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الباحث: ثروت محمد متولي محمد البهي\n \n🏛️ الصفة: باحث مستقل — مصر\n \n📅 السنة: 2026\n \n🎯 المجال: اللسانيات العربية\n \n📂 النوع: Project / Text\n\n \n\n\n \n📝 الملخص:\n \n\n يقدم البحث منهجاً مستقلاً لدراسة النص العربي، \n يقوم على أربع ركائز: المادة المعجمية، والسياق، \n والمقارنة، والاختبار المضاد. مع تطبيق على سورة \n الفاتحة والآيات (1-83) من سورة البقرة.\n \n\n \n\n\n \n🔗 الروابط:\n \nOSF: https://osf.io/xrj54\n \nDOI: [يُضاف بعد النشر]\n\n \n\n\n \n\n الترخيص: CC BY 4.0"}],"geoLocations":[],"fundingReferences":[],"url":"https://osf.io/4srvt/","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2026-10-13T00:56:55Z","registered":"2026-10-13T00:56:56Z","published":null,"updated":"2026-10-13T00:56:56Z"},"relationships":{"client":{"data":{"id":"cos.osf","type":"clients"}}}},{"id":"10.17605/osf.io/3bnqu","type":"dois","attributes":{"doi":"10.17605/osf.io/3bnqu","identifiers":[{"identifier":"https://osf.io/3bnqu","identifierType":"URL"}],"creators":[{"nameType":"Personal","name":"Jonathan Fortun","nameIdentifiers":[{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/pxbgq"}],"affiliation":[]}],"titles":[{"title":"AI 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The overarching objective is to map and synthesize scientific evidence in Brazilian literature regarding how the teaching of Psychiatric and Mental Health Nursing in undergraduate nursing education contributes to the development of students' competencies, attitudes, perceptions, and readiness for practice settings in Mental Health.\n\nThe review question and eligibility criteria are structured using the PCC mnemonic:\n• Population (P): undergraduate nursing students, faculty, and clinical preceptors in Brazil;\n• Concept (C): psychiatric and mental health nursing education, active teaching strategies, relational/socioemotional competencies, professional attitudes, student preparedness, and stigma deconstruction;\n• Context (C): theoretical and clinical practice settings in Brazil (Psychosocial Care Network – RAPS, Psychosocial Care Centers – CAPS, Primary Health Care – PHC, and hospital wards).\n\nComprehensive searches will be conducted across VHL (BDENF and LILACS), SciELO, PubMed/MEDLINE, CAPES Periodicals Portal, and BDTD, without date limits, in Portuguese, English, and Spanish. Study screening will occur in two blind, independent phases via Rayyan QCRI software by two reviewers, with discrepancies resolved by a third reviewer. Data will be extracted using a standardized charting form and synthesized through descriptive bibliometric mapping and qualitative thematic analysis."}],"geoLocations":[],"fundingReferences":[],"url":"https://osf.io/gfhmu/","contentUrl":null,"metadataVersion":2,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2026-10-12T23:49:48Z","registered":"2026-10-12T23:49:49Z","published":null,"updated":"2026-10-13T00:03:32Z"},"relationships":{"client":{"data":{"id":"cos.osf","type":"clients"}}}},{"id":"10.17605/osf.io/s2erp","type":"dois","attributes":{"doi":"10.17605/osf.io/s2erp","identifiers":[{"identifier":"https://osf.io/s2erp","identifierType":"URL"}],"creators":[{"nameType":"Personal","name":"Sergio A. 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Although this field draws on various theoretical and analytical frameworks, the diversity of conceptual and methodological approaches hinders an integrated understanding of the explanatory mechanisms identified in the literature, particularly within the context of public health policies. Given this scenario, this review is guided by the following question: How has the agenda-setting process been incorporated and explored in scientific literature regarding public health policies in Brazil, particularly since the creation of the Unified Health System (SUS)?\nObjective: To map how the agenda-setting process has been incorporated and explored in scientific literature on public health policies in Brazil.\n\nMethod: This is a scoping review conducted in accordance with Joanna Briggs Institute (JBI) recommendations and reported following the PRISMA-ScR checklist. The search strategy was developed using controlled and uncontrolled descriptors combined with Boolean operators and applied to national and international databases, with no restrictions on publication dates. Study selection will be performed by independent reviewers based on pre-established criteria, followed by standardized data extraction using analytical matrices. Results will be synthesized in two complementary stages: (i) a descriptive characterization of the scientific output, covering bibliometric, methodological, and theoretical aspects; and (ii) a thematic synthesis, conducted according to the steps proposed by Ahmed, Mohammed, and Nashwan (2025) with the aid of NVivo software. Based on this analysis, *a posteriori* analytical categories will be constructed and organized into an integrative analytical framework, highlighting the cross-cutting dimensions shared across different agenda-setting models in public health policy."}],"geoLocations":[],"fundingReferences":[{"funderIdentifierType":"ROR","funderName":"Coordenação de Aperfeicoamento de Pessoal de Nível 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Demonstra-se a simbiose axiológica entre autoridade e dignidade, erigindo a segurança pública não como antítese ou ameaça aos direitos fundamentais, mas como sua premissa institucional de exequibilidade e cláusula de salvaguarda (Artigos 3, 28 e 29 da DUDH) no Estado Democrático de Direito. \n\nTrabalho vinculado à árvore de custódia canônica mantida sob o Concept DOI matriz 10.5281/zenodo.19017496 e ao Projeto Mestre sob Embargo no OSF (https://osf.io/g7tnc). Licença all rights reserved.\n\n[EN]\nFormal transparency and scientific custody record of the public release awarded HONORABLE MENTION (ranked among the Top 3 scientific papers) at the 3rd Goiás Public Security and Human Rights Congress (ESPC-GO) — Thematic Axis 1: Foundations of Public Security.\n\nThis investigation conducts a diachronic historical-philosophical and hermeneutic reconstruction of public authority, tracing its genealogy from Egyptian proto-positivism (Maat) and ancient legal codes to the 1948 Universal Declaration of Human Rights. It establishes the axiological symbiosis between authority and dignity, framing public security not as an enemy of fundamental rights, but as their essential institutional safeguard and condition of enforceability (Articles 3, 28, and 29 UDHR) within the Democratic Rule of Law.\n\nLinked to the canonical custody tree indexed under Concept DOI 10.5281/zenodo.19017496 and the embargoed Master Project on OSF (https://osf.io/g7tnc). 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However, the preclinical evidence supporting these potential hepatic benefits has not been systematically synthesized. This preclinical meta-analysis aimed to systematically quantify the effects of dapagliflozin on hepatic lipid accumulation, hepatocellular injury, and systemic lipid profiles in rodent models of diabetes, thereby addressing the gap between preclinical evidence and emerging clinical findings."