# Steel Frequency–Position–Mode Co-Design — Phase 0B

**Decision:** `advance_selected_actuator_to_retained_plasticity_phase0c`  
**Selected candidate:** `optimized_three_internal_tripole_fem` at `kL=5.30π`, `x/L=0.500`  
**Scope:** Scalar damped Helmholtz FEM transfer screen; not a steel, transducer, or plasticity prediction.

## Frozen question

Can a calibration-selected internal-source frequency, position, and local mode
shape retain S-bend allocation when the physical design and complex source
commands are frozen and replayed on 32 unseen transfer plants?

## Calibration selection

| Physical arm | Frequency | Position | Mode | Qualified calibration plants | Median S error |
|---|---:|---:|---|---:|---:|
| optimized_two_end_fem | 5.00π | 0.500 | none | 0/5 | 0.770 |
| optimized_three_internal_monopole_fem | 3.00π | 0.500 | monopole | 5/5 | 0.132 |
| optimized_three_internal_dipole_fem | 6.30π | 0.500 | dipole | 5/5 | 0.108 |
| optimized_three_internal_tripole_fem | 5.30π | 0.500 | tripole | 5/5 | 0.088 |

Only seeds `(17001, 17002, 17003, 17004, 17005)` participated in these choices. Held-out seeds
`28001–28032` were opened after frequency, position,
focus source vectors, and dwell coefficients were frozen. Confirmatory seeds
`64001–64032` remain untouched.

## Held-out S-bend result

| Arm | Qualified | Median error | Projection | Gross write | Focus selectivity |
|---|---:|---:|---:|---:|---:|
| optimized_two_end_fem | 0/32 | 0.764 | 0.239 | 1.204 | 0.210 |
| selected_three_channel_candidate | 29/32 | 0.144 | 0.907 | 1.204 | 0.317 |
| selected_third_channel_zeroed_replay | 0/32 | 0.789 | 0.218 | 1.204 | 0.199 |
| selected_transfer_reidentified_diagnostic | 32/32 | 0.092 | 0.986 | 1.204 | 0.333 |

The transfer-reidentified arm is a diagnostic upper bound. It re-solves the
controller on each held-out transfer map and cannot earn the frozen-policy gate.

### Multi-target scope

| Target | Two-end baseline | Frozen selected candidate | Reidentified diagnostic |
|---|---:|---:|---:|
| arc_positive | 32/32 | 32/32 | 32/32 |
| s_bend_positive | 0/32 | 29/32 | 32/32 |
| localized_left_positive | 0/32 | 0/32 | 0/32 |
| localized_right_positive | 0/32 | 0/32 | 0/32 |

The selected design is an S-bend-specific control result, not broad
programmability. Neither localized target passes even when the transfer is
reidentified on each plant, so no state-coverage or qProgT result is available.

## Decision gates

- Robust S-bend 27/32: **PASS**.
- Median error reduction ≥25% versus optimized two-end FEM: **PASS** (`81.2%`).
- Third-channel-null success-rate drop ≥0.25: **PASS** (`0.906`).
- 96-to-192-element allocation change ≤0.10: **PASS** (`0.0086`).
- Matrix solve residual ≤1e-9: **PASS** (`1.32e-14`).

## Scientific audit of the positive transfer result

- The selected frozen policy qualifies 29/32 plants; its Wilson 95% interval is `0.758–0.968`.
- The three failures are `[{'heldout_seed': 28001, 'normalized_error': 0.2571247266953485, 'projection_ratio': 0.7666389544140825, 'error_gate_pass': False, 'projection_gate_pass': False}, {'heldout_seed': 28005, 'normalized_error': 0.23861718247905844, 'projection_ratio': 0.775184200712903, 'error_gate_pass': True, 'projection_gate_pass': False}, {'heldout_seed': 28015, 'normalized_error': 0.20699442822736747, 'projection_ratio': 0.7975869326604194, 'error_gate_pass': True, 'projection_gate_pass': False}]`. Two miss only the projection floor; one misses both projection and error.
- The third channel carries `0.791–0.929` of the normalized focus-vector input power, so the result is not a small perturbation of the two-end plant.
- The selected third channel is a fixed `[1,-2,1]` three-lobe internal patch. It is one commanded mode but more spatially structured hardware than one point emitter; Phase-0C must include an equal-complexity conventional patch baseline.
- All held-out plants remain inside one scalar Helmholtz model family. This is parametric robustness, not cross-model or material validation.
- Unit output-allocation normalization removes transduction magnitude. The result establishes spatial allocation, not energy efficiency or qProgT.
- Multi-target scope is `{'arc_positive': 32, 's_bend_positive': 29, 'localized_left_positive': 0, 'localized_right_positive': 0}` for the frozen candidate. This is targeted S-bend authority, not a general programmable state set.

The decision therefore authorizes only a retained-plasticity **simulation**.
It does not promote a steel, hardware, electromagnetic, athermal, or
programmability-per-joule claim.

## Interpretation boundary

- The FEM state is an effective scalar generalized-stress transfer coordinate, not calibrated displacement or Cauchy stress.
- Unit allocation normalization tests spatial controllability and deliberately removes assumed transducer strength. No energy-efficiency or qProgT conclusion is available.
- Frozen replay tests open-loop robustness; reidentification diagnoses whether failure comes from inadequate architecture or transfer uncertainty.
- No plasticity, retained shape, damage, heat, temporary softening, Maxwell field, magnetostriction, or electron-cyclotron mechanism is included.

## Requirements carried into Phase-0C

- drive a retained elastoplastic strip with the computed space-time wave stress rather than prescribing curvature or internal moment.
- model the signed sequential mechanical-bias schedule explicitly instead of treating signed dwell coefficients as direct retained curvature.
- compare optimized moving-contact forming, a conventional equal-complexity internal patch, two-end waves, and the selected structured patch under common delivered-energy and damage ceilings.
- restore raw incident/transducer-strength and absorption sweeps rather than unit output-allocation normalization.
- include third-channel-zero, phase/dwell shuffle, transfer-detune, state-coupling-null, stress-superposition-null, and thermal-equal controls.
- keep access, common-state process quality, and matched-coverage qProgT as separate outcomes.

## Audit

- Calibration physical designs: `144`.
- Held-out records: `512`.
- Calibration, held-out, and reserved seeds are disjoint.
- Every focus allocation is normalized and every scalar is finite (verified separately).

## Reproduction

```bash
MPLCONFIGDIR=/tmp/steel-codesign-mpl python3 scripts/active/verify_steel_frequency_position_codesign.py
MPLCONFIGDIR=/tmp/steel-codesign-mpl python3 scripts/active/steel_frequency_position_codesign.py --run
```
