Graceful Extensibility | EAS Resilience Evaluation

UNIT OF ANALYSIS

Family stress boundary

Eligible episode–seed curves form task-family stress cells; the complete family curve determines \(M_f(\lambda)\), \(\lambda_f^*\), and \(\mathrm{AUDC}_f\).

\[ r_{\mathrm{PC}}=\frac{\mathrm{PC}_f}{\max(\tau_{\mathrm{PC}},\varepsilon)}, \quad r_Q=\frac{Q_f}{\max(\tau_Q,\varepsilon)}, \]

\[ r_{\mathrm{safe}}=\frac{\tau_{\mathrm{safe}}+\varepsilon}{\max(u_f,0)+\varepsilon}, \quad r_{\mathrm{rec}}=\frac{\tau_{\mathrm{rec}}+\varepsilon}{\max(C_{\mathrm{rec},f},0)+\varepsilon}, \quad r_{\mathrm{stab}}=\frac{\tau_{\mathrm{stab}}+\varepsilon}{\max(v_{\mathrm{stab},f},0)+\varepsilon}. \]

\[ \widetilde r_j=\min\!\left\{\max(r_j,0),r_{\max}\right\}, \qquad \varepsilon=10^{-6},\quad r_{\max}=2. \]

\[ M_f^{\mathrm{soft}}(\lambda)= \frac{1}{K}\sum_{j\in\operatorname{bottomK}(\widetilde{\boldsymbol r}_f(\lambda))} \widetilde r_{f,j}(\lambda)-1, \quad K=2, \]

\[ M_f^{\mathrm{hard}}(\lambda)=\min_j\widetilde r_{f,j}(\lambda)-1, \qquad M_f^{\mathrm{hard}}(\lambda)\ge-\delta_h, \quad \delta_h=0.05. \]

\[ \lambda_{f,\mathrm{soft}}^*=\max\!\left\{\lambda_k: M_f^{\mathrm{soft}}(\lambda_i)\ge0\;\;\forall i\le k\right\}, \]

\[ \lambda_{f,\mathrm{hard}}^*=\max\!\left\{\lambda_k: M_f^{\mathrm{soft}}(\lambda_i)\ge0,\; M_f^{\mathrm{hard}}(\lambda_i)\ge-\delta_h\;\;\forall i\le k\right\}. \]

\[ y_i=\max\!\left(M_f^{\mathrm{soft}}(\lambda_i)+1,0\right), \qquad \mathrm{AUDC}^{\mathrm{norm}}_f= \frac{\sum_{i=1}^{m}\tfrac{y_i+y_{i-1}}{2}(\lambda_i-\lambda_{i-1})} {\lambda_m-\lambda_0}. \]

A ratio of \(1\) is the operational boundary. The reported headline \(\lambda_f^*\) is the longest continuously passing soft-margin prefix from \(\lambda=0\); a later re-entry cannot extend it. Higher \(\lambda_f^*\) and \(\mathrm{AUDC}^{\mathrm{norm}}_f\) indicate greater capacity. The statistical unit is one task-family stress cell, aggregated from eligible episode–seed paths.

Five channels

PC: task completion; Q: proposition satisfaction; Safety: safety burden; Rebound: recovery cost; Stability: formal perturbation sensitivity. The two weakest channels determine the soft margin, and the weakest channel determines the hard margin.

ALL TEN EPISODES

Eligible episode–seed paths

Official paths require six shared stress rows; strict paths also require valid critic-lifecycle and reward-shaper evidence at all six levels.

