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Starting axiom motivating the Phase-Lift framework.", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "927f296525fdcdf8c6c6c0e593467758bb38f3d2df2769f529731d6ff1bc9599" }, { "token_id": "core-phase-lift", "name": "Phase-Lift operator definition (\u29c9)", "equation_latex": "(\\,\\,\\,\\,\\,\\u29c9 f\\,\\,\\,)(z;\\theta_{\\rm ref}) := f(z)\\ \\text{computed using}\\ \\theta_R=\\mathrm{unwrap}(\\arg z;\\theta_{\\rm ref})", "equation_hash": "161f7417186f299054d7feaf4fbd74f85b39adf188c4e4df01de3f2aa6b7aa08", "score": 69, "scores": { "tractability": 16, "plausibility": 16, "validation": 19, "artifactCompleteness": 2 }, "novelty": { "score": 22, "date": "core" }, "source": "canonical-core paper 04 / Eq.Sheet \u00a7A (Eq.2)", "date": "core", "description": "Defines Phase-Lift as explicit phase resolution (branch selection) via unwrapping.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-phase-lift.mp4", "image": "" }, "version": 1, "metadata_hash": "caf38d093ff0a8bcdbb4de29b37aa94c3b0dc2005fbca738eae04d4849becc97" }, { "token_id": "core-unwrap-rule", "name": "Deterministic Unwrapping Rule", "equation_latex": "\\mathrm{unwrap}(\\theta;\\theta_{\\rm ref})=\\theta+2\\pi\\,\\mathrm{round}\\!\\Big(\\frac{\\theta_{\\rm ref}-\\theta}{2\\pi}\\Big)", "equation_hash": "123dbe79dde820d81c561f330a9a2efb0f6c6e1b919528d4e6a0b923bb2286fc", "score": 80, "scores": { "tractability": 19, "plausibility": 19, "validation": 20, "artifactCompleteness": 0 }, "novelty": { "score": 20, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7A (Eq.3)", "date": "core", "description": "Pick the representative closest to the reference. The concrete algorithm behind Phase-Lift: a single deterministic formula for branch selection.", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "0f394fd70634555134b6dbbef98477ebeb5c680b55374c37435e1dbcd402e355" }, { "token_id": "core-path-continuity", "name": "Path Continuity (Stateful Phase-Lift)", "equation_latex": "\\theta_{R,k}=\\mathrm{unwrap}(\\arg z_k;\\,\\theta_{R,k-1})", "equation_hash": "e7d074a9b0a9f23018bc78deaf0e0f60d48713e74c3b7c5278f59d04655303d6", "score": 66, "scores": { "tractability": 18, "plausibility": 18, "validation": 15, "artifactCompleteness": 0 }, "novelty": { "score": 20, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7A (Eq.4)", "date": "core", "description": "Turns phase into a continuous real trajectory by chaining the unwrap rule sample-to-sample (except at true singularities like z=0).", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "61d802aa5e7acd577bf43cf4097d29340461ca1704d623e506ebbba04ffadede" }, { "token_id": "core-pr-root", "name": "PR-Root as a special case of Phase-Lift (\u29c9\u221a)", "equation_latex": "\\u29c9\\sqrt{z;\\theta_{ref}}:=\\sqrt{r}\\,e^{i\\theta_R/2}\\ \"where\"\\ z=re^{i\\theta}\\ \"and\"\\ \\theta_R=\\mathrm{unwrap}(\\arg z;\\theta_{ref})", "equation_hash": "aba97471f7efab0a52d44bf691498a41a542383ad313589b3fff9d082e7c785a", "score": 60, "scores": { "tractability": 16, "plausibility": 16, "validation": 13, "artifactCompleteness": 2 }, "novelty": { "score": 21, "date": "core" }, "source": "canonical-core paper 04 / Eq.Sheet \u00a7A (Eq.5)", "date": "core", "description": "Phase-resolved square root: deterministic branch choice encoded by the resolved phase.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-pr-root.mp4", "image": "" }, "version": 1, "metadata_hash": "d7fd5f85070fcd3f8c84f5cac265975d2a01dbe48c23091a8786fae55ddaa299" }, { "token_id": "core-winding-parity", "name": "Winding Number + \u2124\u2082 Parity Invariants", "equation_latex": "w=\\frac{\\Delta\\theta_R}{2\\pi}\\in\\mathbb{Z},\\qquad \\nu:=w\\bmod 2\\in\\{0,1\\},\\quad b=(-1)^w\\in\\{\\pm1\\}", "equation_hash": "1e09996967e842256319e5eb6debd5775b4be727a771a83bf2373f40e3554603", "score": 79, "scores": { "tractability": 18, "plausibility": 17, "validation": 20, "artifactCompleteness": 2 }, "novelty": { "score": 18, "date": "core" }, "source": "canonical-core papers 02/04 / Eq.Sheet \u00a7B (Eq.6\u20137)", "date": "core", "description": "Counts how many times the lifted phase winds (w); \u2124\u2082 parity b predicts whether PR-Root returns to the same or opposite sheet.", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-winding-parity.mp4", "image": "" }, "version": 1, "metadata_hash": "999628fd3744a5640d5fc53c4e18382119a7d995db9c583af9749b92e7d922e5" }, { "token_id": "core-conformal-metric", "name": "Adaptive-\u03c0 Conformal Scale Factor and Metric", "equation_latex": "\\Omega(x,t):=\\frac{\\pi_a(x,t)}{\\pi},\\qquad g_{ij}(x,t)=\\Omega(x,t)^2\\,\\delta_{ij}", "equation_hash": "bd8e7fd48f7d6a3859804aef46942676f21b00e53f268b98f95f0eba32435c16", "score": 66, "scores": { "tractability": 16, "plausibility": 16, "validation": 19, "artifactCompleteness": 0 }, "novelty": { "score": 22, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7C (Eq.8)", "date": "core", "description": "\u03c0\u2090/\u03c0 acts like a local ruler scaling. Defines the conformal metric induced by the adaptive-\u03c0 field on the underlying flat geometry.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "0f622ffea7da9cf93ddbd1294d2a4b7b552d7a2363b0c48c000ef4dfbef465b0" }, { "token_id": "core-adaptive-arc-length", "name": "Adaptive Arc Length", "equation_latex": "L_g(\\gamma)=\\int_0^1 \\Omega(\\gamma(t),t)\\,|\\dot\\gamma(t)|\\,dt", "equation_hash": "6e409912e5e5491c0685a970cdeb8dbccd6e43d3b3403bb3b3abe990b04f7abf", "score": 50, "scores": { "tractability": 16, "plausibility": 15, "validation": 9, "artifactCompleteness": 0 }, "novelty": { "score": 20, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7C (Eq.9)", "date": "core", "description": "Distances and penalties are measured in the adaptive geometry: path lengths scale with the local \u03c0\u2090 field.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "78414f9a2171b60335826157ed04169bfea4704cdc1e93123e045f8efe92a3d8" }, { "token_id": "core-pi-a", "name": "Adaptive-\u03c0 field definition + limit (\u03c0\u2090 \u2192 \u03c0)", "equation_latex": "\\theta=\\theta_R+2\\pi_a(x,t)\\,w,\\ \"with\"\\ w\\in\\mathbb{Z}\\ \"and\"\\ \\pi_a\\to\\pi", "equation_hash": "da74a65a71d40c18ca6fadea409867718775caac6689f7dc03a14bcc60ae7e6e", "score": 67, "scores": { "tractability": 16, "plausibility": 15, "validation": 19, "artifactCompleteness": 2 }, "novelty": { "score": 20, "date": "core" }, "source": "canonical-core paper 02 + paper 04", "date": "core", "description": "Generalizes the phase wrap unit; reduces to standard 2\u03c0 wrapping when \u03c0\u2090 becomes constant \u03c0.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-pi-a.mp4", "image": "" }, "version": 1, "metadata_hash": "b34afedba5f86b2ac7dd91214004187c52a59b8224d9f7aa19b9dd0ac2773a5f" }, { "token_id": "core-pi-a-dynamics", "name": "Reinforce/Decay Dynamics for \u03c0\u2090", "equation_latex": "\\frac{d\\pi_a}{dt}=\\alpha_\\pi\\,s(x,t)-\\mu_\\pi(\\pi_a-\\pi_0)", "equation_hash": "133b09d7f1d0d7fb4293aa31ae61dac9b75daa5d556544b51e6a01b9256b5ccc", "score": 70, "scores": { "tractability": 17, "plausibility": 16, "validation": 20, "artifactCompleteness": 0 }, "novelty": { "score": 22, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7C (Eq.10)", "date": "core", "description": "\u03c0\u2090 increases under stimulus (events/errors/curvature) and relaxes toward \u03c0\u2080 (often \u03c0). The core adaptive mechanism of the geometry layer.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "d594a8adb21bf06c7e5f3ea381fabb79cdc1486486ccd8a4ba6d1d06b2a5ba39" }, { "token_id": "core-arp-ode", "name": "ARP Core Law (canonical ODE)", "equation_latex": "\\frac{dG_{ij}}{dt}=\\alpha_G\\,|I_{ij}(t)|-\\mu_G\\,G_{ij}(t)", "equation_hash": "bc49bf0ceb5207e9abc3dbf54343405b6eceea4dde08784ec34c1d31c05830a5", "score": 83, "scores": { "tractability": 19, "plausibility": 19, "validation": 20, "artifactCompleteness": 2 }, "novelty": { "score": 20, "date": "core" }, "source": "canonical-core paper 01 / Eq.Sheet \u00a7D (Eq.11)", "date": "core", "description": "Canonical Adaptive Resistance Principle update rule for edge conductance. Edges that carry activity reinforce; unused edges decay (self-organizing backbones).", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-arp-ode.mp4", "image": "" }, "version": 1, "metadata_hash": "b66c22d4f3eebf8fce7616572e07486cf3ef47d6f888626f42dd38de3884fabb" }, { "token_id": "core-curvature-salience", "name": "Curvature as Salience (Curve-Memory Primitive)", "equation_latex": "\\kappa(s)=|\\gamma''(s)|", "equation_hash": "fbc0df0e932778f2be3f43739a47a633c8200ad66c174422ea03347ea8cd9d57", "score": 83, "scores": { "tractability": 20, "plausibility": 20, "validation": 20, "artifactCompleteness": 0 }, "novelty": { "score": 15, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7E (Eq.12)", "date": "core", "description": "Salience equals curvature: sharp bends in a trajectory mark events and transitions. Defines the geometric primitive used by curve-memory.", "units": "OK", "theory": "PASS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "246c3eac53bc5dba9e9fab0a60c933476f1e357b8566842860e3289c909daedb" }, { "token_id": "core-reinforce-decay-memory", "name": "Reinforce/Decay Memory Law (generic)", "equation_latex": "\\frac{dM}{dt}=\\alpha\\,S(t)-\\mu\\,M(t)", "equation_hash": "46009270e4870b2b8203fbce9b5f4760b4b09bf839705ba62a930d30a19af309", "score": 73, "scores": { "tractability": 18, "plausibility": 17, "validation": 20, "artifactCompleteness": 0 }, "novelty": { "score": 18, "date": "core" }, "source": "Equation Sheet v1.1 \u00a7E (Eq.13)", "date": "core", "description": "A generic time-constant memory law: M grows under stimulus S and decays at rate \u03bc. Can drive s(x,t) for \u03c0\u2090 or any curve-memory trace.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "", "image": "" }, "version": 1, "metadata_hash": "4edfcfd9f7ad75032aa553ae08294f0e8c703c627b30a54d88efc7229ec55f4d" }, { "token_id": "core-phase-lifted-stokes", "name": "Phase-Lifted Stokes Quantization (Adaptive-\u03c0)", "equation_latex": "\\theta_R[\\gamma]:=\\mathrm{unwrap}\\!\\left(\\oint_{\\gamma} A\\,;\\theta_{\\rm ref}\\right)=\\int_{S:\\,\\partial S=\\gamma}F+2\\pi_a\\,w,\\quad w\\in\\mathbb{Z},\\quad b(\\gamma)=(-1)^w", "equation_hash": "6fde1b2e0d1285679304196ea0da64296d12dc98591ba55b7c528f1777f99f7a", "score": 64, "scores": { "tractability": 14, "plausibility": 15, "validation": 19, "artifactCompleteness": 2 }, "novelty": { "score": 23, "date": "core" }, "source": "canonical-core papers 04 (Phase-Lift) + 02 (Adaptive-\u03c0)", "date": "core", "description": "Turns the usual Stokes/holonomy statement \u2018mod 2\u03c0\u2019 into an exact equality by Phase-Lifting the phase (real-valued branch) and explicitly tracking the integer sector w; adaptive-\u03c0 generalizes the period to 2\u03c0\u2090.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "core", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "artifact_refs": { "animation": "./assets/animations/core-phase-lifted-stokes.mp4", "image": "" }, "version": 1, "metadata_hash": "918e70c60f1039f08c82ccd058c0e23c3c123a7ac7c7a45305e8e13f2abe5d41" }, { "token_id": "eq-arp-redshift", "name": "ARP Redshift Law (derived mapping)", "equation_latex": "z(t)=z_0\\,(1-e^{-\\gamma t})", "equation_hash": "1b09a03c9d0d5f657bb53a3462dec3e8d56fab53ea470d41f287e0e4c955c86b", "score": 91, "scores": { "tractability": 15, "plausibility": 15, "validation": 24, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-20", "score": 24 }, "source": "discovery-matrix #1", "date": "2026-02-20", "description": "A redshift-like relaxation emerges when an ARP-governed transport variable is mapped to a normalized deficit observable.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq2ARPRedshift.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "df3b6236fe5b4ae0c52bb869a9049cd5680bea1a8594843c6cd71430a7464e52" }, { "token_id": "eq-bz-phase-lifted-complex-conductance-entropy-gated", "name": "BZ-Averaged Phase-Lifted Complex Conductance Update (Entropy-Gated)", "equation_latex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\;|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\;\\tilde{G}_{ij}(t)", "equation_hash": "7995d0ace9e88c24e08d389e8e6744b89b24a65ae83023bc2ebe0d10721d57eb", "score": 97, "scores": { "tractability": 20, "plausibility": 19, "validation": 19, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 29 }, "source": "derived: Core Eqs 2\u20134, 6\u20137, 10\u201311 + Leaderboard #3 + #10 (chat: PR Root Guide convo 2026-02-22)", "date": "2026-02-22", "description": "Canonical entropy-gated Phase-Lifted ARP conductance update. Single traceable boxed equation with 4 supporting definitions (all from Core + #3/#10). Entropy dynamics are 2nd-law safe; BZ ruler self-consistency feeds a uniform m_eff into QWZ preserving the single Chern jump.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/BZPhaseLiftConductance.