# Physical Validation Cases
## Purpose
This file stores exact evidence for material physical-model findings.
The investigation log records the requirement and decision. This file records the reproducible technical case.
Do not remove a failed case after a correction. Change its status and retain the original evidence.
## Required Case Data
Each case must contain these items:
- A stable case identifier.
- The requirement identifiers that the case tests.
- The exact input values and physical units.
- The input-admission result from the production engine.
- The reference method and its error bound.
- The production result and its error values.
- The physical effect of the error.
- The reproduction command.
- The artifact location and SHA-256 digest.
- The correction status and permanent test name.
- Contrary evidence and unresolved uncertainty.
## Case Index
| `PVC-001` | `RP-013`, `RP-023`, `RP-027`, `RP-033` | Spherical envelope selects the wrong local maximum | Certified replacement and bounded asset admission pass |
| `PVC-002` | `RP-012`, `RP-016`, `RP-028`, `RP-030` | Finalized-page publication exceeds the fixed cache | Confirmed; worker correction implemented |
| `PVC-003` | `RP-008`, `RP-013`, `RP-028` | Midpoint active-mode solve misses a rapid-reversal callback deadline | Confirmed; core tail improved, complete path still fails |
| `PVC-004` | `RP-009`, `RP-013`, `RP-023`, `RP-033` | One wall has two separated near-equal envelope maxima | Confirmed; tracer rejects unresolved height order |
| `PVC-005` | `RP-003`, `RP-005`, `RP-034` | Named scratch presets need canonical timing and production integration | Stab and Chirp topology passes; calibration remains open |
| `PVC-006` | `RP-008`, `RP-020`, `RP-024` | Passive cartridge magnetic-loss invariants | Implemented and green; cartridge measurement remains open |
| `PVC-007` | `RP-008`, `RP-013`, `RP-026`, `RP-033`, `RP-035`, `RP-036`, `RP-038`, `RP-039` | Passive groove compliance, coupled patches, and vector friction | Zero-speed regularization passes; identified tangential state and compliance remain open |
| `PVC-008` | `RP-008`, `RP-013`, `RP-028`, `RP-033` | One contact solve skips rapid groove events | Bounded sweep improves the reduced phono reference; callback deadline fails |
| `PVC-009` | `RP-003`, `RP-013`, `RP-040` | Real picture record for complete player validation | Structural decode passes; PCM and physical-player tests remain open |
## PVC-001: Spherical Envelope Global-Maximum Failure
### Claim Under Test
The integer scan and one golden-section refinement find the global spherical stylus envelope.
### Result
- **Result**: A production wall-velocity source disproves the claim.
- **Observation date**: 2026-08-01.
- **Mechanism**: Competing local maxima occur inside the spherical tracing radius.
- **Former behavior**: The local scan refined only the interval with the best sampled value.
### Primary Exact Fixture
- **Source API**: `GrooveAsset::from_stereo_wall_velocity_m_s`.
- **Source kind**: `StereoWallVelocity`.
- **Source frames**: 96.
- **Left wall**: The artifact stores every `f32` bit pattern.
- **Right wall**: Each value is the exact negation of the left value.
- **Scalar wall**: The vertical channel produces the stored scalar wall bit patterns.
- **Center**: 40.37 source frames.
- **Spatial step**: 0.0000015 meters per source frame.
- **Tracing radius**: 0.000018 meters.
- **Nominal groove radius**: 0.08250592249883855 meters.
- **Nominal speed**: 33.333333333333336 revolutions per minute.
- **Internal sample rate**: 192000 hertz.
The artifact stores exact `f64` bit patterns for the center, spatial step, and tracing radius.
### Cutter Admission
- **Peak wall velocity**: 0.05951165407896042 meters per second.
- **Wall velocity RMS**: 0.03214256359840551 meters per second.
- **Declared sine peak**: 0.07071067811865477 meters per second.
- **Maximum construction frequency**: 44001.59700186703 hertz.
- **Seed cutter bandwidth**: 50000 hertz.
- **Peak vertical displacement**: 0.0000004604399854912308 meters.
- **Generated wall peak displacement**: 0.0000003348184079767935 meters.
- **Generated wall peak slope**: 0.20000236690167938.
- **Generated wall peak velocity**: 0.05760068166768366 meters per second.
- **Generated wall peak acceleration**: 24051.018030149862 meters per second squared.
- **Final vertical drift**: 0.0000004151899796497212 meters.
- **Final program radius**: 0.08250588813830614 meters.
- **Admission result**: The production constructor accepts the source.
The source peak is below the declared sine peak.
The constructed frequencies are below the seed cutter bandwidth.
The source contains less than one revolution.
Therefore, the cutter does not test adjacent-turn clearance.
The seed API does not enforce acceleration or material limits.
Therefore, this case does not prove that a physical lathe can cut the source.
### Former Local-Scan Result
- **Center displacement**: 0.000000023518275394551524 meters.
- **Contact offset**: -0.0000018168345260679822 meters.
- **Groove displacement**: 0.00000011544433359703447 meters.
- **Groove slope**: -0.1014059134888234.
- **Tangent residual**: 0.00004746066926979153.
The executable tests do not call the former local scan.
### Replacement Checkpoint Result
- **Center displacement**: 0.00000004480387360042518 meters.
- **Center bits**: `0x3e680dcc28e73c00`.
- **Contact offset**: 0.00000227562708721225 meters.
- **Contact bits**: `0x3ec316df38b006c0`.
- **Groove displacement**: 0.00000018922991587098192 meters.
- **Groove bits**: `0x3e8965e3f14d4cbe`.
- **Groove slope**: 0.12744631410894303.
- **Slope bits**: `0x3fc050292b8bfcd8`.
- **Tangent residual**: Positive zero.
- **Residual bits**: `0x0000000000000000`.
The replacement height and position are inside the outward reference bounds.
The secondary spatial fixture also passes its outward bounds.
- **Spatial center displacement**: 0.00000023703855781733048 meters.
- **Spatial center bits**: `0x3e8fd09534544c00`.
- **Spatial contact offset**: 0.0000008345522958362521 meters.
- **Spatial contact bits**: `0x3eac00bfe811942b`.
- **Spatial groove displacement**: 0.0000002563955641556607 meters.
- **Spatial groove bits**: `0x3e9134d79dcc3768`.
- **Spatial groove slope**: 0.046413929475732675.
- **Spatial slope bits**: `0x3fa7c3910a5aafc5`.
- **Spatial tangent residual**: -0.00000000000000625888230132432.
- **Spatial residual bits**: `0xbcfc300000000000`.
### Earlier High-Rate Audit Observation
- **Status**: Retained observation without a recovered exact artifact.
- **Reported mechanism**: One realistic Catmull-Rom groove produced two envelope maxima.
- **Reported former result**: The scan-plus-golden tracer selected the lower maximum.
- **Reported height error**: Approximately 7.07 micrometers.
- **Reported contact-position error**: Approximately 3.74 micrometers.
- **Interpretation**: Both reported errors are physically material if the values and units are correct.
- **Uncertainty**: The current fixtures do not reproduce the reported 7.07-micrometer height error.
- **Possible explanation**: The observation can refer to a different fixture or a different height convention.
- **Possible explanation**: The reported height unit can be wrong.
- **Requirement**: Keep this observation until an exact fixture confirms or disproves it.
- **Requirement**: Do not use these approximate values as acceptance bounds.
- **Decision**: The independently reproduced fixtures already require a certified global tracer.
### Constant-Segment Catmull-Rom Micro-Kink
- **Status**: Confirmed numerical defect and corrected production arithmetic.
- **Observation date**: 2026-08-02.
- **Input point value**: `2.0205539476957236e-10` meters.
- **Input point bits**: `0x3debc5313380da57`.
- **Input topology**: All four Catmull-Rom points have that exact value.
- **Required result**: The cubic is constant, and all three derivative coefficients are positive zero.
- **Former cubic coefficient `a`**: `-1.2924697071141057e-26` meters.
- **Former coefficient `a` bits**: `0xba90000000000000`.
- **Former cubic coefficient `b`**: `6.462348535570529e-26` meters.
- **Former coefficient `b` bits**: `0x3ab4000000000000`.
- **Former cubic coefficient `c`**: Positive zero.
- **Mechanism**: The former absolute-value formula canceled repeated nonzero terms after separate multiplication.
- **Effect**: The false derivative prevented a bit-exact constant-edge C1 certificate.
- **Physical significance**: The coefficient size is not a material groove displacement.
- **Engineering significance**: A false kink can reject valid data or hide a real fixture discontinuity.
- **Correction**: Form the cubic from `y0-y1`, `y2-y1`, and `y3-y1` differences.
- **Correction result**: Coefficients `a`, `b`, and `c` are bit-exact positive zero.
- **Consistency requirement**: The production tracer and admission code must use identical arithmetic.
- **Permanent test**: `constant_catmull_points_produce_bit_exact_zero_derivatives`.
### Certified Class-A Checkpoint
- **Status**: The isolated class-A tracer tests pass before complete player integration.
- **Flat test**: `certified_concave_trace_matches_exhaustive_flat_and_clamped_edges`.
- **Flat centers**: 0.0, 64.375, 127.0 source frames.
- **Flat spatial step**: 0.000002 meters per frame.
- **Flat tracing radius**: 0.000018 meters.
- **Tone test**: `certified_concave_trace_matches_exhaustive_inner_groove_tone`.
- **Tone source**: 1024 `f32` samples at 192000 hertz.
- **Tone frequency**: 3000 hertz.
- **Tone displacement peak**: 0.000001 meters.
- **Tone edge treatment**: A 128-frame raised-cosine fade has two exact-zero endpoint samples.
- **Tone spatial step**: 0.0000011 meters per frame.
- **Tone centers**: 128.0, 255.375, 511.875, 800.125 source frames.
- **Maximum positive Catmull-Rom curvature**: 8035.028657119736 per meter.
- **Sphere-curvature limit**: 55555.555555555555 per meter.
- **Available strict margin**: 47520.52689843582 per meter.
- **Curvature use**: 14.4631 percent of the sphere-curvature limit.
- **Supplied test margin**: 23760.26344921791 per meter.
- **Height agreement**: The certified and exhaustive results differ by no more than one picometer.
- **Position agreement**: The results differ by no more than 0.1 nanometers.
- **Slope and residual agreement**: The results differ by no more than 0.000001.
- **Allocation test**: The successful certified trace makes no allocator calls.
- **Failure test**: A forced numerical-bound failure reads one cubic and returns `StationaryContactNotIsolated`.
- **Fallback result**: The forced failure does not call the exhaustive tracer.
### Trace-Admission Coverage Checkpoint
- **Status**: Class A does not cover the rapid seed fixture or `PVC-004`.
- **Permanent test**: `trace_admission_classifies_seed_and_pvc_004_without_rejecting_asset_construction`.
- **Rapid source**: `SEED_DOMAIN_WALL_VELOCITY_BITS` supplies the left wall.
- **Rapid right wall**: Each sample is the exact negation of the left-wall sample.
- **Rapid outer radius**: 0.08250592249883855 meters.
- **Rapid inner radius**: 0.060325 meters.
- **Rapid speed**: 33.333333333333336 revolutions per minute.
- **Rapid sample rate**: 192000 hertz.
- **Rapid cut**: The fixture uses the default cut and an 18-micrometer sphere.
- **Rapid class**: `FixedCapPiecewise`.
- **Rapid maximum signed wall curvature**: 270051.552 per meter.
- **Rapid maximum absolute wall slope**: 0.491673.
- **Rapid runtime support**: One wall trace examines at most 28 source-frame spline pieces.
- **Rapid aggregate count**: 950 pieces across two walls and five representation levels.
- **PVC-004 source**: `pvc_004_inner_groove_sine()` supplies `spatial_asset_for_wall`.
- **PVC-004 class**: `FixedCapPiecewise`.
- **PVC-004 maximum signed wall curvature**: 236652.492 per meter.
- **PVC-004 maximum absolute wall slope**: 0.440791.
- **PVC-004 runtime support**: One wall trace examines at most 28 source-frame spline pieces.
- **PVC-004 aggregate count**: 40950 pieces across two walls and five representation levels.
- **Clamp result**: Neither fixture has a C1 record clamp.
- **Control result**: Flat, clamped-flat, and tapered 3-kilohertz data pass class A.
- **Requirement**: Keep both realistic assets constructible when class A does not apply.
- **Requirement**: Use the bounded class-B tracer for active height-order decisions.
Run the coverage test:
```sh
cargo test --lib physical::rapid_scratch_reference::trace_admission_classifies_seed_and_pvc_004_without_rejecting_asset_construction -- --exact --nocapture
```
### Trace-Ingress Adversarial Cases
The WASM facade has one fixed staging buffer.
The former facade permitted one reservation for each page slot.
Therefore, two slots could receive the same pointer at the same time.
The second write could replace the first slot's uncommitted samples.
The correction permits only one global WASM staging reservation.
The permanent test uses two live page slots.
The second reservation must return `ChunkReservationActive`.
After commit, the facade copies samples into private cache storage.
A stale JavaScript view can then change only the staging buffer.
Precomputed spatial levels now have a separate canonical-content test.
A supplied level can be finite and trace-admissible without being canonical decimation.
A page hash proves identity for those supplied bytes.
It does not prove that the canonical filter produced those bytes.
The adversarial page changes one spatial level.
It also recomputes the trace certificate and page identity.
The cache rejects that page as `NoncanonicalSpatialPyramid`.
The comparison uses fixed work and leaves the page invisible.
Class-B certificates must contain the four exact runtime capacities.
The certificate validator rejects zero and canonical-minus-one values.
This rejection also applies after a caller refreshes the certificate digest.
Full and incremental certification produce equal certificates and work totals.
The test covers unaligned interior pages and both record boundaries.
The test uses one-unit and varied work-budget partitions.
A rejection-class certificate cannot enter the real-time `Ready` state.
This rule applies to expected certificates and Rust-owned raw ingress.
The failure is `TraceAdmissionNotAdmitted`.
It occurs before hashing, seam validation, or publication.
The resident manifest has a separate claim limit.
It binds resident ranges, page identities, and certificate identities.
It does not bind every page in one authoritative full-record catalog.
Until that catalog exists, the page producer remains an explicit trust boundary.
### Reference Method
The test oracle examines every Catmull-Rom segment inside the spherical radius.
The oracle uses outward-rounded interval arithmetic for each cubic and spherical arc.
Branch-and-bound stops at a 0.0000000001-meter height tolerance.
The contact-position cells have a maximum width of 0.0000000001 meters.
The oracle visits 2704 cells for the primary fixture.
A 262145-point dense search falls inside the interval result.
The reversed source produces an overlapping reflected contact-position enclosure.
The method is a test-only oracle.
It is not a production page certificate.
### Reference Result and Error
- **Center displacement lower bound**: 0.00000004480387360039807 meters.
- **Height error bound**: 0.00000000009863312151685398 meters.
- **Chosen contact offset**: 0.0000022756274414062537 meters.
- **Contact enclosure lower bound**: 0.0000022618029785156293 meters.
- **Contact enclosure upper bound**: 0.000002289451904296879 meters.
- **Height miss lower bound**: 0.000000021285598205846543 meters.
- **Height miss upper bound**: 0.000000021384231327363397 meters.
- **Position separation lower bound**: 0.0000040786375045836115 meters.
- **Position separation upper bound**: 0.000004106286430364061 meters.
- **Height enclosure width**: 0.09863312151685398 nanometers.
- **Contact enclosure width**: 27.6489257812497 nanometers.
### Physical Significance
The height error can change the wall constraint position.
The contact-position error can change the groove slope and tangential reaction.
These changes can alter force, torque, contact loss, recapture timing, and cartridge voltage.
The case is valid for the production source domain.
It is not a measured physical-record case.
### Required Correction
Replace the local optimum search with a bounded global envelope method.
The method must examine all applicable segments and competing maxima.
The method must return declared height and position bounds.
The real-time path must have a fixed work limit.
Asset admission must reject content that cannot satisfy that work limit.
Each page identity must include the exact certificate digest.
The certificate must cover each representation level and page seam.
The renderer must fail transactionally when the certificate is absent, stale, corrupt, or too costly.
### Acceptance Tests
- The permanent tests execute only the replacement tracer.
- The replacement tracer must contain the outward interval enclosure.
- Forward and reversed input must produce reciprocal contact positions.
- Contiguous, paged, and real-time paged sources must produce the same result.
- The test must cover each spatial-pyramid level used by rapid scratching.
- Separated contenders with unresolved height order must return `GlobalContactNotIsolated`.
- A failed trace must leave player state and output unchanged.
### Reproduction
Run the focused suite:
```sh
cargo test --lib physical::rapid_scratch_reference::pvc_001_ -- --test-threads=1
```
Print the offline results:
```sh
cargo test --release --lib physical::rapid_scratch_reference::report_rapid_scratch_reference_metrics -- --ignored --nocapture --test-threads=1
```
The primary test is `pvc_001_replacement_contains_seed_domain_global_envelope`.