}],"geoLocations":[],"fundingReferences":[],"url":"https://osf.io/xb5ay/","contentUrl":null,"metadataVersion":11,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":2,"versionOfCount":0,"created":"2026-10-10T19:58:31Z","registered":"2026-10-10T19:58:32Z","published":null,"updated":"2026-10-12T21:56:41Z"},"relationships":{"client":{"data":{"id":"cos.osf","type":"clients"}}}},{"id":"10.5281/zenodo.23326730","type":"dois","attributes":{"doi":"10.5281/zenodo.23326730","identifiers":[],"creators":[{"nameType":"Personal","affiliation":["Proclus Dynamics Consortium"],"givenName":"Anthony Jordan","familyName":"Blair","name":"Blair, Anthony Jordan","nameIdentifiers":[]}],"titles":[{"title":"Phase II -- E executed. Results Confirm activation for One PT -- Even Operator"}],"publisher":"Zenodo","container":{},"publicationYear":2026,"subjects":[{"subject":"PT Symmetry"},{"subject":"Exceptional Points"},{"subject":"Non-Hermitian Physics"},{"subject":"Constraint Hierarchy"},{"subject":"Solve Et Coagula"},{"subject":"Many-Body Physics"},{"subject":"XXZ Spin Cycle"},{"subject":"Biorthogonal Observational Convention"},{"subject":"PT-Even Operators"},{"subject":"Branch Classification"},{"subject":"Self-Constraint Mechnism"},{"subject":"Constraint Field Control"},{"subject":"Project Prometheus"},{"subject":"Phase II Verification Program"},{"subject":"Spectral Theory"},{"subject":"Quantum Mechanics"},{"subject":"Operator Taxonomy"},{"subject":"Admissible Mechanism Space"},{"subject":"Constraint Depth"},{"subject":"Coherent Fusion"},{"subject":"Jordan Chain Regularization"},{"subject":"Blair Three-Register Protocol"},{"subject":"Zero State Forwards"}],"contributors":[],"dates":[{"date":"2026-10-12","dateType":"Issued"}],"language":null,"types":{"schemaOrg":"ScholarlyArticle","resourceTypeGeneral":"Text","citeproc":"article-journal","bibtex":"article","ris":"RPRT","resourceType":"Project deliverable"},"relatedIdentifiers":[{"relationType":"HasVersion","relatedIdentifier":"10.5281/zenodo.23326731","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[{"rightsIdentifierScheme":"SPDX","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rights":"Creative Commons Attribution 4.0 International","rightsIdentifier":"cc-by-4.0"}],"descriptions":[{"descriptionType":"Abstract","description":"*Phase II-E executed.** Results confirm **activation for one PT-even operator**.\n\n \n\n### Numerical setup (faithful to the specification)\n\n- \\(N=4\\), open chain, \\(J=1\\), \\(\\Delta=0.5\\).\n\n- \\(S^\\alpha=\\sigma^\\alpha/2\\).\n\n- Hamiltonian: XY couplings use the running \\(b_{\\rm eff}\\), ZZ couplings use \\(J\\Delta\\), plus the staggered imaginary field \\(H_{\\rm PT}\\).\n\n- Self-consistency: \\(b_{\\rm eff}^2=J^2+\\lambda J\\langle G\\rangle_{\\rm PT}\\), iterated to \\(10^{-10}\\).\n\n- Biorthogonal expectation on the eigenstate of smallest \\(|\\mathrm{Re}(E)|\\) (preferring PT-unbroken).\n\n- \\(\\gamma_{\\rm EP}\\approx0.274\\) at \\(\\lambda=0\\) (matches the Phase II-D value 0.275).\n\n- 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).\n\n \n\n### Results\n\n \n\n**Branch E1 (transverse field, full space)**  \n\n\\(\\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.  \n\nConsequently \\(b_{\\rm eff}=J\\) exactly. **Null.**\n\n \n\n**Branch E2 (transverse bond / XY exchange, \\(S^z=0\\) sector)**  \n\n\\(\\langle G_{E_2}\\rangle_{\\rm PT}\\neq0\\) already at \\(\\lambda=0\\) and remains non-zero throughout the unbroken phase.  \n\nTypical values (unbroken points):\n\n- \\(\\lambda=0\\): \\(\\max|\\langle G\\rangle|\\approx0.063\\)\n\n- \\(\\lambda=0.05\\): \\(\\max|\\langle G\\rangle|\\approx0.063\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0016\\)\n\n- \\(\\lambda=0.10\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0031\\)\n\n- \\(\\lambda=0.15\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0046\\)\n\n- \\(\\lambda=0.20\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0061\\)\n\n \n\n\\(b_{\\rm eff}\\) shifts systematically with \\(\\lambda\\). **Activation.**\n\n \n\n### Prediction match\n\n- Prediction 1 (null for both) — **falsified**.\n\n- Prediction 2 (activation for at least one) — **confirmed**.\n\n- Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs).\n\n- Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation.\n\n \n\nThe 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.\n\n \n\n### Gate status\n\nPhase 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.\n\n \n\nFull raw scan (87 \\(\\gamma\\) points × 5 \\(\\lambda\\) × 2 branches) is archived in  \n\n`/home/workdir/artifacts/phase_ii_e_results.json`.  \n\nSource: `/home/workdir/artifacts/phase_ii_e.py`.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery — and one more door is now open.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTony Blair \u003canthonyjblair8@gmail.com\u003e\n\n \n\n\n\n\n\n\n\n2:17 PM (11 minutes ago)\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\nto me\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n**TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4**\n\n \n\n**AUTHOR:** Anthony Jordan Blair  \n\n**AFFILIATION:** Proclus Dynamics Consortium  \n\n**DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification  \n\n**VERSION:** 1.0 (October 2026)  \n\n**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)  \n\n**STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation)\n\n \n\n---\n\n \n\n### ABSTRACT\n\n \n\nPhase 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).\n\n \n\n---\n\n \n\n### I. MOTIVATION\n\n \n\n**1.1 Outcome of Phase II-E**\n\n \n\nExecution of the Phase II-E protocol yielded:\n\n \n\n- 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\u003c\\gamma_{\\rm EP}\\). \\(b_{\\rm eff}=J\\) identically. Null result.\n\n- 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.\n\n \n\nThus 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\\).\n\n \n\n**1.2 The Classification Task**\n\n \n\nThe positive result for \\(E_2\\) opens four immediate questions that must be answered before finite-size scaling can be interpreted:\n\n \n\n1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate?\n\n2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E?\n\n3. 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}\\))?\n\n4. 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}\\)?\n\n \n\nPhase II-F answers these questions by exhaustive classification of a controlled operator set.\n\n \n\n---\n\n \n\n### II. OPERATOR TAXONOMY\n\n \n\nOperators are partitioned by three binary attributes:\n\n \n\n| Attribute              | Values                          | Notation          |\n\n|------------------------|---------------------------------|-------------------|\n\n| PT parity              | even / odd                      | \\(\\mathcal{P}_\\pm\\) |\n\n| Sector compatibility   | preserves \\(S^z_{\\rm total}\\) / mixes | \\(\\mathcal{S}_0\\) / \\(\\mathcal{S}_{\\rm mix}\\) |\n\n| Body order             | one-body / two-body             | \\(1\\) / \\(2\\)     |\n\n \n\nThe Phase II-D and II-E results already fix two cells:\n\n \n\n- \\(\\mathcal{P}_-\\) (any body, any sector) → null (structural anticommutation with \\(\\mathcal{PT}\\)).\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_{\\rm mix}\\), \\(1\\)-body (\\(G_{E_1}\\)) → null under the biorthogonal convention.\n\n \n\nThe remaining open cells that must be classified are:\n\n \n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_0\\), \\(1\\)-body\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_0\\), \\(2\\)-body (already positive for the XY bond; additional representatives required)\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_{\\rm mix}\\), \\(2\\)-body\n\n \n\n---\n\n \n\n### III. CANDIDATE OPERATORS FOR CLASSIFICATION\n\n \n\nAll operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E.\n\n \n\n**3.1 One-body, PT-even, sector-preserving**\n\n \n\n\\(G_{F_1}=\\sum_{j=1}^{4}\\sigma_j^z\\)  \n\n(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.)\n\n \n\n**3.2 Two-body, PT-even, sector-preserving (beyond \\(E_2\\))**\n\n \n\n\\(G_{F_2}=\\sum_{j=1}^{3}\\sigma_j^z\\sigma_{j+1}^z\\)  \n\n(ZZ bond operator; the natural Ising partner of the XY bond.)