Code and results for: Counting-field obstructions and symmetry conditions for Yang–Lee-type branch points in stochastic charge dynamics

Rongzhen Dai

Published Aug 25, 2026• 0 citations

Abstract

We identify and separate a preliminary gauge obstruction from three logically distinct source requirements for a counting field to activate a Yang–Lee-compatible charge-odd direction in the class of local charge processes studied here. Gate 0 asks whether the counted jump observable is only a boundary term: if its weights are exact, the tilted generator is related to the untilted one by a similarity transformation and the long-time spectrum is unchanged. Once a non-boundary count exists, three further conditions concern the source itself: a charge-odd source must be symmetry-allowed, real analyticity together with charge conjugation must pin it to the imaginary axis on the counting line, and the zero mode to which it couples must be dynamically live rather than a c-number in a fixed charge sector. The last statement is a state-source inertness condition and is algebraically distinct from Gate 0. A 3×3 factorial test of source permission against sector openness contains nine conditions, exactly four of which are live; they are precisely the four carrying both ingredients. A symbolic elimination independently reproduces the source structure and its phase. We then construct a tilted three-state chain in which these requirements are realised. Its tilted generator has an exact antiunitary symmetry, from which we derive and verify that the real part of the charge expectation vanishes on the unbroken branch and splits into an equal-and-opposite pair after the finite-size branch point. Finite-time Lee–Yang zeros of the moment generating function approach that certified branch point as 1/t²; a spectral-side prediction of the convergence amplitude agrees with the zero-side measurement to 2.7–5.2%, improving with size, and the second-order Jordan secular term is observed directly. Two independent finite-size procedures return a common critical-field estimate within 0.02%. We do not promote that agreement to proof of thermodynamic branch-point accumulation, which remains unresolved in the available size window. The same calculations establish, equally sharply, the boundary of what these signatures can decide. The 1/t² law and the associated fixed-size zero-density exponent −1/2 are generic kinematics of a second-order branch point and therefore cannot identify a Yang–Lee universality class. On a published benchmark with a known answer, the dynamical-exponent estimator remains 7.7–11.8% from the truth at the largest comparable sizes and a linear 1/L extrapolation overshoots by 17%. An observable-specific correction-amplitude zero is stable across sizes, explaining why the best-looking finite-size working point can be the worst point from which to extrapolate; a separate rank-one residual test is reserved for deciding whether it is an effective-action property. We therefore keep two inference levels separate. Source-level/static Yang–Lee compatibility does not by itself fix the stochastic-time dynamical exponent. The stronger spectral-CFT route used for a Virasoro-tower, Casimir-energy and central-charge reconstruction additionally requires z=1. Conditional on that gate, three spectral anchors would overdetermine the low-energy tower through target-value-free closure identities; only if those identities close are the lowest weight, effective central charge and central charge to be read out blind, with the velocities and tilt-induced drift cancelling.

DOI: 10.5281/zenodo.22091433

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Two Minuses, One Plus, and a Bare Trace Under Oath: Ordered Closed Traces and Whole-Ball Response Curvature

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With the identity matrix on the stand, the bare trace reports $252$ and the originally normalized prepared state reports $1$: the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed $S_{01}/dx^1$ external source with the co-directional readout operator, zero output momentum, double Laplace parameters $z=w=6c(1-i)$, the same preparation, and an incoming/observing two-ball window of radius $B$, we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the $P_6$ grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight $K$ and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from $289$ fields through auxiliary, Ward and symmetry reductions to a $103$-field system, counts matter feedback once, and constructs a rational $55$-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank $42$; keeping six light matter directions yields a $61$-dimensional second-order system whose constant term has rank $56$, and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical $g_{00}$ source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in $[-1.355200,-1.309214]$, $[2.140271,2.186257]$ and $[-1.742600,-1.696614]$ times $10^{-20}$ in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full $289\times289$ quantum tensor $\Pi_T$, and the finite integral of the mother-Euler amplitude derivative is exactly $[W_T H_{289}-\Pi_T]a$. Coincident clocks give $W_T=T$; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.

