Can a vector edit meet its target, preserve what matters, and carry evidence that another checker can recompute?
We began with an exact area edit, tested asymmetric shapes, then extended the model to connected Bézier segments. A fresh-input comparison now tests the value of verification feedback, cross-checked by a second exact implementation.
This labbook connects the questions, mathematics, results and limitations to the underlying artifacts. The data and figures are embedded; source-file links work from the original lab folder.
Read the findings ↘
Study 04: 80 witnessed-feasible requests, 20 unknown, 10 impossible and 10 unsupported. Refinement wins on 18 feasible cases against nested sampling, loses on zero, and both fail on 30. Methods share initial candidates; records are not independent extra trials.
Increase area by 50%, from 30,000 to 45,000 coordinate units², while retaining four endpoints and the centroid.
A symmetric pair of cubic curves gives an algebraic edit. With endpoint half-height a = 50 and interior half-height h, the area is A = 300(a + h). Choosing h = 100 reaches the target.
The comparison scales the entire shape vertically. Both edits reach the area; only the interior-control edit preserves the four anchors. The checker parses the actual SVG bytes and recomputes area, centroid and separation.
The area-only edit moves each endpoint vertically by 25 units. The constrained edit has zero area error and preserves the centroid exactly in this symmetric family.
Evidence: constrained export →| Artifact | Area | Anchor error | Area target | Full request |
|---|
The original input is checked against the requested edited area, so its target check correctly fails. This is one demonstration, not an estimated success rate. Area tolerance: 10⁻⁷; centroid and anchors: exact equality.
Ten requests introduce unequal boundary controls, translated shapes, narrow sections and strong contractions. Three numerical configurations face the same declared preservation requirements.
| Configuration | What drives the edit | What remains independently checked |
|---|---|---|
| Target-only | Area error plus a small movement regularizer. | Area, centroid, endpoint anchors, vertical gap, simple closed geometry and control bounds. All checks are applied to serialized SVG coordinates. |
| Penalty | Adds finite penalties for preservation and sampled gap shortfalls. | |
| Constrained | Minimizes movement with explicit equalities and sampled gap constraints. |
At area tolerance 0.001 units², the constrained configuration passes all seven requests for which a feasible result is established. The penalty configuration comes close but misses the strict area target; it passes all seven when only the allowed area error changes to 0.1.
Centroid and anchor tolerances stay at 0.001 coordinate units, and the required gap stays at 10. The larger tolerance rechecks the same bytes; it does not rerun or retune the optimizer.
| Area tolerance | Target-only | Penalty | Constrained |
|---|
Endpoint anchors and the minimum gap impose a necessary area floor. A solver cannot recover a request below that floor without changing the shape family or requirements.
Read the lower-bound derivation →One penalty weight and one tolerance cannot establish superiority of an entire method family. These are deterministic synthetic fixtures, with three local starts and fixed settings. There is no held-out dataset, population success estimate or state-of-the-art comparison.
Full protocol, scales and selection rules →Three, four or six cubic segments per boundary add freedom. Smooth interior joins and optional protected landmarks make preservation more demanding.
The sampled smooth controller passes six of eight feasible requests. In the other two it meets the area, centroid, anchor and join constraints, yet dips below the minimum gap between samples.
The checker returns exact rational violation locations. Adding those points to the numerical constraints fixes each edit in one repair round. Every rejected candidate and the accepted repair remain in the record.
Two other requests have independent impossibility proofs. Refinement stops on those bounds, so no invalid edit is promoted to a certificate.
Eight feasible requests. Position-only deliberately omits the slope constraint. Refinement includes the initial smooth solve and additional work; this is not an equal-budget optimizer comparison.
Two requests excluded by the single-cubic model become achievable with four segments. Subdivision initially preserves the source curve; subsequent editing uses the added freedom. The input equivalence bound is recorded separately from the edit.
| Case | Target | Single cubic floor | Four-segment floor |
|---|
Bounds are necessary, not sufficient. Certified final exports provide the positive feasibility evidence.
120 new input files, a second exact implementation, and three follow-up policies from the same initial candidate. The new evaluation tests whether verification feedback adds practical value.
Eighty requests have saved, independently checked feasible witnesses. Twenty have unknown feasibility. Ten impossible targets and ten deliberately unsupported SVGs stop at preflight.
Every policy shares the initial sampled solve and receives at most two further solves, each capped at 150 iterations. The oracle and feasible witness shapes do not supply information to the numerical controller. Settings, fixtures and computation sources are bound in a local freeze manifest.
Witness refinement certifies 50/80 feasible requests; nested sampling certifies 32/80, and restart 21/80. Refinement succeeds on 18 cases where nested sampling fails. The reverse occurs on zero cases.