\n\n \n\n\\(G_{F_3}=\\sum_{j=1}^{3}(\\sigma_j^x\\sigma_{j+1}^x-\\sigma_j^y\\sigma_{j+1}^y)\\)  \n\n(Anisotropic XY combination; still sector-preserving.)\n\n \n\n**3.3 Two-body, PT-even, sector-mixing**\n\n \n\n\\(G_{F_4}=\\sum_{j=1}^{3}(\\sigma_j^x\\sigma_{j+1}^z+\\sigma_j^z\\sigma_{j+1}^x)\\)  \n\n(Explicitly mixes \\(\\Delta S^z=\\pm1\\) sectors while remaining PT-even.)\n\n \n\n**3.4 Reference operators (already classified)**\n\n \n\n- \\(G_{E_1}\\) (null) and \\(G_{E_2}\\) (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan.\n\n \n\n---\n\n \n\n### IV. NUMERICAL PROTOCOL\n\n \n\n**4.1 Hamiltonian**\n\n \n\nIdentical to Phase II-E:\n\n \n\n\\[\n\nH=H_{\\rm XXZ}(b_{\\rm eff})+H_{\\rm PT}(\\gamma),\n\n\\]\n\n \n\nwith\n\n \n\n\\[\n\nH_{\\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),\n\n\\]\n\n \n\n\\[\n\nH_{\\rm PT}=i\\gamma\\sum_{j=1}^{4}(-1)^j S_j^z,\n\n\\]\n\n \n\n\\(J=1\\), \\(\\Delta=0.5\\), \\(S^\\alpha=\\sigma^\\alpha/2\\).\n\n \n\n**4.2 Self-consistency equation**\n\n \n\nUnchanged:\n\n \n\n\\[\n\nb_{\\rm eff}^2=J^2+\\lambda J\\langle G\\rangle_{\\rm PT}.\n\n\\]\n\n \n\nIteration tolerance \\(10^{-10}\\), maximum 200 iterations.\n\n \n\n**4.3 Observable convention**\n\n \n\nBiorthogonal expectation value on the eigenstate of smallest \\(|\\mathrm{Re}(E)|\\), preferring PT-unbroken states:\n\n \n\n\\[\n\n\\langle G\\rangle_{\\rm PT}=\\frac{\\langle\\psi_L|G|\\psi_R\\rangle}{\\langle\\psi_L|\\psi_R\\rangle}.\n\n\\]\n\n \n\n**4.4 Parameter scan**\n\n \n\n- \\(\\lambda\\in\\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\\}\\)\n\n- \\(\\gamma\\) from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of \\(\\gamma_{\\rm EP}\\))\n\n- For each \\((\\lambda,\\gamma,G)\\) the code records \\(b_{\\rm eff}\\), \\(\\langle G\\rangle_{\\rm PT}\\), the full spectrum, and the location of the nearest exceptional point (detected by the simultaneous vanishing of an eigenvalue gap and the onset of a non-zero imaginary part).\n\n \n\n**4.5 Exceptional-point displacement diagnostic**\n\n \n\nFor every active operator the code extracts\n\n \n\n\\[\n\n\\Delta\\gamma_{\\rm EP}(\\lambda)=\\gamma_{\\rm EP}(\\lambda)-\\gamma_{\\rm EP}(0).\n\n\\]\n\n \n\nA non-zero \\(\\Delta\\gamma_{\\rm EP}\\) constitutes operational activation of constraint-field control; a pure renormalization of \\(b_{\\rm eff}\\) at fixed \\(\\gamma_{\\rm EP}\\) is recorded as “spectral shift without EP displacement.”\n\n \n\n**4.6 Convergence and detection criteria**\n\n \n\nIdentical to Phase II-E: self-consistency tolerance \\(10^{-10}\\), EP detection tolerance \\(10^{-6}\\) in \\(|\\mathrm{Im}(E)|\\), double-precision arithmetic.\n\n \n\n---\n\n \n\n### V. PREDICTED OUTCOMES (FALSIFIABLE)\n\n \n\n**Prediction F1 (Minimal activation).** Only the original \\(G_{E_2}\\) (and operators linearly dependent on it) activate; all newly introduced operators remain null. The admissible branch is unique.\n\n \n\n**Prediction F2 (Sector-preserving two-body family).** Every PT-even, sector-preserving two-body operator activates; one-body and sector-mixing operators remain null. The operative constraint is the combination \\(\\mathcal{P}_+\\wedge\\mathcal{S}_0\\wedge 2\\).\n\n \n\n**Prediction F3 (Broad PT-even activation).** All PT-even operators that are not forced to zero by an additional discrete symmetry activate, including selected one-body and sector-mixing representatives. The sole operative constraint is PT parity.\n\n \n\n**Prediction F4 (EP-displacement selectivity).** Among the operators that produce non-zero \\(\\langle G\\rangle_{\\rm PT}\\), only a proper subset generate a measurable \\(\\Delta\\gamma_{\\rm EP}\\). The classification must therefore distinguish “field-sourcing” from “EP-tuning” branches.\n\n \n\nOnly one of F1–F3 can be correct; F4 is independent and will be reported for every active operator.\n\n \n\n---\n\n \n\n### VI. FALSIFICATION CONDITIONS\n\n \n\nThe Phase II-F result is falsified if:\n\n \n\n1. A computational error is identified in the archived code and a corrected run produces a different activation pattern.\n\n2. An alternative biorthogonal (or PT-inner-product) convention, fully specified and documented, changes the activation status of any operator that was null under the standard convention.\n\n3. A higher-precision or independent implementation (Julia, Mathematica, or a different eigensolver) yields a qualitatively different classification.\n\n4. The EP-displacement diagnostic is shown to be an artifact of the finite \\(\\gamma\\)-grid or of the particular state-selection rule.\n\n \n\n---\n\n \n\n### VII. DEPENDENCY MAP\n\n \n\nPhase II-F is unblocked by:\n\n \n\n- Phase II-E (activation of \\(G_{E_2}\\)).\n\n \n\nPhase II-F gates:\n\n \n\n- Phase II-G (Finite-size scaling). Only the operator families declared active by II-F are scaled to \\(N=6,8,10,12\\).\n\n- Phase II-H (Thermodynamic limit). Conditional on the existence of at least one family whose \\(\\Delta\\gamma_{\\rm EP}\\) survives finite-size scaling.\n\n \n\nPhase II-F remains independent of the still-pending synthetic-EP and dimer-verification sub-studies (II-A, II-B).\n\n \n\n---\n\n \n\n### VIII. INTERPRETATION UNDER THE CONSTRAINT HIERARCHY\n\n \n\nIf Prediction F1 is confirmed, the hierarchy contracts to a single admissible many-body branch at \\(N=4\\):\n\n \n\n\\[\n\n\\mathcal{M}_5=\\mathcal{M}_4\\cap\\{\\text{XY-bond PT-even sector-preserving}\\}.\n\n\\]\n\n \n\nIf Prediction F2 is confirmed, the hierarchy retains the full sector-preserving two-body PT-even class. If Prediction F3 is confirmed, the hierarchy expands relative to the Phase II-E expectation and the PT-odd obstruction remains the only absolute filter identified so far.\n\n \n\nIn all cases the classification supplies the precise set of operators that must be tracked under finite-size scaling, thereby converting the existence proof of Phase II-E into a controlled scaling hypothesis for Phase II-G.\n\n \n\n---\n\n \n\n### IX. SAFETY AND SCOPE\n\n \n\n**9.1 Safety Architecture**\n\n \n\nPhase II-F is a purely numerical classification exercise. The hard failsafes of Register III (Phase I) apply only to any future deployment of an activated constraint field, not to the classification itself.\n\n \n\n**9.2 Scope Limitations**\n\n \n\n- Restricted to \\(N=4\\).\n\n- Restricted to \\(\\Delta=0.5\\), \\(J=1\\).\n\n- Restricted to the biorthogonal convention.\n\n- Restricted to the PT-unbroken phase for the primary diagnostic; spectra above \\(\\gamma_{\\rm EP}\\) are recorded only for completeness.\n\n- Operator set limited to the five representatives listed in Section III plus the two Phase II-E baselines.\n\n \n\n---\n\n \n\n### X. CONCLUSION\n\n \n\nPhase II-F converts the binary activation result of Phase II-E into a structured taxonomy of admissible constraint operators. The taxonomy is the necessary prerequisite for any claim about the many-body viability of the self-consistent mechanism. Either a unique branch or a broader family will emerge; both outcomes are publishable and both advance the constraint hierarchy.