### Artifacts
- `tests/fixtures/pvc_001_wall_velocity_global_envelope.json`
- SHA-256: `4b016372341b86e4c9cfd47d07a3a4a407f9f8ed942723cd10172974f7536e92`
- `tests/fixtures/pvc_001_catmull_rom_global_envelope.json`
- SHA-256: `c2af509dcd93b303d877e51369fb8e6e42fc8d0e5a55e3e20384ceddf30d5528`
### Correction Status
- **Production fix**: The bounded global tracer passes both exact fixtures.
- **Permanent replacement tests**: Implemented.
- **Outward interval oracle**: Implemented for uniform Catmull-Rom test data.
- **Page-bound certificate**: Implemented for contiguous, paged, and real-time paged representations.
- **Fixed-work admission**: Implemented for strict-concavity and fixed-cap piecewise classes.
- **Canonical pyramid check**: Implemented with bounded, bit-exact recomputation.
- **Rapid-contact comparison**: Implemented as an offline reduced-system stress test.
- **Correction gate**: The numerical and admission correction passes.
- **Product gate**: Device callback limits and the full-record catalog remain open.
### Uncertainty and Contrary Evidence
- The primary case does not prove cutter-head, lacquer, plating, pressing, or PVC feasibility.
- A short source cannot test adjacent-turn clearance.
- The public sample-closure helper is not an admitted product source.
- The production sources require a private validated admission token.
- A correct result for this case will not prove correctness for all groove data.
The accepted numeric-domain stress fixture now returns a certified result.
The result lies inside the outward height and position enclosures.
A 1,048,576-point dense search corroborates the result.
The rapid-scratch reference fixture now completes without a trace error.
Its passing result does not prove the callback deadline.
### Open-Loop Electrical Observer Evidence
- **Observation date**: 2026-08-02.
- **Rate sweep**: Signed integer rates from `1x` through `20x`.
- **Candidate step**: One mechanical contact solve at 192000 hertz.
- **Reference step**: At most 0.125 source frames.
- **Electrical model**: Concorde MkII Scratch seed cartridge and phono configuration.
- **Observer rule**: Identical electrical models process candidate and reference motion.
- **Coupling limit**: Cartridge reaction force does not return to either mechanical solve.
- **Claim limit**: These values are not complete coupled-player errors.
The rate sweep gives these normalized RMS errors:
| Cartridge load voltage | `0.5863220800818523` | `0.584662684658089` |
| Phono output voltage | `0.7743228579914596` | `0.7494342423073957` |
The second fixture ramps between stop, `20x` forward, and `20x` reverse motion.
It returns to its initial source position.
It gives these results:
| Cartridge-voltage normalized RMS error | `0.8760014176990278` | `0.8761559698355349` |
| Phono-voltage normalized RMS error | `0.9778097746445279` | `0.979299352824134` |
| Cartridge absolute-integral error | `36.1840212852331%` | `36.100696516125497%` |
| Phono absolute-integral error | `47.997181164469366%` | `48.155275767809436%` |
The electrical result confirms a perceptual risk from the contact-bandwidth defect.
The test does not identify the audible difference on physical hardware.
Permanent tests are:
- `rapid_scratch_reference_covers_signed_rates_impulse_loss_and_retracking`.
- `rapid_stop_and_reversal_fixture_reaches_cartridge_and_phono_outputs`.
Run the report with this command:
```sh
cargo test --release --lib physical::rapid_scratch_reference::report_rapid_scratch_reference_metrics -- --ignored --exact --nocapture
```
The certificate rejects unsupported work, slope, joins, geometry, and stale representation data.
The correction applies at each current production trace boundary.
The page producer remains trusted until an authoritative full-record catalog exists.
The secondary spatial fixture is accepted by `from_displacement_m_with_cut`.
Its peak displacement is 1.6356867675952984 micrometers.
Its maximum dimensionless slope is 0.3000049385757358.
The former local tracer missed the reference height by at least 0.11957397830317671 micrometers.
Its contact position was outside the reference enclosure by at least 3.996725512525235 micrometers.
The source needs approximately 0.0864014223098119 meters per second of wall velocity at the nominal speed.
This value exceeds the seed source's declared sine peak.
Therefore, the fixture is accepted spatial data.
Its physical feasibility is not proved.
The extreme numeric-domain fixture remains a scope warning.
It needs more than 16 meters per second of wall velocity.
It also needs 11 million meters per second squared of acceleration.
That fixture demonstrates an accepted numeric domain.
It does not support a physical-record claim.
## PVC-004: Two Near-Equal Envelope Maxima on One Wall
### Claim Under Test
One spherical wall trace can always prove one physical contact position.
### Exact Fixture
- **Observation date**: 2026-08-01.
- **Samples**: 4096 `f32` displacement samples.
- **Sample rate**: 192000 hertz.
- **Tone frequency**: 8000 hertz.
- **Wall-velocity peak**: 0.05 meters per second.
- **Displacement amplitude**: 0.000000994718394324346 meters.
- **Center**: 258 source frames.
- **Nominal groove radius**: 0.060 meters.
- **Nominal speed**: 33.333333333333336 revolutions per minute.
- **Spatial step**: 0.0000010908307824964558 meters per frame.
- **Tracing radius**: 0.000018 meters.
The artifact stores the exact generator inputs and the applicable `f32` support bits.
### Reference Method
The outward interval oracle searches the negative and positive half-radii independently.
Each search has a 0.0000000000001-meter height tolerance.
Each contact-position cell has a maximum width of 0.0000000001 meters.
The two height intervals overlap.
This overlap does not prove that the two exact heights are equal.
The two position intervals do not overlap.
### Reference Result
- **Common height lower bound**: -0.000000993385894471199 meters.
- **Common height upper bound**: -0.0000009933857944761103 meters.
- **Left position interval**: -1.8021611096566203 to -1.7042233605689562 micrometers.
- **Right position interval**: 1.7068199426317835 to 1.8004966339753207 micrometers.
- **Position separation**: 3.4110433032007397 to 3.6026577436319407 micrometers.
A 1000001-point dense search corroborates the result.
Its two heights differ by 0.000000000006776263578034403 nanometers.
### Production Result
The scalar replacement tracer returns `GlobalContactNotIsolated`.
This result prevents an arbitrary single-position force calculation.
The contact-set API returns `ContactHeightOrderNotIsolated`.
It must not convert interval overlap into simultaneous rigid contact.
Contact sets now propagate through contiguous, paged, and realtime-paged player paths.
Public mechanical inputs reject multiple contacts without an internal qualification.
Serialized inputs cannot create this qualification.
A test-only symmetry qualification exercises fixed-capacity force distribution.
No production tracer emits that qualification.
### Physical Scope
The source uses a nominal wall velocity inside the seed profile range.
The case does not prove cutter, lacquer, plating, pressing, or PVC feasibility.
The source is synthetic and not a measured record.
### Acceptance and Remaining Work
- The tracer must keep the two position intervals separate.
- The contact-set tracer must return `ContactHeightOrderNotIsolated`.
- The scalar API must return `GlobalContactNotIsolated`.
- A failed player step must preserve state and output.
- A future rigid multi-contact solver requires a certified common-height equality.
- A compliant solver requires certified height gaps and measured compliance.
- Hardware measurements must determine the prevalence of this topology.
The first three requirements pass.
The player-level transaction test and force distribution remain open.
Mechanical rejection is transactional and preserves its state.
Rigid symmetric force sharing passes test-only mechanics and replay tests.
That result does not qualify conventional near-co-contact for production playback.
### Reproduction
```sh
cargo test --lib physical::rapid_scratch_reference::pvc_004_inner_groove_sine_has_unresolved_separated_height_candidates -- --test-threads=1
```
### Artifact
- `tests/fixtures/pvc_004_inner_groove_height_order_ambiguity.json`
- SHA-256: `c5c3464cfe52d3d00e2f0552d8ef6a09922efcd92a3cbc68c76ed11d9bebd1ad`
## PVC-005: Scratch Technique Topology and Prediction
### Claim Under Test
The existing Rust presets faithfully represent their named scratch techniques.
The existing predictor also remains stable during coalesced input and rapid reversals.
### Audit Result
- **Result**: Code inspection and exact fixtures disprove the claim.
- **Observation date**: 2026-08-01.
- **Ownership defect**: The browser duplicated the preset catalog and defaults.
- **Integration defect**: Only `ScratchAcousticDsp` applied the Rust gate.
- **Timing defect**: The previous integration held one rendered rate across a host block.
- **Prediction defect**: The first stroke used an unverified seed span.
### Former Technique Counterexamples
- Stab stayed open for the complete forward stroke.
- Flare and Orbit used equal gate equations.
- Crab was a Transform duty variant with another default click count.
- Drum differentiated stepwise host intent.
- Therefore, event packetization could create a Drum attack.
- Drum could open before the record completed its outgoing motion.
- Click patterns stopped after the predicted endpoint.
- Therefore, continued record motion could leave the crossfader in one state.
### Replacement Technique Fixture
- **Fixture schema version**: 4.
- **Algorithm version**: 10.
- **Gate snapshot version**: 4.
- **Performance snapshot version**: 4.
- **Sample rate**: 48,000 hertz.
- **Supported record-rate range**: -20 through 20.
- **Supported click count**: 1 through 8.
- **Maximum frame interval**: 0.000125 seconds.
- **Stab forward-open start**: 0.04 stroke.
- **Stab forward-open duration**: Until reversal intent or a physical direction crossing.
- **Stab reverse target**: Muted.
- **Chirp edge-mute distance**: 0.04 stroke in each physical direction.
- **Chirp reversal target**: Muted after opposite intent.
- **Chirp rest target**: Muted while hand contact remains active.
- **Transform open fraction**: 0.24.
- **Flare notch half-width**: 0.07 stroke.
- **Flare notched direction**: Forward only.
- **Orbit notched directions**: Forward and reverse.
- **Crab burst centers**: 0.18 through 0.72 stroke.
- **Crab maximum pulse half-width**: 0.035 stroke.
- **Drum acceleration trigger**: 6.0 record-rate units per second.
- **Drum refractory interval**: 0.045 seconds.
- **Drum maximum opening**: 0.055 seconds.
These values define the current deterministic model.
They are provisional calibration values, not measured physical facts.
### Prediction Fixture
- Initial endpoint confidence is zero.
- The first completed stroke has a span of 0.123 source seconds.
- The model uses that span immediately.
- Confidence becomes 0.25 after that observation.
- The opposite direction keeps its seed span and zero confidence.
- Forward and reverse observations update independent estimates.
- Confidence becomes 1.0 after four observations.
- The permitted learned span is 0.04 through 0.8 source seconds.
No code can know an unseen first endpoint without a supplied or trained motion model.
Therefore, this case does not claim first-stroke endpoint accuracy.
### Packetization Fixture
- The fixture contains 8,000 frames at 48,000 hertz.
- Rendered rate changes from 0.5 to 1.5 at frame 3,000.
- Intent alternates between 0.35 and 0.85.
- One path updates intent every frame.
- The other path updates intent every 64 frames.
- Both paths produce exact target, gate, and audible-gain equality.
### Rapid-Reversal Fixture
- The fixture contains 32 strokes.
- Each stroke contains 400 frames.
- Record rate alternates between plus 8 and minus 8.
- Each confirmed endpoint has a closed automatic target.
- Every output stays finite and inside the unit interval.
The maximum-rate fixture also covers plus and minus 20 record rate.
### Maximum-Rate Confirmation Counterexample
- **Record rate**: `20x`.
- **Former onset delay**: `0.004` seconds.
- **Former onset travel**: `0.080` source seconds.
- **Former reversal delay**: `0.006` seconds.
- **Former reversal travel**: `0.120` source seconds.
- **Stab seed span**: `0.22` source seconds.
- **Former onset ratio**: `36.4` percent of the Stab seed span.
- **Former reversal ratio**: `54.5` percent of the Stab seed span.
- **Failure**: Former onset confirmation could skip the complete first Stab pulse.
- **Failure**: Former reversal confirmation could skip early click events.
- **Correction**: Same-sample physical motion confirms its current direction immediately.
- **Intent rule**: Opposite intent cannot reverse the gate during outgoing physical motion.
- **Accepted result**: The first Stab pulse remains present at `20x`.
- **Accepted result**: Transform, Flare, Crab, and Orbit retain the selected event count at `20x`.
- **Observed rate drift**: One exact-`20x` physical result divided to `20.0000000000003268`.
- **Former result**: Input validation rejected that physical frame.
- **Correction**: Accept `1e-10` rate roundoff and clamp it to exact `20x`.
- **Limit**: Values above the roundoff allowance still reject.
### Physical Travel Fixture
- Technique phase uses exact signed record-angle travel.
- Source travel seconds equal signed record-angle change divided by nominal angular velocity.
- Final rendered rate does not approximate travel across a reversal.
- Supported timing checks use 44.1, 48, and 96 kilohertz.
### Permanent Tests
- `stab_keeps_the_forward_stroke_audible_and_mutes_the_return`
- `chirp_mutes_each_physical_direction_edge`
- `chirp_opens_after_each_edge_and_closes_before_the_turn`
- `intent_cannot_commit_reversal_before_rendered_rate_crosses_zero`
- `outgoing_motion_keeps_the_existing_stroke_until_physical_reversal`
- `flare_is_one_sided_while_orbit_repeats_the_notch_on_return`
- `crab_is_a_clustered_finger_burst_not_a_transform_duty_variant`
- `transform_has_a_closed_baseline_with_brief_uniform_taps`
- `click_driven_techniques_repeat_for_continuous_record_motion`
- `first_stroke_seed_has_zero_confidence_until_one_stroke_is_observed`
- `reversal_learns_only_the_completed_direction_span`
- `asymmetric_direction_spans_clock_clicks_independently`
- `drum_is_invariant_to_intent_event_packetization_for_same_physical_trajectory`
- `drum_reversal_does_not_spend_its_hit_on_outgoing_motion`
- `baby_uses_manual_gain_and_automatic_presets_own_the_output`
- `invalid_input_and_snapshot_restore_are_transactional`
- `performance_snapshot_restore_repeats_every_output_frame`
- `performance_render_path_does_not_allocate`
- `rapid_physical_reversals_remain_bounded_and_hide_each_endpoint`
- `maximum_rate_reversals_remain_finite_and_bounded`
- `physical_maximum_rate_onset_preserves_the_first_stab_attack`
- `physical_maximum_rate_onset_preserves_every_early_click_event`
- `maximum_rate_roundoff_clamps_to_the_exact_supported_boundary`
- `outgoing_physical_motion_rejects_predicted_reversal_until_the_crossing_sample`
- `performance_timing_is_sample_rate_invariant`
- `pvc_005_fixture_matches_canonical_constants_and_claim_limits`
- `stab_and_chirp_change_phono_audio_at_one_eight_and_twenty_times`
- `scratch_gate_reversal_commits_on_the_rendered_motion_crossing`
### Result and Claim Limit
The focused `scratch_gate` suite passes 60 tests.
The wider scratch-filtered suite passes 89 tests.
Two offline evidence reports remain ignored in the wider suite.
The tests validate topology, ownership, deterministic state, packetization behavior, sample-rate behavior, and numeric bounds.
The tests do not validate timing against measured DJ motion or crossfader traces.
The tests do not model a specific crossfader's curve, cut-in, latency, bleed, bounce, or noise.
The tests do not prove blinded expert acceptance.
Physical-player tests verify sample-timed gain on phono voltage.
C ABI tests verify controls, telemetry, validation, and native layout.
Native Swift tests verify that Rust gain changes rendered audio.
The tests do not prove WASM or browser integration.
The presets do not generate record motion.
The Stab and Chirp topology corrections change real phono voltage.
The technique fractions and crossfader envelope values remain unmeasured.
### Integration Decision
Keep the physical player and native Swift on the canonical Rust implementation.
Calibrate the technique timing before a faithful-technique claim.
Do not retain browser technique equations after the Rust API becomes active.
### Skipproof Research Disposition
- **Source**: Hansen and Bresin, [*The Skipproof Virtual Turntable for High-Level Control of Scratching*](https://doi.org/10.1162/comj.2010.34.2.39), 2010.
- **Applicable result**: One record-motion control can generate a coordinated crossfader target.
- **Applicable result**: Precise coordination is necessary to preserve the technique character.
- **Applicable method**: Use expert-recorded trajectories as resampled lookup tables.
- **Rejected behavior**: Do not permit preset motion beyond current hardware limits.
- **Missing artifact**: The paper does not include the complete source trajectory tables.
- **Next evidence**: Find the original GPL tables or record new expert performances.
- **Later catalog candidates**: Rolltear, Forward, Uzi, and Twiddle.
- **Existing coverage**: Baby supplies the open-fader behavior for Tear and Scribble.
- **Internal fader primitive**: Silent Back can mute a user-controlled return stroke.
- **Catalog review**: Compare Chop with Stab before adding a second overlapping preset.
- **Implementation rule**: A preset must not replace or reshape user record motion.
- **Physics rule**: The crossfader must follow rendered physical motion.
- **Research rule**: Recorded motion can identify crossfader landmarks only.
- **Manual rule**: Direct record and crossfader control must remain available.
- **Current decision**: Do not add a preset from this paper during the current pass.
The paper supports the high-level controller design.
It does not calibrate the current preset fractions.
It does not validate current hardware timing or crossfader response.