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

Anthony jordan Blair

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

Oct 12, 20260 citations

Code and results for: Counting-field obstructions and symmetry conditions for Yang–Lee-type branch points in stochastic charge dynamics

Rongzhen Dai

Published Aug 25, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.22091433

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Conserved Records and Finite-Time Dynamical Lee–Yang Zeros in Open Quantum Processes: An Object Ledger and Two No-Go Results

Rongzhen Dai

In a compression-based account of charge, five distinct objects share the single name “charge”: a global conserved charge , an einselected reduced record , a -odd Yang–Lee ordering operator , a decaying Gaussian occupation , and a trajectory charge-transfer count . We separate them inside an instrumented, charge-conserving open quantum process and ask which relations are invariants of the full process and which are only diagnostics of a monitoring scheme or of the no-jump conditional slice. In minimal completely-positive models we establish: (i) [THM] if every jump commutes with and , charge-diagonal monitoring makes the reduced record an exact fixed point, and [CALC] its survivor -algebra is the two-dimensional diagonal (charge-sector) algebra; a charge-violating perturbation makes it a finite-lifetime metastable record whose lifetime rises with the monitoring rate in the strong-monitoring (Zeno) branch. 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We do not derive , a dynamic exponent , or any universality class.In a compression-based account of charge, five distinct objects share the single name “charge”: a global conserved charge , an einselected reduced record , a -odd Yang–Lee ordering operator , a decaying Gaussian occupation , and a trajectory charge-transfer count . We separate them inside an instrumented, charge-conserving open quantum process and ask which relations are invariants of the full process and which are only diagnostics of a monitoring scheme or of the no-jump conditional slice. In minimal completely-positive models we establish: (i) [THM] if every jump commutes with and , charge-diagonal monitoring makes the reduced record an exact fixed point, and [CALC] its survivor -algebra is the two-dimensional diagonal (charge-sector) algebra; a charge-violating perturbation makes it a finite-lifetime metastable record whose lifetime rises with the monitoring rate in the strong-monitoring (Zeno) branch. 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We do not derive , a dynamic exponent , or any universality class.

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Rongzhen Dai

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Aug 7, 20260 citations

When is the Yang–Lee edge an exceptional point?

Rongzhen Dai

Non-Hermitian criticality is often treated as a single phenomenon, yet two distinct objects travel under that name. An exceptional point is a finite-N spectral object—the coalescence of eigenvectors of a single operator; a Lee–Yang edge is a thermodynamic-limit c-number object—the accumulation of partition-function zeros. We show that which of these an imaginary coupling produces is set by the direction the coupling occupies. When the coupling multiplies a conserved charge, [Q, H] = 0, the deformation H − iµQ is normal for every complex µ: its spectrum is entire, its eigenvectors are µ-independent, and no exceptional point exists at any coupling strength or volume, while Lee–Yang zeros persist and can carry an edge. When the coupling lies on a symmetry-invariant but non-conserved direction, exceptional points appear, but the associated universality is the analytic continuation of the Hermitian thermal transition, not the Yang–Lee edge. Only on the order-parameter (symmetry-odd) direction do the two coincide, at the M(2, 5) edge. The direction of an imaginary coupling therefore sets how much genuine non-unitarity its spectrum acquires. We give the theorem, a three-arm criterion, exactly solvable and exact-diagonalization checks, and falsifiable kill conditions.Non-Hermitian criticality is often treated as a single phenomenon, yet two distinct objects travel under that name. An exceptional point is a finite-N spectral object—the coalescence of eigenvectors of a single operator; a Lee–Yang edge is a thermodynamic-limit c-number object—the accumulation of partition-function zeros. We show that which of these an imaginary coupling produces is set by the direction the coupling occupies. When the coupling multiplies a conserved charge, [Q, H] = 0, the deformation H − iµQ is normal for every complex µ: its spectrum is entire, its eigenvectors are µ-independent, and no exceptional point exists at any coupling strength or volume, while Lee–Yang zeros persist and can carry an edge. When the coupling lies on a symmetry-invariant but non-conserved direction, exceptional points appear, but the associated universality is the analytic continuation of the Hermitian thermal transition, not the Yang–Lee edge. Only on the order-parameter (symmetry-odd) direction do the two coincide, at the M(2, 5) edge. The direction of an imaginary coupling therefore sets how much genuine non-unitarity its spectrum acquires. We give the theorem, a three-arm criterion, exactly solvable and exact-diagonalization checks, and falsifiable kill conditions.