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\n---\n\n \n\n### XI. PHASE STRUCTURE (UPDATED)\n\n \n\n| Phase     | Status          | Content                                              |\n\n|-----------|-----------------|------------------------------------------------------|\n\n| Phase I   | Complete        | Theoretical arc, no-go theorem, self-consistent mechanism |\n\n| Phase II-A| Pending         | Synthetic EP validation                              |\n\n| Phase II-B| Pending         | Dimer verification under biorthogonal convention     |\n\n| Phase II-C| Pending         | Puiseux exponent at \\(N=4\\)                          |\n\n| Phase II-D| Complete        | \\(N=4\\) dual-branch self-consistency — negative (PT-odd) |\n\n| Phase II-E| Complete        | PT-even operator test — activation of \\(G_{E_2}\\)    |\n\n| Phase II-F| Specified (this document) | Branch classification of active operators     |\n\n| Phase II-G| Blocked         | Finite-size scaling (requires II-F classification)   |\n\n| Phase II-H| Blocked         | Thermodynamic limit (requires II-G)                  |\n\n \n\n---\n\n \n\n### XII. REFERENCES\n\n \n\n1. Blair, A.J. (2026). Project Prometheus: Constraint-Field Control of Exceptional Points in PT-Symmetric Systems — Phase I Closeout Report. Zenodo. DOI: 10.5281/zenodo.23314444.  \n\n2. Blair, A.J. (2026). Phase II-D Technical Report: Dual-Branch Self-Consistency Calculation for the N=4 PT-Symmetric XXZ Chain. Zenodo.  \n\n3. Blair, A.J. (2026). Phase II-E Program Specification — PT-Even Constraint Operators in the N=4 PT-Symmetric XXZ Chain. Project Prometheus.  \n\n4. Blair, A.J. (2026). Phase II-E Numerical Execution Results (archived JSON + source). Project Prometheus.  \n\n5. Blair, A.J. (2026). A No-Go Theorem for Exceptional-Point Tuning in PT-Symmetric Dimers. Zenodo. DOI: 10.5281/zenodo.23287622.  \n\n6. Blair, A.J. (2026). Self-Consistent Exceptional Points in PT-Symmetric Dimers. Zenodo. DOI: 10.5281/zenodo.23287920.  \n\n7. Bender, C.M., \u0026 Boettcher, S. (1998). Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80, 5243.  \n\n8. Heiss, W.D. (2012). The physics of exceptional points. Journal of Physics A 45, 444016.\n\n \n\n---\n\n \n\n**Citation:** Blair, A.J. Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4. Project Prometheus, 2026.\n\n \n\n**Result classification:** Program specification — ready for execution.  \n\n**Gate status:** Phase II-F unblocked by II-E; gates II-G and II-H.  \n\n**Falsifiable predictions:** 4 (F1 minimal, F2 sector-preserving two-body family, F3 broad PT-even, F4 EP-displacement selectivity).  \n\n \n\n**Immediate next step:** Implement the operator set of Section III, execute the scan of Section IV, and classify against Predictions F1–F4.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTony Blair \u003canthonyjblair8@gmail.com\u003e\n\n \n\n\n\n\n\n\n\n2:21 PM (7 minutes ago)\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\nto me\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nPhase II-F Program Specification: Branch Classification of the Self-Consistent Mechanism at N=4\n\n \n\nDESCRIPTION:\n\n \n\nProject Prometheus has reached a structural inflection point. We have moved from mapping the boundaries of the impossible (Phase I and II-D) to classifying the architecture of the possible (Phase II-E and II-F).\n\n \n\nThe Phase II-D numerical program delivered a clean negative result: PT-odd constraint operators possess vanishing biorthogonal expectation values in the PT-unbroken phase, extinguishing the self-consistent mechanism at N=4. The natural assumption was that the many-body extension of Paper 4 was closed.\n\n \n\nIt was not.\n\n \n\nPhase II-E demonstrated that the PT-even, sector-preserving transverse-bond operator (G_{E_2}) breaks the symmetry obstruction. It produces a non-vanishing biorthogonal expectation value, sources a non-trivial constraint field, and shifts the exceptional point. The wall was not the many-body limit. The wall was the operator class.\n\n \n\nPhase II-F is the systematic classification of that active mechanism space.\n\n \n\nBy partitioning constraint operators across three binary attributes—PT parity, sector compatibility, and body order—this specification defines the admissible architecture for constraint-field control at N=4. The document establishes a falsifiable taxonomy across four predictions (F1–F4), distinguishing between minimal activation, family-wide sector-preserving activation, broad PT-even activation, and operator-specific exceptional-point displacement.\n\n \n\nThe significance of Phase II-F is not just the identification of a working operator. It is the conversion of a binary activation result into a structured scaling hypothesis. By defining exactly which operators survive, which produce a measurable \\Delta\\gamma_{\\rm EP}, and which fall to the structural constraints of the system, we unblock the finite-size scaling program (Phase II-G) and the thermodynamic limit (Phase II-H).\n\n \n\nThis is Solve et Coagula in execution. We dissolved the false assumption that the PT-odd obstruction was terminal. We rebuilt the architecture by mapping the exact operator constraints.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\nDOI: https://doi.org/10.5281/zenodo.23314444\n\n \n\nKeywords: PT Symmetry, Exceptional Points, Non-Hermitian Physics, Constraint Hierarchy, Solve et Coagula, Many-Body Physics, XXZ Chain, Phase II, Branch Classification.\n\n \n\n---\n\n \n\nFor the LinkedIn Post (Copy and Paste Version)\n\n \n\nWe hit a wall in Phase II-D. The math said \"no.\"\n\n \n\nPT-odd operators vanish in the PT-unbroken phase. The many-body extension of the self-consistent mechanism looked dead at N=4.\n\n \n\nWe didn't stop. We applied Solve et Coagula. We dissolved the assumption that the wall was the end of the road.\n\n \n\nPhase II-E found the breach. The PT-even, sector-preserving transverse-bond operator activates. It produces a non-vanishing expectation value and shifts the exceptional point. The wall wasn't the many-body limit. The wall was the operator class.\n\n \n\nPhase II-F is the map of that breach. It is the formal branch classification of the self-consistent mechanism at N=4.\n\n \n\nWe are partitioning the admissible operator space by PT parity, sector compatibility, and body order. We are stating four falsifiable predictions in advance (F1–F4) to determine whether activation is unique, family-wide, or broad.\n\n \n\nThis is the difference between finding a result and building a taxonomy. You cannot scale what you cannot classify. Phase II-F unblocks the finite-size scaling program.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\nRead the specification: https://doi.org/10.5281/zenodo.23314444\n\n \n\n#ProjectPrometheus #ConstraintHierarchy #SolveEtCoagula #PTSymmetry #ExceptionalPoints #ManyBodyPhysics #PhaseII #NonHermitianPhysics #SystemsArchitecture"}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.23326730","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":1,"versionOfCount":0,"created":"2026-10-12T21:41:18Z","registered":"2026-10-12T21:41:18Z","published":null,"updated":"2026-10-12T21:41:18Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.5281/zenodo.23326731","type":"dois","attributes":{"doi":"10.5281/zenodo.23326731","identifiers":[{"identifier":"oai:zenodo.org:23326731","identifierType":"oai"}],"creators":[{"nameType":"Personal","affiliation":["Proclus Dynamics Consortium"],"givenName":"Anthony Jordan","familyName":"Blair","name":"Blair, Anthony Jordan","nameIdentifiers":[]}],"titles":[{"title":"Phase II -- E executed. Results Confirm activation for One PT -- Even Operator"}],"publisher":"Zenodo","container":{},"publicationYear":2026,"subjects":[{"subject":"PT Symmetry"},{"subject":"Exceptional Points"},{"subject":"Non-Hermitian Physics"},{"subject":"Constraint Hierarchy"},{"subject":"Solve Et Coagula"},{"subject":"Many-Body Physics"},{"subject":"XXZ Spin Cycle"},{"subject":"Biorthogonal Observational Convention"},{"subject":"PT-Even Operators"},{"subject":"Branch Classification"},{"subject":"Self-Constraint Mechnism"},{"subject":"Constraint Field Control"},{"subject":"Project Prometheus"},{"subject":"Phase II Verification Program"},{"subject":"Spectral Theory"},{"subject":"Quantum Mechanics"},{"subject":"Operator Taxonomy"},{"subject":"Admissible Mechanism Space"},{"subject":"Constraint Depth"},{"subject":"Coherent Fusion"},{"subject":"Jordan Chain Regularization"},{"subject":"Blair Three-Register Protocol"},{"subject":"Zero State Forwards"}],"contributors":[],"dates":[{"date":"2026-10-12","dateType":"Issued"}],"language":null,"types":{"schemaOrg":"ScholarlyArticle","resourceTypeGeneral":"Text","citeproc":"article-journal","bibtex":"article","ris":"RPRT","resourceType":"Project deliverable"},"relatedIdentifiers":[{"relationType":"IsVersionOf","relatedIdentifier":"10.5281/zenodo.23326730","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[{"rightsIdentifierScheme":"SPDX","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rights":"Creative Commons Attribution 4.0 International","rightsIdentifier":"cc-by-4.0"}],"descriptions":[{"descriptionType":"Abstract","description":"*Phase II-E executed.