### Reproduction
```sh
cargo test --lib scratch_gate::tests --no-fail-fast -- --test-threads=1
```
### Artifact
- `tests/fixtures/pvc_005_scratch_semantics.json`
- SHA-256: `081760f9d4ff294be88c51950abeb53ffe0bae0111de90450784a7aab676771c`
- `src/scratch_gate.rs` SHA-256: `70e768b0f1bf06ea8ff0a67305127920b5c6afafa4205d376b821a3462db735b`
## PVC-002: Finalized-Page Publication Exceeds the Fixed Cache
### Claim Under Test
The browser can publish every finalized page into the fixed AudioWorklet cache without demand paging.
### Exact Fixture
- **Observation date**: 2026-08-01.
- **Source format**: Stereo signed 16-bit PCM.
- **Source sample rate**: 48,000 Hz.
- **Source frame count**: 4,609 frames.
- **Source byte count**: 18,436 bytes.
- **PCM generator seed**: `0x13579bdf`.
- **PCM generator**: The permanent Node test defines the exact integer generator and channel equations.
- **Output sample rate**: 192,000 Hz.
- **Output frame count**: 18,432 frames.
- **Page core length**: 2,048 frames.
- **Tracing halo**: 39 frames.
- **Finalized page count**: 9 pages.
- **Default fixed-cache capacity**: 8 page slots.
The production WASM cutter accepts this source.
It produces nine contiguous canonical pages with exact final identities.
### Prior Publication Policy
The rejected draft policy published every finalized page during initial loading.
It did not evict a page between these initial publications.
The test fully ingests, validates, and publishes the first eight pages.
The ninth `beginRealtimePagedPagePrecomputed()` call returns `PhysicalRealtimePagedStatus.NoEmptySlot`.
`PhysicalRealtimePagedStatus.NoEmptySlot` has numeric value `9`.
The result comes from the production Rust fixed cache through the WASM interface.
### Reference Method
The reference is an exact capacity count.
Nine resident pages cannot fit in eight fixed slots without eviction.
The page-count and slot-count error bound is zero.
No numeric approximation affects this result.
### Physical and Product Effect
The ninth page cannot become render-visible.
A later seek or fast scratch can then produce a page miss.
The error can stop audio or delay a transport change.
It does not directly change the physical equations.
### Correction
The browser worker stores every canonical page in durable IndexedDB storage.
Each page key contains the asset identity, generation, and core range.
The worker awaits the storage transaction before it transfers page buffers.
It commits the complete asset manifest after all page writes succeed.
The default initial warm-up publishes one page.
Seeks and prefetch requests load later pages on demand.
Each demand request returns a fresh transferable page value.
The AudioWorklet must still validate identity, range, and overlapping seams.
### Acceptance Tests
- The real WASM fixture must always produce 18,432 output frames and nine pages.
- The real default cache must report eight slots.
- The ninth un-evicted page must return `NoEmptySlot`.
- One-chunk and irregular-chunk cuts must produce identical pages and identities.
- A distant seek must retrieve the last page without a new cut.
- A reverse boundary request must select the preceding page.
- An evicted page must load again with fresh transferable buffers.
- A stale generation must fail before publication.
- A storage failure must remove raw and partial new-asset pages.
- The publication protocol must permit only one unacknowledged page.
### Reproduction
Run this command:
```sh
cd /Users/jamie/wavey.ai/vin.yl.player
npm run test:physical-groove
```
The permanent production-WASM test is:
```text
the default fixed cache rejects a ninth page without eviction
```
### Artifacts
- **Fixture and test**: `vin.yl.player/test/physical-groove-worker-pipeline.test.mjs`.
- **SHA-256**: `73659fa340ebe9af7af2173d7958d01fcc46e3b572376415c6c70ea7330fa8fc`.
- **Worker design**: `vin.yl.player/PHYSICAL_GROOVE_WORKER.md`.
### Evidence Limits
The cutter, materializer, and fixed-cache overflow test use production WASM.
The Node durable-store test uses a filesystem-backed store double.
It does not execute browser IndexedDB.
The quota test injects a synthetic `QuotaExceededError`.
It does not force a real browser quota failure.
The one-publication lifecycle test uses a synthetic worker.
It does not execute an AudioWorklet.
### Correction Status
- **Worker durable store**: Implemented.
- **One-page warm-up**: Implemented as the default.
- **Demand retrieval**: Implemented for page, seek, and reverse requests.
- **Stored-asset attachment**: Implemented.
- **Production AudioWorklet activation**: Not implemented.
- **Browser IndexedDB smoke test**: Not implemented.
- **Minimum-device deadline test**: Not implemented.
### Uncertainty and Contrary Evidence
- A larger cache can hold this short fixture, but it cannot hold an arbitrary record.
- An eviction policy can fail when browser scheduling exceeds the prefetch horizon.
- IndexedDB can reject writes because of quota, privacy mode, or browser policy.
- The current tests do not prove timely page delivery during sustained 20-times scratching.
- The current tests do not prove safe page publication on each supported browser.
## PVC-003: Midpoint Solver Deadline Miss
### Claim Under Test
The bounded midpoint solver can complete each 128-frame callback during rapid reversal.
### Result
- **Result**: Disproved on the identified test computer.
- **Observation date**: 2026-08-01.
- **Internal sample rate**: 192,000 Hz.
- **One-sample budget**: Approximately 5,208.33 nanoseconds.
- **128-frame deadline**: Approximately 666,666.67 nanoseconds.
- **Observed reversal misses**: 2 of 512 callback blocks.
- **Observed reversal maximum**: 812,750 nanoseconds.
- **Release status**: The midpoint replacement is not production-ready.
### Test Computer
- **Model**: MacBook Air `MacBookAir10,1`.
- **Processor**: Apple M1 with four performance cores and four efficiency cores.
- **Memory**: 16 GB.
- **Operating system**: macOS 26.5, build `25F71`.
- **Rust compiler**: `rustc 1.96.0 (ac68faa20 2026-05-25)`.
- **LLVM**: 22.1.2.
- **Cargo**: `cargo 1.96.0 (30a34c682 2026-05-25)`.
- **Target**: `aarch64-apple-darwin`.
- **Build**: Cargo default release profile with no `RUSTFLAGS` value.
The test did not reserve a real-time core.
The test did not disable other operating-system work.
These limits can add timing noise. They cannot make the observed deadline miss acceptable.
### Core-Solver Fixture
The core test starts the platter and record at one normalized playback rate.
The normal fixture changes both wall displacement and wall slope with deterministic sine functions.
The normal fixture processes 8,192 sequential samples.
The reversal fixture uses the same deterministic wall input.
The hand target alternates between plus and minus 20 normalized rates every 32 samples.
The hand normal force is 5 N. The contact radius is 0.12 m.
The constructed search uses these record rates:
```text
-20, -1, 0, 1, 20
```
It combines seven wall-slope pairs, five displacement pairs, and four prior tangential modes.
The production input checks accepted every reported sequential sample.
### Failed Baseline
The table shows `minimum / p50 / p95 / p99 / maximum`.
| Normal | `1 / 1 / 13 / 13 / 41` | `1 / 1 / 13 / 13 / 41` | `916 / 1,000 / 8,666 / 8,792 / 49,500` | 1,905 |
| Reversal | `1 / 1 / 97 / 289 / 369` | `1 / 1 / 85 / 265 / 333` | `708 / 833 / 42,708 / 144,083 / 212,125` | 7,397 |
The constructed search found one successful case with 311 branches and 311 linear solves.
The formal algorithmic ceiling remains 1,296 branches and 1,296 linear solves.
### Continuation Optimization Checkpoint
The solver now orders hand and slip modes with a torque predictor.
It keeps every active mode in the fallback search.
The solver also starts with the previous wall-contact mask.
It requires positive normal force for a kinetic tangential mode.
| Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `666 / 1,000 / 1,125 / 1,209 / 55,208` | 1,009 |
| Reversal | `1 / 1 / 1 / 1 / 118` | `1 / 1 / 1 / 1 / 109` | `625 / 916 / 959 / 2,875 / 69,541` | 1,182 |
The constructed search found one successful case with 255 branches and 255 linear solves.
These results improve the common path. They do not close the callback deadline failure.
### Static-Slip Prediction Checkpoint
The solver estimates the torque that would remove predicted slip during one sample.
It tries static slip first when that torque is inside the applicable static limit.
This predictor changes only the search order. The complete fallback remains available.
The table shows `minimum / p50 / p95 / p99 / maximum`.
| 1 | Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `791 / 1,167 / 2,625 / 2,709 / 73,875` | 1,342 |
| 1 | Reversal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `708 / 1,042 / 1,125 / 1,542 / 34,791` | 1,102 |
| 2 | Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `667 / 1,000 / 1,042 / 1,125 / 56,583` | 992 |
| 2 | Reversal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `625 / 917 / 959 / 1,083 / 27,209` | 932 |
| 3 | Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `791 / 1,083 / 1,125 / 1,625 / 39,083` | 1,118 |
| 3 | Reversal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `708 / 1,041 / 1,125 / 1,167 / 51,042` | 1,041 |
| 4 | Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `791 / 1,083 / 1,167 / 2,291 / 75,917` | 1,124 |
| 4 | Reversal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `708 / 959 / 1,042 / 2,417 / 25,167` | 1,033 |
| 5 | Normal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `791 / 1,167 / 1,209 / 1,292 / 34,958` | 1,185 |
| 5 | Reversal | `1 / 1 / 1 / 1 / 28` | `1 / 1 / 1 / 1 / 28` | `750 / 1,083 / 1,167 / 1,250 / 24,750` | 1,084 |
Each run processed 8,192 normal samples and 8,192 reversal samples.
All five constructed searches found the same successful maximum of 255 branches and 255 solves.
The allocation guard detected no allocator call during one complete active-mode solve.
This checkpoint closes the observed sequential core branch tail. It does not prove the complete callback deadline.
### Complete Player Callback Fixture
The complete player uses the seed profile and a deterministic 3 kHz sine groove.
Each player renders 1,024 warm-up frames before measurement.
Each measurement contains 512 blocks. Each block contains 128 frames.
The reversal fixture changes the hand target every 64 frames.
The hand target alternates between plus and minus 20 normalized rates.
| Normal | `477,458 / 495,250 / 551,458 / 567,292 / 602,125` | 500,997 | 0 of 512 |
| Reversal | `476,834 / 507,375 / 592,750 / 621,417 / 812,750` | 524,789 | 2 of 512 |
### Provisional Cross-Track Callback Measurement
A later shared checkpoint included changing certified-tracer work.
The complete-player benchmark measured that checkpoint once.
| Normal | `1,997,166 / 2,256,167 / 2,421,583 / 2,501,500 / 2,638,667` | 2,250,275 | 512 of 512 |
| Reversal | `2,002,791 / 2,249,375 / 2,422,542 / 2,660,709 / 5,076,833` | 2,242,475 | 512 of 512 |
The approximately 1-microsecond core solve did not dominate this complete-player result.
This result identifies a tracer-dominated cross-track risk. It is not a final certified-tracer checkpoint.
Do not use this provisional result to replace the stable callback checkpoint above.
### Ambiguity-Safe Certified-Tracer Checkpoint
This checkpoint includes the safe height-order ambiguity contract.
It also includes the spatial-blend endpoint fast path.
The table reports p50, p95, p99, maximum, and deadline misses.
| Normal | 1,130,958 | 1,195,584 | 1,219,625 | 1,248,417 | 512 of 512 |
| Reversal | 1,103,084 | 1,191,667 | 1,258,042 | 1,833,125 | 512 of 512 |
The fixture uses 1,024 warm-up frames.
It then measures 512 blocks of 128 frames at 192 kilohertz.
The reversal target alternates between plus and minus 20 every 64 frames.
A sampling profile attributed approximately 55 percent of render samples to the tracer.
The hottest tracer operations were circle-height bounds, cubic-slope bounds, circle-slope bounds, and root isolation.
One specialized interval-multiplication attempt preserved correctness but increased release time by approximately seven percent.
That optimization was rejected and reverted.
Shared contact edits and operating-system noise can move later measurements.
Retain this exact distribution as one identified source-state checkpoint.
### Validated-Token Callback Checkpoint
This checkpoint removes repeated immutable certificate hashing from active tracing.
Asset admission still recomputes and validates the complete certificate.
The validated token checks only the supplied stylus geometry during rendering.
The table contains the middle result from three consecutive release runs.
| Normal | 1,639,292 | 2,158,958 | 2,209,459 | 2,252,708 | 1,812,518 | 512 of 512 |
| Reversal | 4,685,292 | 8,603,000 | 8,700,625 | 8,889,417 | 5,208,811 | 512 of 512 |
The immediate baseline used the same source and release command.
Its normal mean was 2,506,968 nanoseconds.
Its reversal mean was 7,901,920 nanoseconds.
The normal mean decreases by approximately 27.7 percent.
The reversal mean decreases by approximately 34.1 percent.
The physical output equations and proof limits do not change.
Both fixtures still miss every 666,666-nanosecond deadline.
The remaining profile is dominated by certified root isolation and the coupled contact solve.
This checkpoint does not authorize product activation.
### Reference Method
`std::time::Instant` measures each core step and each complete player block.
The deadline equals the block frame count divided by the internal sample rate.
This comparison has no model error. Operating-system scheduling adds measurement variation.
Repeat tests must report the full distribution and all deadline misses.
### Physical and Product Effect
A callback miss can cause an audio underrun on a host with no remaining timing margin.
An underrun breaks the continuous physical output even when the state equations remain valid.
The risk increases during fast hand reversals and active-mode changes.
### Required Correction
Reduce the practical branch tail without removing a physically admissible mode.
Keep deterministic continuation, complementarity checks, passivity checks, and residual validation.
Measure the complete callback on every minimum supported device.
Do not call the path real-time safe until all declared callback deadlines pass.
### Reproduction
Run the core benchmark:
```sh
cd /Users/jamie/wavey.ai/record-player
cargo test --release --lib physical::contact::tests::midpoint_release_work_benchmark -- --ignored --nocapture
```
Run the complete player benchmark:
```sh
cd /Users/jamie/wavey.ai/record-player
cargo test --release --lib physical::player::tests::midpoint_player_release_block_benchmark -- --exact --ignored --nocapture --test-threads=1
```
### Artifacts
- **Measured result record**: `tests/fixtures/PVC-003-midpoint-deadline-m1.txt`.
- **Measured result SHA-256**: `8955beb7e63f1250ba50fa28453a7ba156e7a1ef079e73c29c7deb74d623a4e8`.
- **Validated-token result record**: `tests/fixtures/PVC-003-validated-token-deadline-m1.txt`.
- **Validated-token result SHA-256**: `f40775453ac7d7fbe7588ea4af6591d2c9f47c4fa203142994e1231c41d23fb8`.
- **Validated-token source SHA-256**: `fa8b9ce0298f7b57559e8dc241b17431d7733c7386d763f393cccd07032bee69`.
- **Core fixture**: `src/physical/contact.rs`.
- **Core fixture SHA-256 at this checkpoint**: `dce3ab59d2b32a64d6c922fca2a91f41364d32df3b61723df55e1b7cac652bb6`.
- **Player fixture**: `src/physical/player.rs`.
- **Player fixture SHA-256 at this checkpoint**: `00df3f834505495842b231c9940809395b55bb4a9f2402d0ef512239ccbc441e`.
The source digests identify the checkpoint before later tail-reduction work.
- **Current static-slip mechanics SHA-256**: `e812f74edc616b2652ccd41bcaf6e2a0e05822bc451f89f933f7255afb9425a7`.
- **Current static-slip contact SHA-256**: `2fc537815e1cb990384e76361c8124b367a61a78114d9b6ddfdb6674970ac862`.
- **Provisional tracer SHA-256**: `4d62068171287ea1dd6f4c2a1e1c4ce4daf9363587e59bd99842c0aaa2428ce7`.
The provisional tracer digest identifies only the measured cross-track checkpoint.
Formatting and an allocation test followed the five core measurements.
The production solver tokens did not change. The current hashes do not identify those measured files byte-for-byte.
### Correction Status
- **Failed baseline retained**: Complete.
- **Continuation ordering**: Implemented.
- **Contact-mask continuation**: Implemented.
- **Static-slip prediction**: Implemented.
- **Scaled residual validation**: Implemented.
- **Active-mode allocation test**: Passed.
- **Complete callback deadline**: Failed.
- **Minimum-device validation**: Not started.
### Uncertainty and Contrary Evidence
- Normal playback met the measured 128-frame deadline on this computer.
- The optimized reversal p99 was below the deadline.
- Two optimized reversal blocks still missed the deadline.
- A later changing tracer checkpoint missed all measured callback deadlines.
- The later measurement is provisional and tracer-dominated.
- The five-run tail record lacks a byte-for-byte source digest.
- A faster computer cannot establish the minimum-device requirement.
- A non-isolated computer can add scheduling noise.
- No test in this case proves hard real-time behavior on macOS.
## PVC-006: Passive Cartridge Magnetic-Loss Invariants
### Claim Under Test
A frequency-dependent coil-loss network can preserve passivity, reciprocity, fixed work, and exact zero-network behavior.
### Exact Fixture
- **Observation date**: 2026-08-02.