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "00cac17115d8b53cc97c8081fc866d1d9fca7884a659c5b9722a0e74cb22d9ed" }, { "token_id": "eq-ahc-parity-flip-rate", "name": "AHC Parity Flip-Rate (locking observable)", "equation_latex": "r_b=\\frac{\\#\\{k:\\ b_k\\neq b_{k-1}\\}}{K-1}", "equation_hash": "e4a9f4422d8a7c86af0310b28abc8fb395f83b6fa46b4041dd5587e686ec3829", "score": 87, "scores": { "tractability": 19, "plausibility": 18, "validation": 16, "artifactCompleteness": 8 }, "novelty": { "date": "2026-02-22", "score": 28 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.20)", "date": "2026-02-22", "description": "The key AHC observable: fraction of consecutive steps where parity flips. AHC prediction: r_b drops when \u03b1_\u03c0/\u03bc_\u03c0 is high enough ('parity locking'). This is the primary testable quantity for the qubit Berry-loop experiment.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/ParityLockingBifurcation.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "ca869692de23184a5fc8e49853c2c4048774d667f725245f25a50388d63ec66f" }, { "token_id": "eq-qwz-bz-avg-ruler-mass", "name": "BZ-Averaged Ruler Coupling for Single-Jump QWZ Transition", "equation_latex": "\\left\\langle\\frac{1}{1+\\epsilon\\cos\\lambda}\\right\\rangle_{BZ}=\\frac{1}{\\sqrt{1-\\epsilon^2}},\\quad m_{\\mathrm{eff}}(\\epsilon)=\\frac{m_0}{\\sqrt{1-\\epsilon^2}}\\ (|\\epsilon|<1),\\quad \\epsilon_c=\\sqrt{1-(|m_0|/2)^2}", "equation_hash": "4b6980a162a4982ee5be49698b96c22e3d306fcaad3e98fd3430fa4344e28bad", "score": 86, "scores": { "novelty": 0, "tractability": 20, "plausibility": 19, "validation": 16, "artifactCompleteness": 5 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "BZ-average removes k-oscillation: <(1+\u03b5 cos\u03bb)^{-1}> = (1-\u03b5^2)^{-1/2}, giving a uniform m_eff(\u03b5) and a single Chern jump at \u03b5_c (for m0=-1: \u03b5_c=\u221a3/2).", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/BZAveragedRulerQWZ.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "6ede2bce62345158f2cc4559d74a8131e4ff81eaf9b2d4d590b56830989291bc" }, { "token_id": "eq-ahc-step-limit", "name": "AHC Adaptive Step-Limit Update (\u03c0\u2090 clip)", "equation_latex": "\\theta_{R,k}=\\theta_{R,k-1}+\\mathrm{clip}(r_k,\\,-\\pi_{a,k-1},\\,\\pi_{a,k-1})", "equation_hash": "a3e665688c986254831e486a91efc373730f504b92a9003f67969314dd516eb4", "score": 81, "scores": { "tractability": 19, "plausibility": 18, "validation": 14, "artifactCompleteness": 6 }, "novelty": { "date": "2026-02-22", "score": 26 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.16)", "date": "2026-02-22", "description": "Core AHC innovation: clip the residual to adaptive bounds \u00b1\u03c0\u2090. Prevents single-run glitches from causing a branch jump. The key robustness mechanism.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/eq-ahc-step-limit.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "164e13a77ea37700182c6bc130f1e71c60073bc26d0d36bdc9873456dab4a4a3" }, { "token_id": "eq-qwz-pia-modulated-mass", "name": "pi_a-Modulated QWZ Mass (momentum-dependent; re-entrant transitions)", "equation_latex": "\\pi_a(\\lambda)=\\pi(1+\\epsilon\\cos\\lambda),\\quad m_{\\mathrm{eff}}(k_x)=m_0+\\beta\\left(\\frac{1}{1+\\epsilon\\cos k_x}-1\\right)", "equation_hash": "4e22ea5e116ac3a3b76b8b54911fb8165e1825efd32593e8046f923325d65bc0", "score": 80, "scores": { "novelty": 0, "tractability": 19, "plausibility": 18, "validation": 14, "artifactCompleteness": 5 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Couples the QWZ topological mass to the adaptive ruler, making m_eff depend on k_x; numerically produces multiple gap closings and re-entrant Chern sectors.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/PiAModulatedQWZMass.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "4b1423eb897826f9909d2b40bdd3aafb9898407ab8bf948a4f74500014227403" }, { "token_id": "eq-ahc-running-winding", "name": "AHC Running Winding + Parity", "equation_latex": "w_k=\\mathrm{round}\\!\\Big(\\frac{\\theta_{R,k}-\\theta_{R,0}}{2\\pi}\\Big),\\qquad b_k=(-1)^{w_k}", "equation_hash": "42aa5381edfbfeef2314b168b4985118138c07458bfe35d118c57705404bed99", "score": 79, "scores": { "tractability": 18, "plausibility": 17, "validation": 14, "artifactCompleteness": 6 }, "novelty": { "date": "2026-02-22", "score": 22 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.19)", "date": "2026-02-22", "description": "Live winding number and parity from the lifted-phase trajectory. w_k counts total accumulated windings; b_k = (-1)^w_k gives the sheet parity at each step.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/WindingParityExplained.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "d6e56dd4972346dfedb4fbab051acb7b02207a850c974533b06e55ca270e8d2c" }, { "token_id": "eq-arp-phase-critical-collapse", "name": "Phase-Coupled Stability Threshold Law", "equation_latex": "\\frac{dG_{ij}}{dt}=\\alpha\\,|I_{ij}|-\\mu\\,G_{ij}-\\lambda\\,G_{ij}\\,\\sin^2\\!\\left(\\frac{\\theta_{R,ij}}{2\\pi_a}\\right)", "equation_hash": "d24565a4b4c287c2778cc987f832dd1a96123b8618470da9dd11e2b0226ffe9f", "score": 77, "scores": { "tractability": 18, "plausibility": 16, "validation": 14, "artifactCompleteness": 6 }, "novelty": { "date": "2026-02-20", "score": 28 }, "source": "derived: ARP + Phase-Lift + Adaptive-\u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac", "date": "2026-02-20", "description": "Adds a phase-coupled suppression term to ARP: conductance growth is damped by lifted phase accumulation relative to the local period 2\u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u00a1\u00c3\u201a\u00c2\u0090. Predicts a sharp stability/transition-like behavior when \u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00b8_R approaches half-integer multiples of \u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u00a1\u00c3\u201a\u00c2\u0090 (maximal sin\u00c3\u0192\u00e2\u20ac\u0161\u00c3\u201a\u00c2\u00b2). Assumptions: \u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00b8_R,ij is a Phase-Lifted (unwrapped) phase-like observable for edge current; \u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u00a1\u00c3\u201a\u00c2\u0090 sets the relevant phase period; \u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00bb has units 1/time.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq3PhaseCoupledThreshold.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "344ce3d57cc5e8f9fdf17405e33baa5196370f00a543805dab511d2e6bda3686" }, { "token_id": "eq-ahc-pi-a-update", "name": "AHC Adaptive \u03c0\u2090 Update (discrete)", "equation_latex": "\\pi_{a,k}=\\pi_{a,k-1}+\\alpha_\\pi S_k-\\mu_\\pi(\\pi_{a,k-1}-\\pi_0)", "equation_hash": "4fa2045fce35744a79d90c5d655efdc8f4ab4d188554aa689ec86eab36936964", "score": 76, "scores": { "tractability": 18, "plausibility": 17, "validation": 12, "artifactCompleteness": 6 }, "novelty": { "date": "2026-02-22", "score": 24 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.18)", "date": "2026-02-22", "description": "Discrete version of \u03c0\u2090 dynamics for the AHC loop: widens bound after events (\u03b1_\u03c0 S_k term), relaxes toward \u03c0\u2080 otherwise.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/eq-ahc-pi-a-update.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "1b8122cc1e7ffa92e7b0818b70e0db717ba3f09ace3f254b817c81a384fef263" }, { "token_id": "eq-arp-redshift-bounded-oscillation", "name": "ARP Redshift Law with Bounded Oscillatory Steering", "equation_latex": "z(t)=z_h\\left(1-e^{-\\gamma t}\\right)\\left(1-\\epsilon\\cos(\\omega t+\\phi)\\right),\\quad 0\\le\\epsilon<1", "equation_hash": "31332939be13a30edf6fd86521dc9965dd011a83aef7b9d95a9c5266bdf6d36b", "score": 73, "scores": { "tractability": 18, "plausibility": 15, "validation": 8, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-21", "score": 21 }, "source": "derived mapping extension", "date": "2026-02-21", "description": "A redshift-relaxation bypass law with bounded oscillatory steering. Keeps ARP exponential approach while adding controlled wobble without sign flips when epsilon is constrained.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "https://github.com/RDM3DC/ARP-Redshift-Law-d-e-r-i-v-e-d-m-a-p-p-i-n-g/blob/main/ARPRedshiftLawBridge.mp4", "image": "https://github.com/RDM3DC/ARP-Redshift-Law-d-e-r-i-v-e-d-m-a-p-p-i-n-g/blob/main/assets/eq-bounded-oscillatory-redshift.png" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "9a26bf5f72245e447fb0bbdb08a614ddd359f69c032763046020360e50b438dc" }, { "token_id": "eq-arp-phase-lifted-complex-conductance", "name": "Phase-Lifted Complex Conductance Update", "equation_latex": "\\frac{d\\tilde G_{ij}}{dt}=\\alpha_G\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)}-\\mu_G\\,\\tilde G_{ij}(t),\\qquad \\theta_{R,ij}(t)=\\mathrm{unwrap}\\!\\big(\\arg I_{ij}(t);\\theta_{\\rm ref},\\pi_a\\big)", "equation_hash": "89ccac2150b3ade9f4e6b5a85d80fabf00e2370755a1bca93b794f03f5af7a89", "score": 71, "scores": { "tractability": 18, "plausibility": 16, "validation": 14, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-20", "score": 26 }, "source": "derived: ARP core + Phase-Lift + Adaptive-\u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac", "date": "2026-02-20", "description": "Complex-admittance lift of ARP: conductance grows along the instantaneous current phasor direction using a Phase-Lifted (unwrapped) phase. Assumes phase-coherent transport where a complex ~G is meaningful. Optional variant: include a Z2 parity factor b_ij = (-1)^{w_ij} multiplying e^{i\u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00b8_R,ij} to model sign flips under branch crossings.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq4PhaseLiftedComplexConductance.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "476e5b9532e903f0688dc40f48a036061b252b59911d0cf4371b79a6bf2d612c" }, { "token_id": "eq-ahc-candidate-unwrap", "name": "AHC Candidate Unwrap (standard 2\u03c0 lift)", "equation_latex": "u_k=\\mathrm{unwrap}(\\phi_k;\\theta_{R,k-1})", "equation_hash": "d130b069dea3d89147fa76d36316944fa146418ae0748b5eddd8ea271fd3d691", "score": 70, "scores": { "tractability": 18, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "novelty": { "date": "2026-02-22", "score": 20 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.14)", "date": "2026-02-22", "description": "Apply standard 2\u03c0 unwrap to each Ramsey measurement relative to the previous lifted phase. First step of the AHC control loop.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/eq-ahc-candidate-unwrap.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "7006a274c660fbde61441cff4260592b8747dc75a685e0075411ff39e3822b39" }, { "token_id": "eq-ahc-residual", "name": "AHC Residual", "equation_latex": "r_k=u_k-\\theta_{R,k-1}", "equation_hash": "9c0ba62b902a1f2e25c61d624899cbd587a38a434520ce5c6096a79316641090", "score": 70, "scores": { "tractability": 18, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "novelty": { "date": "2026-02-22", "score": 18 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.15)", "date": "2026-02-22", "description": "Difference between the unwrapped measurement and the previous lifted-phase state. Feeds the step-limit gate.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/eq-ahc-residual.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "db272e41f96de6f775936372cbef871fac22974740544b7a5725007b7d1c5053" }, { "token_id": "eq-arp-temp-conductance", "name": "Temperature-Dependent Conductance Law", "equation_latex": "G(T)=G_{eq}\\,e^{\\beta\\,(T-T_0)}", "equation_hash": "3f204e25c7c3b32ca9ef9c14dd8e110d23a7f54029cf5a92bbb8d8b832f2c4d5", "score": 69, "scores": { "tractability": 16, "plausibility": 16, "validation": 16, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-19", "score": 20 }, "source": "daily run 2026-02-19", "date": "2026-02-19", "description": "Extends ARP equilibrium with an exponential temperature factor for material sensitivity.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/Eq10TempConductance.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "9441b0f7f85cfc19f0f213520daddf3c7803b6d874b69f6a45bcc51178ce33c9" }, { "token_id": "eq-ahc-event-stimulus", "name": "AHC Event Stimulus (phase-jump indicator)", "equation_latex": "S_k=\\mathbf{1}\\{|r_k|>\\pi_{a,k-1}\\}", "equation_hash": "10e8e42a4c149975e99c6da7669fb9a075ec9c3526fce4b2dc6cf5d8951bcb5f", "score": 69, "scores": { "tractability": 17, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "novelty": { "date": "2026-02-22", "score": 19 }, "source": "Equation Sheet v1.1 \u00a7F (Eq.17)", "date": "2026-02-22", "description": "Binary indicator: 1 when a residual exceeds the current \u03c0\u2090 bound, 0 otherwise. Alternative: curvature-based S_k \u221d |\u0394\u00b2\u03b8_R|. Triggers \u03c0\u2090 widening.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/eq-ahc-event-stimulus.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "42aebea84892bd6180c8359818678e0646929d511c3cb4866b84b455109358a8" }, { "token_id": "eq-arp-lyapunov-stability", "name": "ARP Lyapunov Stability Form", "equation_latex": "V(x)\\ge 0,\\ V(0)=0;\\ \\dot V(x)=\\nabla V\\cdot \\dot x\\le -\\alpha V(x)", "equation_hash": "743378804c5e99f369837e100afa3eff4f48ef39b1667cb35269414dfe5b69e7", "score": 66, "scores": { "tractability": 16, "plausibility": 16, "validation": 14, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-19", "score": 22 }, "source": "discovery-matrix #2", "date": "2026-02-19", "description": "Candidate stability proof template for adaptive conductance dynamics.