# Double-Pendulum Operational Atlas ## A nonlinear benchmark for Operational Information Geometry ## 1. Research question Animations of a large grid of double pendulums often use the grid coordinates to specify two initial angles. Each cell then evolves an independent dynamical system. Bands and islands of similar colour can be visually striking, but the image alone does not say what has become similar, whether that similarity is stable, or whether the cells are physically synchronized. This project turns that picture into a declared experimental object: > Which finite observations distinguish initial conditions, parameters, or > model hypotheses in a double pendulum; which distinctions survive > preparation error, calibration nuisance, numerical uncertainty, and output > noise; and which launches, sensors, and observation times should be selected > under a finite experimental budget? The atlas is intended as a nonlinear benchmark for the proof-producing OIG protocol engine. It is not evidence for a fundamental physical geometry, an infinite-time classification of the double pendulum, or synchronization between uncoupled simulations. --- ## 2. Three claim tiers The work is divided by what its evidence actually proves. 1. **Reproducible numerical atlas.** A declared ODE, finite horizon, grid, integrator, tolerance ladder, observation map, and thresholds produce a finite numerical portrait with energy and refinement diagnostics. 2. **Validated nonlinear enclosure.** Interval or otherwise outward-validated integration encloses trajectories, event times, and variational responses over declared point or cell inputs. 3. **End-to-end exact operational certificate.** Outward response, secant, tangent, and nuisance enclosures are passed to the existing exact OIG engines. Exact rational elimination and \(LDL^T\) tests then certify a finite-library design or robust separation claim for the physical model. An exact algebraic certificate over *assumed* rational response boxes is useful before Tier 2, but remains conditional. It becomes a Tier-3 pendulum result only after a validated transfer proves that the nonlinear model lies in those boxes. That composition is now complete at two sharply bounded targets at \(u=v=0\): the two active scalar parameter protocols without nuisance, and the selected five-output A+B mixture after profiling one shared local clock column. It remains incomplete for preparation error, the other grouped/nuisance models, parameter neighbourhoods, empirical model adequacy, and hardware calibration. A Tier 1 result may discover a candidate for Tier 2 or Tier 3. It is not itself an interval or physical theorem. --- ## 3. Declared mechanical model The first model consists of two point masses on massless rigid rods in a uniform gravitational field. Both angles are absolute and measured from the downward vertical. The state convention is frozen as \[ z=(\theta_1,\theta_2,\omega_1,\omega_2). \tag{3.1} \] For positive masses and lengths, the angular mass matrix is \[ M(\theta)= \begin{pmatrix} (m_1+m_2)\ell_1^2 & m_2\ell_1\ell_2\cos(\theta_1-\theta_2)\\ m_2\ell_1\ell_2\cos(\theta_1-\theta_2) & m_2\ell_2^2 \end{pmatrix}. \tag{3.2} \] Its determinant is \[ \det M=m_2\ell_1^2\ell_2^2 \left(m_1+m_2\sin^2(\theta_1-\theta_2)\right)>0, \tag{3.3} \] so solving the two-by-two mass system is nonsingular for every configuration. The initial numerical benchmark is conservative and unforced. Damping, forcing, coupling between different pendulums, and flexibility are different model families and will never be introduced silently. The initial atlas uses the two-dimensional slice \[ q=(\theta_1(0),\theta_2(0))\in\mathbb T^2, \qquad z_0(q)=(\theta_1(0),\theta_2(0),0,0). \tag{3.4} \] This is a slice of the four-dimensional phase space, not the whole state space. Its cells are independent launches unless a later model explicitly introduces coupling. --- ## 4. Topology and observations The angle domain is a torus. The numerical grid is cell-centred and half-open, so the equivalent endpoints \(-\pi\) and \(\pi\) are not duplicated. Opposite grid edges are adjacent. The default phase observation is seam-free: \[ \chi(z)= \left( \sin\theta_1,\cos\theta_1, \sin\theta_2,\cos\theta_2, \omega_1/s_\omega,\omega_2/s_\omega \right), \tag{4.1} \] where \(s_\omega>0\) is a declared output calibration. For protocol \[ P=(h,\{t_k\}_{k=1}^m,W,c), \tag{4.2} \] the observation map is \[ Y_P(q,p,\nu) =\operatorname{stack}_{k=1}^m h_k\!