- **Sample rate**: 192000 hertz.
- **Declared self-inductance**: 0.850 henry.
- **First relaxation inductance**: 0.020 henry.
- **First loss resistance**: 4000 ohms.
- **Second relaxation inductance**: 0.180 henry.
- **Second loss resistance**: 1800 ohms.
- **Residual self-inductance**: 0.650 henry.
- **Unused slots**: Two zero-valued slots.
- **Energy-test coupling ratios**: -0.72, 0.0, and 0.72 of the residual self-inductance.
- **Energy-test length**: 20000 alternating rapid reversals for each coupling ratio.
The two active relaxation times are strictly increasing.
### Network
The coil has a residual coupled inductor in series with direct-current coil resistance.
Each active relaxation section contains a resistor in parallel with an inductor.
The self impedance is:
```text
Z(s) = Rdc + s * (L0 - sum(Lk)) + sum(Rk * s * Lk / (Rk + s * Lk))
```
The signed mutual term is `s * M` in the off-diagonal entries.
Validation requires both residual modal inductances to remain positive.
### Zero-Network Result
`zero_magnetic_loss_slots_are_bit_identical_to_the_legacy_coupled_solver` uses 10000 samples.
The test compares each affine bias and matrix coefficient with `f64::to_bits`.
All compared values are bit-identical to the `6791e90` equations.
The existing uncoupled 10000-sample end-to-end bit test also passes.
### Energy and Reciprocity Result
`magnetic_loss_network_closes_the_exact_discrete_energy_balance` covers all three coupling ratios.
For each sample, the test compares stored-energy change with interval port energy and all resistor losses.
The accepted residual is:
```text
8e-11 * max(abs(energy change), abs(port energy), 1e-30) + 3e-27 J
```
`magnetic_loss_network_is_reciprocal_in_the_same_sample_without_allocation` covers 10000 coupled steps.
It checks generator power against the opposite mechanical reaction power.
It also checks that the coupled processing path allocates no memory.
### Frequency-Domain Result
`passive_magnetic_loss_impedance_has_the_required_low_and_high_frequency_limits` checks the limiting impedance.
At low frequency, the direct inductance approaches 0.850 henry.
At high frequency, the direct inductance approaches 0.650 henry.
At high frequency, both active loss resistances add to the direct coil resistance.
The test also checks positive common-mode and differential-mode resistance and reactance.
`magnetic_loss_transfer_matrix_agrees_with_the_complex_circuit_equations` checks five frequencies.
`time_domain_magnetic_loss_response_matches_the_warped_complex_transfer` checks 1000 and 12000 hertz.
The time-domain comparison includes the trapezoidal rule's bilinear frequency warp.
### State and Configuration Result
`magnetic_loss_json_snapshot_and_rollback_preserve_all_branch_state` covers JSON continuation and failed-step rollback.
`magnetic_loss_configuration_enforces_bounds_slots_and_canonical_order` checks incomplete, gapped, unordered, and excessive networks.
`magnetic_loss_realtime_steps_allocate_no_memory` checks the active cartridge path.
The seed profile keeps all four branch slots at zero and marks them `Estimated`.
Each branch parameter requires `CartridgeMagneticLossLevelAndTemperature` calibration evidence.
### Test Results
Debug results:
- Cartridge tests: 32 passed.
- Electromechanical tests: 11 passed.
- Profile tests: 21 passed.
Release results:
- Cartridge tests: 32 passed.
- Electromechanical tests: 11 passed.
- Profile tests: 21 passed.
### Reproduction
Run the focused debug tests:
```sh
cargo test physical::cartridge::tests -- --nocapture
cargo test physical::electromechanical::tests -- --nocapture
cargo test physical::profile::tests -- --nocapture
```
Run the focused release tests:
```sh
cargo test --release physical::cartridge::tests -- --nocapture
cargo test --release physical::electromechanical::tests -- --nocapture
cargo test --release physical::profile::tests -- --nocapture
```
### Artifacts
- **Commit**: `49cd9424b129d79ee7d3ec40f8d5bf11543e30a2`.
- **Cartridge source SHA-256**: `682f0e1832c49772e3c5f9e2669550b0f6a7b8bea2b6962e8d5bba58e2e25d5d`.
- **Electromechanical source SHA-256**: `3c54358f987934f1edc7820a4ec051162a1c83445b6ef255607f56361da38365`.
- **Profile source SHA-256**: `1919b02933869b5a200a535ca4a612e86e0f4c3779000610d071b47b4a6ad903`.
### Claim Limit
The tests prove the implemented network's numerical and passive invariants.
They do not identify real Concorde magnetic-loss values.
The network does not model saturation, nonlinear hysteresis, temperature drift, or channel-asymmetric loss.
The four-pole limit can omit additional magnetic modes.
Loaded response alone cannot separate coil loss from cable and preamplifier loading.
Hardware measurements remain necessary before a calibrated cartridge claim.
## PVC-007: Passive Groove Compliance and Coupled Contact
### Status
- **Observation date**: 2026-08-02.
- **Implementation status**: Open.
- **Validation status**: The model contract and rejected alternatives are registered.
- **Claim status**: No calibrated groove-compliance or multiple-patch claim is permitted.
### Current Production Reduction
`StylusTraceContactSet` stores as many as eight positions for each wall.
The production mechanics still solves one normal multiplier for each wall.
It averages same-wall slopes and normal-force scales.
It then divides one wall multiplier equally between retained positions.
That equal division is a minimum-norm arithmetic rule.
It is not an identified contact-pressure law.
The solver has no candidate height interval, gap interval, footprint, or maximum-penetration certificate.
The solver also has no material state at an absolute groove coordinate.
Therefore, a reversal cannot recover deformation from an earlier visit.
### Contact Port and Complementarity
For wall `w`, let `n_w` be its inward 45-degree normal.
Candidate `i` has rigid sphere-center threshold `h_i` and along-groove slope `p_i`.
Let `y_w = n_w dot q_tip`.
Let positive `u_i` mean wall recession or indentation.
Use the projected wall multiplier `lambda_i` as the material force port.
The unilateral contact equations are:
```text
g_i = y_w - h_i + u_i
lambda_i >= 0
g_i >= 0
lambda_i * g_i = 0
```
For active contact, `u_i = h_i - y_w`.
The material power is `lambda_i * u_dot_i`.
Candidate slope `p_i` supplies the opposite record and stylus generalized forces.
The complete discrete power balance must include the material port.
### Registered Passive Material Topology
Use a generalized Kelvin creep realization on a groove-coordinate material field.
Let `B(s)` map candidate patch forces into fixed groove cells.
Define the cell load as:
```text
f = B(s) * lambda
```
Use one instantaneous compliance matrix `C_0`.
Use a fixed number of retardation branches `C_r` and `tau_r`.
```text
d_0 = C_0 * f
tau_r * x_dot_r + x_r = C_r * f
d = d_0 + sum(x_r)
u = transpose(B) * d
```
Require `tau_r > 0`.
Require symmetric positive-semidefinite `C_0` and `C_r` matrices.
Store each matrix through a positive-semidefinite factor or a certified reciprocal kernel.
The fitted creep compliance is:
```text
J(s) = C_0 + sum(C_r / (1 + s * tau_r))
```
This topology has finite recoverable static compliance `J(0)`.
Reject a fit that requires a negative branch.
Do not add an unmeasured Hertz spring to repair a rejected fit.
### Measured Nonlinear Elastic Extension
A recoverable static penetration curve can be nonlinear without plastic memory.
For one qualified patch, let measured equilibrium penetration be `u_eq(lambda)`.
After fitting positive retardation weights, define:
```text
u_0(lambda) = u_eq(lambda) - sum(C_r) * lambda
```
Accept this curve only when `u_0(0) = 0`.
Also require `du_0 / d(lambda) > 0` across the declared load range.
Then the elastic force derives from a convex stored-energy potential.
The retardation branches continue to supply nonnegative loss.
At preload `lambda_0`, the small-signal compliance is:
```text
du_0 / d(lambda) at lambda_0
+ sum(C_r / (1 + s * tau_r))
```
A fixed monotone piecewise-linear curve can provide bounded nonlinear elasticity.
Split a sample at each crossed breakpoint.
Alternatively, use a discrete-gradient force that closes the registered energy identity.
For multiple patches, require a measured convex complementary potential `Psi_0(f)`.
Its Hessian must be positive semidefinite and reciprocal.
Reject a fit when relaxation subtraction makes the instantaneous curve non-monotone.
Keep permanent deformation in a separate later model.
### Exact Trapezoidal State Update
Let `h = dt / 2`.
Let `alpha_r = h / tau_r`.
Let `beta_r = alpha_r / (1 + alpha_r)`.
The endpoint branch update is:
```text
x_r,n = (1 - 2 * beta_r) * x_r,p
+ beta_r * C_r * (f_p + f_n)
```
For a fixed contact map, condense endpoint material compliance into:
```text
A = transpose(B) * (C_0 + sum(beta_r * C_r)) * B
```
The previous branch state supplies an explicit gap bias.
Solve this bias and `A` inside the same normal complementarity step.
Do not decay the state first and clamp a tensile result later.
### Discrete Energy Identity
On each active positive-semidefinite subspace, stored energy is:
```text
E = 0.5 * transpose(f) * C_0 * f
+ sum(0.5 * transpose(x_r) * inverse(C_r) * x_r)
```
The required trapezoidal identity is:
```text
transpose(f_bar) * (d_n - d_p)
= E_n - E_p
+ sum((tau_r / dt)
* transpose(dx_r) * inverse(C_r) * dx_r)
```
The final sum must be nonnegative.
Divide this loss energy by `dt` to report loss power.
Complementarity must select `lambda = 0` for a tensile trial force.
The retardation state must then recover under zero load.
The same sample can reactivate contact when recovery closes the gap.
### Moving Footprint Requirement
A moving contact changes `B`.
Exact power then includes work from `B_n - B_p`.
That work contributes to the record and stylus tangential reaction.
The first implementation can freeze midpoint geometry only with a work-conjugate discrete map.
The validation must measure the residual from this frozen map.
A scalar filter attached to the stylus moves memory with the stylus.
It does not represent groove material during a reversal or revisit.
### Candidate Admission Contract
Define measured maximum penetration as `delta_max`.
The tracer must retain every candidate within `delta_max` of the certified rigid global height.
Each retained row must include these values:
- Absolute groove-material coordinate.
- Stable wall identity.
- Ordered contact-position interval.
- Certified height interval.
- Certified gap interval.
- Slope bounds.
- Curvature or footprint data.
The certificate must prove that every omitted candidate is farther than `delta_max`.
The certificate must also bind the fixed candidate capacity.
Reject force, pressure, indentation, uncertainty, or candidate count outside the measured domain.
Do not truncate a candidate set.
Keep `PVC-004` rejected until the compliant multiple-contact gate passes.
### Coupled Patch Requirement
The groove wall is one continuous body.
Nearby footprints require reciprocal cross-compliance `C_ij`.
Independent springs can count the same deforming volume more than once.
Do not enable independent local springs as calibrated multiple-patch physics.
Do not average candidate slopes.
Do not split force equally.
Use each candidate's multiplier, slope, footprint, friction vector, and cross terms.
Merge columns only when a certificate proves that they represent one equivalent contact.
Report only the unique resultant when the measured operator remains rank deficient.
Do not report an arbitrary multiplier division as pressure.
### Fixed-Cap Solve
The frictionless endpoint system has the form:
```text
g = q + W * lambda
```
`W` contains the mechanical Delassus compliance and material matrix `A`.
Positive-semidefinite material matrices make this a monotone linear complementarity problem.
A positive instantaneous contact compliance can make the result unique.
Material memory and cross-coupling can produce non-prefix active sets.
Therefore, the proposed `N + 1` height-prefix shortcut has no current proof.
Use a deterministic fixed-cap principal-pivot or active-set solver.
Alternatively, exhaust every active set within a declared small capacity.
Sliding and sticking remain mixed complementarity modes.
Candidate-specific friction can remove matrix symmetry.
Verify passivity for every accepted mode.
### Groove-Coordinate State
Add a per-player `GrooveDeformationState`.
Keep it separate from immutable cut geometry and irreversible damage.
Key each cell by source identity, generation, wall, and canonical absolute groove coordinate.
Record direction and stylus position must not move the state.
Forward and reverse visits to one cell must see the same state.
Separate player instances must not share deformation.
Use fixed page-aligned slots and a fixed relaxation count.
Use fixed kernel support and a fixed contact capacity.
Store branch states, prior distributed load, last-update sample, slot identity, and energy.
Do not silently evict a cell with recoverable energy.
Reject a profile when measured recovery exceeds its bounded residency horizon.
Snapshots must store all material identities and states.
Rollback must prepare changes in a fixed journal.
Publish the journal only after all same-sample systems succeed.
### Friction Vector Counterexample
Let `e_t` be the along-groove unit vector.
The surface normal is proportional to `n_w - p * e_t`.
One longitudinal unit tangent is:
```text
(e_t + p * n_w) / hypot(p, 1)
```
For projected wall force `lambda`, the physical normal magnitude is `lambda * hypot(p, 1)`.
The Coulomb friction magnitude is `mu * lambda * hypot(p, 1)`.
Its along-groove component has magnitude `mu * lambda`.
Its wall-coordinate component has magnitude `mu * lambda * p`.
The former code used the complete Coulomb magnitude as the along-groove force.
At `abs(p) = 0.5`, the record-torque factor is `1.118033988749895`.
The possible torque error is approximately `11.8034` percent.
The former reduction also omitted the wall-coordinate friction component.
This finding does not apply if `mu` is an identified effective horizontal coefficient.
No current measurement establishes that alternative definition.
### Sliding Vector Correction Checkpoint
- **Observation date**: 2026-08-02.
- **Implementation scope**: The correction covers nonzero sliding velocity.
- **Projected normal force**: The contact multiplier is `lambda`.
- **Record-tangent force**: The value is `-sign(v) * mu * lambda`.
- **Wall-coordinate force**: The value is the record-tangent force multiplied by `p`.
- **Local friction power**: The value is `-mu * lambda * (1 + p * p) * abs(v)`.
- **Coupling result**: Both force components are inside the same pickup and deck solve.
- **Force accounting result**: Telemetry uses the complete per-wall resultant.
- **Zero-slope result**: The corrected branch is bit-identical to the former scalar branch.
- **Reversal result**: Both friction components change sign for nonzero forward and reverse velocities.
- **Local passivity result**: Local friction power is negative for both sliding directions.
- **Focused debug suite**: 39 tests pass, zero tests fail, and one test is ignored.
- **Focused release suite**: 39 tests pass, zero tests fail, and one test is ignored.
- **Command**: `cargo test --lib physical::contact::tests`.
### Skating-Port Power Audit
- **Observation date**: 2026-08-02.
- **Status**: Confirmed scope gap and corrected reduced-model coupling.
- **Definition**: Let `K` be the skating-force conversion for one newton of along-groove force.
- **Slip velocity**: `v_slip = v_record - K * v_body_lateral`.
- **Contact velocity**: `gamma_dot_i = v_slip + p_i * v_wall_i`.
- **Correction**: Joint normal rows include the reciprocal `p_i * K` body-velocity term.
- **Correction**: The flat sticking row uses the same power-conjugate arm mapping.
- **Correction**: Sliding direction uses the solved contact velocity.
- **Correction**: Mixed nonzero contact directions reject without state change.
- **Correction**: Telemetry sums record, body, and cross-plane friction power.
- **Snapshot result**: Restore rejects impossible Coulomb modes, signs, and magnitudes.
- **Test**: `midpoint_sliding_closes_record_body_and_cross_plane_coulomb_power`.
- **Test**: `standalone_sliding_uses_the_reciprocal_body_velocity_and_all_force_ports`.
- **Limit**: This result applies to the reduced rigid-longitudinal sliding model.
- **Limit**: It does not prove coupled uniqueness over the complete profile domain.
### Same-Wall Reduction Power Audit
- **Status**: Confirmed limit for the test-qualified multiple-contact path.
- **Finding**: The momentum solve uses the same-wall mean slope `p_bar`.
- **Finding**: The telemetry sums each contact's squared slope.
- **Solve power term**: The reduced term is `f_total * (1 + p_bar * p_bar) * v`.
- **Reported power term**: The detailed term uses `f_i * (1 + p_i * p_i) * v`.
- **Counterexample**: A reflected pair with slopes `+p` and `-p` has `p_bar = 0`.
- **Effect**: Its reduced solve has no wall-friction component, but detailed telemetry reports one.
- **Current protection**: Production rejects same-wall sets without a physical qualification.
- **Requirement**: Keep that rejection until the solve represents each accepted contact port.
The solve needs a strict inward force-direction margin.
For each certified wall, require:
```text
mu * maximum_abs_slope < 1 - 1e-6
```
The current general trace cap permits `maximum_abs_slope = 16`.
The default friction coefficient is `0.25`.
Therefore, the general cap and default coefficient are not compatible by themselves.
The player must use each certificate's actual maximum absolute slope.
The rapid fixture product is approximately `0.12291825`.
The `PVC-004` product is approximately `0.11019775`.
Both products are inside the required margin for the default coefficient.
This scalar test is necessary for the local wall-force direction.