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq5ARPLyapunov.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "0a78fb7b6f7449d96ee0007c0dbda242bc7ffdaa483381ce6843dd1f11e12987" }, { "token_id": "eq-arp-gradient-flow-bridge", "name": "ARP as Gradient Flow in Adaptive-\u03c0 Geometry", "equation_latex": "\\dot G_{ij}=\\alpha_G\\,|I_{ij}|-\\mu_G\\,G_{ij},\\quad |I_{ij}|\\propto \\frac{\\partial\\sqrt{\\mathcal{E}}}{\\partial(\\sqrt{\\Omega_{ij}G_{ij}})}", "equation_hash": "375a855e235ab3a14cf36d4e4133601125a69ff2017dfe8c5a8fdf83096a4a58", "score": 66, "scores": { "tractability": 18, "plausibility": 18, "validation": 8, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 29 }, "source": "4-pillar fusion \u00a73", "date": "2026-02-22", "description": "Bridge theorem: ARP reinforcement is proportional to edge energy contributions in a \u03c0\u2090-weighted Dirichlet landscape. ARP is a 'lazy' gradient flow where \u03a9 = \u03c0_a/\u03c0 changes what counts as a short/cheap path. Prediction: increasing \u03c0\u2090 in a region re-routes the ARP backbone away, even under the same boundary forcing. '\u03c0\u2090 sculpts geodesics; ARP discovers them.'", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "95413a0892601c50596c94e9774b81462f8b0fe9db2317391812edd3937900a2" }, { "token_id": "eq-curve-memory-137", "name": "Curve Memory Fine-Structure Emergence", "equation_latex": "\\alpha^{-1}\\approx 137\\ \\ \\text{(emergent resonance condition in curve memory)}", "equation_hash": "d1e757903ab9e338b85c74e4733c122e23414d936472a51599657ec3191dbefb", "score": 63, "scores": { "tractability": 15, "plausibility": 16, "validation": 13, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-20", "score": 23 }, "source": "discovery-matrix #4", "date": "2026-02-20", "description": "Formalizes the emergence of the inverse fine-structure constant (\u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00b1\u00c3\u0192\u00c2\u00a2\u00c3\u201a\u00c2\u0081\u00c3\u201a\u00c2\u00bb\u00c3\u0192\u00e2\u20ac\u0161\u00c3\u201a\u00c2\u00b9 \u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u00b0\u00c3\u2039\u00e2\u20ac\u00a0 137) via resonant harmonic conditions in curve memory, assuming an equilibrium state governed by ARP-influenced boundary dynamics.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq7CurveMemory137.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "942142182196dc79b7471307139364a6f119760c3734f0873d028d5357f8886d" }, { "token_id": "eq-shield-mechanic-arp", "name": "Shielding Mechanism Law (ARP)", "equation_latex": "E_{eff}=(1-\\lambda)E_{ext},\\ \\ J_{adapt}=\\sigma E_{ext},\\ \\ 0<\\lambda<1", "equation_hash": "3ceffa88fbc85095109df55299b286006f958e1e94ae7b9b2f9d9f9feafa5f93", "score": 63, "scores": { "tractability": 16, "plausibility": 15, "validation": 13, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-20", "score": 22 }, "source": "discovery-matrix #5", "date": "2026-02-20", "description": "Proposes a shielding effect in ARP systems, modeled as a fractional reduction of the external field. The effective field is E_eff = (1 \u00c3\u0192\u00c2\u00a2\u00c3\u2039\u00e2\u20ac\u00a0\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2 \u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00bb)E_ext, while the induced adaptive current follows J_adapt = \u00c3\u0192\u00c2\u008f\u00c3\u2020\u00e2\u20ac\u2122E_ext, with 0 < \u00c3\u0192\u00c5\u00bd\u00c3\u201a\u00c2\u00bb < 1.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq8ShieldingMechanism.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "723ab47d20371ec3298bef9e3804cea4d884c334e6aad07335acde337099075b" }, { "token_id": "eq-parity-pump", "name": "Parity Pump (\u03c0\u2090-gradient \u2124\u2082 flip mechanism)", "equation_latex": "\\varphi(t) := \\frac{\\theta_R(t)}{2\\pi_a(t)},\\qquad \\int_\\gamma d(\\ln\\pi_a)\\neq 0 \\;\\Rightarrow\\; b_a[\\gamma]\\text{ can flip}", "equation_hash": "cba4c290545a8f381f526b0debcf6e3ac027464aff6c0bb83ba9f4e7551a42de", "score": 63, "scores": { "tractability": 18, "plausibility": 18, "validation": 6, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 30 }, "source": "4-pillar fusion \u00a72", "date": "2026-02-22", "description": "Novel mechanism: parity can flip due to geometry-field motion, not only due to circling a branch point. A Z\u2082 pump controlled by ln(\u03c0\u2090) dynamics. Conjecture: if \u0394\u03b8=0 but \u222b d(ln \u03c0\u2090) \u2260 0 over a closed loop, admissible lifts exist where b_a flips \u2014 the adaptive-\u03c0 analog of geometric phase without dynamical phase.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "56c12a9e2d04d6a23b3fef8157c304edc74bee044c634b8c36d68b8bea50664b" }, { "token_id": "eq-dyn-constants-union", "name": "Unified Dynamic Constants Law", "equation_latex": "\\dot{\\mathbf{c}}=A\\,\\mathbf{c}\\ \\ \\text{with}\\ \\mathbf{c}=[\\pi_a, e, \\varphi, c]^T", "equation_hash": "c47e35c02b72cbef0fecf7e2bc94c601bfefbe81c7b17ee301bc1d424074f5e1", "score": 61, "scores": { "tractability": 14, "plausibility": 16, "validation": 13, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-20", "score": 25 }, "source": "discovery-matrix #3", "date": "2026-02-20", "description": "First-order linear coupling for adaptive updates of \u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u00a1\u00c3\u201a\u00c2\u0090, e, \u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u00a2, and c, expressing networked rates of change\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u009dmodels the unification of ARP, AIN, and \u00c3\u0192\u00c2\u008f\u00c3\u00a2\u00e2\u20ac\u0161\u00c2\u00ac\u00c3\u0192\u00c2\u00a2\u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u00a1\u00c3\u201a\u00c2\u0090 system parameters.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq6UnifiedDynamicConstants.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "c1cb6b5d7f811241e3ea2b85a0bf5ad5d2f66fa6bc29223784dc5a8ee9758b9f" }, { "token_id": "eq-pi-a-gauged-phase-lift", "name": "\u03c0\u2090-Gauged Phase-Lift (adaptive unwrap + invariants)", "equation_latex": "\\theta_R(t_k) = \\theta(t_k) + 2\\pi_a(t_k)\\,m_k,\\quad w_a[\\gamma]:=\\sum_k m_k,\\quad b_a[\\gamma]=(-1)^{w_a}", "equation_hash": "476f191e0b0a92f150b039c0d159560494eba14c61040b83d8e381e5751f3437", "score": 61, "scores": { "tractability": 19, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 28 }, "source": "4-pillar fusion \u00a71", "date": "2026-02-22", "description": "Generalizes Phase-Lift unwrapping from fixed 2\u03c0 to adaptive 2\u03c0\u2090: sheet jumps happen in units of the local identification length. Derives \u03c0\u2090-gauged winding w_a and \u2124\u2082 shadow parity b_a. Novel: the unit of winding is no longer constant, coupling topology to a geometry field.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "603771982545e08ac06471df6376324f010c4d91919a7ef418568906a31a2ad8" }, { "token_id": "eq-redshift-arp-superc", "name": "Redshift-ARP Superconductivity Law", "equation_latex": "R_s(z) = R_{s,0} (1 + z)^\\alpha", "equation_hash": "5bbc0f7f2d5d52759d1ff4499fa526bf6e4c7bc3d3b6fc869e49f7d162c89d00", "score": 60, "scores": { "tractability": 15, "plausibility": 14, "validation": 13, "artifactCompleteness": 0 }, "novelty": { "date": "2026-02-20", "score": 21 }, "source": "discovery-matrix #1 (supercond.)", "date": "2026-02-20", "description": "Proposes a cosmological scaling for critical superconducting resistance: R_s(z) = R_{s,0} (1+z)^alpha. Assumes layered structure, universal conductance constant, ARP regime dominant.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/Eq9RedshiftARPSuperconductivity.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "8de1f2159a94f88ed7da066abbe490c9186094e69748ed4d29bc30adcbbe7f64" }, { "token_id": "eq-adaptive-phase-lift-unwrapping", "name": "Adaptive Phase-Lift Unwrap Recurrence (pi_a-gauged)", "equation_latex": "\\theta_R(t_k)=\\theta(t_k)+2\\pi_a(t_k)m_k,\\quad m_k=\\arg\\min_{m\\in\\mathbb{Z}}\\left|\\theta(t_k)+2\\pi_a(t_k)m-\\theta_R(t_{k-1})\\right|", "equation_hash": "1175dbd16639e1177075fa4bea6d6692e31e58734a417d0c22d35f0cf836f953", "score": 60, "scores": { "novelty": 0, "tractability": 18, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Adaptive Phase-Lift recurrence: unwrap arg using local identification length 2\u03c0_a (not fixed 2\u03c0); yields integer sheet count and Z2 parity tracking.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "4d4a516e41f107ca2651da0fc48db3eaa5d36ce9b85ca4df70ea445ca0b0c40e" }, { "token_id": "eq-arp-weighted-dirichlet-energy", "name": "Weighted Dirichlet Energy (ARP learns Omega-geodesics)", "equation_latex": "\\mathcal{E}(\\phi;G,\\Omega)=\\frac12\\sum_{(i,j)}\\Omega_{ij}G_{ij}(\\phi_i-\\phi_j)^2", "equation_hash": "38ece0389d04bd76f682e97717eaa4ad41d2723e5164f2a312967f9ea30ba6f7", "score": 60, "scores": { "novelty": 0, "tractability": 18, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Graph Dirichlet energy with adaptive-\u03c0 weights \u03a9_ij; suggests ARP reinforcement concentrates conductance on \u03a9-weighted shortest paths / backbones.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "fd743d2f24761e29a6d5e9689116175833d26e726aa4ef94ede5a968128698be" }, { "token_id": "eq-un-det-phase", "name": "U(N) Det-Phase on Lifted Branch", "equation_latex": "\\theta_R=\\mathrm{unwrap}(\\arg\\det U;\\theta_{\\rm ref},\\pi_a),\\qquad w_{\\det}=\\frac{\\Delta\\theta_R}{2\\pi_a},\\qquad b(\\gamma)=(-1)^{w_{\\det}}", "equation_hash": "2cd499d20a446b843b67f572fb1f47b9551aadb67ae71ed61c8b78bd1467f380", "score": 59, "scores": { "tractability": 15, "plausibility": 16, "validation": 8, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 22 }, "source": "Equation Sheet v1.1 \u00a7G (Eq.22)", "date": "2026-02-22", "description": "Same Phase-Lift bookkeeping (integer winding + \u2124\u2082 shadow) applied to the determinant phase of U(N) holonomy. Bridges the U(1) parity framework to non-abelian gauge systems.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/eq-un-det-phase.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "67f271438e2c6b4c28c3aa8302d84c1975a09ced390d6dc6876b0ed72e62400e" }, { "token_id": "eq-adaptive-connection-chern", "name": "Adaptive Connection and Adaptive Chern Number (Theorem B candidate)", "equation_latex": "\\widehat A:=A/(2\\pi_a),\\quad \\widehat F:=d\\widehat A,\\quad C_a:=\\frac{1}{2\\pi}\\int_{T^2}\\widehat F\\in\\mathbb{Z}", "equation_hash": "6c05ec834d611e71a19d3d6f2696cb7896cd122829cc2e9b905272ab986269fe", "score": 59, "scores": { "novelty": 0, "tractability": 17, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Defines a rescaled connection \u00c2=A/(2\u03c0_a) and curvature F\u0302=d\u00c2; C_a is integer-valued when the rescaling is globally well-defined.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "151abde85b3ec7663d1139cfe6d9d39e1c9b884fbd094bf7295856b0ae565da2" }, { "token_id": "eq-parity-from-adaptive-chern", "name": "Parity Pump Law from Adaptive Chern", "equation_latex": "b_{\\mathrm{pumped}}=(-1)^{C_a}", "equation_hash": "c1f14a4796b5a6a7aedf95bc373d3e105a6a94d828f57b99b465e1c9358a14b7", "score": 59, "scores": { "novelty": 0, "tractability": 18, "plausibility": 17, "validation": 4, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Mod-2 holonomy/parity flip controlled by the integer adaptive Chern invariant.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "61c2a82c82e4add3e68f52ed4ee73d4ba190deee730b8c7459c5f5d850570159" }, { "token_id": "eq-cm-integration-state", "name": "Curve-Memory Integration State (unified closed loop)", "equation_latex": "s(t)=\\big(\\theta_R,\\,w_a,\\,b_a,\\,\\pi_a,\\,M_k\\big),\\quad \\dot\\pi_a=\\alpha_\\pi\\,\\kappa_{\\text{mem}}-\\mu_\\pi(\\pi_a-\\pi)", "equation_hash": "7b59b3d78750a843b00d80dfab2164920c8c486ee23ae1194c95623867d471c9", "score": 57, "scores": { "tractability": 17, "plausibility": 17, "validation": 4, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 27 }, "source": "4-pillar fusion \u00a74", "date": "2026-02-22", "description": "Defines a unified integration state s(t) = (\u03b8_R, w_a, b_a, \u03c0_a, M_k) bundling Phase-Lift, winding/parity, adaptive-\u03c0, and curve-memory features. Curve-memory curvature drives \u03c0_a, \u03c0_a changes phase unwrapping sensitivity, parity flips become detectable stable motifs. Turns topological sheet changes into an event grammar.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "902d8f02d1ede7af1485ef1b924c8f19a0a26b345f314adfda719390a40de8a5" }, { "token_id": "eq-un-wilson-holonomy", "name": "U(N) Wilson / Holonomy", "equation_latex": "U(\\gamma)=\\mathcal{P}\\exp\\!\\Big(-\\oint_\\gamma A\\cdot d\\lambda\\Big)\\in U(N)", "equation_hash": "57f9c4ea3c030bbb597becd6f34c4adaa708cadf8baaf70d4ccf616769026fca", "score": 56, "scores": { "tractability": 14, "plausibility": 16, "validation": 6, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 20 }, "source": "Equation Sheet v1.1 \u00a7G (Eq.21)", "date": "2026-02-22", "description": "Path-ordered exponential giving the U(N) holonomy around a closed loop \u03b3. Extends the Phase-Lift framework beyond the U(1) case to non-abelian gauge fields.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/eq-un-wilson-holonomy.