The initial solve passes 21/80. Refinement recovers 29 of the remaining 59 within two extra solves.
| Policy | Certified / feasible | Recovered / initial failures | Certified / unknown |
|---|---|---|---|
| Restart | 21 / 80 | 0 / 59 | 1 / 20 |
| Nested sampling | 32 / 80 | 11 / 59 | 3 / 20 |
| Witness refinement | 50 / 80 | 29 / 59 | 9 / 20 |
The unit is an input/request pair. Common starts, method records and repair histories are paired observations. Ten unsupported and ten proven-impossible controls are excluded from optimizer success denominators. No population confidence intervals are inferred from these synthetic cases.
| Stratum | Initial / 20 | Restart | Nested | Witness |
|---|---|---|---|---|
| expand | 20 | 20 | 20 | 20 |
| contract | 1 | 1 | 5 | 14 |
| near-margin | 0 | 0 | 4 | 8 |
| landmarks | 0 | 0 | 3 | 8 |
| unknown | 1 | 1 | 3 | 9 |
Twenty cases per optimized stratum. The first four strata are witnessed feasible; the unknown group is kept separate. Inputs vary asymmetric gaps, segment widths, translations, slopes and landmarks, but retain an affine common centerline.
The new oracle evaluates every endpoint and stationary point using exact quadratic-radical signs. Direct Bernstein strip integrals cross-check the earlier Green-integral moments. It imports none of the old parser, subdivision or moment functions.
In development, 2,022 polynomial decisions and 42 earlier export entries revealed no contradictory decisive result. Seven bounded-subdivision decisions remained unresolved where the oracle was decisive.
Derivations and implementation boundaries →Both implementations replayed 564 candidate paths, 120 input paths and 80 feasible witness paths. All candidate areas and centroids matched exactly; no contradictory decisive result appeared.
This is separate implementation checking by the same author, with shared Python/XML dependencies and model assumptions. It is not formal verification or external review.
Complete replay record →| Policy | Median seconds | P90 seconds | Maximum seconds |
|---|---|---|---|
| Restart | 0.04305 | 0.08997 | 0.10336 |
| Nested sampling | 0.03800 | 0.07553 | 0.08920 |
| Witness refinement | 0.03080 | 0.05440 | 0.06419 |
Equal solve ceilings do not mean identical CPU work. Decision-path times include optimization, export and both checkers; some setup/bookkeeping outside the timed boundaries is excluded. The report records this post-run scope clarification without changing the frozen protocol. Subsecond measurements and retrospective deadline truncations are descriptive.
These are established geometric and numerical tools. The derivations below explain the local mechanism; the linked dossiers connect them to the research library.
For upper and lower graphs u(x), l(x), let g(x) = l(x) − u(x).
The producer uses Gaussian quadrature. The original checker independently integrates the closed polynomial boundary using Green’s identities and rational arithmetic. Study 04 cross-checks it with exact Bernstein strip integrals.
Derivation and measurement scope →With shared linear x, a segment’s gap is a cubic Bézier polynomial whose coefficients are lower minus upper y controls. Its convex hull bounds the whole interval.
If this sufficient bound fails, exact subdivision tightens it. A low control coefficient is not a violating curve point. Only an evaluated witness establishes a pointwise violation.
Certificate outcomes and challenge →For adjacent cubic intervals of widths h₀, h₁, position continuity plus equal global derivatives gives C¹.
Equal handle lengths do not imply equal derivatives when interval widths differ. The global parameter in this prototype is x.
Farin & Hansford, chapter 9 →The cubic midpoint identity is
For a fixed partition and gap ≥ m, summing segment bounds gives
Conservative endpoint gaps include anchor tolerance. These bounds do not prove feasibility when satisfied.
Assumptions and derivation →Every boundary is a continuous graph over increasing x. Segment interiors occupy disjoint x intervals, positive gaps separate upper and lower curves even at the joins, and caps lie only at the extreme x positions. Together these properties establish a simple closed outline. This is a sufficient argument for this representation, not a general curve-intersection algorithm.