\left(\varphi_{t_k}^{p}(z_0(q,\nu))\right), \tag{4.3} \] with output-noise precision \(W\succ0\), cost \(c>0\), physical/model parameters \(p\), and declared preparation or calibration nuisance \(\nu\). Cartesian bob positions and velocities are alternative sensor maps. Every atlas must name its map, horizon, cadence, metric, and nuisance semantics. --- ## 5. Four different geometries Several attractive but incorrect identifications are avoided by keeping four notions separate. ### 5.1 Finite-time dynamical regularity Examples include a tangent-flow gain, finite-time Lyapunov exponent, recurrence, or winding statistic. A sampled net excursion of \(2\pi\) is only a descriptive event unless dense-output or interval event tracking proves that no crossing was missed. ### 5.2 Operational coherence A cell \(C\) is operationally coherent at radius \(\eta\) only when a uniform bound proves \[ \operatorname{diam}_{W}Y_P(C)\le\eta. \tag{5.1} \] A grid-edge finite difference is a useful Tier 1 diagnostic, but it is not a bound over the whole cell and not the largest tangent singular value. ### 5.3 Local observability Where differentiable, the pullback information field is \[ G_P(q)=D_qY_P(q)^T W D_qY_P(q). \tag{5.2} \] Relative to a declared source metric \(S\succ0\), \[ \sqrt{\lambda_{\min}(G_P,S)} \quad\text{measures weakest local identifiability,} \qquad \sqrt{\lambda_{\max}(G_P,S)} \quad\text{measures strongest local output amplification.} \tag{5.3} \] Low amplification may indicate regular motion or merely a blind sensor. It is not automatically desirable experimental information. ### 5.4 Synchronization Synchronization is reserved for a declared phase-lock or frequency-lock predicate, or for a model with actual coupling/shared forcing. Similar colours from independent conservative trajectories mean only finite-resolution output coherence under the chosen display. --- ## 6. Fibres, confusability, and islands Exact observational fibres are level sets of \(Y_P\). Let \(\mathcal T_P(C)\) denote the allowed output/nuisance tube *before* measurement noise is added. Robust separation at equal measurement-noise radius \(\varepsilon\) requires \[ \operatorname{dist}_{W} \left(\mathcal T_P(C),\mathcal T_P(C')\right)>2\varepsilon. \tag{6.1} \] Failure to prove (6.1) is **undecided**, not proof of physical confusability. Even actual pairwise confusability need not be transitive: \(A\) can be confusable with \(B\), and \(B\) with \(C\), while \(A\) and \(C\) remain separable. Consequently, connected components of thresholded cells are display regions, not quotient classes or mutual-indistinguishability classes. The current atlas primitives therefore use the deliberately limited phrase **connected low-stretch, no-sampled-full-turn region** for a connected component whose cells have low grid-edge feature stretch and no sampled full-turn excursion. This does not prove libration: a trajectory can rotate less than one turn within the horizon, or cross and return between samples. --- ## 7. Query-directed experimental design An initial condition that the experimenter deliberately selects cannot also be treated as the unknown source in the same design statement. Two OIG modes are therefore separated. ### 7.1 Local state-recovery atlas At a fixed launch, the unknown source is a small perturbation of the initial state. Protocols choose sensors and observation times. The atlas maps local response singular values and robust tangent bounds over the angle grid. ### 7.2 Parameter/model identification The candidate protocol selects a launch, sensor set, and observation schedule. The unknown source is an identifiable dimensionless parameter or finite model hypothesis. A useful first parameter source is \[ u=\left(\delta\log(m_2/m_1),\delta\log(\ell_2/\ell_1)\right). \tag{7.1} \] Common mass scaling is exactly blind in the ideal unforced angle dynamics and is retained as a falsification control. Gravity, length, and clock scaling can also be confounded without external calibration. For a candidate protocol \(i\), the local response and nuisance matrices are \[ H_i=\frac{\partial Y_i}{\partial u}, \qquad B_i=\frac{\partial Y_i}{\partial\nu}. \tag{7.2} \] Validated entrywise enclosures