T-S01 exact analytic family

Versioned result record syn07-ts01-20261014-e22d9c2aeb05, sourced from commit e22d9c2aeb05dc7567038801e6a21b8780184c9e. Strategy csl.synthesis.ts01-analytic version 1; profile ts01_exact_integrator.

Strict admitted profile

Discrete-time, two-step scalar system with times (0,1,2).
Known initial state x0=0; unrestricted one-dimensional state at every stage.
Both transitions are exactly x[t+1]=x[t]+u[t], with zero offsets and no external channels.
Independent closed control intervals u0,u1 in [0,1].
Nominal open-loop information; no observations, temporal constraints, or other constraints.
Terminal quadratic objective has Q=2, linear coefficient -2, and offset 1: J(x2)=x2^2-2*x2+1.
Requested target is exact_optimal_relation, representation csl.polyhedron.h, numerical assurance, no materialization limit, and empty transformation lineage.

Complete canonical relation for this profile

For x₀=0, xₜ₊₁=xₜ+uₜ, and u₀,u₁∈[0,1], the terminal objective is J(x₂)=x₂²−2x₂+1=(x₂−1)². Since x₂=u₀+u₁, its minimum is zero and is attained exactly when u₀+u₁=1. Thus every optimal joint history, and only such a history, is

(x₀,x₁,x₂,u₀,u₁) = (0,s,1,s,1−s),  0 ≤ s ≤ 1.

Analytic expected minimum: 0. Scenario SHA-256: 1137cd7638206310d13f0bf635c06450d0036d83f2dc3550368f256f3f5fc146; input SHA-256: 816162785675a960051adeef7027c85489cc9adee6267c140227e942b99a0d30.

The canonical coordinates are (x0,x1,x2,u0,u1). The exact H-representation is saved in records.json.

The entire optimal control relation is the closed line segment; the canonical states are x0=0, x1=u0, x2=1.

Exact binary64 membership examples

The public relation query uses exact dyadic arithmetic for finite binary64 coordinates.

Example	Coordinates	Result	Why
s=0	`[0,0,1,0,1]`	member	Closed-segment endpoint; exact equations and bounds hold.
s=1/4	`[0,0.25,1,0.25,0.75]`	member	Interior dyadic parameter; exact equations and bounds hold.
s=1	`[0,1,1,1,0]`	member	Closed-segment endpoint; exact equations and bounds hold.
s=1-2^-53	`[0,0.9999999999999999,1,0.9999999999999999,1.1102230246251565e-16]`	member	Adjacent representable dyadic coordinates sum to exactly one.
feasible_nonoptimal_zero	`[0,0,0,0,0]`	not a member	Feasible controls, but x2 is not one and objective is one.
feasible_nonoptimal_endpoint	`[0,1,2,1,1]`	not a member	Feasible controls, but x2 is two and objective is one.
off_relation_dyadic_perturbation	`[0,1,1,1,5e-324]`	not a member	Exact binary64 sum is greater than one; no tolerance is applied.

Assurance and completeness boundary

The relation is complete for the narrowly admitted analytic T-S01 profile. The outer API still reports numerical assurance; validated assurance is unavailable. This is not a general strategy or a proof-replayable serialized capability.

The complete relation is established analytically only for the strict profile listed above; admission rejects mutations outside that profile.
The outer API labels assurance numerical. Validated assurance is unavailable in the shared registry contract.
The proof capability is internally issued and non-forgeable by clients; serialized decoding creates an untrusted capability which publication rejects until a reviewed proof replay/exporter exists.
This does not establish completeness for any generic synthesis strategy, another profile, or a decoded artifact.
This report records the profile theorem and exact membership examples; it is not an independent mathematical review of the source implementation.

Observed StrategyPort result

Termination completed; mathematical outcome optimal; objective bounds observed from the registered StrategyPort: lower 0, upper 0. Residuals were unavailable because the StrategyPort result exposes no residual fields. These observations are distinct from the analytic expected minimum above.

Performance

Exploratory single-strategy observations. One excluded warmup preceded seven measured repetitions; all seven returned the expected complete H-relation and passed exact-membership checks. Timing scope: Fresh problem construction and resolution, strategy registration/selection/preparation, exclusive workspace creation, public bound-workspace run, and independent complete-H-relation plus exact-membership validation. Excludes process startup, JSON serialization, build, collection, and rendering.

Median 29000 ns; range 28600–40400 ns (width 11800 ns). No comparison or speedup is claimed. Performance conclusion: inconclusive. Limiter profile: No usable process/call-stack profile was collected; the limiting function is unknown. A separate Windows Performance Recorder CPU-stack capture was attempted after the timed run but failed before recording: wpr.exe -start CPU -filemode returned HRESULT 0xc5585011 (Failed to enable the policy to profile system performance). No profile-mode solver run was made.

Raw data: measurement JSONL · timing CSV · measurement summary.

Reproduction

git checkout e22d9c2aeb05dc7567038801e6a21b8780184c9e
cmake -S . -B build/ts01-report -G 'Visual Studio 17 2022' -A x64 -DBUILD_TESTING=ON -DCSL_BUILD_SYN07_REPORT=ON
cmake --build build/ts01-report --config Release --target csl_syn07_ts01_report --parallel 4
build/ts01-report/Release/csl_syn07_ts01_report.exe  # one warmup plus seven measured repetitions; raw JSONL
ctest --test-dir build/ts01-report -C Release --output-on-failure -R strategy_ts01_analytic_profile_tests
python reports/algorithm-results/tooling/ts01/report.py --check reports/algorithm-results/syn07-ts01-20261014-e22d9c2aeb05/records.json

Membership rows and their raw JSONL/CSV projections are version-controlled with this standalone page. The HTML embeds all styling and figures and needs no network access.

Publication syn07-ts01-5f46eaee678c04b251c4bd9a78dcfb5ea45ce4f417c6bff7b8ba6225e04a6d66; native source syn07-ts01-20261014-e22d9c2aeb05. Versioned envelope · Publication manifest