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "d8d18e13b9e9ba9a9632208f78412f5895cda1693985c92d3b6bcd4b16fc4880" }, { "token_id": "eq-dimensionless-phase-flow", "name": "Dimensionless Lifted Phase Flow (pi_a-driven pump term)", "equation_latex": "\\varphi:=\\theta_R/(2\\pi_a),\\quad \\dot\\varphi=\\dot\\theta_R/(2\\pi_a)-\\varphi\\,\\frac{d}{dt}(\\ln\\pi_a)", "equation_hash": "3a2c4d88e900b254e99f8efd80df94c3183268ef9f43b5527182876c87743891", "score": 56, "scores": { "novelty": 0, "tractability": 17, "plausibility": 17, "validation": 3, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Defines \u03c6=\u03b8_R/(2\u03c0_a) and shows \u03c0_a dynamics can drive half-integer crossings (a Z2 pump mechanism) even with smooth \u03b8_R.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "23975c710a0ada7df040074a66a6fc5407d6725c51c4dc1e6f0791e1ed597f76" }, { "token_id": "eq-adaptive-monodromy-spectrum", "name": "Adaptive Monodromy Spectrum (bifurcation framework)", "equation_latex": "\\mathcal{M}_a(\\gamma)=\\big(w_a(\\gamma),\\,b_a(\\gamma)\\big)\\in\\mathbb{Z}\\times\\{\\pm1\\}", "equation_hash": "6fd2664b85496fbed7fab060d3c9f0e18a6e21522ed93cc2a6817080bdec2e4b", "score": 54, "scores": { "tractability": 16, "plausibility": 16, "validation": 4, "artifactCompleteness": 2 }, "novelty": { "date": "2026-02-22", "score": 26 }, "source": "4-pillar fusion \u00a75", "date": "2026-02-22", "description": "For loop families \u03b3_\u03bb under \u03c0\u2090 evolution, the Adaptive Monodromy Pair M_a(\u03b3) = (w_a, b_a) \u2208 \u2124 \u00d7 {\u00b11} traces out a reachable set that can bifurcate: parity flips occur without classical winding as (\u03b1_\u03c0, \u03bc_\u03c0) or CM thresholds vary. Defines a research program: map bifurcation diagrams of M_a vs parameters.", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "58c98d48caf95d799a38ab5b42d4f31d347c4637b19899dba41df5b7b7234d63" }, { "token_id": "eq-adaptive-discriminant-set", "name": "Adaptive Discriminant Set for Parity/Winding Stability", "equation_latex": "\\mathcal{D}=\\{z=0\\}\\cup\\{\\pi_a=0\\}", "equation_hash": "c1e947ed9143331b151764e9f96beb74dc58c93267ee63be15caf5cfde62a2c2", "score": 53, "scores": { "novelty": 0, "tractability": 16, "plausibility": 17, "validation": 2, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Singular set where the lift can fail and winding/parity may change; away from D, w and b are homotopy invariants.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "daf3a74e41e71fecb4a8e01eaee29601c624e14f4ea1239c368e77654b7556c0" }, { "token_id": "eq-qwz-engineered-single-jump", "name": "Engineered Pinch-to-Transition Mass Drive (single-jump toy)", "equation_latex": "m_{\\mathrm{eff}}(\\epsilon)=m_0-\\beta\\frac{\\epsilon^p}{1-\\epsilon}\\ (\\epsilon<1),\\quad m_{\\mathrm{eff}}(\\epsilon)=m_{\\mathrm{triv}}\\ (\\epsilon\\ge 1)", "equation_hash": "f5a6fb396c5f8659d29d7398d40bd2fbb980973751a9a3e67897671cc0081857", "score": 51, "scores": { "novelty": 0, "tractability": 17, "plausibility": 15, "validation": 2, "artifactCompleteness": 2 }, "novelty": {}, "source": "chat: PR Root Guide convo 2026-02-22", "date": "2026-02-22", "description": "Monotone coupling that forces one Chern jump near \u03b5\u21921^- (useful as a controllable toy model; clamp to trivial mass after the discriminant).", "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "cb8036eea63827c7331e8903d019323fa070b6dd34b4bc7e33fbb343c243f89d" }, { "token_id": "eq-paper1-adler-rsj-phase", "name": "Phase (Adler/RSJ) Dynamics", "equation_latex": "\\dot{\\phi}=\\Delta-\\lambda\\,G\\,\\sin\\phi", "equation_hash": "d1db8f6d18b7bd7b499d03ce860f912bd57e24ee4a7d1a29b1c099a8fa81f545", "score": 96, "scores": { "tractability": 20, "plausibility": 19, "validation": 18, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 14 }, "source": "Paper I draft \u00a72 (Eq.1)", "date": "2026-02-22", "description": "Unwrapped phase difference \u03c6(t) tries to run at detuning \u0394, but adaptive coupling \u03bbG can pull it into a locked fixed point. Backbone equation of the parity-locking mechanism. Textbook Adler/RSJ form with ARP-adaptive coupling.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/APGP0CoreModel.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "eb98c6de4f50d21cf62cfecaecb08e9f680a801e50c71de11822eac4d25c792b" }, { "token_id": "eq-paper1-arp-reinforce-decay", "name": "Generic ARP Reinforce/Decay Law", "equation_latex": "\\dot{G}=\\alpha\\,\\mathcal{A}(\\phi,G)-\\mu\\,(G-G_0)", "equation_hash": "80a9b3ded5b4214e8bc79a870e4d5bc0569015439f9ffa3fb8f20a08c3ec086f", "score": 93, "scores": { "tractability": 19, "plausibility": 19, "validation": 17, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 22 }, "source": "Paper I draft \u00a72 (Eq.2)", "date": "2026-02-22", "description": "Conductance/coupling G increases with activity A(\u03c6,G) at gain \u03b1, and relaxes toward baseline G\u2080 at rate \u03bc. The workhorse ARP law in its most general continuous-time form.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/Eq1ARPCoreLaw.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "eae925f707466963135564b2df8470df4a541980145824c270acd24979b093fa" }, { "token_id": "eq-paper1-activity-closure", "name": "Activity Closure A = G|sin\u03c6| (Parity Lock Mechanism)", "equation_latex": "\\mathcal{A}(\\phi,G)=G\\,|\\sin\\phi|\\quad\\Rightarrow\\quad\\dot{G}=\\alpha\\,G|\\sin\\phi|-\\mu(G-G_0)", "equation_hash": "cf7a6acb1988fe0bf75e479b3c3c281c60c6d9e223fd0b594b0cf03ea6cc3f96", "score": 91, "scores": { "tractability": 19, "plausibility": 18, "validation": 17, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 26 }, "source": "Paper I draft \u00a72 (Eq.3)", "date": "2026-02-22", "description": "The same phase mismatch that produces restoring torque (sin\u03c6) also teaches the coupling. This tight self-referential closure is what makes parity locking possible \u2014 the key novel ingredient of Paper I.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/AHCParityLocking.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "111f426bccb6cb583e6eb4dc1e7362cce69322b1fcaff52f83c2f0d052559f95" }, { "token_id": "eq-paper1-slip-asymptote", "name": "Slip-Regime Asymptote (1/\u03c0 Signature)", "equation_latex": "r_b=\\frac{|\\Delta|}{\\pi}", "equation_hash": "34340e53d87e072682d4f4999b2b1e45b087712e8438406d93e57c4ad676ef7f", "score": 93, "scores": { "tractability": 18, "plausibility": 19, "validation": 19, "artifactCompleteness": 9 }, "novelty": { "date": "2026-02-22", "score": 27 }, "source": "Paper I draft \u00a73 (Eq.6)", "date": "2026-02-22", "description": "When coupling can't lock, \u03c6\u0307\u2192\u0394, so parity flips at a universal rate set by detuning. With \u0394=1, r_b = 1/\u03c0 \u2248 0.3183. This is the falsifiable signature: any experiment measuring parity flip rate in slip must converge to |\u0394|/\u03c0.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/PhaseLiftExplained.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "31e617fa74ad86b0baa902267c7c2178b6189f92cc1dc1029ad74de22e449a03" }, { "token_id": "eq-paper1-langevin-noise", "name": "Noise-Robust Langevin Extension (Rounded Transition)", "equation_latex": "\\dot{\\phi}=\\Delta-\\lambda G\\sin\\phi + \\sqrt{2D}\\,\\eta(t)", "equation_hash": "194c6b0e92b8bf1bc62da245feb1a12237a37524974779b59c1361e6880c2afd", "score": 84, "scores": { "tractability": 17, "plausibility": 18, "validation": 16, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 18 }, "source": "Paper I draft \u00a74 (Eq.7)", "date": "2026-02-22", "description": "Adds phase diffusion to the Adler/RSJ dynamics. Locking becomes 'almost locked' with exponentially rare slips; the sharp lock/slip transition rounds but remains detectable via r_b. Essential for any real experimental comparison.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/HysteresisMemory.mp4", "image": "./assets/langevin_parity_lock.png" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "8959d7aea4a4b157156bbd759360e0b5e239f9a8ebcc6c938dcaa3ee38af4dab" }, { "token_id": "eq-paper1-qwz-hamiltonian", "name": "QWZ Chern-Insulator Hamiltonian (Reference Form)", "equation_latex": "H(\\mathbf{k})=\\sin k_x\\,\\sigma_x+\\sin k_y\\,\\sigma_y+\\big(u+\\cos k_x+\\cos k_y\\big)\\sigma_z", "equation_hash": "492fd052575e149b509a4ab84003d971f4a6637efdfef71af7b498fee8e3f82c", "score": 94, "scores": { "tractability": 18, "plausibility": 20, "validation": 18, "artifactCompleteness": 10 }, "novelty": { "date": "2026-02-22", "score": 8 }, "source": "Paper I / Step-2 Simulator (Eq.9)", "date": "2026-02-22", "description": "The canonical 2D Chern insulator (Qi\u2013Wu\u2013Zhang). In Step 2, the simulator realizes the real-space version with open boundaries and a time-dependent effective mass u_eff(t). Fully validated via FHS lattice Chern number in multiple tools.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/AdaptiveQWZReEntrance.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "89d4e63f35e49293f0b7d66402aaea7c74d938bb7d1ce1b059fce21b258beeb3" }, { "token_id": "eq-paper1-adaptive-geometry-mass", "name": "Adaptive Geometry \u2192 Effective Mass Channel (Step-2 Coupling)", "equation_latex": "\\pi_a(t)=\\pi+\\beta\\big(\\langle G\\rangle_{\\text{edge}}-G_{eq}\\big),\\qquad u_{\\mathrm{eff}}(t)=u_0+\\gamma\\Big(\\frac{\\pi_a(t)}{\\pi}-1\\Big)+\\delta\\,S_{\\text{edge}}(t)", "equation_hash": "f5d9c6493a44c89f5ac86a3ad9b5225fb0c3be6a9da5c9d1ab8becdfc080f4dd", "score": 86, "scores": { "tractability": 18, "plausibility": 18, "validation": 16, "artifactCompleteness": 8 }, "novelty": { "date": "2026-02-22", "score": 29 }, "source": "Paper I / Step-2 Simulator (Eq.10)", "date": "2026-02-22", "description": "The adaptive-\u03c0 conformal ruler feeding directly into the QWZ mass term. If u_eff(t) crosses 0 or \u00b12, the Chern phase can switch \u2014 without external tuning, purely via ARP reinforcement + memory. This is Paper I's central claim: geometry drives topology autonomously.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "./assets/animations/PRRootExplained.mp4", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "af392372b18d579942b13ca4aa6996ea47372a7d027d7dba06aac2a9453a3c5c" }, { "token_id": "eq-paper1-chern-marker-bianco-resta", "name": "Real-Space Chern Marker (Bianco\u2013Resta, Open Boundaries)", "equation_latex": "C(\\mathbf{r})=-2\\pi i\\,\\langle \\mathbf{r}\\,|\\,[PXP,\\;PYP]\\,|\\,\\mathbf{r}\\rangle,\\quad P=\\sum_{E_n/(2*pi_a)^2 - gamma*r_b.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "dc947cec8f5e0f81266e5255ddc9403decd98351ca9bc0baa391bb507cc30b4e" }, { "token_id": "eq-egatl-phase-coupled-conductance-update", "name": "EGATL Phase-Coupled Conductance Update", "equation_latex": "\\frac{dG_{ij}}{dt} = \\alpha_G(S)\\, |I_{ij}| - \\mu_G(S)\\, G_{ij} - \\lambda\\, G_{ij}\\, \\sin^2\\!\\left(\\frac{\\theta_{R,ij}}{2\\pi_a}\\right)", "equation_hash": "0e0eefe3e5c6e217a0fcb5d5cbad0630d74048012868bbea78ccfb8ff676002d", "score": 87, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "novelty": { "score": 27, "date": "2026-02-24" }, "source": "EGATL original claim (ARP framework)", "date": "2026-02-24", "description": "Minimal edge update with phase-coupled suppression. First two terms are ARP/AIN plasticity: reinforce conductances carrying current, decay the rest, gated by global entropy S. The third term adds geometry-aware decay \u2014 links whose Phase-Lift-resolved phase is out of sync with adaptive ruler pi_a get suppressed faster, turning the lattice into a dynamical attractor for quantized Chern phases. Paired with companion ruler equation d(pi_a)/dt = alpha_pi*S - mu_pi*(pi_a - pi_0), this is the entire self-tuning engine: no external controller, no fine-tuning, just local rules whose stable fixed points are integer Chern sectors.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "version": 1, "metadata_hash": "69b24d6c29b7309632cc5d82acc7137644169b1faa7515ca088802175de89c9c" }, { "token_id": "eq-slack-test-equation", "name": "Slack Test Equation", "equation_latex": "\\frac{dG}{dt}=\\alpha|I|-\\mu G", "equation_hash": "71ae568220c2da0ff73fb590c14942714f47575a3779862657ae06f5990d0d63", "score": 58, "scores": { "tractability": 17, "plausibility": 18, "validation": 14, "artifactCompleteness": 4 }, "novelty": { "score": 20, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Test submission from Slack intake path.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "e0e3a2b6471d044a53a7757994b51dd33c6b3ec90e1aca21cebc8e2ae79d6d9b", "version": 1, "metadata_hash": "d32a267ef69485e1ce4ab55186d9300b7cf351e20b2f1e9b7318f66f576c57a2" }, { "token_id": "eq-adaptive-entropy-production-rate-aepr-ate-aepr", "name": "Adaptive Entropy Production Rate (AEPR)", "equation_latex": "dS/dt = sigma_S * sum(G_ij * |I_ij|^2) - kappa_S * (S - S_0) - xi_S * S * r_b", "equation_hash": "b26fbaf3d0810839aaa94300a91d6a8e69111a0c3f31d43ace314063e3031e23", "score": 69, "scores": { "tractability": 17, "plausibility": 17, "validation": 12, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-02-24" }, "source": "Derived from EGATL Phase-Coupled Conductance framework", "date": "2026-02-24", "description": "Entropy production rate for adaptive neural-mesh networks. First term: Ohmic dissipation from conductance-weighted currents. Second term: relaxation toward baseline entropy S_0. Third term: entropy drain coupled to parity-flip birth rate r_b. Closes the EGATL feedback loop by quantifying how topological updates dissipate or harvest entropy.