for \(H_i\) can be consumed directly by `EnclosedProtocol` and `design_enclosed_protocols`. The exact OIG engine then chooses a cost-weighted finite mixture and proves a robust information floor for every response inside the declared rational boxes. The outward Picard--Taylor transfer now supplies this premise for the two active responses at the nominal parameter point. Responses for the other five candidates and all parameter-neighbourhood claims remain Tier-1 or conditional. --- ## 8. Nuisance semantics The first design contract distinguishes: - physical parameters or camera calibration shared across every launch; - preparation error shared within one launch but independently re-fitted between launches; - sensor gain or offset shared across all times from that sensor; - temporally correlated measurement noise represented by the full precision matrix rather than counted as independent samples; - numerical/model remainder carried as an outer response or output tube. Correlation is part of the physics. Replacing a common coefficient by independent coordinate boxes can destroy or invent apparent distinguishability. The existing structured-nuisance zonotope interface preserves finite shared coefficients, while the query engine already distinguishes globally shared from independently refitted nuisance. A general group-incidence assembler is a later engine target. The present response-box design certificate and the structured/query nuisance certificates are separate ledgers. They may be composed only after a fixed nuisance projection or a new joint theorem has been validated for the same uncertain response family; running both modules does not by itself certify uncertain \(H\) and grouped nuisance simultaneously. --- ## 9. Numerical acceptance ledger The Tier 1 simulator is accepted only with all declarations serialized and the following controls exercised: 1. the downward equilibrium remains fixed; 2. the two small-angle normal frequencies match the analytic linearization; 3. energy drift is reported and bounded at accepted steps, midpoints, and observation times; 4. a tolerance and maximum-step refinement compares trajectories on the same output grid; 5. angle differences used for diagnostics respect the torus seam; 6. a deliberately loose solver fails the same materially meaningful accuracy gate that the production configuration passes; 7. grid side, half-cell shift, and cadence refinements are compared; 8. horizon is continued as a physical protocol coordinate, not required to preserve a finite-time classification; 9. initial energy is published so zero-velocity energy contours cannot be mistaken for an independent stability mechanism; 10. raw scalar fields and masks, not a colour palette, remain authoritative; 11. points whose convergence or enclosure gate fails are `unresolved`, never assigned zero uncertainty. Energy conservation is necessary but insufficient: chaotic trajectory error can grow while energy remains accurate. Refinement agreement is also empirical, not an outward error proof. --- ## 10. Current executable foundation The implementation now contains: - `double_pendulum_dynamics.py`: declared mechanics, adaptive DOP853 integration, energy audit, and same-grid state refinement; - `oig_double_pendulum_atlas.py`: seam-free trajectory features, sampled turn excursions, grid-edge operational stretch, and toroidal regions; - `oig_double_pendulum_lab.py`: an all-cell Tier-1 runner and strict internal-consistency verifier; - `oig_double_pendulum_render.py`: an SVG rendering derived from the verified raw report; - `double_pendulum_variational.py`: the analytic vector-field Jacobian, joint state/tangent integration, general source injection, and local OIG Grams; - `oig_double_pendulum_variational_atlas.py`: a two-angle, RMS-matched weakest/strongest-gain atlas with coarse/fine tangent, response, Gram, and direct gain gates; - `oig_double_pendulum_persistence.py`: resolution, half-cell-shift, threshold, cadence, horizon, and energy ledgers on a declared common torus comparison grid; - `oig_double_pendulum_parameter_sensitivity.py`: analytic inhomogeneous sensitivities for the two active dimensionless parameter protocols, with multiscale finite-difference and physical-similarity audits; - `oig_double_pendulum_protocol.py`: numerical parameter-response discovery, explicit