It is not a sufficient coupled Painlevé or uniqueness proof.
The coupled normal response is the Delassus matrix `W = J * A^-1 * G`.
The response also depends on skating geometry, masses, damping, deck inertia, and both contacts.
The accepted profile domain needs a P-matrix proof or an equivalent bounded proof.
### Confirmed Coupled Normal-Response Counterexample
- **Observation date**: 2026-08-02.
- **Status**: The canonical production operator confirms that the scalar gate admits a negative minor.
- **Timestep**: Use `1 / 192000` seconds.
- **Record inertia**: Use `1e-7` kilogram square meters.
- **Stylus moving mass**: Use `0.01` kilograms.
- **Friction coefficient**: Use `0.25`.
- **Wall slope**: Use `-0.125`.
- **Groove radius**: Use `0.14605` meters.
- **Groove pitch**: Use `125e-6` meters per revolution.
- **Tonearm**: Use the default tonearm geometry and axis values.
- **Generator coefficient**: Use a small valid positive value, such as `1e-12` volt seconds per meter.
- **Deck mode**: The deck bearing and slipmat slide.
- **Hand mode**: The hand is separated.
- **Pickup mode**: The pickup bearing sticks.
- **Tangential mode**: The stylus uses positive sliding.
- **Scalar admission**: `mu * abs(p)` is `0.03125`.
- **Deck contribution**: The value is `-0.007866686227009534`.
- **Pickup contribution**: The value is `0.000536859604648429`.
- **One-wall minor**: `W_00` is `-0.007329826622361105`.
- **Confirmed conclusion**: The one-contact Delassus operator is not positive for this witness.
- **Possible physical effect**: A one-dimensional complementarity problem can have two solutions or no solution.
- **Canonical builder**: Production and the witness use the same midpoint `H` and `G` coefficients.
- **Order protection**: Separate types identify equation-row and velocity-column coordinate orders.
- **Signed-zero test**: `contact_operator_writers_preserve_the_signed_zero_layout` protects bit-level assembly parity.
- **Permanent test**: `admitted_midpoint_configuration_has_a_negative_normal_minor` reproduces `W_00`.
- **Former mode count**: The 48 signed families cover only groove-wall sliding.
- **Mechanical basis**: There are 24 nominal mobility classes after kinetic signs share one left-hand side.
- **Land addition**: Record-land sliding adds 48 labeled families.
- **Sticking addition**: Groove and land sticking add 24 labeled families for each surface.
- **Boundary addition**: Held-boundary groove contact has a different spiral-origin row.
- **Separated addition**: The solver also evaluates separated labels for active masks.
- **Groove catalog**: Four stylus labels, two origin laws, and 24 mechanical classes give 192 families.
- **Land catalog**: Four stylus labels and 24 mechanical classes give 96 families.
- **Contacting total**: The structurally different labeled catalog contains 288 families.
- **Contact-mask total**: Nonempty principal masks give 672 contacting mode-mask combinations.
- **Zero-mask total**: Actual separated zero-mask branches increase this count to 744.
- **Unrelated sample count**: The bounded-box test uses 768 samples from 96 families and eight corners.
- **Runtime maximum**: One hand-active groove sample evaluates 1,053 current branches.
- **Registered bound**: The 1,296 bound is safe, but it is not the exact current fallback count.
- **Implementation status**: The crate generates the versioned 24-class and 288-label catalog.
- **Point builder**: It reuses the production deck, pickup, constraint, `H`, and `G` assembly.
- **KKT parity**: A permanent test compares every active point coefficient by exact `f64` bits.
- **Coverage result**: Every labeled family builds a finite point mobility and contact response.
- **Diagnostic result**: Each build reports rank, dependency, scaled pivots, and maximum backward error.
- **Coordinate test**: Six distinct values protect the different equation and velocity orders.
- **Witness result**: The builder reproduces `W_00 = -0.007329826622361105` for the negative witness.
- **Current limit**: The builder supplies point values, not outward interval bounds.
- **Admission limit**: The player does not bind these results to a loaded source.
- **Required proof**: Prove both diagonal minors and the determinant for every family.
- **Required domain**: Bind radius, both slopes, pitch, tonearm geometry, damping, profile, source, and generation.
- **Required failure**: Reject an uncertified operator before any state change.
- **Current uncertainty**: The test confirms one point. It does not yet certify a bounded source domain.
- **Required method**: Use outward interval bounds over radius, slope, and every configuration-dependent coefficient.
### Fixed-Mode and Hybrid Uniqueness Scope
- **Fixed-mode theorem**: A P-matrix gives one solution for every right-hand side of one fixed LCP.
- **Scope limit**: Different fixed systems can each have one solution and still overlap.
- **Exact production overlap**: Set the stylus friction coefficient to zero.
- **Contact state**: Use a loaded contact with positive relative slip.
- **Sliding result**: `SlidingPositive` uses the zero-friction normal operator.
- **Separated result**: `Separated` uses the same augmented system and passes its current force check.
- **Consequence**: Both mode labels can describe the same accepted mechanical state.
- **Tolerance-band correction**: Positive-friction separation admits summed normal force through `TANGENTIAL_FORCE_TOLERANCE_N`.
- **Current tolerance**: `TANGENTIAL_FORCE_TOLERANCE_N` equals `1e-10 N`.
- **Clamp correction**: Small negative active normal forces can clamp to zero.
- **Proof action**: Certify the separated operator. Do not skip it because its force range is small.
- **Additional overlap**: Static limits above kinetic force can make stick and slide branches both feasible.
- **Deck witness start**: Set platter and record velocity to zero.
- **Deck witness control**: Separate the hand and stylus, and make the slipmat stick.
- **Deck witness torque**: Apply `0.00020 N m` of motor torque.
- **Default bearing limits**: Kinetic torque is `0.00018 N m` and static torque is `0.00024 N m`.
- **Sticking result**: Zero platter speed and `0.00020 N m` bearing torque pass the static limit.
- **Sliding result**: Positive sliding gives `3.88313554466e-9 rad/s`.
- **Sliding slipmat torque**: The result is `1.50976309976e-6 N m`.
- **Consequence**: Both branches pass and produce different next states.
- **Other overlaps**: The slipmat, hand, and pickup bearing also have static limits above kinetic limits.
- **Mask witness**: Let one free gap be `-delta`, where `0 < delta <= 1e-11 m`.
- **Inactive result**: The inactive mask accepts this penetration through `CONTACT_TOLERANCE_M`.
- **Active result**: The active mask can close the gap with a positive normal force.
- **Negative-force case**: The solver accepts forces down to `-1e-10 N` and clamps them to zero.
- **Theorem mismatch**: These tolerance bands are not the exact complementarity law used by the P-matrix theorem.
- **Current selection**: Branch order and prior-state hints select the first accepted mode.
- **Interpretation**: This is a deterministic algorithmic selection law.
- **Claim limit**: It is not evidence for one unique physical hybrid mode.
- **Requirement**: Add a constitutive transition law or prove that valid mode interiors cannot overlap.
- **Alternative**: Form one global mixed complementarity problem and prove a global uniqueness property.
- **Alternative**: Evaluate all valid candidates and reject materially different results.
- **Common requirement**: Use outward-certified guards and define redundant static-force selection.
- **Reference**: The fixed-LCP P-matrix result is at <https://doi.org/10.1137/0120041>.
- **Reference**: Rigid frictional contact limits are reviewed at <https://arxiv.org/abs/1601.03545>.
### Midpoint Constraint-Rank Counterexample
- **Former projection**: Hand sticking and stylus sticking both used the deck vector `[0, 1]`.
- **Production stylus row**: The full row also contains `-2 * K / r` times lateral body velocity.
- **Former effect**: The selector could reject unequal targets before a solve.
- **Former effect**: Equal targets could remove a physically independent stylus constraint.
- **Correction**: Keep both rows when the pickup bearing slides and `K` is nonzero.
- **Dependent case**: Deduplicate the rows when the pickup bearing sticks or `K` is zero.
- **Permanent rank test**: `hand_and_stylus_sticking_keep_the_independent_body_constraint` covers both cases.
- **Permanent branch test**: `joint_branch_enforces_independent_and_dependent_sticking_equalities` covers both cases.
### Coupled Contact Certificate Design Record
- **Owner**: `PhysicalRecordPlayer` must own the validated certificate with the loaded source.
- **Exclusion**: Do not store this profile-dependent proof in the profile-independent trace certificate.
- **Config binding**: Hash every exact `PhysicalPlaybackConfig` field with versioned tags.
- **Source binding**: Bind the complete `PhysicalGrooveSourceIdentity` and current generation.
- **Domain binding**: Bind radius bounds, slope bounds, origin regime, surface, and family catalog.
- **Algorithm binding**: Bind certificate, operator, family-set, and numerical-solver versions.
- **Margins**: Store lower bounds for both diagonal minors and the determinant.
- **Land margin**: Store the one-dimensional land response lower bound.
- **Numerical margin**: Store a solve-conditioning margin in addition to physical minors.
- **Load gate**: Build the proof before `load_source` changes player state.
- **Replacement gate**: Build a prospective proof before a paged cache replacement commits.
- **Publication gate**: Realtime page publication must update source identity and proof atomically.
- **Render gate**: Compare cached source and certificate identities once for each render block.
- **Sample gate**: Check only radius and slope enclosure bounds in the sample loop.
- **Snapshot gate**: Bind and verify the certificate identity before restore changes state.
- **Thread rule**: Run interval subdivision and family enumeration outside the audio thread.
- **Realtime limit**: Resident-page slope maxima do not bound an unpublished whole record.
- **Requirement**: Add an authoritative whole-record domain or certify each publication transactionally.
### Player-Owned Source Admission Boundary
- **Observation date**: 2026-08-02.
- **Status**: Read-only mutation-path audit complete. Integration remains open.
- **Source owner**: `PhysicalRecordPlayer` privately owns every loaded `PhysicalGrooveSource`.
- **Contiguous ownership**: A contiguous source uses an immutable `Arc<GrooveAsset>`.
- **Paged ownership**: A paged source uses an immutable `Arc<PagedGrooveCache>`.
- **Realtime ownership**: A realtime source uses one owned `Box<RealtimePagedGrooveCache>`.
- **Load boundary**: `PhysicalRecordPlayer::load_source` is the central source-load gate.
- **Paged boundary**: `replace_loaded_paged_cache_snapshot` is the immutable cache-replacement gate.
- **Realtime identity changes**: Only publication and eviction change the published realtime source identity.
- **Staging result**: Begin, ingest, reserve, commit, cancel, finish, advance, and discard do not change that identity.
- **Manifest bound**: Publication and eviction scan at most 64 published slots.
- **Manifest result**: Each scan updates the stored representation identity and maximum certified slope.
- **Bypass one**: `PhysicalGrooveSource::as_realtime_paged_cache_mut` bypasses player proof ownership.
- **Bypass two**: `PhysicalRecordPlayer::realtime_paged_cache_mut` bypasses player proof ownership.
- **Bypass three**: `PhysicalHostRenderer::realtime_paged_cache_mut` bypasses player proof ownership.
- **Required replacement**: Add a non-`DerefMut` `LoadedRealtimePagedGrooveSession`.
- **Session rule**: The session can forward staging methods without exposing the mutable cache.
- **Publication preflight**: Read the Ready page trace certificate before the cache changes.
- **Slope preflight**: Reject a Ready page when its maximum slope exceeds the profile proof envelope.
- **Failure rule**: Do not run a fallible admission check after publication changes the manifest.
- **Publication commit**: Publish the page, end the cache borrow, then read the complete new source identity.
- **Token commit**: Calculate and assign the fixed-size source-binding token before the method returns.
- **Atomicity result**: The exclusive player borrow hides the cache-and-token transition from callers.
- **Eviction commit**: Refresh the token after successful eviction.
- **Eviction domain**: Eviction cannot increase the published maximum slope.
- **Player representation**: Store source and token in one private `AdmittedPhysicalGrooveSource`.
- **Profile representation**: Store one profile-wide fixed-contact proof separately in the player.
- **Token binding**: Bind config identity, proof identity, algorithm versions, radius, slope, and source identity.
- **Load transaction**: Build the prospective token before source installation changes player state.
- **Replacement transaction**: Build the prospective token before `mem::replace` changes the paged cache.
- **Block gate**: Compare live source identity, token identity, and proof identity before each render block.
- **Sample gate**: Check traced radius and each returned slope against the proof envelope.
- **Audio-thread rule**: Do not run interval proof work during render, publication, or eviction.
- **Snapshot binding**: Store the profile proof identity and optional loaded admission identity.
- **Player version**: Increase snapshot version 10 to 11 during integration.
- **Renderer version**: Increase snapshot version 4 to 5 during integration.
- **Comment defect**: The renderer snapshot comment does not match the current exact manifest identity check.
- **WASM impact**: WASM uses the mutable renderer accessor for realtime page operations.
- **WASM rule**: Keep JavaScript method names and route their work through the session.
- **C impact**: The C facade currently loads contiguous sources only.
- **C version rule**: Internal enforcement does not require a C ABI version change.
- **Rust impact**: Removing the mutable accessors is an intentional Rust API compatibility break.
- **Test requirement**: Verify identical proof identities for identical profiles.
- **Test requirement**: Verify identity changes after one exact configuration-bit change.
- **Test requirement**: Reject out-of-envelope loads and replacements without mutation.
- **Test requirement**: Verify staging leaves source and admission identities unchanged.
- **Test requirement**: Verify accepted publication changes both identities.
- **Test requirement**: Keep a rejected page Ready and keep all published state unchanged.
- **Test requirement**: Verify eviction refreshes the binding and failed eviction changes nothing.
- **Test requirement**: Reject stale or forged snapshot proof identities transactionally.
- **Test requirement**: Verify player publication and eviction allocate no memory.
- **Existing test change**: Move invalid realtime slope rejection from render time to publication time.
- **Claim limit**: This boundary only enforces the domain that the completed proof certifies.
- **Multiple-contact limit**: Unqualified same-wall multiple contact remains outside this proof.
- **Standalone limit**: Direct pickup APIs remain outside the player source gate.
### Verified Mobility Proof Plan
- **Observation date**: 2026-08-02.
- **Status**: Bounded algebraic contact evaluation is implemented. Production admission does not use it yet.
- **Base systems**: Build one exact KKT system for each of 24 mechanical mobility classes.
- **Builder result**: All 24 mechanical classes build with the shared production KKT writers.
- **Family result**: All 288 contacting labels build with shared production contact operators.
- **Parity result**: `fixed_mode_builder_reuses_the_production_lhs_and_hg_operators` verifies exact assembly bits.
- **Coordinate result**: `typed_joint_coordinates_keep_equation_and_velocity_orders_distinct` guards coordinate mappings.
- **Diagnostic result**: The builder reports scaled pivots, backward error, rank, and dependent equalities.
- **Point-only limit**: The current builder does not enclose coefficient or solution rounding error.
- **Cartridge helper**: Config and duration now determine reciprocal electromagnetic damping without state history.
- **Canonical formula**: Runtime and proof helpers share one damping derivation.
- **Runtime result**: The affine path reuses its current-response matrix and avoids a second circuit solve.
- **State test**: Five configurations keep exact damping bits across 257 advanced cartridge states each.
- **Transform test**: Mechanical-coordinate damping matches the existing coil transform by exact bits.
- **Base verification**: All 24 point KKT systems now have outward solve enclosures.
- **Scale replay**: The verifier uses the production row and column scales.
- **Pivot replay**: The verifier uses the deterministic production pivot schedule.
- **Pivot gate**: The complete pivot interval must exclude zero and clear the production tolerance.
- **Bit parity**: Each point replay matches production mobility by exact `f64` bits.
- **Mobility enclosure**: Each outward interval contains the matching production coefficient.
- **Residual enclosure**: Each original KKT residual interval contains zero.
- **Seed pivot lower bound**: The smallest verified scaled pivot is `0.08165190678245332`.
- **Seed robustness floor**: The current initial floor is `6.4e-9`.
- **Seed mobility width**: The largest raw dynamic coefficient width is `2.664535259100376e-15`.
- **Seed residual bound**: The largest raw residual absolute bound is `2.7222668563808843e-13`.
- **Units warning**: These raw mobility and residual diagnostics contain mixed physical units.
- **Exact oracle**: Independent dyadic rational solves check representative scaled systems.
- **Bounded evaluator**: Outward intervals enclose the exact reduced algebraic operators over radius and slope boxes.
- **Config binding**: Point-mobility certificate version 2 requires the exact playback configuration identity.
- **Point coverage**: All 288 point labels verify complete production KKT mobility and contact response containment.
- **Bounded coverage**: Tests sample 768 groove corners and 288 land points against production response.
- **Rank coverage**: Tests confirm 18 independent and six dependent sticking classes.
- **Mask coverage**: An asymmetric-slope test prevents family-wide sticking rejection.
- **Zero-`K` coverage**: Tests accept zero skating-factor intervals for all 216 non-sticking families.
- **Negative witness**: The known negative `W_00` interval fails the strict positive predicate.
- **Positive seed result**: All 192 groove and 96 land labels pass raw strict predicates at the zero-slope seed point.
- **Bounded fixture result**: Interior, held, and land fixture groups each pass all 96 raw strict predicates.