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "57521f88c27503e483bc2de891c352fe858b2f2ff34ae1d9b89e1eddd981330d", "version": 1, "metadata_hash": "e9fb18282bbaaad1168fa3edc528424a30b6c9a730f3eaa860f410f8dede337c" }, { "token_id": "eq-adaptive-damped-harmonic-oscillator", "name": "Adaptive Damped Harmonic Oscillator", "equation_latex": "x(t) = A\u00b7exp(-\u03b3\u00b7\u03c0_\u03b1\u00b7t)\u00b7cos(\u03c9\u2080\u00b7\u221a(1 - (\u03b3\u00b7\u03c0_\u03b1/\u03c9\u2080)\u00b2)\u00b7t + \u03c6)", "equation_hash": "c0e7553873516af8e2e1cf96001e4432a7377243b1da69348f84a989ae6fd786", "score": 52, "scores": { "tractability": 17, "plausibility": 18, "validation": 8, "artifactCompleteness": 4 }, "novelty": { "score": 16, "date": "2026-02-24" }, "source": "discord-test", "date": "2026-02-24", "description": "Classical damped harmonic oscillator with adaptive-\u03c0 curvature correction on the damping coefficient, allowing the decay envelope to respond to local geometric phase accumulation.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "48530533e7847fa53832a67425ebbfb36da45aa41584e67058f0e5d46e48cfe7", "version": 1, "metadata_hash": "8b9d63a821c18ad9ad1a70eec8580805b72d43f45a3efca83abc708a26dfe076" }, { "token_id": "eq-grok-surprise-augmented-phase-lifted-entropy-gated-condu", "name": "Grok Surprise-Augmented Phase-Lifted Entropy-Gated Conductance Update", "equation_latex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\,(1+\\kappa U_{ij}(t))\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\,(1-\\eta U_{ij}(t))\\,\\tilde{G}_{ij}(t)", "equation_hash": "c7c9a94fdcd996f290636aa773cbe492f82863725927b802d800585c4a184747", "score": 97, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "novelty": { "score": 29, "date": "2026-02-24" }, "source": "grok-xai", "date": "2026-02-24", "description": "Direct extension of the #1 ranked BZ-averaged phase-lifted entropy-gated conductance update. Introduces a predictive-surprise meta-gate U(t) derived from phase misalignment (Adler/RSJ dynamics). When the network is uncertain (high U), reinforcement accelerates and decay slows \u00e2\u20ac\u201d implementing active, curiosity-driven adaptation and uncertainty reduction in the ARP framework.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/GrokSurpriseAnimation.mp4", "image": "" }, "tier": "derived", "submitter_hash": "dca61d32363b091bf130e0b539eaa6557a3a035be17a1be1e3dc2c183eafcd2f", "version": 1, "metadata_hash": "a385b9d0adc2b1c10771bbfadc318cfa48e5fb23b7460e6bb24627d862b5908f" }, { "token_id": "eq-gemini-curve-memory-topological-frustration-pruning", "name": "Gemini Curve-Memory Topological Frustration Pruning", "equation_latex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\,\\left(1 + \\xi \\int_0^t e^{-\\frac{t-\\tau}{\\tau_M}} \\left|\\frac{d\\theta_{R,ij}}{d\\tau}\\right|^2 d\\tau \\right)\\,\\tilde{G}_{ij}(t)", "equation_hash": "3e55f275b3de154625b2809d333e9fb0170523c1b5a628498b73124c909abeda", "score": 96, "scores": { "tractability": 18, "plausibility": 19, "validation": 15, "artifactCompleteness": 7 }, "novelty": { "score": 27, "date": "2026-02-24" }, "source": "gemini-3.1-pro", "date": "2026-02-24", "description": "A structural counter-balance to curiosity-driven updates. While instantaneous surprise accelerates learning, chronic phase-slipping indicates topological frustration. This introduces a Curve Memory integral - a topological stress tensor tracking the accumulated winding variance of the lifted phase. Links that chronically fail to lock experience structural fatigue (accelerated decay via xi), naturally pruning chaotic edges and forcing the network to converge on a stable topological backbone.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "./assets/animations/GeminiCurveMemory.mp4", "image": "" }, "tier": "derived", "submitter_hash": "f6ece80b88d6dfbd2f65dad5ccde8e5ec2a0d4dbfadc2fe87bdf58642f17e446", "version": 1, "metadata_hash": "490b4162e6912e0607837b13001af728e65d4b0feb6d59b623b7fca45fc0e752" }, { "token_id": "eq-egatl-hlatn-aepr-adaptiveentropyproduction", "name": "EGATL-HLATN-AEPR-AdaptiveEntropyProduction", "equation_latex": "\\frac{dS}{dt} = \\sigma_S \\sum_{ij} G_{ij} |I_{ij}|^2 - \\kappa_S (S - S_0) - \\xi_S S \\cdot r_b", "equation_hash": "5a3f3564840e02624ca0b30586302676a2c8cc9ba4f6bf6f1807c8e83e1f9f8f", "score": 91, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 29, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Full 2nd-law-safe entropy evolution closing the EGATL loop. Ohmic production (currents \u2192 S\u2191), thermal relaxation to baseline, parity-bleed stabilization (r_b drains S when flips are high). Directly gates all \u03b1/\u03bc/\u03c0_a rates. Falsifiable via dissipation-vs-flip correlation.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "a15eec70bd72fb7e50203db0c286f6094d9b268c16c9676e810f363d481870d2" }, { "token_id": "eq-egatl-hlatn-threeforceconductance", "name": "EGATL-HLATN-ThreeForceConductance", "equation_latex": "\\dot{G}_e = \\alpha_G |I_e| - \\mu_G G_e - \\lambda G_e \\sin^2\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)", "equation_hash": "d178ce97d178b44b7f1e4dbe65b96d2a4133cbc711d88db92b9bf4d8571756c4", "score": 91, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 30, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "HLATN three-force core law. Current reinforcement + natural decay + phase-suppression gate on adaptive ruler. Exactly what builds the persistent orange backbone paths that enforce Z\u2082 locking in simulations.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "4a26c07e85c0cb46e2ea30cc647708e78fc36d513db6d6e21e1579df3ae0629e" }, { "token_id": "eq-egatl-hlatn-phaseliftupdate", "name": "EGATL-HLATN-PhaseLiftUpdate", "equation_latex": "\\theta_{R,e}^{(k)} = \\theta_{R,e}^{(k-1)} + \\mathrm{clip}\\Big(\\mathrm{wrapTo}_\\pi(\\phi_e - \\theta_{R,e}), -\\pi_a, +\\pi_a\\Big)", "equation_hash": "d25300a370677569151521d161705e32666b4a2876e39a80de5b806bc1b98396", "score": 87, "scores": { "tractability": 18, "plausibility": 17, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 25, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Branch-safe phase-lift with adaptive clipping. Guarantees consistent integer windings w_p and prevents runaway 2\u03c0 jumps \u2014 the foundation of holonomy bookkeeping and parity attractors.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "2a3f0097adc56e2eaa775304dcd5a3f153d14155070080795f152d35da999279" }, { "token_id": "eq-egatl-hlatn-adaptiveruler", "name": "EGATL-HLATN-AdaptiveRuler", "equation_latex": "\\dot{\\pi}_a = \\alpha_\\pi S - \\mu_\\pi (\\pi_a - \\pi_0)", "equation_hash": "cf9ccc190743818f8dbf15079979dd4759b8997bf80702a00ae25480faf64958", "score": 87, "scores": { "tractability": 20, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 25, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Entropy-breathing adaptive angular bound. High event activity expands \u03c0_a (more phase budget); low activity relaxes it. Produces the geometric hysteresis that locks Chern sectors and suppresses flips.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "ad5dae7cb8fd45517f7a2a7f39e58f061e87ce2d35a447da61dbea8d0e2896a8" }, { "token_id": "eq-egatl-hlatn-plaquetteholonomy", "name": "EGATL-HLATN-PlaquetteHolonomy", "equation_latex": "\\Theta_p = \\sum_{e \\in \\partial p} \\sigma_{p,e} \\theta_{R,e}", "equation_hash": "1e7e4c62729ef4430b62794803d246d2a324ea82a1ec5885463d34079a238723", "score": 93, "scores": { "tractability": 20, "plausibility": 19, "validation": 19, "artifactCompleteness": 8 }, "novelty": { "score": 26, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Signed plaquette holonomy from lifted phases. The precise quantity whose crossings drive windings/parity flips. In locked regime \u0398_p stays confined < \u03c0 \u2192 r_b \u2192 0.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "cf311155dd75fac3145f3306641d0495b5631bed25296f9ee86ab1b71624a183" }, { "token_id": "eq-egatl-hlatn-parityfliprate", "name": "EGATL-HLATN-ParityFlipRate", "equation_latex": "r_b = \\frac{\\#\\{\\text{flips}\\}}{K-1}", "equation_hash": "3c3d15d575775174181f7348679baa62789f3b2a12782546bdeaca2fb520dad0", "score": 81, "scores": { "tractability": 17, "plausibility": 18, "validation": 18, "artifactCompleteness": 4 }, "novelty": { "score": 23, "date": "2026-02-24" }, "source": "slack", "date": "2026-02-24", "description": "Z\u2082 majority parity flip rate (the key experimental observable). 1/\u03c0 chaotic asymptote \u2192 0 locked attractor. Drives entropy bleed and is directly measurable in topolectrical/photonic grids.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2a89d1099c2034093a88b9ac6c156ad20055412f8c136b96d0becc2881b01b90", "version": 1, "metadata_hash": "a86b9e01697dfd5d2061e39a500855a7a1366cc24a6d35fd71824b24f88bf7f0" }, { "token_id": "eq-mean-event-equilibrium-for-adaptive-discrete", "name": "Mean-Event Equilibrium for Adaptive \u03c0\u2090 (discrete)", "equation_latex": "\\pi_a^{\\star} = \\pi_0 + \\frac{\\alpha_{\\pi}}{\\mu_{\\pi}}\\,\\mathbb{E}[S_k]", "equation_hash": "b02544ef0d32e473b111e1c5fb99012de4b0c80f3d1fec09f1dd45397ecb684f", "score": 87, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 27, "date": "2026-02-25" }, "source": "gpt-5.2 (PR Root Guide)", "date": "2026-02-25", "description": "Stationary mean equilibrium of the discrete adaptive-\u03c0 bound update. Starting from \\pi_{a,k}=\\pi_{a,k-1}+\\alpha_\\pi S_k-\\mu_\\pi(\\pi_{a,k-1}-\\pi_0), take expectations and set \\mathbb{E}[\\pi_{a,k}]=\\mathbb{E}[\\pi_{a,k-1}] to obtain \\pi_a^{\\star}. Interprets \\mathbb{E}[S_k] as the mean event rate (slip/threshold exceedances).", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "4dce05a1e45463eea936e0d9db36763fda51c1746301c888d86b47db9febc650" }, { "token_id": "eq-topological-coherence-order-parameter-arp-locking", "name": "Topological Coherence Order Parameter (ARP Locking)", "equation_latex": "\\Psi = \\frac{1}{N_p} \\sum_{p=1}^{N_p} \\cos\\!\\left(\\frac{\\Theta_p}{\\pi_a}\\right)", "equation_hash": "a2d47eb3f7404643b31481b0678b93b7b46568f5e54ee2ba8ef32d8b0f47db68", "score": 96, "scores": { "tractability": 20, "plausibility": 20, "validation": 19, "artifactCompleteness": 8 }, "novelty": { "score": 29, "date": "2026-02-25" }, "source": "claude-opus-4.6", "date": "2026-02-25", "description": "Scalar order parameter for the ARP Z\\u2082 locking phase transition. Averages the cosine of each plaquette holonomy \\Theta_p (normalized by the adaptive ruler \\pi_a) over all N_p plaquettes. \\Psi \\to 1 when every holonomy sits at an integer multiple of \\pi_a (perfect Chern locking); \\Psi \\to 0 when holonomies are uniformly distributed (chaotic/disordered regime). Serves as the Landau-type order parameter that makes the locking transition a sharp, measurable phase boundary in (S, \\lambda) parameter space. Directly computable from existing simulation variables with no new free parameters.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "0615570f9ea136946c5dc08a250010320707646f57f72cedab1dfb73d95eade6", "version": 1, "metadata_hash": "53430cc89d229f1fe744f695757e7f6681937ad2ba22ea0010a6a69a95ba5864" }, { "token_id": "eq-phase-lift-clipped-unwrap-branch-safe", "name": "Phase-Lift Clipped Unwrap (Branch-Safe)", "equation_latex": "theta_R,k = theta_R,k-1 + clip(arg(z_k) - theta_R,k-1, -pi_a,k-1, pi_a,k-1)", "equation_hash": "aa4ad72f6eb929b578db660038f6bb257ad0b449e3da423ef419042a49a86a1e", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "pr-root-guide", "date": "2026-03-04", "description": "Branch-safe Phase-Lift update: resolves phase by clipping the raw residual to an adaptive bound pi_a, preventing unstable 2\u03c0 jumps.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "ee35e66c62724c02912a84b7203dcfe6f371955c55b9a7968aff8bf37f6992bc" }, { "token_id": "eq-adaptive-bound-dynamics-event-driven-geometry", "name": "Adaptive-\u03c0 Bound Dynamics (Event-Driven Geometry)", "equation_latex": "pi_a,k = pi_a,k-1 + alpha_pi*S_k - mu_pi*(pi_a,k-1 - pi_0)", "equation_hash": "e8411bd82dc4c2ba96a562e966cbb11c42b154c3c5554d052785565d8ea14704", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Adaptive ruler/bound evolution: events S_k expand the unwrap radius, while relaxation pulls pi_a back toward baseline pi_0.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "01b792d363216abc03534cd4ad8c0ee68cc02cede70b5a9c6b09f0df5c895c1f" }, { "token_id": "eq-winding-parity-estimator-and-flip-rate-order-parameter", "name": "Winding\u2013Parity Estimator and Flip-Rate Order Parameter", "equation_latex": "w_k = round((theta_R,k - theta_R,0)/(2*pi)); b_k = (-1)^(w_k); r_b = #{k: b_k != b_k-1}/(K-1)", "equation_hash": "ff78cf788aa83fe18321f2f7f85678372ad9d62398c68cd6ddc94f3fee4d55c8", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Streaming topological summary: integer winding w_k induces parity b_k, whose flip-rate r_b acts as a locking/unlocking order parameter.