** Results confirm **activation for one PT-even operator**.\n\n \n\n### Numerical setup (faithful to the specification)\n\n- \\(N=4\\), open chain, \\(J=1\\), \\(\\Delta=0.5\\).\n\n- \\(S^\\alpha=\\sigma^\\alpha/2\\).\n\n- Hamiltonian: XY couplings use the running \\(b_{\\rm eff}\\), ZZ couplings use \\(J\\Delta\\), plus the staggered imaginary field \\(H_{\\rm PT}\\).\n\n- Self-consistency: \\(b_{\\rm eff}^2=J^2+\\lambda J\\langle G\\rangle_{\\rm PT}\\), iterated to \\(10^{-10}\\).\n\n- Biorthogonal expectation on the eigenstate of smallest \\(|\\mathrm{Re}(E)|\\) (preferring PT-unbroken).\n\n- \\(\\gamma_{\\rm EP}\\approx0.274\\) at \\(\\lambda=0\\) (matches the Phase II-D value 0.275).\n\n- 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).\n\n \n\n### Results\n\n \n\n**Branch E1 (transverse field, full space)**  \n\n\\(\\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.  \n\nConsequently \\(b_{\\rm eff}=J\\) exactly. **Null.**\n\n \n\n**Branch E2 (transverse bond / XY exchange, \\(S^z=0\\) sector)**  \n\n\\(\\langle G_{E_2}\\rangle_{\\rm PT}\\neq0\\) already at \\(\\lambda=0\\) and remains non-zero throughout the unbroken phase.  \n\nTypical values (unbroken points):\n\n- \\(\\lambda=0\\): \\(\\max|\\langle G\\rangle|\\approx0.063\\)\n\n- \\(\\lambda=0.05\\): \\(\\max|\\langle G\\rangle|\\approx0.063\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0016\\)\n\n- \\(\\lambda=0.10\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0031\\)\n\n- \\(\\lambda=0.15\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0046\\)\n\n- \\(\\lambda=0.20\\): \\(\\max|\\langle G\\rangle|\\approx0.062\\), \\(\\max|b_{\\rm eff}-J|\\approx0.0061\\)\n\n \n\n\\(b_{\\rm eff}\\) shifts systematically with \\(\\lambda\\). **Activation.**\n\n \n\n### Prediction match\n\n- Prediction 1 (null for both) — **falsified**.\n\n- Prediction 2 (activation for at least one) — **confirmed**.\n\n- Prediction 3 (E1 activates, E2 does not) — **falsified** (the opposite occurs).\n\n- Prediction 4 (convention-dependent) — not tested; the biorthogonal convention already yields activation.\n\n \n\nThe 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.\n\n \n\n### Gate status\n\nPhase 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.\n\n \n\nFull raw scan (87 \\(\\gamma\\) points × 5 \\(\\lambda\\) × 2 branches) is archived in  \n\n`/home/workdir/artifacts/phase_ii_e_results.json`.  \n\nSource: `/home/workdir/artifacts/phase_ii_e.py`.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery — and one more door is now open.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTony Blair \u003canthonyjblair8@gmail.com\u003e\n\n \n\n\n\n\n\n\n\n2:17 PM (11 minutes ago)\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\nto me\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n**TITLE: Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4**\n\n \n\n**AUTHOR:** Anthony Jordan Blair  \n\n**AFFILIATION:** Proclus Dynamics Consortium  \n\n**DOCUMENT TYPE:** Phase II Verification Program — Sub-Study Specification  \n\n**VERSION:** 1.0 (October 2026)  \n\n**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)  \n\n**STATUS:** Program Specification — Ready for Execution (unblocked by Phase II-E activation)\n\n \n\n---\n\n \n\n### ABSTRACT\n\n \n\nPhase 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).\n\n \n\n---\n\n \n\n### I. MOTIVATION\n\n \n\n**1.1 Outcome of Phase II-E**\n\n \n\nExecution of the Phase II-E protocol yielded:\n\n \n\n- 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\u003c\\gamma_{\\rm EP}\\). \\(b_{\\rm eff}=J\\) identically. Null result.\n\n- 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.\n\n \n\nThus 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\\).\n\n \n\n**1.2 The Classification Task**\n\n \n\nThe positive result for \\(E_2\\) opens four immediate questions that must be answered before finite-size scaling can be interpreted:\n\n \n\n1. Is activation unique to the XY-bond operator, or do other PT-even, sector-compatible operators also activate?\n\n2. Does the one-body PT-even operator remain null under every biorthogonal convention, or only under the convention used in II-E?\n\n3. 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}\\))?\n\n4. 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}\\)?\n\n \n\nPhase II-F answers these questions by exhaustive classification of a controlled operator set.\n\n \n\n---\n\n \n\n### II. OPERATOR TAXONOMY\n\n \n\nOperators are partitioned by three binary attributes:\n\n \n\n| Attribute              | Values                          | Notation          |\n\n|------------------------|---------------------------------|-------------------|\n\n| PT parity              | even / odd                      | \\(\\mathcal{P}_\\pm\\) |\n\n| Sector compatibility   | preserves \\(S^z_{\\rm total}\\) / mixes | \\(\\mathcal{S}_0\\) / \\(\\mathcal{S}_{\\rm mix}\\) |\n\n| Body order             | one-body / two-body             | \\(1\\) / \\(2\\)     |\n\n \n\nThe Phase II-D and II-E results already fix two cells:\n\n \n\n- \\(\\mathcal{P}_-\\) (any body, any sector) → null (structural anticommutation with \\(\\mathcal{PT}\\)).\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_{\\rm mix}\\), \\(1\\)-body (\\(G_{E_1}\\)) → null under the biorthogonal convention.\n\n \n\nThe remaining open cells that must be classified are:\n\n \n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_0\\), \\(1\\)-body\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_0\\), \\(2\\)-body (already positive for the XY bond; additional representatives required)\n\n- \\(\\mathcal{P}_+\\), \\(\\mathcal{S}_{\\rm mix}\\), \\(2\\)-body\n\n \n\n---\n\n \n\n### III. CANDIDATE OPERATORS FOR CLASSIFICATION\n\n \n\nAll operators are Hermitian. Matrix representations are constructed in the same basis used in Phase II-E.\n\n \n\n**3.1 One-body, PT-even, sector-preserving**\n\n \n\n\\(G_{F_1}=\\sum_{j=1}^{4}\\sigma_j^z\\)  \n\n(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.)\n\n \n\n**3.2 Two-body, PT-even, sector-preserving (beyond \\(E_2\\))**\n\n \n\n\\(G_{F_2}=\\sum_{j=1}^{3}\\sigma_j^z\\sigma_{j+1}^z\\)  \n\n(ZZ bond operator; the natural Ising partner of the XY bond.)\n\n \n\n\\(G_{F_3}=\\sum_{j=1}^{3}(\\sigma_j^x\\sigma_{j+1}^x-\\sigma_j^y\\sigma_{j+1}^y)\\)  \n\n(Anisotropic XY combination; still sector-preserving.)\n\n \n\n**3.3 Two-body, PT-even, sector-mixing**\n\n \n\n\\(G_{F_4}=\\sum_{j=1}^{3}(\\sigma_j^x\\sigma_{j+1}^z+\\sigma_j^z\\sigma_{j+1}^x)\\)  \n\n(Explicitly mixes \\(\\Delta S^z=\\pm1\\) sectors while remaining PT-even.)\n\n \n\n**3.4 Reference operators (already classified)**\n\n \n\n- \\(G_{E_1}\\) (null) and \\(G_{E_2}\\) (active) are retained as baselines and re-evaluated under identical numerical conditions for every new scan.