Strict: included in official and strict estimates Official only: included in the official estimate Excluded: omitted from the estimator

Official and strict support status for 30 candidate episode×seed paths across 10 episodes
Family	Episode	Seed 0	Seed 1	Seed 2	Official	Strict
Placement	75	Strict	Strict	Excluded	2/3	2/3
	78	Strict	Official only	Strict	3/3	2/3
	98	Strict	Strict	Strict	3/3	3/3
Mixed	239	Excluded	Strict	Strict	2/3	2/3
	258	Excluded	Strict	Excluded	1/3	1/3
	290	Strict	Excluded	Strict	2/3	2/3
	304	Excluded	Excluded	Excluded	0/3	0/3
Spatial	131	Excluded	Excluded	Excluded	0/3	0/3
	144	Excluded	Excluded	Strict	1/3	1/3
	174	Excluded	Excluded	Excluded	0/3	0/3

Denominator definition: Among 30 candidate paths, official shared-complete=14 (Placement 8 + Mixed 5 + Spatial 1) and strict-complete=13. The single complete Spatial path remains in the support audit; its clean reference has n=1<min_samples=2, so its family cells remain N/A. Operative support for estimable families is therefore 13 official paths and 12 strict paths.

OFFICIAL BOUNDARY

First stress boundary

Placement and Mixed first fail at \(\lambda=.2\); Spatial remains N/A because its clean reference has \(n=1<2\).

Three-panel GE research figure showing Placement and Mixed soft and hard margins, five channel ratios for Placement at lambda 0.2, and support denominators for the three task families. — **Fig. G1 | Family-level Graceful Extensibility under the six-level stress sweep.** a, Placement and Mixed soft and hard margins, with the first cliff at λ=.2; b, five-channel Placement ratios at λ=.2 on a logarithmic x-axis, identifying Stability as the hard bottleneck and Rebound as the second-weakest channel; c, official-complete episode×seed support, including Spatial at 1/9 paths and capacity N/A. The curves are provisional family estimates under the reported support and threshold calibration.

PLACEMENTλ*=0

AUDC=0.4668; official 8/9, strict 7/9, across 3/3 episodes.

MIXED TEMPORALλ*=0

AUDC=0.5027; official/strict 5/12, across 3/4 episodes.

SPATIAL RELATIONN/A

1/9 path; the clean-reference min_samples criterion is unmet.

FIRST CLIFFλ=.2

Both estimable families cross the boundary at the smallest positive stress.

CASE STUDY

Placement boundary case

Placement combines the family-level boundary estimate with contributing execution traces across Episodes 75, 78, and 98.

Five capability channels converge on a shared capacity corridor, with the coral Stability channel contracting at the first positive stress.

IMAGEGEN · FAMILY-BOUNDARY CONCEPT · CONCEPTUAL, NON-NUMERICAL

PLACEMENT · MULTI-ROLLOUT ESTIMATOR

Stability-limited first cliff

Placement covers eight official-complete paths across Episodes 75, 78, and 98. At λ=0, the soft margin is +0.0556. At λ=.2, r_PC=r_Q=1.111, while r_stab=9.12×10⁻⁷ and r_rec=0.727 reduce the soft margin to −0.6367 and the hard margin to approximately −1.

Image evidence boundary
The ImageGen panel communicates the conceptual geometry of five channels sharing a boundary while the weakest channel contracts early. Numerical claims derive from GE cells and support CSV files.

OFFICIAL SUPPORT8 / 9

STRICT SUPPORT7 / 9

AUDC0.467

FIRST CLIFF0.2

ENGLISH ASPECT VIDEO · EVIDENCE-ALIGNED

Episode 78 at the boundary

This trace-driven reconstruction compares Episode 78 / seed 1 clean and λ=.2 contributing rollouts. M_soft, λ*, and normalized AUDC remain Placement-family summaries (n=8 per cell; 8/9 complete paths), and λ*=0 is interpreted as a provisional threshold-calibration diagnostic.

GE aspect-video poster showing the Episode 78 clean and λ=.2 Placement trajectories. — This trace-driven reconstruction compares Episode 78 / seed 1 clean and λ=.2 contributing rollouts. M_soft, λ*, and normalized AUDC remain Placement-family summaries (n=8 per cell; 8/9 complete paths), and λ*=0 is interpreted as a provisional threshold-calibration diagnostic.