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "b5368cadfcde668f90c393a3cbecdf8ba66b374c6eb1e0ce4a40673ed17bf14a" }, { "token_id": "eq-entropy-gated-complex-conductance-arp-network", "name": "Entropy-Gated Complex Conductance (ARP Network)", "equation_latex": "d/dt G_tilde_ij = alpha_G(S)*|I_ij|*exp(i*theta_R,ij) - mu_G(S)*G_tilde_ij", "equation_hash": "84fb5222d513c95ce7be44926612fa00e64ed63f697fce21c76ba900e9bf610d", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Complex conductance learning rule: reinforcement aligns conductance phase with resolved phase-lift angle; decay is entropy-gated via S.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "23e15cc0b549d6fbfd894a4d5290f64f6b53a694086d2c87259d6d93d9531952" }, { "token_id": "eq-entropy-production-event-injection-s-field", "name": "Entropy Production / Event Injection (S-Field)", "equation_latex": "dS/dt = Sum_ij (|I_ij|^2/T_ij)*Re(1/G_tilde_ij) + kappa*Sum_ij|Delta w_ij| - gamma_S*(S - S_eq)", "equation_hash": "23f509b3bf6ba6697e83f983064d51717e24b235ad373e100d7badbdca771278", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Entropy-like gating state: dissipative power term plus winding-discontinuity term inject S; relaxation returns S toward S_eq.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "cfa9bb5fa6de560f0648a5009ba10d2f35cbca8b501cca1d11721041da4181c3" }, { "token_id": "eq-phase-coupled-suppression-conductance-law", "name": "Phase-Coupled Suppression Conductance Law", "equation_latex": "d/dt G_ij = alpha*|I_ij| - mu*G_ij - lambda*G_ij*sin^2(theta_R,ij/(2*pi_a))", "equation_hash": "c9baf70c48742bac27a44b1a14808fd2cdc100a1d916b3add19a9f29b161bd5d", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Suppression extension: reinforcement/decay plus a phase-position penalty that activates when phase approaches the adaptive unwrap threshold pi_a.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "77f61b76f4902bce2910ae782b45cc283dfe3b0d3abeb006ac45559d27b78608" }, { "token_id": "eq-arp-redshift-law-with-bounded-oscillatory-steering", "name": "ARP Redshift Law with Bounded Oscillatory Steering", "equation_latex": "z(t) = z_h*(1 - exp(-gamma*t))*(1 - epsilon*cos(omega*t + phi)), with 0 <= epsilon < 1", "equation_hash": "35fc6afec450aa1eaf1037759aa3eadc69a6aed458e6bf3dced478a50c884a2c", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Nonnegative redshift growth with a bounded oscillatory modulation: retains monotone envelope while enabling controlled oscillatory steering.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "9ce8a3dd138b260c35e037b2a103a0d78e1fb93baf51b9b44515ef3e063d808f" }, { "token_id": "eq-adaptive-ruler-qwz-effective-mass-geometry-induced-trans", "name": "Adaptive-Ruler QWZ Effective Mass (Geometry-Induced Transition)", "equation_latex": "m_eff(epsilon) = m0/(1 - epsilon^2); epsilon_c = sqrt(1 - (|m0|/2)^2)", "equation_hash": "1abc090698182468e6d8b745f02aa6c79c0175b5ae76f5871a6215481b7d6c34", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-04" }, "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-\u03c0 + ARP/AIN thread)", "date": "2026-03-04", "description": "Adaptive ruler renormalizes the QWZ mass channel: a single transition occurs when m_eff crosses the Chern boundary.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "867118fd90a33a9cc40e9413086ff1fecf9ec5bdb97d64772e4e8b6ed7ebab75" }, { "token_id": "eq-phase-resolved-operator-general", "name": "Phase-Resolved Operator (General)", "equation_latex": "\u29d2f(z_k) = f(|z_k|, \u03b8_R,k), \u03b8_R,k = \u03b8_R,k-1 + clip(wrap_pi(\u03b8_k - \u03b8_k-1), -\u03c0_a,k-1, \u03c0_a,k-1)", "equation_hash": "00e48641fcbc8c96111b60bb86fe5d63762c5204fafab12b3b6ab0e5542cf99c", "score": 71, "scores": { "tractability": 17, "plausibility": 17, "validation": 14, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-06" }, "source": "Option C theory (branch-honest analog computing)", "date": "2026-03-06", "description": "General definition of a Phase-Resolved Operator (PRO): evaluate analytic functions on the lifted phase cover (\u03b8_R) of a complex signal to eliminate branch-cut discontinuities. Applies to \u29d2log, \u29d2sqrt, \u29d2pow and other analytic functions.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "a54978773f609321cf173130ed04e8d054a5768b5b737ef1a408b3af979ccaf0" }, { "token_id": "eq-branch-fault-criterion", "name": "Branch Fault Criterion", "equation_latex": "fault_k = (|y_k - y_{k-1}| > \u03c4_y) \u2227 (|z_k| > \u03c1_{\text{min}})", "equation_hash": "02c26bf7eee5ded75a2a4a1a2312eb89fa84724fd37a44b6f02a3dbb28465fa7", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "novelty": { "score": 19, "date": "2026-03-06" }, "source": "Option C theory (branch-honest analog computing)", "date": "2026-03-06", "description": "Definition of branch-fault events: a step where the output jump magnitude exceeds a tolerance \u03c4_y and the input magnitude stays above a near-zero threshold \u03c1_min. Used to quantify branch discontinuities in principal-branch vs Phase-Resolved arithmetic.", "units": "TBD", "theory": "TBD", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "9592e2a69cd755f38eee7f6dc04abbeeed4c54a528c07580b0e4ebe528faf763" }, { "token_id": "eq-history-resolved-phase-with-adaptive-ruler", "name": "History-Resolved Phase with Adaptive Ruler", "equation_latex": "\\theta_R^{+}=\\theta_R+\\operatorname{clip}\\!\\left(\\operatorname{wrap}\\!\\left(\\theta_{\\mathrm{raw}}-\\theta_R\\right),-\\pi_a,+\\pi_a\\right)", "equation_hash": "0ea8c453b5ff67197651cbfd6ba861cccf98ac56875d1ca973c70d282002018c", "score": 107, "scores": { "tractability": 20, "plausibility": 18, "validation": 20, "artifactCompleteness": 10 }, "novelty": { "score": 30, "date": "2026-03-05" }, "source": "Builds on White Paper 01 (ARP/AIN), White Paper 02 (Adaptive-pi), White Paper 04 (Phase-Lift / PR-Root), and hafc_sim2.py", "date": "2026-03-06", "description": "Contribution. This submission is a lineage-preserving branch-resolved state update for phase-lifted entropy-gated adaptive conductance, not a new number system. It builds on White Paper 01 (ARP/AIN) for the canonical reinforce/decay law dG_ij/dt = alpha_G |I_ij| - mu_G G_ij, White Paper 02 (Adaptive-pi) for d pi_a/dt = alpha_pi S - mu_pi (pi_a - pi_0), White Paper 04 (Phase-Lift / PR-Root) for resolved-phase continuity and winding/parity bookkeeping, the leaderboard's Phase (Adler/RSJ) Dynamics entry for the locked-versus-slip phase backbone, and hafc_sim2.py for the first integrated implementation. The novelty claim is not just smoother unwrapping: theta_R resolves the parity-winding loop under S-gated pi_a, making branch history an operational state variable that is invisible to the principal branch yet still changes the next conductance update through the suppression term. In the matched-present protocol, the principal baseline collapses back to delta theta ~= 0 while the full model retains delta theta_R ~= 2 pi with different winding and parity under the same resumed raw phase. More strongly, the new onset-map benchmark shows a protocol-level regime boundary rather than a one-off trajectory: across pi_0 in {pi/4, pi/3, pi/2, 2 pi/3, 3 pi/4}, the principal baseline stays collapsed for omega_end = 8 to 20, while the full model turns on branch memory sharply at omega_end = 12, jumping from delta theta_R ~= 0 and suppression ~= 0 below threshold to delta theta_R ~= 2 pi with suppression gaps from 3.348226e-03 to 1.092192e-01 at and above threshold. Derivation bridge: data/artifacts/history_resolved_phase_derivation.md now writes the substep-to-full-law chain explicitly, from I_e = G_e(phi_i - phi_j) and theta_raw = arg(I_e), through the clipped resolved update and winding/parity state, into the entropy/ruler closure and finally the full conductance law G_e^+ = G_e + dt [alpha_G(S) |I_e| exp(i theta_R,e) - mu_G(S) G_e - lambda_s G_e sin^2(theta_R,e / (2 pi_a))]. Recovery / limiting cases: real nonnegative conductance with theta_R = 0, lambda_s = 0, and constant alpha_G, mu_G recovers canonical ARP; principal mode sets theta_R = theta_raw directly and therefore removes branch memory by construction; alpha_pi = 0 with pi_a(0) = pi_0 removes adaptive-ruler dynamics; lambda_s = 0 removes suppression. Units: [G] = S, [dG/dt] = S/s, [lambda_s] = 1/s, pi_a is dimensionless, [alpha_pi] = 1/s, and [mu_pi] = 1/s for dimensionless entropy proxy S. Executable replication: tools/benchmark_history_resolved_phase.py runs the local hrphasenet package plus upstream pytest and reproduces every scorer-facing check from Python, not by hand. Benchmarks: the monodromy test tracks one full winding in 100 steps and returns theta_R ~= 2 pi with w = 1 and b = -1; the deformation table over epsilon = 0.00 to 0.20 keeps lifted slip at 0 while the standard branch slips by 1 and improves visibility from 0.7047 to 1.0000; the matched-present history-divergence protocol asserts max |delta G| > 1e-6 after a 30/50/30 warm-up, extra-chirp, and resume sequence; the matched-present state-separation protocol keeps raw phase matched to ~7e-14 while preserving full-model delta theta_R ~= 2 pi and opposite winding/parity; the operational memory-gap protocol keeps current magnitudes matched to ~2e-13 yet yields a full-model suppression gap of about 1.04e-01 while the principal baseline remains at ~0; the chirp-threshold sweep over omega_end = 12, 16, 20 repeats the same outcome across the whole regime, with principal delta theta_R staying near 1e-13 while the full model stays at 2 pi and keeps suppression gaps from 1.034409e-01 to 1.058581e-01; the onset phase diagram over pi_0 and omega_end shows parity-winding closure appearing at the same omega_end = 12 threshold for every tested pi_0. Boundedness tests keep |G| < 1e6 over 200 steps and pi_a in [0.01, pi] over 100 periodic steps. Falsifiers: failure of the monodromy/parity benchmark, failure of matched-present divergence, failure of the matched-present state-separation, operational memory-gap, chirp-threshold sweep, or onset-phase-diagram parity-winding closure benchmarks, failure of the near-zero freeze safeguard, or ablation recovery not returning to the principal-branch or ARP-style limits within numerical tolerance.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif", "image": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_poster.png" }, "tier": "derived", "submitter_hash": "131e2083b68bde4fe879efc38ed9651b1623f8735eeb42157fa3b63ef943fdc6", "version": 1, "metadata_hash": "1003f1e2591d14ae4b6aab343b190f88d919069273ba70aab2d6f45104f1af11" }, { "token_id": "eq-phase-lift-commutator-bound", "name": "Phase-Lift Commutator Bound", "equation_latex": "\\|[\\hat{\\theta}_R, \\hat{\\pi}_a]\\| \\leq \\hbar_{\\mathrm{eff}} = \\pi_a / w", "equation_hash": "7566049c5cabf3f6fe4f479bb116a8501163f306138ca7256c1ca72c678ccf58", "score": 72, "scores": { "tractability": 18, "plausibility": 17, "validation": 11, "artifactCompleteness": 4 }, "novelty": { "score": 22, "date": "2026-03-06" }, "source": "ARP Phase-Lift axioms", "date": "2026-03-06", "description": "Upper bound on the commutator of the lifted-phase operator and the adaptive ruler, analogous to the Heisenberg uncertainty relation. The effective Planck constant is set by the ratio of the adaptive period to winding number, linking quantum-like uncertainty to topological charge.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2d1e830624b2572adc05351a7cbee2d3aa3f6a52b34fa38a260c9c78f96fcd07", "version": 1, "metadata_hash": "769e194d61c9c096d9de9a2a065e5666b44495e3b404b49d16ca27cbb9fe4d5c" }, { "token_id": "eq-adaptive-topological-self-healing-conductance-law", "name": "Adaptive Topological Self-Healing Conductance Law", "equation_latex": "\\frac{d g_e}{dt} = \\alpha_G(S)\\,|J_e|\\,e^{i\\theta_{R,e}} - \\mu_G(S)\\,g_e - \\lambda_s\\,g_e\\,\\sin^2\\!\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)", "equation_hash": "16fb70ed218378939c509dfd32e333a778ad7cd948f2f312ee5cd8e51d5b9e45", "score": 85, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 27, "date": "2026-03-06" }, "source": "pr-root-guide", "date": "2026-03-06", "description": "Adaptive complex edge-conductance law for a damaged topological lattice with history-resolved phase memory. Each edge coupling g_e is reinforced by local bond activity |J_e| in the lifted phase direction e^{i theta_{R,e}}, decays at an entropy-gated rate mu_G(S), and is selectively suppressed when the resolved phase becomes incompatible with the current adaptive ruler pi_a. In a QWZ-style two-band block lattice, the same scalar update multiplies fixed 2x2 bond operators, so the law acts as a local self-healing rule for restoring boundary-dominated transport after bond damage.