\n\n \n\n---\n\n \n\n### IV. NUMERICAL PROTOCOL\n\n \n\n**4.1 Hamiltonian**\n\n \n\nIdentical to Phase II-E:\n\n \n\n\\[\n\nH=H_{\\rm XXZ}(b_{\\rm eff})+H_{\\rm PT}(\\gamma),\n\n\\]\n\n \n\nwith\n\n \n\n\\[\n\nH_{\\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),\n\n\\]\n\n \n\n\\[\n\nH_{\\rm PT}=i\\gamma\\sum_{j=1}^{4}(-1)^j S_j^z,\n\n\\]\n\n \n\n\\(J=1\\), \\(\\Delta=0.5\\), \\(S^\\alpha=\\sigma^\\alpha/2\\).\n\n \n\n**4.2 Self-consistency equation**\n\n \n\nUnchanged:\n\n \n\n\\[\n\nb_{\\rm eff}^2=J^2+\\lambda J\\langle G\\rangle_{\\rm PT}.\n\n\\]\n\n \n\nIteration tolerance \\(10^{-10}\\), maximum 200 iterations.\n\n \n\n**4.3 Observable convention**\n\n \n\nBiorthogonal expectation value on the eigenstate of smallest \\(|\\mathrm{Re}(E)|\\), preferring PT-unbroken states:\n\n \n\n\\[\n\n\\langle G\\rangle_{\\rm PT}=\\frac{\\langle\\psi_L|G|\\psi_R\\rangle}{\\langle\\psi_L|\\psi_R\\rangle}.\n\n\\]\n\n \n\n**4.4 Parameter scan**\n\n \n\n- \\(\\lambda\\in\\{0.00,0.05,0.10,0.15,0.20,0.30,0.50\\}\\)\n\n- \\(\\gamma\\) from 0 to 0.40 with step 0.002 (dense coverage of the unbroken region and the immediate vicinity of \\(\\gamma_{\\rm EP}\\))\n\n- For each \\((\\lambda,\\gamma,G)\\) the code records \\(b_{\\rm eff}\\), \\(\\langle G\\rangle_{\\rm PT}\\), the full spectrum, and the location of the nearest exceptional point (detected by the simultaneous vanishing of an eigenvalue gap and the onset of a non-zero imaginary part).\n\n \n\n**4.5 Exceptional-point displacement diagnostic**\n\n \n\nFor every active operator the code extracts\n\n \n\n\\[\n\n\\Delta\\gamma_{\\rm EP}(\\lambda)=\\gamma_{\\rm EP}(\\lambda)-\\gamma_{\\rm EP}(0).\n\n\\]\n\n \n\nA non-zero \\(\\Delta\\gamma_{\\rm EP}\\) constitutes operational activation of constraint-field control; a pure renormalization of \\(b_{\\rm eff}\\) at fixed \\(\\gamma_{\\rm EP}\\) is recorded as “spectral shift without EP displacement.”\n\n \n\n**4.6 Convergence and detection criteria**\n\n \n\nIdentical to Phase II-E: self-consistency tolerance \\(10^{-10}\\), EP detection tolerance \\(10^{-6}\\) in \\(|\\mathrm{Im}(E)|\\), double-precision arithmetic.\n\n \n\n---\n\n \n\n### V. PREDICTED OUTCOMES (FALSIFIABLE)\n\n \n\n**Prediction F1 (Minimal activation).** Only the original \\(G_{E_2}\\) (and operators linearly dependent on it) activate; all newly introduced operators remain null. The admissible branch is unique.\n\n \n\n**Prediction F2 (Sector-preserving two-body family).** Every PT-even, sector-preserving two-body operator activates; one-body and sector-mixing operators remain null. The operative constraint is the combination \\(\\mathcal{P}_+\\wedge\\mathcal{S}_0\\wedge 2\\).\n\n \n\n**Prediction F3 (Broad PT-even activation).** All PT-even operators that are not forced to zero by an additional discrete symmetry activate, including selected one-body and sector-mixing representatives. The sole operative constraint is PT parity.\n\n \n\n**Prediction F4 (EP-displacement selectivity).** Among the operators that produce non-zero \\(\\langle G\\rangle_{\\rm PT}\\), only a proper subset generate a measurable \\(\\Delta\\gamma_{\\rm EP}\\). The classification must therefore distinguish “field-sourcing” from “EP-tuning” branches.\n\n \n\nOnly one of F1–F3 can be correct; F4 is independent and will be reported for every active operator.\n\n \n\n---\n\n \n\n### VI. FALSIFICATION CONDITIONS\n\n \n\nThe Phase II-F result is falsified if:\n\n \n\n1. A computational error is identified in the archived code and a corrected run produces a different activation pattern.\n\n2. An alternative biorthogonal (or PT-inner-product) convention, fully specified and documented, changes the activation status of any operator that was null under the standard convention.\n\n3. A higher-precision or independent implementation (Julia, Mathematica, or a different eigensolver) yields a qualitatively different classification.\n\n4. The EP-displacement diagnostic is shown to be an artifact of the finite \\(\\gamma\\)-grid or of the particular state-selection rule.\n\n \n\n---\n\n \n\n### VII. DEPENDENCY MAP\n\n \n\nPhase II-F is unblocked by:\n\n \n\n- Phase II-E (activation of \\(G_{E_2}\\)).\n\n \n\nPhase II-F gates:\n\n \n\n- Phase II-G (Finite-size scaling). Only the operator families declared active by II-F are scaled to \\(N=6,8,10,12\\).\n\n- Phase II-H (Thermodynamic limit). Conditional on the existence of at least one family whose \\(\\Delta\\gamma_{\\rm EP}\\) survives finite-size scaling.\n\n \n\nPhase II-F remains independent of the still-pending synthetic-EP and dimer-verification sub-studies (II-A, II-B).\n\n \n\n---\n\n \n\n### VIII. INTERPRETATION UNDER THE CONSTRAINT HIERARCHY\n\n \n\nIf Prediction F1 is confirmed, the hierarchy contracts to a single admissible many-body branch at \\(N=4\\):\n\n \n\n\\[\n\n\\mathcal{M}_5=\\mathcal{M}_4\\cap\\{\\text{XY-bond PT-even sector-preserving}\\}.\n\n\\]\n\n \n\nIf Prediction F2 is confirmed, the hierarchy retains the full sector-preserving two-body PT-even class. If Prediction F3 is confirmed, the hierarchy expands relative to the Phase II-E expectation and the PT-odd obstruction remains the only absolute filter identified so far.\n\n \n\nIn all cases the classification supplies the precise set of operators that must be tracked under finite-size scaling, thereby converting the existence proof of Phase II-E into a controlled scaling hypothesis for Phase II-G.\n\n \n\n---\n\n \n\n### IX. SAFETY AND SCOPE\n\n \n\n**9.1 Safety Architecture**\n\n \n\nPhase II-F is a purely numerical classification exercise. The hard failsafes of Register III (Phase I) apply only to any future deployment of an activated constraint field, not to the classification itself.\n\n \n\n**9.2 Scope Limitations**\n\n \n\n- Restricted to \\(N=4\\).\n\n- Restricted to \\(\\Delta=0.5\\), \\(J=1\\).\n\n- Restricted to the biorthogonal convention.\n\n- Restricted to the PT-unbroken phase for the primary diagnostic; spectra above \\(\\gamma_{\\rm EP}\\) are recorded only for completeness.\n\n- Operator set limited to the five representatives listed in Section III plus the two Phase II-E baselines.\n\n \n\n---\n\n \n\n### X. CONCLUSION\n\n \n\nPhase II-F converts the binary activation result of Phase II-E into a structured taxonomy of admissible constraint operators. The taxonomy is the necessary prerequisite for any claim about the many-body viability of the self-consistent mechanism. Either a unique branch or a broader family will emerge; both outcomes are publishable and both advance the constraint hierarchy.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\n---\n\n \n\n### XI. PHASE STRUCTURE (UPDATED)\n\n \n\n| Phase     | Status          | Content                                              |\n\n|-----------|-----------------|------------------------------------------------------|\n\n| Phase I   | Complete        | Theoretical arc, no-go theorem, self-consistent mechanism |\n\n| Phase II-A| Pending         | Synthetic EP validation                              |\n\n| Phase II-B| Pending         | Dimer verification under biorthogonal convention     |\n\n| Phase II-C| Pending         | Puiseux exponent at \\(N=4\\)                          |\n\n| Phase II-D| Complete        | \\(N=4\\) dual-branch self-consistency — negative (PT-odd) |\n\n| Phase II-E| Complete        | PT-even operator test — activation of \\(G_{E_2}\\)    |\n\n| Phase II-F| Specified (this document) | Branch classification of active operators     |\n\n| Phase II-G| Blocked         | Finite-size scaling (requires II-F classification)   |\n\n| Phase II-H| Blocked         | Thermodynamic limit (requires II-G)                  |\n\n \n\n---\n\n \n\n### XII. REFERENCES\n\n \n\n1. Blair, A.J. (2026). Project Prometheus: Constraint-Field Control of Exceptional Points in PT-Symmetric Systems — Phase I Closeout Report. Zenodo. DOI: 10.5281/zenodo.23314444.  \n\n2. Blair, A.J. (2026). Phase II-D Technical Report: Dual-Branch Self-Consistency Calculation for the N=4 PT-Symmetric XXZ Chain. Zenodo.  \n\n3. Blair, A.J. (2026). Phase II-E Program Specification — PT-Even Constraint Operators in the N=4 PT-Symmetric XXZ Chain. Project Prometheus.  \n\n4. Blair, A.J. (2026). Phase II-E Numerical Execution Results (archived JSON + source). Project Prometheus.  \n\n5. Blair, A.J. (2026). A No-Go Theorem for Exceptional-Point Tuning in PT-Symmetric Dimers. Zenodo. DOI: 10.5281/zenodo.23287622.  \n\n6. Blair, A.J. (2026). Self-Consistent Exceptional Points in PT-Symmetric Dimers. Zenodo. DOI: 10.5281/zenodo.23287920.  \n\n7. Bender, C.M., \u0026 Boettcher, S. (1998). Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80, 5243.  \n\n8. Heiss, W.D. (2012). The physics of exceptional points. Journal of Physics A 45, 444016.\n\n \n\n---\n\n \n\n**Citation:** Blair, A.J. Phase II-F Program Specification — Branch Classification of the Self-Consistent Mechanism at N=4. Project Prometheus, 2026.\n\n \n\n**Result classification:** Program specification — ready for execution.  \n\n**Gate status:** Phase II-F unblocked by II-E; gates II-G and II-H.  \n\n**Falsifiable predictions:** 4 (F1 minimal, F2 sector-preserving two-body family, F3 broad PT-even, F4 EP-displacement selectivity).  \n\n \n\n**Immediate next step:** Implement the operator set of Section III, execute the scan of Section IV, and classify against Predictions F1–F4.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTony Blair \u003canthonyjblair8@gmail.com\u003e\n\n \n\n\n\n\n\n\n\n2:21 PM (7 minutes ago)\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\nto me\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nPhase II-F Program Specification: Branch Classification of the Self-Consistent Mechanism at N=4\n\n \n\nDESCRIPTION:\n\n \n\nProject Prometheus has reached a structural inflection point. We have moved from mapping the boundaries of the impossible (Phase I and II-D) to classifying the architecture of the possible (Phase II-E and II-F).\n\n \n\nThe Phase II-D numerical program delivered a clean negative result: PT-odd constraint operators possess vanishing biorthogonal expectation values in the PT-unbroken phase, extinguishing the self-consistent mechanism at N=4. The natural assumption was that the many-body extension of Paper 4 was closed.\n\n \n\nIt was not.\n\n \n\nPhase II-E demonstrated that the PT-even, sector-preserving transverse-bond operator (G_{E_2}) breaks the symmetry obstruction. It produces a non-vanishing biorthogonal expectation value, sources a non-trivial constraint field, and shifts the exceptional point. The wall was not the many-body limit. The wall was the operator class.\n\n \n\nPhase II-F is the systematic classification of that active mechanism space.\n\n \n\nBy partitioning constraint operators across three binary attributes—PT parity, sector compatibility, and body order—this specification defines the admissible architecture for constraint-field control at N=4. The document establishes a falsifiable taxonomy across four predictions (F1–F4), distinguishing between minimal activation, family-wide sector-preserving activation, broad PT-even activation, and operator-specific exceptional-point displacement.\n\n \n\nThe significance of Phase II-F is not just the identification of a working operator. It is the conversion of a binary activation result into a structured scaling hypothesis. By defining exactly which operators survive, which produce a measurable \\Delta\\gamma_{\\rm EP}, and which fall to the structural constraints of the system, we unblock the finite-size scaling program (Phase II-G) and the thermodynamic limit (Phase II-H).\n\n \n\nThis is Solve et Coagula in execution. We dissolved the false assumption that the PT-odd obstruction was terminal. We rebuilt the architecture by mapping the exact operator constraints.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\nDOI: https://doi.org/10.5281/zenodo.23314444\n\n \n\nKeywords: PT Symmetry, Exceptional Points, Non-Hermitian Physics, Constraint Hierarchy, Solve et Coagula, Many-Body Physics, XXZ Chain, Phase II, Branch Classification.\n\n \n\n---\n\n \n\nFor the LinkedIn Post (Copy and Paste Version)\n\n \n\nWe hit a wall in Phase II-D. The math said \"no.\"\n\n \n\nPT-odd operators vanish in the PT-unbroken phase. The many-body extension of the self-consistent mechanism looked dead at N=4.\n\n \n\nWe didn't stop. We applied Solve et Coagula. We dissolved the assumption that the wall was the end of the road.\n\n \n\nPhase II-E found the breach. The PT-even, sector-preserving transverse-bond operator activates. It produces a non-vanishing expectation value and shifts the exceptional point. The wall wasn't the many-body limit. The wall was the operator class.\n\n \n\nPhase II-F is the map of that breach. It is the formal branch classification of the self-consistent mechanism at N=4.\n\n \n\nWe are partitioning the admissible operator space by PT parity, sector compatibility, and body order. We are stating four falsifiable predictions in advance (F1–F4) to determine whether activation is unique, family-wide, or broad.\n\n \n\nThis is the difference between finding a result and building a taxonomy. You cannot scale what you cannot classify. Phase II-F unblocks the finite-size scaling program.\n\n \n\nThe mechanism was the vessel. The constraint was the discovery. Classification is the map.\n\n \n\nRead the specification: https://doi.org/10.5281/zenodo.23314444\n\n \n\n#ProjectPrometheus #ConstraintHierarchy #SolveEtCoagula #PTSymmetry #ExceptionalPoints #ManyBodyPhysics #PhaseII #NonHermitianPhysics #SystemsArchitecture"}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.23326731","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2026-10-12T21:41:17Z","registered":"2026-10-12T21:41:17Z","published":null,"updated":"2026-10-12T21:41:17Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.17605/osf.io/2wqjd","type":"dois","attributes":{"doi":"10.17605/osf.io/2wqjd","identifiers":[{"identifier":"https://osf.io/2wqjd","identifierType":"URL"}],"creators":[{"nameType":"Personal","name":"Silas Marques","nameIdentifiers":[{"nameIdentifierScheme":"ORCID","schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0000-0001-5141-8755"},{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/rbpy5"}],"affiliation":[]}],"titles":[{"title":"Inspiratory and Expiratory Muscle Training in the Decannulation Process of Tracheostomized Patients"}],"publisher":"OSF","container":{},"publicationYear":2026,"subjects":[{"subject":"Medicine and Health Sciences","subjectScheme":"bepress Digital Commons Three-Tiered Taxonomy"},{"subject":"Analytical, Diagnostic and Therapeutic Techniques and Equipment","subjectScheme":"bepress Digital Commons Three-Tiered Taxonomy"},{"subject":"Rehabilitation and Therapy","subjectScheme":"bepress Digital Commons Three-Tiered Taxonomy"},{"subject":"Expiratory Muscle Training"},{"subject":"Inspiratory Muscle Training"},{"subject":"Respiratory Muscle Training"},{"subject":"Tracheostomy"},{"subject":"Tracheostomy Decanulation"}],"contributors":[{"nameType":"Organizational","name":"Center for Open