TRACE-LEVEL CASE STUDY

Boundary and trajectory evidence

The trajectory browser applies two fixed comparisons to all 180 candidate runs. The longitudinal analysis follows Episode 78 / seed 1 across six stress levels, and the cross-sectional analysis holds seed 1 / λ=1.0 constant across episodes. GE capacity remains a task-family aggregate.

GRACEFUL EXTENSIBILITY · TWO-LEVEL EVIDENCE

From cells to planning nodes

Each panel reports the family-level margin alongside single-run simulation fields. Trajectories explain channel mechanisms, and capacity is aggregated from the complete eligible stress curves for that family.

LONGITUDINAL COMPARISON

Episode 78 stress trajectories

Six independent rollouts hold episode, seed, task contract, Planner, and Judge settings fixed.

Placement M_soft(λ) is estimated from 8/9 official-complete episode×seed paths. Ep78 / seed 1 provides mechanism evidence; its C_rec, β, and simulation fields remain rollout-level quantities.

Aspect: GE · family stress-response
Primary trajectory: Episode 78 · placement_only
Controlled identity: seed 1 · GPT-5.1 Judge
Family support: 8/9 official · 7/9 strict

λ 0.0M_soft +.0556Ep78: C_rec 25.517 · β 0

λ 0.2 · first cliffM_soft −.6367Ep78: C_rec 48.643 · β 1.436

λ 0.4M_soft −.6002Ep78: C_rec 27.650 · β 1.746

λ 0.6M_soft −.6144Ep78: C_rec 25.527 · β .190

λ 0.8M_soft −.6140Ep78: C_rec 8.968 · β .157

λ 1.0M_soft −.4571Ep78: C_rec 25.527 · β .188

PLANNING NODES · λ 0

Clean: API repair

sim step 0replan 1
Initial parameters omitted
The planner directs toy food and the toy fire truck to chest_of_drawers_37; both Rearrange calls return the five-parameter API requirement.
sim step 1–819Explore
Objects and target furniture resolved
Thought summary: explore bedroom_1, verify the object instances, and select chest_of_drawers_36 as the target.
sim step 973completion .5
Five-parameter Rearrange executes
Agent 0 places toy_food_1 successfully while Agent 1 continues the parallel transfer of toy_fire_truck_0.
sim step 1264success 1
All propositions satisfied
Both objects are on chest_of_drawers_36, with task completion=1.0.

PLANNING NODES · λ .2

First-cliff repair

sim step 0replan 1
Two incomplete Rearrange calls
The observation returns the five-parameter template while completion remains 0.
sim step 819–820replan 3–4
Object resolution and second API repair
Thought summary: after identifying toy_food_1 and toy_fire_truck_0, add the spatial relation and constraint parameters.
sim step 926completion .5
First object transferred
Agent 0 completes the five-argument Rearrange, placing toy_fire_truck_0 on the target furniture.
sim step 1307–1809nav / place failure
Navigation target and placement distance unmet
Agent 1 first reports an inaccessible target; Place then fails because of distance or occlusion, prompting an Explore → Navigate → Place recovery.
sim step 1898–1899success 1
Terminal state satisfied
Place succeeds for the second object and completion rises to 1.0.

family M_soft / M_hard: −.6367 / −.99999909
family ratios: PC 1.111 · Q 1.111 · Safe .903 · Rec .727 · Stab 9.12e−7
sim steps / replans A0: 1899 / 11
single-run C_rec / β: 48.643 / 1.436
collision rate: .0920
outcome: success · completion 1.0

Thought summary

Reconstruct object locations after parameter rejection, then add an explicit Navigate when placement distance is insufficient.

Action

Explore → Rearrange → Explore → Place(fail) → Navigate → Place(success).

Observation

Terminal success co-occurs with a family Stability ratio near zero, motivating separate reporting of endpoint completion and path sensitivity.

The first cliff is an aggregate result for the Placement family cell; the ep78-s1-l0.2 trajectory records one auditable mechanism within that cell.