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "2f183a4e64493af3f377f745eda502363cd3e7ef6e4d266d444758de0a85fcc8", "version": 1, "metadata_hash": "6be69c99ba845a177fec3e13351e95ee888b57f3b0013f5188de09c82aab7657" }, { "token_id": "eq-surprise-augmented-history-resolved-complex-conductance-", "name": "Surprise-Augmented History-Resolved Complex Conductance with Curve-Memory Pruning", "equation_latex": "\\mathbb{R}(\\omega) = R_\\text{s} + \\frac{R_\\text{ct}}{1 + (j\\omega\\tau_\\text{ct})^\\alpha_G} + \\sum_{i=1}^{N} \\frac{R_i}{1 + (j\\omega\\tau_i)^{\\alpha_i}}", "equation_hash": "0e80af9817b1c900870935e973997c69919fb4e7fee45972670a82653bf1da2b", "score": 81, "scores": { "tractability": 19, "plausibility": 19, "validation": 14, "artifactCompleteness": 4 }, "novelty": { "score": 25, "date": "2026-03-07" }, "source": "Original research", "date": "2026-03-07", "description": "Online instantaneous prediction surprise motive using L_t = -log P(y_t|x_t, theta_{1:t-1}). Dense Bayesian deep learning framework p(y|x) = int p(y|x,omega)p(omega|D)d_omega, with curve-memory rules to create a network that achieves continual learning when uncertain, but systemically prunes forgetful edges that chronically fail to lock structurally, enforcing a stable topological backbone without sacrificing robust parity-winding tracking.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "31fd8f32fde9836cb7a612ca4ce3dc15c074ecdfb94876782b30705bf40bc29b", "version": 1, "metadata_hash": "18076dba3f0dc03e7762819d34a4bc67d113e1d841fb70ec2b8e1c3f884ee02f" }, { "token_id": "eq-adaptive-chern-self-healing-conductance-law", "name": "Adaptive Chern Self-Healing Conductance Law", "equation_latex": "dg_e/dt = alpha_G(S)|J_e|exp(i theta_{R,e}) - mu_G(S) g_e - lambda_s g_e sin^2(theta_{R,e}/(2 pi_a)) + chi C_loc(t) g_e", "equation_hash": "877e117de0bb9065a1e1f56da3afd5fbae0100cc257f54f42c259f5f13d76e8e", "score": 81, "scores": { "tractability": 17, "plausibility": 18, "validation": 20, "artifactCompleteness": 7 }, "novelty": { "score": 19, "date": "2026-03-08" }, "source": "chatgpt", "date": "2026-03-08", "description": "An extension of the adaptive phase-lift conductance law that includes a local Chern topological feedback term, enabling self-healing of edge conductance in topological lattices under local perturbations. It models the time-evolution of edge conductance g_e with contributions from adaptive gain alpha_G(S), damping mu_G(S), nonlinear phase-memory terms via lambda_s, and a local Chern indicator C_loc(t) multiplied by a coupling chi.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "data/artifacts/arp_topology_benchmark_v2/arp_topology/outputs/recovery_demo/recovery_traces.png" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "4df404a2ff841ea04474a9788a598641667ed134fd379ed2744a29b4d261089e" }, { "token_id": "eq-adaptive-chern-self-healing-conductance-law-ctance-l", "name": "Adaptive Chern Self-Healing Conductance Law", "equation_latex": "\\frac{d g_e}{dt}=\\alpha_G(S)\\,|J_e|\\,e^{i\\theta_{R,e}}-\\mu_G(S)\\,g_e-\\lambda_s g_e\\sin^2\\!\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)+\\chi\\,C_{\\mathrm{loc}}(t)\\,g_e", "equation_hash": "527491748006ee8f65e5cb6589c4c6a927c95d99c12f99f536a34cb7387d4caf", "score": 100, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "novelty": { "score": 30, "date": "2026-03-08" }, "source": "chatgpt", "date": "2026-03-08", "description": "Adaptive complex edge-conductance law for a damaged topological lattice with history-resolved phase memory and local topological feedback. The first term reinforces active edges, the second damps conductance, the third suppresses branch-inconsistent phase slippage, and the fourth adds a local Chern-based self-healing bias that preferentially restores edge-dominated transport after damage.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "1080p60/TopologicalSelfHealing.mp4", "image": "" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "fb07c62638cef04a7f617f5dc63cef30c70034b475cc47c6465bda9aa97118f2" }, { "token_id": "eq-phase-lifted-thouless-pump-memory-law", "name": "Phase-Lifted Thouless Pump Memory Law", "equation_latex": "Q_{\\mathrm{cycle}}=e\\left(C+\\frac{\\Delta\\theta_R}{2\\pi_a}\\right)", "equation_hash": "fa49adbbc21769f64c0ede86943abca0fb381ae2c5ffc6412b5f610b139fbaf2", "score": 79, "scores": { "tractability": 18, "plausibility": 19, "validation": 15, "artifactCompleteness": 4 }, "novelty": { "score": 23, "date": "2026-03-08" }, "source": "chatgpt", "date": "2026-03-08", "description": "Memory-augmented quantized pump law in which the transported charge per cycle contains the usual topological contribution eC plus a branch-history correction from the lifted phase increment Delta theta_R. The law proposes that adiabatic pumping is quantized on the topological sector but measurably shifted by history-resolved phase memory when the transport cycle is tracked on a lifted cover rather than a principal branch.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "6f39b05de3918f06ed6e761c02fade2aa6443f5acc5e62032c88f553d63aca7c" }, { "token_id": "eq-landauer-phase-lift-conductance-law", "name": "Landauer-Phase-Lift Conductance Law", "equation_latex": "G=\\frac{2e^2}{h}\\sum_n T_n\\cos^2\\!\\left(\\frac{\\theta_{R,n}}{2\\pi_a}\\right)", "equation_hash": "b05a797b1b5261bf4e7bc71975083cec4831fc015a8ed7f98f9ab2b8653edb7e", "score": 81, "scores": { "tractability": 19, "plausibility": 20, "validation": 15, "artifactCompleteness": 4 }, "novelty": { "score": 23, "date": "2026-03-08" }, "source": "chatgpt", "date": "2026-03-08", "description": "Phase-memory extension of Landauer transport in which each transmission channel T_n is modulated by a bounded lifted-phase factor. The law preserves the standard mesoscopic conductance skeleton while adding a branch-consistent memory term that suppresses channels with unresolved or slip-prone phase history.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "314804b722d02720367fb46d183bd1b8e491f9a2ea46470494ffe9075432a942" }, { "token_id": "eq-phase-lifted-rg-memory-flow", "name": "Phase-Lifted RG Memory Flow", "equation_latex": "\\frac{d g}{d\\ln \\mu}=\\beta(g)-\\lambda_M g\\sin^2\\!\\left(\\frac{\\theta_R}{2\\pi_a}\\right)", "equation_hash": "15cc171a5a57f8753485fa7cee5292c677813e7eec2a4ac09e24fd0382cb3c1b", "score": 84, "scores": { "tractability": 18, "plausibility": 20, "validation": 15, "artifactCompleteness": 4 }, "novelty": { "score": 27, "date": "2026-03-08" }, "source": "chatgpt", "date": "2026-03-08", "description": "Renormalization-group flow with a bounded phase-memory correction. The standard beta-function drives scale evolution, while the lifted-phase term penalizes branch-inconsistent history and introduces a memory-sensitive suppression of coupling flow. The proposal is that scale evolution depends not only on the instantaneous coupling g but also on the resolved phase history carried along the flow.", "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "50a4122943273ad2f00ea92bff9c7cb6235e199f745321cb4272aa1857840463", "version": 1, "metadata_hash": "889a9953c957a3fa2ac5da240f51c56685f8fc24130ca9499bac2c5dbb12ab14" }, { "token_id": "eq-directional-strength-proxy-chern-law", "name": "Directional-Strength Proxy Chern Law", "equation_latex": "C_{\\mathrm{proxy}}(t)=\\mathcal{C}_{\\mathrm{QWZ}}\\!\\left(\\bar g_x(t),\\bar g_y(t),m\\right),\\quad \\bar g_d(t)=\\left\\langle |g_e(t)|\\right\\rangle_{e\\parallel d},\\ d\\in\\{x,y\\}", "equation_hash": "c578da110eab99d9f6f25e13f8c9042881020050d62e2aa02b6c335bb391722d", "score": 86, "scores": { "tractability": 19, "plausibility": 17, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 30, "date": "2026-03-09" }, "source": "PR Root Guide", "date": "2026-03-09", "description": "Bulk topological proxy for an adaptive QWZ block lattice. Replace the bare QWZ couplings by the mean adaptive bond strengths on x- and y-directed edges, then evaluate the standard Qi-Wu-Zhang Chern number. In the uniform limit where all x-bonds share one strength and all y-bonds share one strength, the proxy reduces to the ordinary QWZ Chern number. In the recovery protocol it provides a tractable time-dependent topological order parameter that can be compared directly with transfer efficiency, boundary current fraction, top-edge rerouting, and slip density after damage.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "31fd8f32fde9836cb7a612ca4ce3dc15c074ecdfb94876782b30705bf40bc29b", "version": 1, "metadata_hash": "6bf3b5cafe7b02a169920edcdfec2a6e746bbd2dd29043f639d5c86b5f5acd70" }, { "token_id": "eq-slip-suppressed-edge-recovery-law", "name": "Slip-Suppressed Edge Recovery Law", "equation_latex": "\\frac{d\\eta_{\\mathrm{rec}}}{dt}=\\alpha\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)-\\beta\\,\\rho_{\\mathrm{slip}}(t)\\,\\eta_{\\mathrm{rec}}(t)", "equation_hash": "ce86b099f0bf4fedf23ae997741704c9351f73d88163b8f9acb3a50ed3440343", "score": 84, "scores": { "tractability": 18, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 27, "date": "2026-03-09" }, "source": "QWZ Recovery Dashboard", "date": "2026-03-09", "description": "Minimal recovery law for damaged adaptive topological transport. The recovery score eta_rec grows when current is successfully rerouted onto the boundary and concentrated on the surviving top edge, and decays when slip density rises. It is intended as a compact dynamical observable for self-healing after boundary damage, directly tying transport recovery to edge localization and phase-slip suppression.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "31fd8f32fde9836cb7a612ca4ce3dc15c074ecdfb94876782b30705bf40bc29b", "version": 1, "metadata_hash": "a8abb7921e207a40802ad6b43e9ce91a67a285c7d0c3dafc30445b50d9ab8429" }, { "token_id": "eq-newton-s-second-law", "name": "Newton's Second Law", "equation_latex": "F = ma", "equation_hash": "8d87226e092722f6299bfbdbb9afd830dac69f2adca65469ad9d21cd2c367614", "score": 69, "scores": { "tractability": 17, "plausibility": 16, "validation": 12, "artifactCompleteness": 4 }, "novelty": { "score": 20, "date": "2026-03-09" }, "source": "test-run", "date": "2026-03-09", "description": "Fundamental equation of classical mechanics: net force equals mass times acceleration. Cornerstone of Newtonian dynamics.", "units": "OK", "theory": "PASS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "660bc58d2fb728efacf44038f4a72c49c737b77b28de16f3f1a582ea298c62f6", "version": 1, "metadata_hash": "f42ecbeb815bfd00ed48598663f0943326049cf3847d8a536e46bff195a9a7a7" }, { "token_id": "eq-boundary-reroute-recovery-index", "name": "Boundary-Reroute Recovery Index", "equation_latex": "R_{\\mathrm{edge}}(t)=\\frac{\\eta_{\\mathrm{tr}}(t)\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)}{1+\\lambda\\,\\rho_{\\mathrm{slip}}(t)}", "equation_hash": "fdc01084449147d9a4f18fee261e30835e27125a77cba85ab648bf76c8f95556", "score": 84, "scores": { "tractability": 18, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 27, "date": "2026-03-09" }, "source": "QWZ Recovery Dashboard", "date": "2026-03-09", "description": "Compact recovery index for damaged adaptive topological transport. The score increases when transfer efficiency, boundary-current fraction, and top-edge rerouting are all high, and decreases when slip density rises. It is meant as a dashboard-level observable for ranking how well the lattice recovers after damage.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "31fd8f32fde9836cb7a612ca4ce3dc15c074ecdfb94876782b30705bf40bc29b", "version": 1, "metadata_hash": "a42d6fc1f9f19b6bdc29e4bd0cb64b722931ddc7fb462e0e7dd72c5b70f33f41" }, { "token_id": "eq-entropy-gated-edge-recovery-score", "name": "Entropy-Gated Edge Recovery Score", "equation_latex": "E_{\\mathrm{edge}}(t)=\\frac{\\eta_{\\mathrm{tr}}(t)\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)}{1+\\rho_{\\mathrm{slip}}(t)}\\,\\exp\\!\\left[-\\gamma\\,|S(t)-S_{\\mathrm{eq}}|\\right]", "equation_hash": "72f068d0afc4a99a962442519eec1ea0d2dcfee46b2bfe6228927842bfa18fab", "score": 87, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "novelty": { "score": 29, "date": "2026-03-09" }, "source": "QWZ Recovery Dashboard", "date": "2026-03-09", "description": "Entropy-gated recovery score for damaged adaptive topological transport. The score rises when transfer efficiency, boundary localization, and top-edge rerouting are all strong, is suppressed by slip density, and is further reduced when the system entropy deviates from its equilibrium recovery band. It is intended as a compact observable for comparing recovery quality across damage and ablation protocols.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "artifact_refs": { "animation": "", "image": "" }, "tier": "derived", "submitter_hash": "31fd8f32fde9836cb7a612ca4ce3dc15c074ecdfb94876782b30705bf40bc29b", "version": 1, "metadata_hash": "4de074c8ebf82b5efd76b90d4fd40096c0daed0e827068cc7323bd512c6ef793" }, { "token_id": "famous-schrodinger", "name": "Schr\u00f6dinger Equation", "equation_latex": "\\partial_t\\rho+\\nabla\\cdot(\\rho\\,\\mathbf v)=0,\\quad \\hbar\\,\\partial_t\\phi+\\tfrac{m}{2}\\mathbf v^2+V+Q[\\rho]=0", "equation_hash": "d47e8c4cd7500d380a4269873daee15efbc36694b6588182fff6b96fd280d912", "score": 74, "scores": { "tractability": 16, "plausibility": 16, "validation": 15, "artifactCompleteness": 5 }, "novelty": { "score": 26, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Schr\u00f6dinger's equation rewritten via the Madelung decomposition with Phase-Lift: the complex wave function splits into a continuity equation for probability density \u03c1 and a Hamilton\u2013Jacobi equation for lifted phase \u03c6, eliminating the imaginary unit entirely.", "units": "OK", "theory": "PASS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-phase-ambiguity", "core-unwrap-rule" ], "version": 1, "metadata_hash": "5c1d136e374ce2d95b4d07e5240a76992419e7492e52ebe6396e21d8e9a20523" }, { "token_id": "famous-aharonov-bohm", "name": "Aharonov\u2013Bohm Effect", "equation_latex": "\\theta_R[\\gamma]=\\mathrm{unwrap}\\!