| Finding | Evidence | Practical implication |
|---|---|---|
| F01 · Hitting the target can violate preservation.Observed | Study 01’s area-only edit moves the anchors by 25 units. Inspect the counterexample | A steering request must state both what changes and what must remain invariant. |
| F02 · Method rankings depend on tolerances.Observed, conditional | The same seven penalty outputs change from 0/7 to 7/7 accepted as area tolerance changes from 0.001 to 0.1. Sensitivity table | Choose precision according to the intended measurement. One strict result does not establish a universal method ranking. |
| F03 · A failed edit is not an impossibility proof.Derived + observed | Independent area bounds explain particular failed requests. Other failures need diagnosis. Necessary bounds | Report violated candidates, unresolved certificates, solver failures and impossible requests separately. |
| F04 · Verification can supply corrective information.Observed | Two sampled smooth edits fail the gap check; each passes after one witness-guided repair. Inspect the full repair history | A geometric receipt can drive a bounded edit–check–repair workflow, with no acceptance until the final export passes. |
| F05 · Representation is a capability choice.Derived + observed | Four-segment edits satisfy two requests excluded by one cubic per boundary. Recovered cases | Separate limits imposed by geometry representation from limits of the optimizer. |
| F07 · Feedback improves this fixed comparison.Observed on fresh inputs | Refinement certifies 50/80 feasible requests versus 32/80 for nested sampling. Paired wins/losses: 18/0. Frozen comparison | Further develop witness feedback in the supported model while retaining the exact acceptance gate. |
| F08 · Feasible failures remain after repair.Observed with constructive witnesses | Thirty requests have valid witness shapes but still fail the gap check after two repair rounds. Full case ledger | Diagnose repair depth and clearance behavior before broadening the geometry family. |
| F06 · Geometry receipts are not attribution evidence.Scope conclusion | No training data, causal influence, signatures, identity or rights assertions were tested. Recorded scope | The current mechanism can support geometric conformance checks and local operation records. Attribution needs a separately defined evidence model and experiments. |
Define a geometric request, generate a supported vector edit, inspect exact checks, retain rejected outputs, repair supported gap violations, and reproduce the result from the saved artifacts.
General outlines need a broader intersection strategy. Perceptual or generative steering needs new measurands and model experiments. Authenticated provenance needs a trust mechanism. Training-data attribution needs causal definitions, access assumptions and validation.
We have not established novelty, broad comparative performance or applicability to arbitrary SVG artwork. These studies are a local engineering result built on established mathematics.
The proposed validation round is complete. It supports further work on feedback, and identifies 30 witnessed-feasible failures that should be understood before generalizing.
A separate exact implementation, a locally frozen 120-input set, matched solve ceilings, paired comparisons, full candidate retention and a two-checker replay audit. All 41 behavioral tests pass.
Use the remaining feasible failures as labeled development cases. Trace changing minimum-gap and witness locations, then freeze a repair-depth/clearance comparison on another untouched set. This follow-up has not been run.
The generator retains an affine common centerline. More general outlines, perception and training-data attribution remain separate studies.
| Study | Cases | Distinct input hashes | Entries | Distinct recorded SVG hashes |
|---|
Studies 01–03 contain 63 entries referring to 55 distinct SVG byte hashes. Distinct bytes do not necessarily imply different curves; exact subdivision can preserve geometry.
Study 04 adds 360 workflow rows, including 60 preflight stops. Its 300 eligible workflow records share 100 initial candidates. The 564 candidate paths, 120 inputs and 80 witnesses are counted separately in its audit.
Study 04 has its own complete 120-case ledger →
Each row opens a traceable record with the original and edited geometry, request, checks, solver attempts, exact witnesses and file links. The 63 rows include the original demonstration’s input; they are not 63 independent trials.
| Study / case | Method | Outcome | Area error | Centroid error | Slope jump | Solver | Evidence |
|---|
Area errors are absolute coordinate units²; centroid errors are maximum-coordinate distances; slope jumps are dy/dx. “—” means not measured or not applicable. A successful solver can still produce a violated request. Impossible flags come from mathematical bounds, not solver messages.
This snapshot reflects saved artifacts, not a complete transcript of every exploratory action. Unsaved earlier runs cannot be reconstructed. The original receipt has no timestamp; it is presented first by documented study sequence. A hash binds bytes to a record but does not authenticate who made it or when.
cd vector-steering-lab
python3 build_labbook.pyThe builder replays the geometry checks, verifies code and file hashes, and regenerates this HTML plus data, CSV and manifest snapshots. It stops on inconsistent study inputs and writes current structural validation. Earlier browser QA remains separately dated and bound to the HTML version it inspected.
Optimization runs replace named experiment outputs. Use the saved protocols to recreate each study with its recorded settings, then rebuild the labbook.
| Source file | Bytes | SHA-256 |
|---|
These are adversarial and proof-budget tests, separate from the optimization denominators. The figures above show actual solver failures; the challenge files deliberately construct further edge cases.
Study 04: exact extrema, Bernstein moments and feasible-witness derivations →
The HTML embeds its data and images and uses no remote scripts, fonts or chart services. Original source links require the accompanying folder; external and library links require their original locations. Portable reading is distinct from carrying the complete executable source archive.