- **Focused result**: Twenty-one fixed-certificate tests pass.
- **Current proof limit**: The raw intervals do not yet prove the complete production scaled-solver margin.
- **Mobility definition**: Store response as velocity rows by equation columns.
- **Equation order**: `[platter, record, tip-x, body-x, tip-z, body-z]`.
- **Velocity order**: `[platter, record, tip-x, tip-z, body-x, body-z]`.
- **Permutation test**: Use six distinct values to detect exchanged coordinates.
- **Contact evaluation**: Evaluate the shared `H` and `G` algebra over radius and slope boxes.
- **Skating algebra**: Production uses only arithmetic and square root for the force factor.
- **Skating domain**: Require `-1 < c < 1` and `1 - c * c > 0`.
- **Endpoint result**: Both program-radius factors are within one ULP of 100-digit calculations.
- **Bit change**: The outer result changed by one ULP. The inner result changed by four ULP.
- **Boundary result**: Exact geometric reach boundaries reject as unreachable.
- **Permanent test**: `algebraic_skating_factor_matches_independent_references` checks 257 radii and three high-precision results.
- **Sticking update**: Use the production tangential force column and equality row.
- **Sticking proof**: Require the nonsymmetric Schur denominator interval to exclude zero.
- **Force column**: The dynamic equation coefficients are `record = -r` and `body-x = K`.
- **Equality row**: The velocity coefficients are `record = 1` and `body-x = -2K/r`.
- **Schur denominator**: Calculate `D = h M c` with typed equation and velocity coordinates.
- **Mobility update**: Calculate `M_stick = M - (M c)(h M)/D`.
- **Symmetry warning**: The force column is not the transpose of the equality row.
- **Rank rule**: Apply the Schur update only when the stylus equality adds rank.
- **Adds-rank result**: Eighteen classes add rank when the skating factor `K` is nonzero.
- **Dependent result**: Six pickup-bearing-stick classes have a dependent stylus equality.
- **Dependency condition**: Hand sticking makes the stylus row dependent in these classes.
- **Dependency condition**: Deck-bearing and slipmat sticking together also make it dependent.
- **Dependent action**: Use base mobility and mark runtime right-hand-side compatibility as required.
- **Zero-denominator meaning**: The dependent case is not a singular Schur failure.
- **Zero-`K` edge**: A box that cannot exclude `K = 0` has an unresolved rank relation.
- **Exact-zero classification**: At `K = 0`, the production selector gives a 12-and-12 split.
- **Held-boundary domain**: Both wall slopes must use exact-zero intervals.
- **Separated operator**: A sloped separated mode retains its modulation reaction.
- **Mask-specific rule**: Positive-friction sticking requires zero slope only on each active wall.
- **Catalog warning**: Family-level reachability can hide a feasible one-wall sticking mask.
- **Catalog**: Generate 288 contacting labels from production mode mappings.
- **Solve-only result**: The versioned catalog contains 24 lowered and 24 cue-supported systems.
- **Solve-only certificate version**: Version 2 stores the complete active KKT inverse for each system.
- **Shared builder**: Production zero-contact branches and certificate responses use one canonical KKT assembler.
- **Lowered result**: All 24 lowered systems match the separated-land bases by exact bits.
- **Cue result**: Cue support changes only the typed vertical-body dynamic coefficient.
- **Verification result**: All 48 systems enclose their production unit-RHS mobility coefficients.
- **Full-inverse result**: All 48 systems replay and enclose every active KKT basis response.
- **Algebraic record count**: The 288 contact labels and 48 solve-only systems give 336 records.
- **Full-KKT correction**: A literal scaled-solver catalog needs 672 active masks and 48 solve-only systems.
- **Full-KKT total**: That alternative numerical catalog contains 720 subjects.
- **RHS limit**: Unit-RHS mobility does not certify live constraint-row right-hand sides.
- **Groove proof**: Require positive lower bounds for both diagonals and the determinant.
- **Land proof**: Require a positive lower bound for its scalar response.
- **Subdivision**: Split the largest normalized radius or slope width when a result is inconclusive.
- **Work cap**: Fail closed when the fixed box or depth limit is exhausted.
- **Radius domain**: Use the closed program-radius interval from the playback configuration.
- **Interior slope domain**: Use independent `[-S, S]` intervals for both groove walls.
- **Held and land slopes**: Use exact-zero intervals for both walls.
- **Candidate ladder**: Test exact powers of two from `2^4` through `2^-40`, then zero.
- **Candidate claim**: Report the largest certified ladder value, not a maximum physical slope.
- **Failed-attempt record**: Keep every larger candidate and its stable failure class.
- **Exhaustion meaning**: Report exhausted work as inconclusive, not as physical invalidity.
- **Stable algebraic order**: Evaluate 288 contact families, 24 lowered systems, then 24 cue-supported systems.
- **No algebraic aggregation**: Keep all 336 algebraic records in certificate version 1.
- **Split selection**: Select the active axis with the smallest accumulated split depth.
- **Split tie order**: Prefer radius, then left slope, then right slope.
- **Child order**: Push the upper child first so the lower child is evaluated next.
- **Unsplittable rule**: Fail closed when the midpoint equals either endpoint.
- **Depth limit**: Version 1 permits 36 subdivisions on one subject path.
- **Subject work limit**: Version 1 permits 8,191 evaluated boxes for one subject.
- **Candidate work limit**: Version 1 permits 1,048,576 evaluated boxes for one candidate.
- **Physical margin rule**: Keep dimensional minor bounds separate from numerical conditioning margins.
- **Identity input**: Bind every subject, domain, result, margin, attempt, limit, and exact float bit.
- **Identity encoding**: Use explicit little-endian integers and explicit enum and option tags.
- **Multiple-contact gate**: Reject a runtime wall with more than one represented contact.
- **Scope note**: Certificate version 1 proves only fixed-mode, single-contact normal response.
- **Arithmetic**: Expand round-to-nearest results with adjacent finite `f64` values.
- **Arithmetic status**: The finite outward interval core is implemented.
- **Supported operations**: It includes add, subtract, multiply, divide, square, square root, negation, and interval queries.
- **Finite rule**: An operation rejects if its outward result would require an infinite bound.
- **Identity rule**: Exact zero, one, and negative-one operations preserve maximum finite operands.
- **Concurrency rule**: Do not change the process rounding mode.
- **Exact oracle**: Decode finite `f64` values into exact dyadic rationals in tests.
- **Square-root oracle**: Verify each returned bound by exact rational squaring.
- **Random oracle**: Check 4,096 deterministic normal and subnormal bit-pattern cases.
- **Focused result**: Thirteen interval tests pass with no failure.
- **Build result**: Workspace, all-target, and WASM checks pass.
- **Conditioning gate**: Require dimensionless solve margins above `6.4e-9` initially.
- **Schur warning**: A raw denominator that excludes zero can still be too weak for the production scaled solver.
- **Minor warning**: A raw positive contact minor does not establish production full-mask KKT conditioning.
- **Bounded mobility audit**: Sampled direct checks found 177 misses among 38,016 production mobility coefficients.
- **Miss shape**: All 177 interval coefficients were structural zero singletons.
- **Maximum residue**: The largest production residue was `5.421010862427522e-20`.
- **Residue ceiling**: No sampled raw discrepancy exceeded `1e-15`.
- **Miss labels**: Twelve added-rank sticking labels from six pickup-bearing-stick mechanical classes contain the misses.
- **Interpretation**: The observed raw residues are small, but mixed units prevent a materiality conclusion.
- **Containment result**: The residues still disprove bounded production-solver containment.
- **Required proof**: Bind normalized margins rigorously to production elimination or verify each scaled active-mask solve.
- **Admission status**: Do not use the interval contact evaluator as a player gate before this link exists.
- **Failure classes**: Distinguish bad minors, singular mobility, singular Schur updates, weak margins, and exhausted work.
- **Scope**: A successful strict P-matrix certificate would prove normal-contact uniqueness for its fixed mode and certified box.
- **Exclusion**: It does not prove unique selection across overlapping hybrid modes.
### Scaled-Solver Replay Audit
- **Audit status**: A read-only prototype tested full production-scaled elimination.
- **Production status**: The crate does not contain or use this prototype.
- **Replay method**: Certify each selected row scale, column scale, pivot row, pivot value, and elimination operation.
- **Exact operations**: Keep pivot self-division equal to one and eliminated pivot coefficients equal to zero.
- **Narrow domain**: Use radius interval `[0.099, 0.101] m` with the seed profile.
- **Narrow result**: All 288 family systems passed the fixed-path replay.
- **Sample count**: Check 31,104 endpoint and midpoint mobility coefficients.
- **Sample result**: All sampled production coefficients stayed inside their intervals.
- **Residue result**: The replay enclosed all 177 previously missed structural-zero residues.
- **Maximum raw width**: The largest interval width was `3.749987697e-4` in its coefficient units.
- **Maximum normalized width**: The largest width divided by its family mobility scale was `3.200882e-4`.
- **Point normalized width**: At radius `0.100 m`, the corresponding ratio was `1.706e-15`.
- **Full domain**: The seed profile spans radius interval `[0.060325, 0.14605] m`.
- **Single-box result**: One fixed replay path failed for all 54 adds-rank labels at a stylus-row scale switch.
- **Half-box result**: Two radius halves left nine labels unresolved in each half.
- **Quarter-box result**: Four uniform radius quarters certified all 288 labels.
- **Quarter one ratio**: Its maximum family-scale width ratio was `0.441` percent.
- **Quarter two ratio**: Its maximum family-scale width ratio was `0.404` percent.
- **Quarter three ratio**: Its maximum family-scale width ratio was `0.474` percent.
- **Quarter four ratio**: Its maximum family-scale width ratio was `1.498` percent.
- **Point residual bound**: Direct interval residual evaluation gave `1.275e-13` at radius `0.100 m`.
- **Point production value**: The sampled production backward error was `5.55e-17`.
- **Box residual bound**: Direct interval residual evaluation gave `1.416` on `[0.099, 0.101] m`.
- **Required threshold**: Production rejects a backward error greater than `1e-10`.
- **Failure cause**: Direct intervals lose matrix-and-solution correlation across a nonzero radius interval.
- **Audit conclusion**: The mobility replay is practical with subdivision. The naive residual proof is not practical.
- **Replacement candidate**: Use verified base solves with canonical sticking and normal-contact Schur updates.
- **Runtime candidate check**: Verify the actual KKT residual and forward-error bound before branch acceptance.
- **Uncertainty**: This audit uses one profile and one prototype. It can be wrong for other valid configurations.
- **Required target test**: Verify residual FMA behavior on native and `wasm32-unknown-unknown` targets.
- **Required switch test**: Cover scale ties, pivot ties, zero skating factor, and unit body-coefficient crossings.
- **Required stress test**: Cover near-threshold pivots, extreme valid profiles, stop, reversal, and rapid signed motion.
### Certified Block-Solver Replacement Hypothesis
- **Status**: The verified base inverses are implemented. The runtime block replacement is not implemented.
- **Goal**: Replace parameterized full-KKT elimination with the same algebra used by the bounded contact evaluator.
- **Base requirement**: Store the verified complete inverse of each fixed base KKT system.
- **Base result**: Certificate version 2 stores point and interval values for each complete active inverse.
- **Catalog scope**: The catalog contains all 24 lowered systems and all 24 cue-supported systems.
- **Path result**: Every right-hand side in one system uses one common verified scale-and-pivot path.
- **Maximum scaled-pivot width**: The catalog-wide value is `6.66133814775094e-15`.
- **Maximum dynamic-solution width**: The catalog-wide value is `2.9198865547641623e-14`.
- **Maximum full-solution width**: The catalog-wide value is `1.1812062439275908e-10`.
- **Maximum residual width**: The catalog-wide value is `2.625029082992115e-10`.
- **Maximum residual absolute bound**: The catalog-wide value is `1.3136514098732735e-10`.
- **Units warning**: Full-solution and residual values mix physical units. They are not normalized robustness margins.
- **Current gap**: Production does not calculate contact branches from these inverses.
- **Base solve**: Multiply the complete base inverse by the actual runtime right-hand side.
- **Sticking update**: Apply the nonsymmetric stylus force column and equality row as one rank-one update.
- **Contact update**: Solve each active one-wall or two-wall normal Schur system directly.
- **Dependent-row rule**: Apply no sticking update when the row is dependent.
- **Compatibility rule**: Keep the existing runtime right-hand-side compatibility test for a dependent row.
For base matrix `A0`, base right-hand side `b`, and base result `u`:
```text
u = inverse(A0) * b
```
For independent sticking force column `c`, equality row `h`, and equality target `s`:
```text
y = inverse(A0) * c
D = h * y
phi = (h * u - s) / D
u = u - y * phi
```
Here, `G` contains physical normal-force right-hand-side columns.
The assembled KKT matrix contains `-G`.
For each active normal column `G_j`, calculate:
```text
Z_j = inverse(A0) * G_j
z_j = h * Z_j / D
Y_j = Z_j - y * z_j
```
When sticking does not add rank, use `Y_j = inverse(A0) * G_j`.
For active gap rows `H` and gap targets `q`, calculate:
```text
W = H * Y
v = q - H * u
lambda = inverse(W) * v
u = u + Y * lambda
phi = phi + z * lambda
```
- **Multiplier update**: Normal contact changes the independent stylus multiplier as well as the base coordinates.
- **Dependent multiplier**: Keep the stylus multiplier at exact zero when its equality is dependent.
- **Former ambiguity**: Earlier notes did not distinguish physical `G` from the negative KKT matrix column.
- **Correction**: This document now uses physical right-hand-side `G` consistently.
Use these direct normal solves:
```text
one wall:
lambda0 = v0 / W00
two walls:
det = W00 * W11 - W01 * W10
lambda0 = (v0 * W11 - W01 * v1) / det
lambda1 = (W00 * v1 - v0 * W10) / det
```
- **Layout rule**: Insert normal multipliers before static multipliers in the production solution layout.
- **Sign rule**: Do not transpose `h` to make `c`. The two operators are not symmetric.
- **Candidate rule**: Keep the existing production candidate checks after the block solve.
- **New guard**: Add one rigorous point residual and forward-error enclosure before branch acceptance.
- **Scale rule**: Give each proof leaf fixed power-of-two row and column scales.
- **Identity rule**: Bind every signed scale exponent to the proof identity.
For `S = R * A * C`, let the proof supply `beta >= norm_inf(inverse(S))`.
At runtime, use the actual finite `A`, `b`, and calculated `x_hat`:
```text
r = b - A * x_hat
rho = norm_inf(R * r)
e_scaled = beta * rho
error_j = abs(C_j) * e_scaled
```
- **Point-interval rule**: Calculate `r` with outward point intervals and explicit operation order.
- **No-RHS-bound result**: This calculation covers the actual right-hand side without a global right-hand-side limit.
- **Finite rule**: Reject overflow, underflow in an exact scale, or an unbounded interval.
- **Backward-error rule**: Require a rigorous upper bound no greater than `1e-10`.
- **Forward-error rule**: Require every component interval `[x_hat - error, x_hat + error]` to remain finite.
- **Point-guard rule**: Every existing branch guard must pass at `x_hat`.
- **Interval-guard rule**: Every branch guard must also pass over the complete solution enclosure.
- **Guard scope**: Check gaps, forces, velocity signs, friction limits, power, and separation tolerances.
- **Ambiguity rule**: Reject a branch when its enclosure crosses a guard boundary.
- **Enumeration rule**: Continue the existing deterministic branch order after one branch rejects.
- **Transaction rule**: Change deck and pickup state only after both point and interval guards pass.
- **Failure rule**: If all branches reject, return the existing no-consistent-mode error.
- **Output rule**: A failed sample must not change state or output.
- **Floating contract**: Require IEEE-754 binary64 with round-to-nearest and ties-to-even.
- **Underflow contract**: Require gradual underflow and reject unsupported target behavior.
- **FMA contract**: Use fused `mul_add` only at specified operations.
- **Proof scope**: This design proves numerical containment for one fixed mode and one-contact model.
- **Exclusion**: It does not prove hybrid uniqueness, multiple contact, tracing uniqueness, or material fidelity.
- **Possible error**: Point residual intervals can still become too wide for extreme valid right-hand sides.
- **Disproof test**: Use huge finite right-hand sides and require certification or transactional rejection.
### Pointwise Candidate Validator Prototype
- **Status**: This read-only prototype does not change production behavior.
- **Scope**: The prototype instrumented the current full-KKT branch solves.
- **Replacement limit**: These results do not validate the proposed block solver.
- **Allocation result**: The instrumented active-mode solve still passes the no-allocation test.
- **Verifier result**: Every instrumented solve passed the pointwise verifier.
- **Maximum normalized residual**: The largest value was less than `2.95e-16`.
- **Production threshold**: The current backward-error threshold is `1e-10`.
- **Required certificate data**: Store an outward bound for every component of the complete KKT inverse.
- **Rejected shortcut**: A six-by-six dynamic mobility cannot bound normal or static multipliers.
For residual interval `R_i`, define its magnitude bound as:
```text
rho_i = max(abs(lower(R_i)), abs(upper(R_i)))
```
Let `C_ij` contain an outward upper bound for `abs(inverse(A)_ij)`.