Jul 12, 20260 citations

Two Minuses, One Plus, and a Bare Trace Under Oath: Ordered Closed Traces and Whole-Ball Response Curvature

Jian Gao

With the identity matrix on the stand, the bare trace reports $252$ and the originally normalized prepared state reports $1$ : the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed $S_{01}/dx^1$ external source with the co-directional readout operator, zero output momentum, double Laplace parameters $z=w=6c(1-i)$ , the same preparation, and an incoming/observing two-ball window of radius $B$ , we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the $P_6$ grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight $K$ and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from $289$ fields through auxiliary, Ward and symmetry reductions to a $103$ -field system, counts matter feedback once, and constructs a rational $55$ -dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank $42$ ; keeping six light matter directions yields a $61$ -dimensional second-order system whose constant term has rank $56$ , and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical $g_{00}$ source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in $[-1.355200,-1.309214]$ , $[2.140271,2.186257]$ and $[-1.742600,-1.696614]$ times $10^{-20}$ in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full $289\times289$ quantum tensor $\Pi_T$ , and the finite integral of the mother-Euler amplitude derivative is exactly $[W_T H_{289}-\Pi_T]a$ . Coincident clocks give $W_T=T$ ; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.With the identity matrix on the stand, the bare trace reports $252$ and the originally normalized prepared state reports $1$: the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed $S_{01}/dx^1$ external source with the co-directional readout operator, zero output momentum, double Laplace parameters $z=w=6c(1-i)$, the same preparation, and an incoming/observing two-ball window of radius $B$, we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the $P_6$ grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight $K$ and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from $289$ fields through auxiliary, Ward and symmetry reductions to a $103$-field system, counts matter feedback once, and constructs a rational $55$-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank $42$; keeping six light matter directions yields a $61$-dimensional second-order system whose constant term has rank $56$, and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical $g_{00}$ source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in $[-1.355200,-1.309214]$, $[2.140271,2.186257]$ and $[-1.742600,-1.696614]$ times $10^{-20}$ in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full $289\times289$ quantum tensor $\Pi_T$, and the finite integral of the mother-Euler amplitude derivative is exactly $[W_T H_{289}-\Pi_T]a$. Coincident clocks give $W_T=T$; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.

Oct 11, 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

With the identity matrix on the stand, the bare trace reports

252

and the originally normalized prepared state reports

1

: the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed

S_{01}/dx^1

external source with the co-directional readout operator, zero output momentum, double Laplace parameters

z=w=6c(1-i)

, the same preparation, and an incoming/observing two-ball window of radius

B

, we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the

P_6

grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight

K

and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from

289

fields through auxiliary, Ward and symmetry reductions to a

103

-field system, counts matter feedback once, and constructs a rational

55

-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank

42

; keeping six light matter directions yields a

61

-dimensional second-order system whose constant term has rank

56

, and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical

g_{00}

source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in

[-1.355200,-1.309214]

[2.140271,2.186257]

and

[-1.742600,-1.696614]

times

10^{-20}

in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full

289\times289

quantum tensor

\Pi_T

, and the finite integral of the mother-Euler amplitude derivative is exactly

[W_T H_{289}-\Pi_T]a

. Coincident clocks give

W_T=T

; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.

*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