Science","nameIdentifiers":[{"nameIdentifierScheme":"URL","nameIdentifier":"https://cos.io/"},{"nameIdentifierScheme":"ROR","schemeUri":"https://ror.org","nameIdentifier":"https://ror.org/05d5mza29"}],"contributorType":"HostingInstitution","affiliation":[]}],"dates":[{"date":"2026-10-10","dateType":"Created"},{"date":"2026-10-12","dateType":"Updated"}],"language":null,"types":{"schemaOrg":"ScholarlyArticle","resourceTypeGeneral":"Text","citeproc":"article-journal","bibtex":"article","ris":"RPRT","resourceType":"Project"},"relatedIdentifiers":[],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[{"rightsIdentifierScheme":"SPDX","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rights":"Creative Commons Attribution 4.0 International","rightsIdentifier":"cc-by-4.0"}],"descriptions":[{"descriptionType":"Abstract","description":"JUSTIFICATION\nDecannulation depends on effective coughing, clearance of secretions, and sufficient respiratory muscle strength. Prolonged ventilation and critical illness often cause respiratory muscle weakness, which may increase the risk of decannulation failure. Inspiratory (IMT) and expiratory (TME) muscle training is an established rehabilitation strategy in other populations, but it is not clear how it has been described and applied in the tracheostomy decannulation process. No existing or ongoing review on the topic was identified in PROSPERO, the Cochrane Database of Systematic Reviews or the OSF Registries.\n\nOBJECTIVE AND QUESTIONS\nMap the literature on IMT and TME interventions used in the decannulation process of tracheostomized patients. Primary question: What IMT and TME interventions have been described as part of the decannulation process? Secondary questions: in which phases (pre-decannulation, weaning, post-decannulation) were they applied; which respiratory outcomes were measured; which populations were studied; whether gains in respiratory muscle strength were associated with successful decannulation; and what evidence gaps exist.\n\nDRAWING\nScope review according to the JBI methodology, reported according to PRISMA-ScR.\n\nELIGIBILITY (PCC)\nPopulation: tracheostomy patients of any age or etiology, in any care environment.\nConcept: TMI and/or TME (resistance devices, PEP therapy, diaphragmatic and chest expansion exercises, biofeedback, cough training, bronchial hygiene devices).\nContext: any phase of the decannulation process.\nSources: original studies, institutional protocols, guidelines, theses/dissertations and gray literature, in Portuguese, English or Spanish, without date restrictions. Excluded: studies without specific TMI/TME interventions, narrative reviews without primary data, animal/simulation studies and sources without access to the full text.\n\nSEARCH STRATEGY\nPubMed/MEDLINE, Scopus, Web of Science, LILACS/VHL, Cochrane Library and CINAHL; gray literature (Google Scholar, BDTD, RCAAP, AARC, ERS, SBPT, AMIB); reference lists and citation tracking. Strategy in three blocks (tracheostomy AND decannulation AND respiratory muscle training), adapted to each base. The full PubMed string is in the protocol supplementary file.\n\nDATA SELECTION AND EXTRACTION\nTitle/abstract and full text screening by two independent reviewers, with a third reviewer for discrepancies (Rayyan or Covidence). Extraction with a standardized form tested in 5 to 10 studies: identification, population, intervention (device, load, progression, frequency, duration, phase), outcomes (PImax/PEmax, peak cough flow, success and time until decannulation, ventilation outcomes, complications), main results, limitations and gaps. One reviewer extracts and a second checks at least 20% of the sources.\n\nSUMMARY\nNo critical evaluation will be carried out (optional in scope reviews). Descriptive synthesis with tables, figures and conceptual map, categorizing interventions into TMI, TME, TMI+TME combined or combined with other interventions, and identifying evidence gaps.\n\nETHICS, DISSEMINATION AND SCHEDULE\nOnly published data will be used; ethics committee approval is not required. The results will be submitted to peer-reviewed journals and conferences. Estimated duration: 5 to 6 months. Any amendments will be dated, justified and reported in a transparent manner.\n\nCONFLICTS OF INTEREST AND FUNDING\nNo conflicts of interest. No financing."}],"geoLocations":[],"fundingReferences":[],"url":"https://osf.io/2wqjd/","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2026-10-12T21:37:08Z","registered":"2026-10-12T21:37:09Z","published":null,"updated":"2026-10-12T21:37:09Z"},"relationships":{"client":{"data":{"id":"cos.osf","type":"clients"}}}},{"id":"10.17605/osf.io/k4uge","type":"dois","attributes":{"doi":"10.17605/osf.io/k4uge","identifiers":[{"identifier":"https://osf.io/k4uge","identifierType":"URL"}],"creators":[{"nameType":"Personal","name":"Italo Fabian Gonzalez","nameIdentifiers":[{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/39xwd"}],"affiliation":[]}],"titles":[{"title":"Effective Biological Time Framework (τβ) and the González Ortiz 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SENSOR "}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.23326331","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":1,"versionOfCount":0,"created":"2026-10-12T21:01:55Z","registered":"2026-10-12T21:01:56Z","published":null,"updated":"2026-10-12T21:01:56Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.5281/zenodo.23326332","type":"dois","attributes":{"doi":"10.5281/zenodo.23326332","identifiers":[{"identifier":"oai:zenodo.org:23326332","identifierType":"oai"}],"creators":[{"nameType":"Personal","givenName":"LUGALIA","familyName":"COLLINS JADEYA","name":"COLLINS JADEYA, LUGALIA","nameIdentifiers":[],"affiliation":[]}],"titles":[{"title":"AUTOMATIC HAND SANITIZER WITH TEMPERATURE 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SENSOR "}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.23326332","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2026-10-12T21:01:52Z","registered":"2026-10-12T21:01:53Z","published":null,"updated":"2026-10-12T21:01:53Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.17605/osf.io/83g7b","type":"dois","attributes":{"doi":"10.17605/osf.io/83g7b","identifiers":[{"identifier":"https://osf.io/83g7b","identifierType":"URL"}],"creators":[{"nameType":"Personal","name":"Sara ibrahim abdelkader","nameIdentifiers":[{"nameIdentifierScheme":"ORCID","schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0009-0002-5877-4722"},{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/pw9nv"},{"nameIdentifierScheme":"URL","nameIdentifier":"https://researchgate.net/profile/https://www.researchgate.net/profile/Sara-Abdelkad"}],"affiliation":[]},{"nameType":"Personal","name":"Omar Ahmed Salem Eid","nameIdentifiers":[{"nameIdentifierScheme":"ORCID","schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0009-0006-8029-701X"},{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/cfve6"}],"affiliation":[]},{"nameType":"Personal","name":"Ashraf Nabhan","nameIdentifiers":[{"nameIdentifierScheme":"URL","nameIdentifier":"http://github.com/ashrafnabhan"},{"nameIdentifierScheme":"URL","nameIdentifier":"http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=6150587"},{"nameIdentifierScheme":"URL","nameIdentifier":"http://researcherid.com/rid/A-1718-2008"},{"nameIdentifierScheme":"ORCID","schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0000-0003-4572-2210"},{"nameIdentifierScheme":"URL","nameIdentifier":"https://osf.io/qr3ay"}],"affiliation":[]}],"titles":[{"title":"Barriers to Valid Medical Research Among Early-Career Researchers in Egypt"}],"publisher":"OSF","container":{},"publicationYear":2026,"subjects":[{"subject":"Barriers"},{"subject":"Delphi"},{"subject":"Research"},{"subject":"Research waste"}],"contributors":[{"nameType":"Organizational","name":"Center for Open 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