rational box declaration, and a sealed exact conditional design; - `oig_double_pendulum_validated_transfer.py`: outward Arb Picard--Taylor integration of the twelve-dimensional state/sensitivity system, strict active-box inclusion, and composition with the exact rational OIG child; - `oig_double_pendulum_structured_nuisance.py`: clock, calibration, and per-launch preparation columns plus exact rank/confounding controls; - `oig_double_pendulum_grouped_protocol.py`: launch-batched multi-output protocols, a complete denominator-12 budget-share search, and exact shared-nuisance query children for the selected rational point models; - `DOUBLE_PENDULUM_GROUPED_NUISANCE_TRANSFER_THEOREM.md`: invariant response- level and augmented-Gram transfer theorems for uncertain query and clock derivatives; - `oig_double_pendulum_grouped_validated_transfer.py`: outward validation of all five selected parameter/clock rows and exact rational interval certification of the clock-profiled floor; - `DOUBLE_PENDULUM_GROUPED_VALIDATED_TRANSFER_AUDIT.md`: independent interval, alternate-partition, wrong-weight, nuisance-incidence, and false-floor attacks on the grouped Tier-3 certificate; - focused mechanics, topology, variational, verifier, protocol, renderer, and committed-artifact tests. The committed provisional Tier-1 portrait is `artifacts/double_pendulum_operational_atlas_stage1.json`, with its derived SVG beside it. The exact conditional protocol artifact is `artifacts/double_pendulum_protocol_design_conditional.json`. The matched persistence and variational reports are `artifacts/double_pendulum_persistence_13x13_t4_tier1.json` and `artifacts/double_pendulum_variational_atlas_13x13_t4_tier1.json`. The active analytic response report is `artifacts/double_pendulum_parameter_sensitivity_tier1.json`. The composed validated-model certificate is `artifacts/double_pendulum_physical_positive_floor.json`. The strict nuisance-obstruction report is `artifacts/double_pendulum_structured_physical_nuisance.json`. The constructive grouped report is `artifacts/double_pendulum_grouped_protocol_tier1.json`. Its composed Tier-3 physical certificate is `artifacts/double_pendulum_grouped_physical_clock_floor.json`. The atlas verifier reconstructs the serialized numerical contract, gates, raw summaries, and masks, but deliberately reports that provenance, ODE-to-array linkage, and physical flow enclosure are unverified. The protocol verifier reconstructs the seven-candidate rational child without rerunning floating numerics and deliberately reports that ODE membership for the full library is unproved. Separately, the validated-transfer verifier recomputes the outward ODE proof for the two active rows and the exact fixed-mixture child. The structured-nuisance verifier reconstructs its local-column relations and exact finite-linear children but explicitly does not replay or outwardly validate the nuisance ODE derivatives. The grouped verifier reruns its Tier-1 state/sensitivity/clock calculations, share-grid search, and exact query children, while keeping ODE enclosure and preparation-error flags false. The grouped validated-transfer verifier then reruns all five outward ODE rows, reconstructs the exact interval augmented Gram, and rechecks both rational \(LDL^T\) floor certificates; its preparation and hardware/model flags remain false. The persistence verifier accepts the source numerical fields as Tier-1 input and reconstructs their periodic comparison ledger; the variational verifier reconstructs the serialized refinement, response/Gram, direct-gain, finite-difference spot, unresolved-mask, and summary relations. Neither verifier replays or outwardly encloses the ODE. --- ## 11. Current results For the canonical equal-unit pendulum, the current portrait uses a cell-centred \(13\times13\) angle grid, horizon \(T=4\), 81 observation times, and nine equal-RMS phase samples. All 169 state-refinement comparisons pass. The maximum scaled state discrepancy is \(6.08645\times10^{-8}\) against a \(2\times10^{-7}\) gate, and the maximum sampled scaled energy drift is \(5.99996\times10^{-10}\) against a \(2\times10^{-9}\) gate. At the diagnostic threshold \(\log s\le0.5\), seven cells are marked in five toroidal components of sizes \((2,2,1,1,1)\). The joint-angle reflection audit passes. This is a finite-resolution observation, not