CROSS-SECTIONAL COMPARISON

Fixed-stress episode comparison

Seven episodes share seed 1 and \(\lambda=1\); Episode 304 is the outcome-associated negative control.

At λ=1, Placement family M_soft=−.4571 and Mixed family M_soft=−.4444. Each trajectory supplies an execution mechanism; the shared family cell is repeated across member panels to make the inferential level explicit.

PLACEMENT · EP75

Concise parameter repair after perception refresh

step 0PerceptionScanTool
Object index refreshed
Thought summary: use a perception scan to locate stuffed_toy_0 and toy_vehicle_1.
step 779API rejection
Simplified Rearrange fails
The target couch_23 is resolved, while the call omits the spatial relation and constraint fields.
step 1016–1120success 1
Both objects placed
Complete Rearrange calls advance completion from 0 to .5 and then to 1.0.

Complete indices: Download the 180-run index, 3115-node planning-event index, and case-selection manifest. Candidate screening covers all 10 episodes; Thought summaries condense only planner rationale explicitly recorded in the trace.

CHANNEL FORENSICS

Origin of \(\lambda^*=0\)

Clean \(\beta=0\) sets \(\tau_{\mathrm{stab}}=0\), inducing the first positive-stress Stability cliff under the current calibration.

\[ \tau_{\mathrm{stab}}=0, \qquad r_{\mathrm{stab}}(\lambda)= \frac{\tau_{\mathrm{stab}}+\varepsilon}{\beta(\lambda)+\varepsilon} =\frac{\varepsilon}{\beta(\lambda)+\varepsilon}, \qquad \varepsilon=10^{-6}. \]

For \(\lambda>0\) and \(\beta(\lambda)>0\), \(r_{\mathrm{stab}}\) contracts to approximately \(10^{-6}\)–\(10^{-5}\), yielding \(M_f^{\mathrm{hard}}\approx-1\). The observed \(\lambda^*=0\) is therefore a threshold-calibration diagnostic; estimating system perturbation capacity requires an independently calibrated Stability threshold.

Family-specific secondary bottlenecks

Placement is principally constrained by Rebound at λ=.2/.6/.8 and by Safety at λ=.4/1.0. For Mixed at λ=.6, Rebound ratio=0.500 yields the deepest soft margin, −0.750. At other stress levels, the second bottleneck shifts among Safety, Rebound, and PC/Q.

MECHANISM-RICH FAMILY CONTRAST

Mixed-family comparison

Mixed \(\lambda^*=0\) uses 5/12 official paths across Episodes 239, 258, and 290; Episode 304 remains an excluded negative control.

Mixed official curve

At λ=0, the soft margin is +0.0556; at λ=.2 it declines to −0.4444. At λ=.6, r_rec=0.500 and a near-zero r_stab produce the curve’s deepest soft margin, −0.7501. AUDC=0.5027 indicates slightly greater integrated retention over the current support set than Placement; both families retain λ*=0.

Episode 304 negative control

task_state_success=0/18; mean completion=0.6667, with 16/18 runs at 0.75 and 2/18 at 0; C_rec valid=0/18. Its exclusion from the Mixed estimator demonstrates the support-selection dependence of PC/Q=1.111 in complete-pair cells.

SUPPORTING TRAJECTORIES

Prespecified mechanism excerpts

Three strict-complete successful paths explain mechanisms while each family estimate retains every eligible path.

PRIMARY FAMILY MECHANISM

Placement · Ep78 / seed 0

λ=.2: C_rec=25.454, β=1.408, unsafe=.0207, 1252 steps, success. Explore resolves the target and a two-agent Rearrange repair follows; terminal success co-occurs with a near-zero Stability ratio.

MIXED FIRST-CLIFF EXCERPT

Mixed · Ep239 / seed 2

λ=.2: C_rec=18.113, β=1.151, unsafe=.0219, 2689 steps, success. A two-parameter Rearrange error is repaired; `next_to,None` then fails, followed by completion with a reference object.

