a fixed representation category and tolerance regime. The resulting directed finitary demand profile supports two distinct mechanisms. Internal divergence occurs when a single strengthened contract contains finite restrictions of arbitrarily large certified demand and can therefore exclude every finite-resource strong realisation in the declared category. Family divergence occurs when every member of a parametrised family remains finitely realisable while the additional resource induced by strengthening grows without bound across the family. Lower recognition is kept logically separate from upper-side existence: bounded certified demand is not treated as an existence theorem, and finite regimes require independent construction, compactness/gluing or another valid upper certificate. Conditional on finite baseline feasibility, the resulting pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unbounded finite separation carried separately as a family-scaling modifier. Several exact witnesses separate these regimes. With three native labels, the same coarse distinction graph K3, representation category, Hilbert-dimension resource and discrimination threshold fixed, changing only interface commonality produces an exact minimum-dimension transition from 2 to 3 throughout 2/3 < τ ≤ 1/2 + √3/4. Thus interface structure can itself be representation-resource-bearing. For the canonical TUS OS target, the complete static Atlas has exact minimum Hilbert dimension 4, whereas recovery-complete selected-line instrument access has exact minimum 64, giving an exact strict-finite transition 4→64. The weaker state-only common-CPTP problem remains unresolved within 32≤d≤64, demonstrating that the stronger interface theorem cannot be transferred backward through a forgetful map. For periodically modulated decay processes, every finite member remains finitely realisable while the contract-induced capacity gap diverges with family size. For the fixed dual-Poisson process, every positive worst-case transparency strengthening eliminates all finite-resource strong realisations in the declared autonomous one-copy classical–quantum category. A synthetic Countable Pairwise Identity Ladder supplies a cross-interface internal-divergence test, while the Fanizza–Lumbreras–Winter result marks the separate outer boundary of baseline category nonrealisability. Together, the results establish that operational observability is a resource-bearing input to quantum realisability without being invariably resource-increasing. Resource transition depends on the typed structure of the operational obligation—not merely native-state count, predictive-state cardinality, distinction count or coarse distinction graph. Supporting General Theory Programme This paper is Paper 7 in the TUS OS quantum-operational and Answerability research programme. Building on Paper 6’s contract-relative resource theory, it develops operational observability and a structural classification of contract-induced resource transitions, distinguishing resource-neutral, strict finite, finite-resource breakdown and unresolved pointwise outcomes, with family divergence carried separately as a family-scaling modifier while separating lower-bound certification from constructive realisability. Papers 8–9 then move to Answerability as a generic mathematical object, developing the resource geometry of answerable inference and the Answerability Calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops the structural classification theory for contract-induced representation-resource transitions. It introduces the Contract Distinction System (CDS) as a representation-independent source description of the additional distinctions, recovery targets, challenge structure and interface/commonality obligations introduced by lawful contract strengthening; semantic challenge refinement and CRC cofinal/refinement equivalence to control presentation dependence; and a Certified Observability Transport Theory (COTT) that maps licensed source-level structure into necessary category- and tolerance-specific resource lower bounds. The resulting directed finitary-demand theory separates internal divergence within one fixed strengthened contract from family divergence across finitely realisable members, while keeping lower recognition logically independent from upper-side construction or compactness/gluing. Conditional on finite baseline feasibility, the pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unbounde

Aug 21, 20260 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