```text
epsilon_i = upward_sum_j(C_ij * rho_j)
```
The exact solution component is then inside this interval:
```text
[x_hat_i - epsilon_i, x_hat_i + epsilon_i]
```
- **Acceptance rule**: Require a normalized residual no greater than `1e-10`.
- **Finite rule**: Require each `epsilon_i` and each solution interval to be finite.
- **Guard rule**: Require the complete solution box to satisfy every active branch predicate.
- **Ambiguity rule**: Reject the branch if one predicate interval crosses its boundary.
- **Scalar fallback**: A verified infinity-norm inverse bound is valid but less precise.
- **Preferred method**: Use componentwise inverse bounds because the KKT variables have different physical units.
| Seed normal | 15 | `2.910e-11` | `2.183e-11` | `1.687e-16` | `5.852e-12` |
| Rapid `+20` | 45 | `2.328e-10` | `1.746e-10` | `2.355e-16` | `3.000e-11` |
| Rapid `-20` | 88 | `2.328e-10` | `1.746e-10` | `2.355e-16` | `3.000e-11` |
| Stop | 119 | `5.551e-17` | `4.163e-17` | `1.769e-16` | `8.930e-17` |
| 256-step reversal | 371 | `1.455e-11` | `1.091e-11` | `2.944e-16` | `1.364e-11` |
- **Seed dynamic bounds**: The six maxima were `5.121e-15`, `5.121e-15`, `3.219e-17`, `2.728e-17`, `5.784e-21`, and `7.879e-21`.
- **Rapid positive dynamic bounds**: The six maxima were `3.744e-14`, `5.614e-14`, `4.783e-15`, `2.985e-15`, `7.123e-19`, and `7.487e-19`.
- **Rapid negative dynamic bounds**: The six maxima matched the positive fixture except for the fifth value, which was `7.182e-19`.
- **Stop dynamic bound**: Every dynamic component bound was no greater than `4.462e-20`.
- **Reversal dynamic bounds**: The six maxima were `2.292e-15`, `2.924e-15`, `7.039e-16`, `5.829e-16`, `3.945e-16`, and `9.874e-19`.
- **Static-multiplier result**: Static multipliers produced the largest forward bounds in the seed and rapid fixtures.
- **Next test**: Repeat this fixture audit on each proposed base-plus-Schur result.
### Runtime Branch Guard Inventory
- **Status**: This read-only audit records every known branch predicate before block-solver integration.
- **Purpose**: A small residual does not certify a branch unless its complete solution enclosure passes these predicates.
- **Normal tolerance**: `CONTACT_TOLERANCE_M = 1.0e-11 m`.
- **Force tolerance**: `TANGENTIAL_FORCE_TOLERANCE_N = 1.0e-10 N`.
- **Sticking velocity tolerance**: `TANGENTIAL_VELOCITY_TOLERANCE_M_S = 1.0e-12 m/s`.
- **Friction-power allowance**: `1.0e-18 W`.
The active solution uses this coordinate order:
| 0 | Platter angular velocity |
| 1 | Record angular velocity |
| 2 | Tip lateral velocity |
| 3 | Tip vertical velocity |
| 4 | Body lateral velocity |
| 5 | Body vertical velocity |
| 6 and later | Active multipliers |
Active multipliers use this order:
1. Active normal multipliers in ascending constraint order.
2. Pickup-bearing multiplier.
3. Deck-bearing multiplier.
4. Slipmat multiplier.
5. Hand multiplier.
6. Stylus sticking multiplier.
- **Mask warning**: For mask `0b10`, solution index 6 belongs to wall 1.
- **Dependent stylus rule**: A missing stylus multiplier means exact zero after the runtime compatibility check.
- **Normal multiplier meaning**: Each normal multiplier is a projected normal force.
- **Stylus multiplier meaning**: The stylus multiplier is the total record-reaction tangential force.
For each raw active normal multiplier `lambda`:
```text
lambda >= -1.0e-10 N
projected_force = max(lambda, 0)
```
- **Clamp warning**: The clamp creates a derivative discontinuity at zero.
- **Enclosure clamp**: Map `[lo, hi]` to `[max(lo, 0), max(hi, 0)]`.
- **Inactive gap guard**: Require `gap >= -1.0e-11 m` for each inactive available constraint.
- **Active gap omission**: The current validator does not check active gaps after solving.
- **KKT role**: The active equality supplies a mathematical zero-gap condition.
The tangential relative velocity is:
```text
gamma = 0.5 * groove_radius * (previous_record_velocity + record_velocity)
- skating_factor * body_lateral_velocity
```
Sticking requires:
```text
normal_force > 1.0e-10 N
abs(gamma) <= 1.0e-12 m/s
abs(total_tangent_force - modulation_force)
<= friction_coefficient * normal_force + 1.0e-10 N
```
Sliding-positive requires `normal_force > 1.0e-10 N` and `gamma > 0`.
Sliding-negative requires `normal_force > 1.0e-10 N` and `gamma < 0`.
- **Sliding sign warning**: Sliding uses a strict zero boundary, not the sticking velocity tolerance.
- **Separated tolerance**: Positive-friction separation permits normal force no greater than `1.0e-10 N`.
- **Zero-friction exception**: A separated branch can carry positive normal load when friction is zero.
- **Friction-cone rule**: The complete interval must remain inside the cone and its force tolerance.
- **Power rule**: The complete friction-power interval must have an upper bound no greater than `1.0e-18 W`.
- **Power warning**: Sliding power is bilinear in projected normal force and relative velocity.
- **Local-power gap**: Production checks total friction power, not each contact's local power.
Pickup-bearing sticking requires:
```text
abs(bearing_multiplier)
<= lateral_static_friction + 1.0e-10 N
```
Pickup-bearing positive requires positive body lateral velocity.
Pickup-bearing negative requires negative body lateral velocity.
Each sticking deck port requires:
```text
abs(relative_velocity) <= 1.0e-6 rad/s
abs(torque) <= static_limit + 1.0e-9 N*m
```
- **Deck sliding rule**: Each sliding mode requires a strict relative-velocity sign.
- **Hand separation rule**: A separated hand has structurally exact-zero torque.
- **Bearing relative velocity**: Use platter angular velocity.
- **Slipmat relative velocity**: Use platter velocity minus record velocity.
- **Hand relative velocity**: Use hand velocity minus record velocity.
Before a solve, production also checks these conditions:
- Reject groove sticking with positive friction when any active wall slope is not exact zero.
- Preserve the existing equality-rank threshold of `1.0e-14`.
- Preserve the dependent right-hand-side tolerance of `1.0e-12` times the applicable scale.
- Reject a system larger than 13 variables.
- Reject inconsistent row and column counts.
- Reject separated deck-bearing or slipmat modes.
The current scaled solver rejects these numerical conditions:
- A nonfinite or zero row scale.
- A nonfinite or zero column scale.
- A nonfinite scaled pivot.
- A scaled pivot no larger than `128 * EPSILON * system_size` in magnitude.
- A nonfinite active solution coordinate.
- A nonfinite normalized backward error.
- A normalized backward error greater than `1.0e-10`.
Post-validation commit adds these important conditions:
- **Stylus-torque limit**: Require `abs(stylus_torque) <= 1.0 N*m`.
- **All-gap rule**: Recalculate and check every groove-wall or land gap.
- **Active-gap result**: The commit gap test includes active constraints.
- **Pickup-time rule**: Commit uses `1 / pickup_sample_rate` for tip displacement.
- **Deck-time rule**: Candidate validation uses the deck time step.
- **Time-step tolerance**: The caller permits the two time steps to differ by 16 machine epsilons.
- **Certificate consequence**: Certify both gap formulas separately.
- **Nonfinite-gap gap**: The current groove commit can ignore a nonfinite gap during its penetration test.
- **Fail-closed requirement**: The certificate must reject every nonfinite gap.
- **Force-accounting rule**: Preserve the production split-and-residual operation order.
- **Force-equality tolerance**: Commit uses relative `1.0e-12` plus absolute `1.0e-18 N`.
- **Energy rule**: Commit finite-checks kinetic and suspension energies, which are quadratic.
- **Counter rule**: A deck step-counter overflow rejects independently of the candidate solution.
After both commits, the caller requires this exact floating equality:
```text
pickup_record_reaction_force * groove_radius == deck_stylus_torque
```
- **Reciprocity rule**: Preserve one canonical expression so this comparison remains structural.
- **Successor-state warning**: The contact booleans use `projected_force > 0`.
- **Successor ambiguity**: A raw multiplier enclosure that crosses zero cannot certify one exact next mode.
- **Strong-successor rule**: Require the raw multiplier interval to lie strictly above zero or no greater than zero.
- **Fallback rule**: Reject an inconclusive certificate branch and continue deterministic enumeration.
- **Transaction rule**: Do not call a commit that can reject after a branch has been selected.
### Playback Configuration Identity Checkpoint
- **Status**: Implemented as a non-gating building block.
- **Identity version**: `1`.
- **Leaf count**: The identity binds all 84 validated playback manifest leaves.
- **Field encoding**: Each leaf includes its stable path and explicit value-type tag.
- **Float encoding**: Each floating-point value contributes its exact `f64` bits.
- **Text encoding**: Each enum value uses a length-delimited byte string.
- **Hash protocol**: The protocol uses a versioned, domain-separated SHA-256 hash.
- **Rejected protocol**: The implementation does not hash JSON text.
- **Validation rule**: An invalid playback configuration cannot create an identity.
- **Permanent test**: Every manifest leaf changes the identity after one controlled value change.
- **Permanent test**: Field or value-type tag changes cause a manifest mismatch.
- **Permanent test**: Serialization and restoration preserve the complete identity.
- **Current limit**: The player does not bind this identity to a contact certificate yet.
This result concerns the reduced rigid sliding model.
It does not resolve sloped sticking or measured material behavior.
### Exact-Zero Sloped Sticking Regularization
- **Status**: Bounded production fallback implemented.
- **Former behavior**: A stopped, modulated groove could return `UnsupportedGrooveWallSticking`.
- **Former effect**: The player could abort the first audio frame after a seek.
- **Threshold**: Relative speed must not exceed `1.0e-9` meters per second.
- **Fallback**: The solver selects zero longitudinal traction for that sample.
- **Normal-contact rule**: The coupled normal contact solve remains active.
- **State rule**: The successor restores the configured friction coefficient.
- **Exit rule**: The next signed-slip sample uses complete kinetic friction.
- **Physical basis**: Ideal Coulomb friction includes zero traction at zero slip.
- **Calibration limit**: The threshold is not a measured PVC or stylus value.
- **Physical reason**: Static friction has one unknown traction for each loaded wall.
- **Constraint count**: The reduced model has only one global along-groove sticking constraint.
- **Consequence**: The static wall-traction distribution is not unique.
- **Rejected rule**: Do not split traction in proportion to projected normal force.
- **Rejected rule**: Do not use a minimum-norm traction split.
- **Reason**: Neither rule is an identified material law.
- **Additional limit**: One new longitudinal pickup coordinate is not sufficient by itself.
- **Requirement**: Add local tangential compliance or another measured history law for each contact.
- **Requirement**: Add an along-groove pickup coordinate and velocity.
- **Requirement**: Store tangential state at stable groove coordinates.
- **Requirement**: Reconcile contact identities when the traced contact set changes.
- **Requirement**: Keep stored-energy, loss, snapshot, rollback, and fixed-work accounting exact.
- **Calibration requirement**: Fit stiffness, loss, and static yield from vector reversal measurements.
### Zero-Speed Regularization Artifact
- **Source**: `src/physical/contact.rs`.
- **SHA-256**: `1c58d18aa61f43d732bce02df1dd2e5e983d8152dca849a74aa815fa9e7d93e4`.
- **Permanent test**: `midpoint_nonzero_slope_zero_speed_uses_bounded_zero_traction`.
- **Native regression**: A short stereo groove starts from frame 12 without a render failure.
- **Claim limit**: Rapid scratch fidelity remains open until exact-zero reversals pass this model.
### Unique-Contact Tangential-State Hypothesis
- **Status**: Working design hypothesis.
- **Confidence**: Medium for passivity and identifiability.
- **Scope**: Permit one uniquely certified contact on each wall.
- **Exclusion**: Keep same-wall multiple-patch playback closed.
- **Reason**: The current certificate does not bind footprint, cross-compliance, gaps, or maximum penetration.
Add one power-conjugate along-groove pickup coordinate `x_t`.
Also add its velocity `v_t` and suspension deflection `d_t`.
Let `rho` be the groove radius.
Let `kappa(rho)` map lateral arm-body velocity to along-groove body velocity.
Freeze `kappa(r)` for one coupled sample.
Then:
```text
d_dot = v_t - kappa(r) * v_body_lateral
F_s = -k_t * d - c_t * d_dot
```
Put `F_s` in the along-groove tip equation.
Put `-kappa(r) * F_s` in the lateral body equation.
This force pair must close the suspension power port.
Do not reuse the lateral suspension parameters without measurements.
For contact `i`, let `r` be the interval slip velocity.
The current deck uses trapezoidal record velocity.
The pickup uses endpoint backward Euler velocity.
Therefore, use:
```text
r = 0.5 * (v_record,previous + v_record,new) - v_t,new
```
Use the same interval slip for travel, force, work, and material identity.
Let `p_i` be the certified groove slope.
Let `f_i` be the friction force on the record along the groove.
The active normal relation is:
```text
v_wall_i = p_i * r
```
Define generalized tangential deformation by:
```text
gamma_dot_i = (1 + p_i * p_i) * r
```
The candidate force components are:
```text
tip wall-coordinate force = lambda_i + p_i * f_i
tip along-groove force = p_i * lambda_i - f_i
record along-groove force = -p_i * lambda_i + f_i
```
The normal and modulation power terms then cancel.
The remaining contact power is `f_i * gamma_dot_i`.
### Proposed Passive Return Map
Use one Jenkins element for each uniquely certified wall contact.
This element combines an elastic tangential spring with a Coulomb slider.
Store generalized elastic displacement `z_i`.
Require measured stiffness `k_i > 0`.
For one endpoint update:
```text
delta_gamma_i = dt * (1 + p_i * p_i) * r_n
z_trial = z_previous + delta_gamma_i
f_trial = -k_i * z_trial
```
Use the elastic branch when `abs(f_trial) <= mu * lambda_i`.
Then set `z_i = z_trial` and `f_i = f_trial`.
For positive sliding, set `f_i = -mu * lambda_i`.
For negative sliding, set `f_i = mu * lambda_i`.
Set `z_i = -f_i / k_i` on either sliding branch.
Validate the plastic-increment sign before commit.
Use one measured coefficient for yield and sliding in the first model.
Separate static and kinetic coefficients need an identified release law.
An instantaneous unmeasured force drop can destroy stored energy.
For stored energy `E_i = 0.5 * k_i * z_i * z_i`, backward Euler gives:
```text
f_i * delta_gamma_i
= -(E_i,new - E_i,previous)
- 0.5 * k_i * delta_z_i * delta_z_i
- plastic_loss_i
```
Each accepted branch must have nonnegative numerical and plastic loss.
Define `delta_z_i = z_i,new - z_i,previous`.
Define `delta_gamma_pl_i = delta_gamma_i - delta_z_i`.
Define `plastic_loss_i = -f_i,new * delta_gamma_pl_i`.
Positive sliding requires `delta_gamma_pl_i >= 0` and `f_i = -mu * lambda_i`.
Negative sliding requires `delta_gamma_pl_i <= 0` and `f_i = mu * lambda_i`.
### Fixed-Work Tangential Modes
Enumerate `Elastic`, `SlidingPositive`, and `SlidingNegative` for each wall.
Two loaded walls give at most nine tangential combinations.
The former Cartesian estimate gave 2,916 candidates.
Mask-aware enumeration gives 1,296 candidates when all hand modes are active.
The four wall masks contribute `1 + 3 + 3 + 9 = 16` material-mode combinations.
Recalculate this cap after the final compliance operator exists.
Validate these properties before commit:
- Each normal multiplier is nonnegative.
- Each elastic force stays inside its yield limit.
- Each plastic increment has the selected sliding sign.
- Force and torque remain reciprocal.
- Stored energy is nonnegative.
- Loss is nonnegative.
- The solve count does not exceed the registered cap.
### Tangential Material Identity
Do not key material state by array order or midpoint floating-point bits.
Return an outward absolute contact-coordinate interval from the certified tracer.
Map this interval into measured canonical material cells.
Use this state key:
```text
(source identity, generation, wall, canonical cell index)
```
Use the same fixed footprint map for deformation and transpose force transfer.
This pairing is necessary for discrete power closure.
Reject an interval that cannot isolate all required cell weights.
Use the typed error `TangentialContactIdentityNotIsolated` for that case.
The certified algorithms now return their outward coordinate intervals.
The result uses an exact `u64` origin and two local `f64` bounds.
This split preserves large origins without one lossy floating-point addition.
Do not reconstruct it from a midpoint, slope, page, or travel direction.
Exclude page identity and representation identity from the material key.
Those identities can change at seams or after immutable cache extension.
Use base spline cells for the first deterministic plumbing slice.