yet a spatially persistent island theorem, and state refinement does not validate tangent matrices. The atlas-matched persistence family surrounds this portrait by resolutions \(N=9,13,17\), three half-cell shifts, 5/9/17/41/81 equal-RMS feature schedules, thresholds, five initial-energy strata, and horizons \(T=2,4,6\). The source-grid marked counts at the reference predicate are \(3/7/11\) over the three resolutions, while the shifted \(N=13\) grids contain \(4,4,8\) marked cells. Thus the marked area remains near four percent, but the low-threshold scalar pattern is not yet shift-stable. Against the 81-sample reference, the threshold-\(0.5\) cadence Jaccard values are \(0.714,0.806,0.886,1\) for 5, 9, 17, and 41 samples; this is a declared-grid trend, not a quadrature enclosure. Horizon continuation changes the operational geometry and is reported without a convergence pass/fail flag. Initial energy is a major covariate: baseline log stretch has Pearson correlation \(0.8314\) and Spearman correlation \(0.8948\) with the zero-velocity initial energy. This supports reporting energy stratification alongside the portrait; it does not identify a causal mechanism. The two-angle variational atlas replaces grid-edge differences by the analytic tangent response of exactly the same RMS feature. On the \(13\times13\), \(T=4\) grid, 167 of 169 cells pass the declared state, tangent, weighted response, Gram, weakest-gain, and strongest-gain refinement gates. The two seam-adjacent cells \((0,12)\) and \((12,0)\) are explicitly unresolved and their selected gains are `null`. Only four declared cells receive independent centred finite-difference spot checks; this is not a claim that all 167 resolved cells were finite-difference validated. For the two active design protocols, analytic inhomogeneous parameter sensitivities now replace finite differences in the selected response calculation: \[ \begin{aligned} R_{\mathrm{A},\theta_2,\tau=1} &=(0.0845025509351,-0.454351245129),\\ R_{\mathrm{B},\nu_1,\tau=5/4} &=(0.469054738020,0.00414012202747). \end{aligned} \] The standard/refined analytic discrepancies are below \(2.4\times10^{-15}\), and the smallest-step comparisons to independently integrated centred differences are below \(7.4\times10^{-9}\). Physical representatives related by the declared time and velocity scaling agree to \(1.33\times10^{-15}\). These Tier-1 controls settle the parameter convention and derivative formula; the separate validated-transfer lane supplies the outward membership proof. The first parameter-design library contains seven launch/sensor/time candidates. The nominal-centre E-design assigns nonzero budget to two: 1. launch A \([4/5,-7/20,0,0]\), sensor \(\theta_2\), \(\tau=1\), budget share \(4589/10000\); 2. launch B \([-3/5,9/10,0,0]\), sensor \(\sqrt{\ell_1/g}\,\omega_1\), \(\tau=5/4\), budget share \(5411/10000\). Within the seven declared rational centres, the certified nominal E-optimal efficiency exceeds \(0.9989907\). For the chosen mixture, outward Arb Picard--Taylor enclosures of both active response rows compose with the exact response-box theorem to prove the declared-model floor \[ \frac{195932905640268820789}{2500000000000000000000} \approx 0.0783732. \] The default ODE enclosures have radii between \(3.36\times10^{-13}\) and \(5.45\times10^{-11}\), and every entry retains more than \(0.00100998\) of exact rational inclusion slack. That unused slack is an available budget for a separately proved discrepancy bound; it is not itself proof of model or calibration error. The result does not prove robust optimality over the boxes, physical selection or efficiency for all seven candidates, a parameter- neighbourhood theorem, empirical model adequacy, or hardware calibration. The next structured-nuisance audit gives a sharp negative boundary. With only the two active scalar observations, any nonzero unrestricted shared clock or common-gain coefficient reduces the profiled source rank from two to one. Exactly one nuisance-invariant query survives. A shared clock together with a common gain generically spans both outputs, while independently refitted initial-angle preparation errors or sensor-specific gains already span each scalar output separately; in those cases no nonzero query survives. The exact query and structured-nuisance engines certify these