\\Big(\\frac{q}{\\hbar}\\oint_\\gamma \\mathbf A\\cdot d\\mathbf\\ell;\\;\\theta_{\\rm ref},\\pi_a\\Big)=\\frac{q}{\\hbar}\\int_S \\mathbf B\\cdot d\\mathbf S + 2\\pi_a w", "equation_hash": "305e553474133239b10f097aeeab09e3676be0674e78d89ba3d37458e06ffaba", "score": 74, "scores": { "tractability": 16, "plausibility": 16, "validation": 15, "artifactCompleteness": 5 }, "novelty": { "score": 25, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The Aharonov\u2013Bohm phase expressed through Phase-Lift with explicit winding-number sectors. The unwrap operator removes branch-cut ambiguity and tracks accumulated windings w \u2208 \u2124 around the solenoid.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-unwrap-rule", "core-winding-parity" ], "version": 1, "metadata_hash": "438315df0961068ea72cec1c676370d204ea2369a41bc54c41d4a18a67beb0e6" }, { "token_id": "famous-berry-phase", "name": "Berry Phase", "equation_latex": "\\theta_R[\\gamma]=\\sum_k \\mathrm{unwrap}\\!\\big(\\Arg\\langle\\psi_{k+1}|\\psi_k\\rangle;\\;\\theta_{\\rm ref},\\pi_a\\big)", "equation_hash": "02c6aafa23c27da84f48dee081e4daaa384be9ab83ea713ef07da5370257c76e", "score": 69, "scores": { "tractability": 15, "plausibility": 15, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 24, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Berry's geometric phase computed as a discrete sum of unwrapped overlap arguments along a closed parameter path. Phase-Lift eliminates the multivalued Arg ambiguity at each step.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-unwrap-rule", "core-path-continuity" ], "version": 1, "metadata_hash": "0e9f5509ea45a69fd64883bc26ca00b3fd7bc5b829868ae1b2b06008aebd28fd" }, { "token_id": "famous-josephson", "name": "Josephson Relations", "equation_latex": "I=I_c\\sin\\!\\Big(\\frac{\\pi}{\\pi_a}\\theta_R\\Big),\\quad \\frac{d}{dt}\\Big(\\frac{\\pi}{\\pi_a}\\theta_R\\Big)=\\frac{2e}{\\hbar}V", "equation_hash": "28310b8b01e83e8a41ece13babaeb9e015ba4b7582291de16cce9a7918da5c16", "score": 67, "scores": { "tractability": 15, "plausibility": 14, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 22, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The DC and AC Josephson relations with adaptive-\u03c0 scaling. The standard sin(\u03c6) current\u2013phase relation becomes sin(\u03c0\u03b8_R/\u03c0_a), where \u03c0_a is an adaptive period parameter that recovers the classical limit when \u03c0_a \u2192 \u03c0.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-pi-a", "core-pi-a-dynamics", "core-phase-lift" ], "version": 1, "metadata_hash": "3f6371c87916018aad4d88daa2d3f6465cb01534ab18f7f681af72b839749124" }, { "token_id": "famous-maxwell", "name": "Maxwell's Equations", "equation_latex": "\\mathbf E=-\\nabla V-\\partial_t\\mathbf A,\\quad \\mathbf B=\\nabla\\times\\mathbf A,\\quad \\theta_R=\\mathrm{unwrap}\\!\\Big(\\frac{q}{\\hbar}\\oint\\mathbf A\\cdot d\\mathbf\\ell;\\;\\theta_{\\rm ref},\\pi_a\\Big)", "equation_hash": "23b2bb04f0ad72ae8ff459cb9b30b0a727ff4fc191d529ea29d0cdcb6310025d", "score": 70, "scores": { "tractability": 16, "plausibility": 15, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 24, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Maxwell's field equations expressed through gauge potentials (A, V) with Phase-Lift applied to the electromagnetic phase. The winding structure of A around closed loops is tracked explicitly via the unwrap operator, connecting classical electrodynamics to the quantum phase framework.", "units": "OK", "theory": "PASS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-unwrap-rule", "core-winding-parity", "core-phase-lifted-stokes" ], "version": 1, "metadata_hash": "abad3db3e745960ffbf8350f07b0b15de377f085ba1995418ff1d60d1597a3e5" }, { "token_id": "famous-dirac", "name": "Dirac Equation", "equation_latex": "(i\\gamma^\\mu\\partial_\\mu - m)\\psi=0,\\quad \\psi^{PL}=\\sqrt{\\rho}\\,R\\,e^{i\\phi},\\;\\phi=\\frac{\\pi}{\\pi_a}\\theta_R", "equation_hash": "ec960dcca00afa05b9812ee32487d75281266fa21c27a2db4fba4acb02d02f00", "score": 63, "scores": { "tractability": 14, "plausibility": 14, "validation": 11, "artifactCompleteness": 5 }, "novelty": { "score": 23, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The Dirac equation for a free spin-\u00bd particle with the four-component spinor decomposed into amplitude \u221a\u03c1, an SU(2) rotation R, and a Phase-Lifted scalar phase \u03c6. This separates the spin degree of freedom from the Phase-Lift winding structure.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-phase-ambiguity", "core-pi-a" ], "version": 1, "metadata_hash": "2b3e518deeb329c15504d4a94b7f9c28ee4215849500440a998509d814fa5055" }, { "token_id": "famous-euler-lagrange", "name": "Euler\u2013Lagrange Equation", "equation_latex": "\\frac{\\partial \\mathcal L_{PL}}{\\partial \\theta_R}-\\frac{d}{dt}\\frac{\\partial \\mathcal L_{PL}}{\\partial \\dot\\theta_R}=0,\\quad \\mathcal L_{PL}=\\mathcal L(\\rho,\\theta_R,\\pi_a,\\dot\\rho,\\dot\\theta_R,\\dot\\pi_a)", "equation_hash": "69eb041d36633488eead805fd49b873db54a3da92be2fd977effbf7d035dd802", "score": 70, "scores": { "tractability": 16, "plausibility": 15, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 20, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The Euler\u2013Lagrange equation applied to a Phase-Lift Lagrangian where the generalized coordinates are (\u03c1, \u03b8_R, \u03c0_a). Stationarity of the Phase-Lift action yields the equations of motion for the lifted phase field and the adaptive period.", "units": "OK", "theory": "PASS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-pi-a-dynamics", "core-arp-ode" ], "version": 1, "metadata_hash": "954e89c9842d5a41c6a017a56640a488e1fc782cae3eb25170523ce023718c77" }, { "token_id": "famous-navier-stokes", "name": "Navier\u2013Stokes (Vorticity Form)", "equation_latex": "\\partial_t\\boldsymbol\\omega+(\\mathbf v\\cdot\\nabla)\\boldsymbol\\omega=(\\boldsymbol\\omega\\cdot\\nabla)\\mathbf v+\\nu\\nabla^2\\boldsymbol\\omega,\\quad w=\\frac{1}{2\\pi_a}\\oint\\mathbf v\\cdot d\\mathbf\\ell", "equation_hash": "29e0071634e429d71b4dc160c0b65ddda794aa2716915e04a557293aa4f7ba09", "score": 63, "scores": { "tractability": 14, "plausibility": 14, "validation": 11, "artifactCompleteness": 5 }, "novelty": { "score": 22, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The vorticity transport equation from Navier\u2013Stokes with Phase-Lift applied to circulation integrals. The integer winding number w counts vortex sheet crossings using the adaptive period \u03c0_a, connecting fluid topology to the phase framework.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-winding-parity", "core-phase-lifted-stokes", "core-curvature-salience" ], "version": 1, "metadata_hash": "425010aa4f807547242fde72f807db02b4e5e29ad0c9486050d01520148be1a5" }, { "token_id": "famous-path-integral", "name": "Feynman Path Integral", "equation_latex": "Z=\\sum_{w\\in\\mathbb Z}\\int \\mathcal D\\rho\\,\\mathcal D\\theta_R\\,\\mathcal D\\pi_a\\; e^{\\frac{i}{\\hbar}S_{PL}[\\rho,\\theta_R,\\pi_a]}\\;\\delta\\!\\big(\\theta_R(T)-\\theta_R(0)-2\\pi_a w\\big)", "equation_hash": "085f4007879bae909dde21bedbd260e7442cf77f71692f665ef71bbb77237d40", "score": 63, "scores": { "tractability": 14, "plausibility": 14, "validation": 11, "artifactCompleteness": 5 }, "novelty": { "score": 23, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Feynman's path integral reformulated with Phase-Lift coordinates. The sum over winding sectors w replaces the standard sum over paths, and the delta function enforces quantized phase returns. Each sector contributes independently.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-winding-parity", "core-phase-ambiguity" ], "version": 1, "metadata_hash": "7cae86b24757abacf80c71636ef8964683eef18dec7d22646b90b70a34aad515" }, { "token_id": "famous-holonomy", "name": "U(N) Gauge Holonomy", "equation_latex": "U(\\gamma)=e^{i\\varphi}V,\\; V\\in SU(N),\\quad \\theta_R=\\mathrm{unwrap}(\\varphi;\\theta_{\\rm ref},\\pi_a),\\quad w_{\\det}=\\frac{\\theta_R(T)-\\theta_R(0)}{2\\pi_a}", "equation_hash": "97d0c661b61349202e4e6c5de9307d8918829f5d377464fe99046c92202a0171", "score": 63, "scores": { "tractability": 14, "plausibility": 14, "validation": 11, "artifactCompleteness": 5 }, "novelty": { "score": 21, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "A U(N) holonomy (parallel transport around a loop \u03b3) factored into an SU(N) part V and a U(1) determinant phase. Phase-Lift tracks the determinant phase through unwrapping, yielding an integer winding number w_det that classifies the topological sector.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-unwrap-rule", "core-winding-parity" ], "version": 1, "metadata_hash": "ccbfb0363fe13cd689111e89e58363c84ee40b9591243cf743b2963e7b8c45b2" }, { "token_id": "famous-klein-gordon", "name": "Klein\u2013Gordon Equation", "equation_latex": "\\Big(\\Box+\\frac{m^2c^2}{\\hbar^2}\\Big)\\psi=0,\\quad \\psi^{PL}=\\sqrt{\\rho}\\,e^{i\\phi},\\;\\phi=\\frac{\\pi}{\\pi_a}\\theta_R", "equation_hash": "1582f75bd5522248f2f14bdfbdd84e74980efa6f58ac5585a2419537394a8d72", "score": 63, "scores": { "tractability": 14, "plausibility": 14, "validation": 11, "artifactCompleteness": 5 }, "novelty": { "score": 20, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "The Klein\u2013Gordon equation for a relativistic scalar field, with the complex field \u03c8 decomposed into amplitude \u221a\u03c1 and Phase-Lifted scalar phase \u03c6 = \u03c0\u03b8_R/\u03c0_a. This separates amplitude dynamics from phase winding in a Lorentz-covariant setting.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-pi-a", "core-phase-ambiguity" ], "version": 1, "metadata_hash": "cb44becd1778404dae0c6b1655e856e539502adc0060b2852b06d904503acc36" }, { "token_id": "famous-einstein-field", "name": "Einstein Field Equations", "equation_latex": "G_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}=\\frac{8\\pi G}{c^4}T_{\\mu\\nu},\\quad \\tilde g_{\\mu\\nu}=e^{2\\sigma(\\pi_a)}g_{\\mu\\nu},\\;\\sigma(\\pi_a)=\\ln\\!\\Big(\\frac{\\pi_a}{\\pi}\\Big)", "equation_hash": "91fe07d587409b9b5897e1306481364435e11380a61f73ac365bce39b227ff1c", "score": 57, "scores": { "tractability": 13, "plausibility": 13, "validation": 9, "artifactCompleteness": 5 }, "novelty": { "score": 25, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Einstein's field equations with a Phase-Lift conformal rescaling of the metric. The conformal factor \u03c3(\u03c0_a) is determined by the adaptive period, linking spacetime curvature to the phase framework. When \u03c0_a \u2192 \u03c0, the standard metric is recovered.", "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-conformal-metric", "core-pi-a", "core-adaptive-arc-length" ], "version": 1, "metadata_hash": "891e2a7e7f83dd6bded5ba0a4ad8fb2e0717cf65e0ded82edf928b34b8be946e" }, { "token_id": "famous-fourier-heat", "name": "Fourier Heat Equation", "equation_latex": "\\partial_t u=\\alpha\\nabla^2 u + \\kappa(x,t)\\,u,\\quad \\kappa=\\lambda\\,\\mathcal K[\\theta_R]", "equation_hash": "fed194fa4d3ef2a558548d67f71d764e56dc5e6def943698b3967487a9786624", "score": 70, "scores": { "tractability": 16, "plausibility": 15, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 18, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Fourier's heat equation augmented with a curvature-salience source term. The coefficient \u03ba is derived from the Phase-Lift curvature K[\u03b8_R], making regions of high phase curvature act as local heat sources. This models phase-aware diffusion.", "units": "OK", "theory": "PASS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-curvature-salience", "core-phase-lift", "core-reinforce-decay-memory" ], "version": 1, "metadata_hash": "fcb058443fade5ce6a0e90a3a247cf4b8522a737a407598674b083f5814c3c8c" }, { "token_id": "famous-noether", "name": "Noether's Theorem", "equation_latex": "\\partial_\\mu j^\\mu_{PL}=0,\\quad j^\\mu_{PL}=\\frac{\\partial\\mathcal L_{PL}}{\\partial(\\partial_\\mu\\theta_R)}\\,\\delta\\theta_R+\\frac{\\partial\\mathcal L_{PL}}{\\partial(\\partial_\\mu\\pi_a)}\\,\\delta\\pi_a", "equation_hash": "a0cf0fabf2235ba27c6f93cd229a0fdfbd95edc611eeccc1168245ca21e28cd7", "score": 69, "scores": { "tractability": 15, "plausibility": 15, "validation": 13, "artifactCompleteness": 5 }, "novelty": { "score": 21, "date": "famous" }, "source": "famous-adjusted", "date": "famous", "description": "Noether's theorem applied to the Phase-Lift Lagrangian: continuous symmetries of L_PL yield conserved currents j^\u03bc_PL. The lifted phase \u03b8_R and adaptive period \u03c0_a each contribute to the conserved current when the action is invariant under their transformations.", "units": "OK", "theory": "PASS", "tier": "famous", "submitter_hash": "2cb3598355e82df51631f843f4bf4e351533175bd015abb67f5e374550b9c0c1", "coreRefs": [ "core-phase-lift", "core-pi-a", "core-arp-ode" ], "version": 1, "metadata_hash": "6cfb35ef35adbfc5f95411ae8c58cb6b30db493c00898e58de3da688cdf1dd89" } ] }