This grid does not claim a measured physical footprint.
Both closed interval bounds must select the same cell.
The final record coordinate belongs to the last real spline cell.
The provisional resolver rejects forged, nonordered, outside-source, and multiple-contact input.
Deserialization removes the private live-trace seal.
Keep the state bank outside `PickupMechanicalState`.
The solver receives only prior values and proposed updates for the two walls.
Commit the updates only after the complete coupled candidate passes.
Use fixed page-aligned or set-associative state slots.
Do not evict a slot that contains recoverable energy.
Return `TangentialStateCapacityExceeded` before any state change.
### Certified Identity Plumbing Checkpoint
- **Observation date**: 2026-08-02.
- **Schema version**: `TANGENTIAL_CONTACT_IDENTITY_VERSION` is `1`.
- **Trace coverage**: Class A, Class B, exhaustive, scalar, and multiresolution paths retain certified bounds.
- **Path coverage**: Immutable pages, real-time pages, player transforms, pickup input, and telemetry preserve the bounds.
- **Large-origin result**: An exactly representable center above `2^53` preserves its `u64` origin.
- **Large-origin limit**: General unrepresentable centers above `2^53` are not supported.
- **Cell rule**: An exact singleton at an internal join selects the right cell.
- **End rule**: Exact frame `N - 1` selects cell `N - 2`.
- **Edge blocker**: Actual outward record-edge intervals still extend outside the source and reject.
- **Seam result**: Page and whole-record traces resolve the same cell at both seam sides.
- **Direction result**: The cell remains equal for advances of `-20` and `+20` frames.
- **Raw-bounds result**: Valid page and whole-record enclosures can have different origins and widths.
- **Test rule**: Compare certified trace limits and the resolved key across representations.
- **Rejected rule**: Do not require raw bound bits to match across different coordinate origins.
- **Allocation result**: Live tracing and key resolution do not allocate memory.
- **Snapshot versions**: Pickup is `5`, player is `10`, and renderer is `4`.
Reproduction commands:
```text
cargo test --lib physical::stylus::tests
cargo test --lib physical::tangential_identity::tests
cargo test --lib physical::paged_groove::tests::bounded_cache_matches_a_monolithic_groove_asset -- --exact
cargo test --lib physical::realtime_paged_groove::tests::traces_match_immutable_pages_at_seams_in_both_directions_and_twenty_times -- --exact
```
### Hard-Cell Liveness Counterexample
- **Input**: Use a flat Class A wall and center the stylus at source frame `64`.
- **Certified result**: The lower bound is below `64`, and the upper bound is above `64`.
- **Cell result**: The interval covers base spline cells `63` and `64`.
- **Resolver result**: The resolver returns `TangentialContactIdentityNotIsolated`.
- **Phase A result**: Rigid playback continues because it does not request a material key.
- **General result**: A continuous trace must cross every finite hard cell boundary.
- **Consequence**: A hard one-cell key cannot guarantee uninterrupted rapid scratching.
- **Rejected workaround**: Do not select a cell from the interval midpoint or travel direction.
- **Requirement**: Add a continuous partition of unity with the same transpose force map.
- **Claim limit**: The current key proves identity plumbing only.
Permanent test:
```text
physical::stylus::tests::class_a_flat_integer_contact_remains_fail_closed_at_a_cell_join
```
### Continuous Material-Map Hypothesis
- **Status**: Design hypothesis only. Production does not use this map.
- **Selected coordinate**: Let `x = k + u`, where `0 <= u <= 1`.
- **Normalization**: Let `d = (1 - u)^2 + u^2`.
- **Left amplitude**: Let `a_0 = (1 - u) / sqrt(d)`.
- **Right amplitude**: Let `a_1 = u / sqrt(d)`.
- **Identity rule**: Derive possible keys from the certified interval only.
- **Weight rule**: Derive amplitudes from a tracer-selected coordinate inside that interval.
- **Boundary capacity**: A narrow interval can require three possible coefficient keys.
- **Reciprocity rule**: Use the same amplitude bits for state scatter and force gather.
The normalized amplitudes satisfy:
```text
a_0^2 + a_1^2 = 1
```
Raw linear amplitudes do not satisfy this condition.
Their effective self stiffness is:
```text
k * ((1 - u)^2 + u^2)
```
This value falls from `k` at a node to `k / 2` at a cell center.
Therefore, raw linear amplitudes are rejected for the uniform Jenkins seed.
Use one scalar Jenkins yield surface for each wall.
For coefficient state `q`, use:
```text
q_trial = q_0 + a * DeltaGamma
f_trial = a dot (-k * q_trial)
DeltaGamma_pl = (f_target - f_trial) / (k * (a dot a))
q_1 = q_trial - a * DeltaGamma_pl
```
The proposed backward-Euler energy identity is:
```text
f * DeltaGamma
= -(E_1 - E_0)
- 0.5 * k * norm_squared(Deltaq)
- D_pl
```
Require `D_pl = -f * DeltaGamma_pl >= 0`.
### Rapid-Sweep Material-State Blocker
- **Admission limit**: One physics sample can advance by `-20` or `+20` source frames.
- **Center-sweep capacity**: A 20-frame move can require at least 23 possible coefficient keys.
- **Root-motion risk**: Contact offset can move within the spherical support during that sample.
- **Branch risk**: The global contact can change envelope branches during that sample.
- **Certificate gap**: No current certificate bounds contact-root travel between physics samples.
- **Consequence**: One midpoint stencil can skip material state during intensive scratching.
- **Requirement**: Add a certified contact-trajectory bound.
- **Requirement**: Use deterministic event substeps or a proved swept return map.
- **Requirement**: Register a fixed work cap for the complete sweep.
- **Reference gate**: Compare against converged microsteps at `+20`, `-20`, stop, and one-sample reversals.
- **Claim limit**: Do not activate Jenkins state until these tests pass.
Cache generation also fails to identify one physical record instance.
Add `GrooveMaterialInstanceId` before state can survive representation changes correctly.
### Tangential-State Persistence Limit
A finite state bank cannot retain energetic cells for an unlimited record.
The model needs a measured recovery law and a bounded residency horizon.
One permitted policy retains every energetic cell until it recovers.
Another policy retires a cell below a registered residual-energy bound.
That policy must add the retired energy to reported loss.
Do not use silent least-recently-used eviction.
Do not attach material history to the moving stylus.
Without recovery measurements, long-play calibrated persistence is not possible.
### Tangential-State Required Tests
- Exact-zero sloped reversal produces unique wall tractions.
- Unequal wall histories produce unequal deterministic tractions.
- The result does not use proportional or minimum-norm splitting.
- Load, hold, reverse, slide, and unload close the energy balance.
- A forward and reverse revisit retrieves the same material state.
- Root reordering and page seams do not swap material state.
- An ambiguous cell map fails without changing state or output.
- Slot exhaustion fails without changing state or output.
- Snapshot restore and render partitioning continue bit-identically.
- Success and failure paths allocate no memory.
- The active-set count stays inside 2,916 candidates.
- The longitudinal natural frequency passes the temporal-resolution gate.
### Tangential-State Required Measurements
1. Measure along-groove moving mass, stiffness, damping, and arm transfer impedance.
2. Measure pre-sliding stiffness against normal load, wall, slope, speed, and temperature.
3. Measure stop and reversal force-displacement hysteresis.
4. Measure yield and sliding traction for the identified PVC and stylus pair.
5. Measure contact footprint, kernel support, and material-cell pitch.
6. Measure unloaded recovery and the residual-energy residency horizon.
7. Measure cross-wall transfer before any coupled two-wall material claim.
8. Measure separated-patch transfer before any same-wall multiple-contact claim.
9. Bound contact-coordinate uncertainty below the material-map tolerance.
The current groove coefficient `0.25` is estimated.
Therefore, a correct tangential-state implementation will still need calibration evidence.
### Non-smooth Normal-Cone Counterexample
A clamp can have zero slope on one side and slope `-0.5` on the other side.
The contact position and height can be unique while the force direction remains non-unique.
The spherical-envelope tangent inside that cone is not a unique physical force direction.
The scalar tracer must return `GrooveSlopeBoundNotMet` for this fixture.
Future mechanics can replace this rejection with explicit nonnegative one-sided multipliers.
### Required Measurements
1. Measure static load, penetration, unloading, and recovery for the identified PVC compound.
2. Record the wall angle, stylus profile, temperature, speed, and preload for each measurement.
3. Identify the recoverable and permanent deformation limits separately.
4. Measure phase-calibrated complex force and velocity impedance across frequency.
5. Repeat the impedance measurement at several preloads, temperatures, and groove speeds.
6. Include forward, reverse, stop, hold, and release motions.
7. Measure the fixture and pickup impedance for de-embedding.
8. Measure two-point and multiple-indenter transfer impedance against separation.
9. Measure contact footprints or pressure when the fixture permits it.
10. Measure transfer between the two 45-degree groove walls.
11. Measure revisit and reversal responses at controlled separations and delays.
12. Measure vector friction against load, speed, direction, temperature, and slope.
13. Perform before-and-after metrology during repeated-pass tests.
### Identification Risks
- Poles outside the measured band can collapse into one aggregate compliance.
- Nearby poles can be strongly correlated.
- Preload and temperature can make a linear fit local.
- Moving speed can confound temporal relaxation with spatial footprint.
- One-point data cannot separate temporal and spatial kernels.
- Height uncertainty can be comparable to indentation.
- Scalar impedance cannot identify patch or cross-wall coupling.
- Permanent deformation cannot use a recoverable branch state.
### Implementation Gates
1. Correct vector friction and state the longitudinal stylus limit.
2. Extend trace certificates with bounded candidates, gaps, and `delta_max`.
3. Keep rigid production behavior unchanged during the certificate extension.
4. Implement one qualified candidate per wall with groove-coordinate passive state.
5. Make zero material branches and `delta_max = 0` bit-identical to rigid contact.
6. Calibrate and validate single-patch compliance inside a declared domain.
7. Enable multiple patches only after spatial transfer measurements identify cross-compliance.
8. Add nonlinear convex elasticity only after measured preload sweeps require it.
9. Keep plastic damage and wear in a separate groove-coordinate overlay.
### Planned Permanent Tests
- `zero_material_branches_and_zero_penetration_are_bit_identical_to_rigid_contact`.
- `groove_material_energy_closes_for_load_hold_reverse_unload_and_revisit`.
- `deformation_stays_with_absolute_groove_coordinates_across_reversal`.
- `candidate_admission_rejects_an_omitted_gap_inside_maximum_penetration`.
- `coupled_patch_compliance_does_not_equal_independent_springs`.
- `friction_vector_closes_force_torque_and_power_for_signed_slopes`.
- `non_smooth_normal_cone_rejects_scalar_contact`.
### Explicit Rejections
- No independent local springs presented as calibrated physics.
- No stylus-attached viscoelastic memory.
- No equal force split presented as pressure.
- No `N + 1` prefix reduction without a theorem for the identified operator.
- No rigid acceptance from overlapping height intervals.
- No silent eviction of material state.
## PVC-008: Rapid Travel Skips Contact Events
### Assertion Under Test
One certified trace and one contact solve are sufficient for each 192-kilohertz sample.
This assertion applies through signed `20x` record motion.
### Counterexample Fixture
- **Source**: Use the deterministic rapid-scratch programme fixture.
- **Rates**: Sweep integer rates from signed `1x` through signed `20x`.
- **Events**: Include contact loss, recapture, an impulse, and reverse travel.
- **Second fixture**: Ramp to a stop, reverse, and return to a stop.
- **Candidate**: Use one contact step for each output sample.
- **Reference**: Use steps of at most `0.125` source frames.
- **Observer**: Use equal cartridge and phono observers for both trajectories.
- **Observer limit**: Do not return cartridge reaction force to either reduced trajectory.
### Disproved Result
The one-step rate sweep gives phono normalized RMS errors of `0.7743228579914596` and `0.7494342423073957`.
The one-step stop and reversal case gives phono normalized RMS errors near `0.98`.
Its phono absolute-integral errors are approximately `48%`.
These values prove that the skipped contact events reach the electrical observer.
### Production Correction
- **Step limit**: Permit as many as four contact steps for each output sample.
- **Travel limit**: Keep predicted travel at or below five source frames for each step.
- **Trace rule**: Run the certified global tracer for each step.
- **Solver rule**: Run the reciprocal deck, pickup, and cartridge solve for each step.
- **Phono rule**: Process the final cartridge endpoint once for each output sample.
- **Allocation rule**: Do not allocate memory in the render loop.
- **Representation rule**: Apply the same rule to contiguous, paged, and real-time paged sources.
The production telemetry field is `swept_contact_substeps`.
Expected values are one at `1x`, two at `8x`, and four at `20x`.
### Acceptance Evidence
| Wall-height normalized RMS | `1.3655193973757973` | `0.5122237543419718` |
| Wall-force normalized RMS | `1.0158164015715798` | `0.4924456991670574` |
| Reaction-torque normalized RMS | `1.0024104080839553` | `0.6568966477312397` |
| Contact-occupancy mean absolute | `0.21010711785380817` | `0.0911330880694634` |
| Left phono normalized RMS | `0.7743228579914596` | `0.43714264111544443` |
| Right phono normalized RMS | `0.7494342423073957` | `0.3959241852448356` |
The permanent stop and reversal test requires lower phono normalized RMS error in both channels.
The test also verifies the fixed four-step work limit.
### Contrary Evidence
The bounded rate-sweep candidate produces 225 contact transitions.
The reference produces 269 contact transitions.
The former one-step candidate produces 265 transitions.
The stop and reversal cartridge absolute-integral error increases to approximately `52.5%`.
These results keep the complete rapid-contact claim open.
### Real-Time Result
The measured 128-frame deadline is `0.667` milliseconds at 192 kilohertz.
The one-step normal path misses 512 of 512 measured deadlines.
The four-step rapid path also misses 512 of 512 measured deadlines.
The measured medians are approximately `2.35` and `7.37` milliseconds.
The current certified runtime trace is not real-time safe.
Precompute certified contact-envelope data before product activation.
Repeated SHA-256 certificate validation was also present in each active trace.
The validated-token correction removes that immutable proof work from rendering.
Normal mean time decreases from 2.507 milliseconds to 1.813 milliseconds.
Rapid-reversal mean time decreases from 7.902 milliseconds to 5.209 milliseconds.
Both corrected fixtures still miss 512 of 512 deadlines.
### Numerical Boundary Result
An exact `20x` render can derive `20.00000000000632383` from record-angle travel.
The mechanics endpoint remains exact `20x` in this case.
Accept at most `1e-10` derived-rate roundoff.
Clamp an accepted value to exact `20x`.
Reject a true rate excess of `0.001x`.
### Permanent Tests
```text
physical::rapid_scratch_reference::bounded_swept_candidate_reduces_stop_and_reversal_phono_error
physical::electromechanical::tests::record_player_midpoint_accepts_bounded_shorter_internal_steps
physical::player::tests::stab_and_chirp_change_phono_audio_at_one_eight_and_twenty_times
scratch_gate::tests::maximum_rate_roundoff_clamps_to_the_exact_supported_boundary
```
### Claim Limit
The bounded sweep is a material improvement against the reduced reference.
It does not prove complete coupled-player accuracy.
It does not pass the current callback deadline.
## PVC-009: Real Picture Record Player Fixture
### Source
- **Observation date**: 2026-08-02.
- **File name**: `0a7ee1f9-3a66-45e1-80a2-c1f3dd7ea0f6.png`.
- **File size**: 1,246,200 bytes.
- **Image size**: 576 by 576 pixels.
- **SHA-256**: `4238fd1b96c6f69eb0f63f28f524e6c3baa5b22840f8df4d73dab4ea38e4c35d`.
- **Title**: `Never`.
- **Artist**: `Lori Asha`.
- **Record profile**: `single45`.
- **Payload encoding**: `toned-v1`.
The file is an external test artifact in the user's Downloads directory.
Do not copy the file into this repository without a separate asset decision.
### Structural Decode Result
The current Bitneedle `record-test` binary accepts the image.
The command returns exit status zero.
All 171 format checks pass.
The decoded stream contains one track and 164 ECDC payload entries.
The programme contains 10,496,000 samples at 48 kilohertz.
The programme duration is approximately 218.667 seconds.
The record has no signed release reference or YL issuance markers.
This signing state does not prevent local player validation.
### Scope Limit
`record-test` decodes the picture-record structure and validates each ECDC packet map.
This run does not prove that all ECDC packets decode to PCM.
Run the native programme decoder before the physical-player test uses this fixture.
### Reproduction Command
```sh
cd /Users/jamie/wavey.ai/bitneedle
cargo build -p test-spin
target/debug/record-test \
/Users/jamie/Downloads/0a7ee1f9-3a66-45e1-80a2-c1f3dd7ea0f6.png
```
### Planned Player Use
1. Verify the exact ECDC-to-PCM decode.
2. Cut the decoded stereo PCM into the canonical virtual groove.
3. Render normal forward playback through the complete physical chain.
4. Render stops, reverse travel, and rapid signed motion.
5. Render each Rust scratch preset with user-controlled record motion.
6. Record output, telemetry, callback time, and failure counts.
The player test must bind its results to the file SHA-256 value.