finite rational rank and confounding statements. The nuisance columns themselves remain Tier-1 variational evaluations, not outward ODE enclosures. The obstruction is nevertheless constructive rather than terminal. Batching the original seven observations by exact launch creates three grouped protocols with \(3\), \(2\), and \(2\) scalar outputs. Launch A alone is the minimum \(m=3\) witness: after profiling one shared clock column, the declared rational point model still has effective query rank two and an exact floor lower bound \(9.78912\times10^{-6}\). A complete denominator-12 search over the three group budget shares selects \[ (p_A,p_B,p_C)=\left(\frac23,\frac13,0\right), \] with conservative additive group costs \(7/2,9/4,5/2\). Its five stacked outputs are profiled against one global clock only after assembly, and the exact query child proves \[ \lambda_{\min}^{\mathrm{profiled}}>0.018026929453263358. \] As an optimistic dimensional stress test, one shared clock plus one shared common gain across all channel types retains rank two using all seven outputs, with a conditional point-model floor above \(0.00218401\). The share-grid ordering is deterministic binary64 discovery, not an exact optimality theorem. All three design statements assume exact preparation, and the search ordering remains Tier 1. For the fixed primary A+B shares, however, outward Arb Picard--Taylor integration now encloses every one of the five parameter rows and its shared-clock derivative. The largest parameter-response half-width is \(7.21765\times10^{-11}\), and the largest clock half-width is \(3.1076\times10^{-11}\). Exact rational interval arithmetic gives \[ b^TWb>1.6380507069296194 \] and a three-pivot interval \(LDL^T\) proof certifies uniformly over the five Cartesian row boxes that \[ \lambda_{\min}^{\mathrm{profiled}} >\frac{180269}{10000000}=0.0180269. \] This is a Tier-3 theorem for the fixed local nonlinear model and one shared clock derivative. It does not transfer the search optimality, bound launch preparation error, cover finite clock-dilation amplitudes, or validate the model and noise metric against hardware. --- ## 12. Research hypotheses The central empirical hypothesis is not that chaotic regions are automatically the best experiments. Instead: > Raw response sensitivity may rise near chaotic or interface regions, while > robust query information peaks at an intermediate finite-time stretching > scale because the same dynamics amplifies preparation and model uncertainty. This will be tested by displaying side by side: 1. weakest and strongest local response gains; 2. finite-time dynamical amplification; 3. nuisance-profiled or response-box-robust information floors; 4. the protocols selected by the exact finite-library design engine. A null result is informative: it would demonstrate that visual regularity, dynamical sensitivity, local observability, and robust experiment quality are genuinely different geometries. --- ## 13. Reproduction Install the repository dependencies, then run the focused suite: ```bash python -m pip install -r requirements.txt python -m unittest -v \ test_double_pendulum_dynamics.py \ test_double_pendulum_variational.py \ test_oig_double_pendulum_atlas.py \ test_oig_double_pendulum_lab.py \ test_oig_double_pendulum_protocol.py \ test_oig_double_pendulum_protocol_artifact.py \ test_oig_double_pendulum_persistence.py \ test_oig_double_pendulum_variational_atlas.py \ test_oig_double_pendulum_parameter_sensitivity.py \ test_oig_double_pendulum_validated_transfer.py \ test_oig_double_pendulum_validated_transfer_audit.py \ test_oig_double_pendulum_structured_nuisance.py \ test_oig_double_pendulum_grouped_protocol.py \ test_oig_double_pendulum_grouped_validated_transfer.py \ test_oig_double_pendulum_grouped_validated_transfer_audit.py \ test_oig_double_pendulum_render.py \ test_double_pendulum_artifacts.py ``` Regenerate and verify the Tier-1 portrait: ```bash python oig_double_pendulum_lab.py \ --side 13 --duration 4 --observations 81 --feature-samples 9 \ --log-stretch-threshold 0.5 \ --output artifacts/double_pendulum_operational_atlas_stage1.json python oig_double_pendulum_lab.py \ --verify artifacts/double_pendulum_operational_atlas_stage1.json python oig_double_pendulum_render.py \ artifacts/double_pendulum_operational_atlas_stage1.json \ artifacts/double_pendulum_operational_atlas_stage1.svg ``` Regenerate and verify the conditional protocol certificate: ```bash python oig_double_pendulum_protocol.py \ --output artifacts/double_pendulum_protocol_design_conditional.json python oig_double_pendulum_protocol.py \ --verify artifacts/double_pendulum_protocol_design_conditional.json ``` Regenerate the bounded smoke report and the atlas-matched persistence study: ```bash python oig_double_pendulum_persistence.py \ --output artifacts/double_pendulum_persistence_tier1.json python oig_double_pendulum_persistence.py --research \ --output artifacts/double_pendulum_persistence_13x13_t4_tier1.json python oig_double_pendulum_persistence.py \ --verify artifacts/double_pendulum_persistence_tier1.json python oig_double_pendulum_persistence.py \ --verify artifacts/double_pendulum_persistence_13x13_t4_tier1.json ``` Regenerate and verify the two-angle variational atlases: ```bash python oig_double_pendulum_variational_atlas.py \ --side 7 --duration 2 \ --output artifacts/double_pendulum_variational_atlas_tier1.json python oig_double_pendulum_variational_atlas.py \ --side 13 --duration 4 \ --output artifacts/double_pendulum_variational_atlas_13x13_t4_tier1.json python oig_double_pendulum_variational_atlas.py \ --verify artifacts/double_pendulum_variational_atlas_tier1.json python oig_double_pendulum_variational_atlas.py \ --verify artifacts/double_pendulum_variational_atlas_13x13_t4_tier1.json ``` Regenerate and recompute the active analytic parameter responses: ```bash python oig_double_pendulum_parameter_sensitivity.py \ --output artifacts/double_pendulum_parameter_sensitivity_tier1.json python oig_double_pendulum_parameter_sensitivity.py \ --verify artifacts/double_pendulum_parameter_sensitivity_tier1.json ``` Regenerate and strictly verify the composed validated-model certificate: ```bash python oig_double_pendulum_validated_transfer.py \ --physical-output artifacts/double_pendulum_physical_positive_floor.json python oig_double_pendulum_validated_transfer.py \ --verify-physical artifacts/double_pendulum_physical_positive_floor.json ``` Regenerate and verify the structured-nuisance obstruction report: ```bash python oig_double_pendulum_structured_nuisance.py \ --output artifacts/double_pendulum_structured_physical_nuisance.json python oig_double_pendulum_structured_nuisance.py \ --verify artifacts/double_pendulum_structured_physical_nuisance.json ``` Regenerate and strictly recompute the constructive grouped protocol report: ```bash python oig_double_pendulum_grouped_protocol.py \ --output artifacts/double_pendulum_grouped_protocol_tier1.json python oig_double_pendulum_grouped_protocol.py \ --verify artifacts/double_pendulum_grouped_protocol_tier1.json ``` Regenerate and strictly verify the grouped Tier-3 shared-clock certificate: ```bash python oig_double_pendulum_grouped_validated_transfer.py \ --output artifacts/double_pendulum_grouped_physical_clock_floor.json python oig_double_pendulum_grouped_validated_transfer.py \ --verify artifacts/double_pendulum_grouped_physical_clock_floor.json ``` --- ## 14. Next research order 1. Add a separately bounded preparation layer or measured preparation channel. Unrestricted independent preparation refits erase scalar-output information, so this requires additional outputs or an external bound. 2. Extend outward validation to all seven scalar candidates and the remaining grouped rows before claiming physical finite-library selection or nominal- efficiency transfer. 3. Extend the persistence study from a thresholded scalar field to uncertainty- aware variational cells, while retaining horizon as a protocol coordinate. 4. Certify cell coherence and pairwise separation with the three honest outcomes `distinct`, `confusable by witness`, and `undecided`. 5. Build a sealed bundle whose verifier reconstructs contracts, metrics, raw hashes, nuisance incidence, response boxes, and exact OIG children. The first end-to-end validated OIG-selected nonlinear model experiment is now closed for the two active nominal responses. A constructive multi-output design now preserves a positive floor after one shared clock nuisance, and its fixed A+B mixture has an outward nonlinear Tier-3 transfer. The immediate proof target is a scientifically declared preparation bound, followed by the larger candidate-library transfer.