MIXED REBOUND BOTTLENECK

Mixed · Ep258 / seed 1

λ=.6: C_rec=75.256, β=.305, unsafe=.0253, 1702 steps, success. After repeated use of invalid `next_to,None`, the saucer is placed first and then used as the reference for the plant.

Mechanism-excerpt selection audit: fixed eligibility, target cell, and observable trace features
Report role	Path / cell	Eligibility	Quantitative anchor	Trace question	Estimator status
Placement first-cliff mechanism	Ep78 / seed 0 / λ=.2	Strict complete; success=1	β=1.408; C_rec=25.454	How does success co-occur with the Stability cliff after parameter repair?	Excerpt only
Mixed first-cliff contrast	Ep239 / seed 2 / λ=.2	Strict complete; success=1	β=1.151; C_rec=18.113	Does the same first-cliff cell contain an auditable error–repair sequence?	Excerpt only
Mixed Rebound bottleneck	Ep258 / seed 1 / λ=.6	Strict complete; success=1	family r_rec=0.500; C_rec=75.256	How does the second Rebound bottleneck accumulate across actions?	Excerpt only

The selection table can be verified against the case-selection audit CSV, trajectory-source CSV, and official family-cells CSV. Trajectory excerpts address how a channel mechanism can arise; family GE remains jointly estimated from all eligible episode×seed curves in the corresponding task family.

REPORTED FAMILY CELLS · ROUNDED

Channel ratios and margins

Ratios use the clean family reference; retained precision resolves \(M_f^{\mathrm{hard}}\) from \(-1\), and invalid Spatial placeholders are omitted.

Placement

λ	PC	Q	Safety	Rebound	Stability	M soft	M hard
0	1.111	1.111	1.250	1.250	1.000	+0.0556	0
.2	1.111	1.111	.903	.727	9.12e−7	−.6367	−.99999909
.4	1.111	1.111	.800	.857	5.97e−7	−.6002	−.99999940
.6	1.111	1.111	1.535	.771	2.95e−6	−.6144	−.99999705
.8	1.111	1.111	1.310	.772	1.55e−6	−.6140	−.99999845
1	1.111	1.111	1.086	1.321	4.45e−6	−.4571	−.99999555

Mixed temporal

λ	PC	Q	Safety	Rebound	Stability	M soft	M hard
0	1.111	1.111	1.250	1.250	1.000	+.0556	0
.2	1.111	1.111	2.000	1.875	8.69e−7	−.4444	−.99999913
.4	1.111	1.111	.781	2.000	1.93e−6	−.6096	−.99999807
.6	1.111	1.111	1.895	.500	3.28e−6	−.7501	−.99999672
.8	1.111	1.111	1.824	1.024	3.36e−6	−.4879	−.99999664
1	1.111	1.111	2.000	1.166	1.37e−5	−.4444	−.99998625

ROBUSTNESS & LIMITATIONS

Inference tiers

Lifecycle-strict analysis narrows support while preserving the original rollouts, exclusions, and qualitative boundary test.

Official

Shared-complete=14; operative curves for estimable families=13. Placement AUDC=.4668 and λ*=0; Mixed AUDC=.5027 and λ*=0.

Lifecycle-strict

Shared-complete=13; operative curves for estimable families=12. Placement AUDC=.4763 and λ*=0, preserving the qualitative conclusion.

Without Stability

In this diagnostic, Mixed λ*=.2 and Placement remains 0. The preregistered formal GE retains the complete five-channel definition.

Required calibration

Independent clean–clean replication must estimate a nonzero noise floor; exact capacity can then be recomputed after preregistration of τ_stab.

Calibrated interpretation: Under the current degenerate threshold induced by clean β=0, formal GE structurally crosses the hard boundary at the first positive-stress cell. This provisional result primarily reflects threshold calibration; physical capacity requires independent calibration.

Graceful Extensibility
Task Family-level stress capacity.