The Vacuity Barrier: Conditional Lower Bounds and Measured Premises for Latent World Model Planning

1. Introduction

Two prior papers established a boundary. Dynamic LeJEPA [1] proved a placement rule for physics constraints in latent world models—maximum-entropy encoder, dynamics in the predictor, physics in the decoder—and measured that offline latent planning can be mistaken for control competence. DEGP [2] proved conditioning guarantees for a decoupled predictor and measured a $0/50$ closed-loop deployment boundary. A rank-r coupling channel between the prediction loss and the encoder was examined within this program as a candidate mechanism; a controlled study measured that it raises contact selectivity only to $1.20\times$ against a $2.0\times$ target, and that a random subspace matches the channel.

This paper provides a conditional response to the boundary those results leave open. The placement rule is correct but insufficient for control under the measured premises, because information preservation does not imply control sufficiency. Certified conditioning is achievable but does not by itself address the binding constraints—control sufficiency, decoder transfer, exploitability, contact-mode discrimination, and planner feasibility. And the coupling channel is not supported by the measurements as a route to the required contact selectivity.

The response is conditional, not rhetorical. Every condition we formalize traces to a measurement in the prior work, and every lower bound we prove takes a measured quantity as premise. The result is a boundary paper: a map of the exact assumptions a control-sufficient latent world model requires, with each assumption’s lower bound instantiated on data where possible, and stated as a conditional premise where not.

1.1. The Research Program and This Paper’s Place in It

This is the third paper in a program of four on latent world models for control. The questions, in order: where does domain knowledge belong in a world model (Paper I [1]); does certified latent conditioning enable closed-loop planning (Paper II [2]—answered negatively, $0/50$); and—this paper—was that failure contingent, repairable by engineering, or necessary under the measured premises? Section 8 and Section 9 specify the fourth paper: the attack on the boundary mapped here.

Each paper is self-contained. Every definition, instrument, and threshold used below is stated in this paper; the predecessors are cited for provenance, not as prerequisite reading. The only inherited artifact is the Push-T corpus (Section 6.1); its release includes a seeded collection script that is documented to regenerate the corpus deterministically [2], though we did not re-verify bit-identity in this run (disclosed limitation, Section 6.1). The full provenance chain—what is closed and what remains open—is recorded in Table 4.

Paper I — Dynamic LeJEPA [1] (extending LeJEPA [9] to sequential prediction): the placement rule and the exploitation confound.

Six theorems and 110 seeded ablation runs across two domains (driving, bimanual manipulation) and two encoder mechanisms established that physics constraints are benign in the observation decoder and destructive in the encoder, yielding the design rule every successor in this program adopts: maximum-entropy encoder, dynamics in the predictor, physics in the decoder (placement ordering significant in all four domain × mechanism arms, joint Holm-corrected $p=0.0078$). One measured caution from that paper governs all evaluation choices here: a CEM planner optimizing a learned decoder’s cost found action sequences roughly a quarter of the action range away from the recorded trajectory that still scored better than replaying the true actions. Offline planning advantage is therefore not evidence of control competence; this paper evaluates only closed-loop, oracle-grounded, or control-grounded quantities.

Paper II — DEGP [2]: certified conditioning, measured deployment failure.

DEGP built the strictest instance of the Paper-I design on Push-T: a variance–covariance encoder trained with no other signal, a residual predictor whose conditioning is certified by theorem, and a leashed planner, under a pre-registered falsification protocol. The certification held in measurement—unrolled-Jacobian conditioning was statistically flat in latent dimension $K\in \{64,128,256\}$ (TOST equivalence, 30 paired seeds) and within the $ϵ·T$ planning budget at every horizon tested. Deployment nonetheless failed completely and was reported as a measured null: $0/50$ episodes in every condition (Clopper–Pearson upper $5.8\%$), block movement at $0.0$–$1.1$ px, while the same planners with oracle dynamics moved the block. A pre-registered 18-condition attribution battery localized the failure to two components: (i) a contact-conditional prediction gap—across four model versions, predicted movement on contact and free strata never differed by more than $1.18\times$ (selectivity $0.98$–$1.18\times$), although the same latents decode contact vs. free at $5.3$–$6.0\times$: the information is present in the representation and unused by every evaluated predictor; and (ii) the probe-decoded cost. Planner family, planning order, perception input, and the leash were excluded as binding factors.

What remained open.

Paper II established that the best-certified instance of the placement rule fails, and where it fails. It could not establish whether the failure was repairable. Distinguishing contingent from necessary requires exactly what neither predecessor provided: a formal statement of what a control-sufficient latent world model must satisfy, and a measurement of where the inherited setup stands against each requirement. That is this paper’s contribution. Section 3 and Section 4 state five conditions and prove conditional lower bounds; Section 6 measures the premises under a stricter protocol than any predecessor (30 paired seeds, 180 encoder–predictor pairs, 150 control models, 90 trained decoders)—and does not merely inherit prior readings, but tightens one: the coupling channel’s earlier $1.20\times$ selectivity reading is not sustained (max $1.04\times$, $p=0.96$). The oracle experiments separate planner feasibility from model error ($13/50$ terminal-pose cost, $2/50$ coverage-shaped; the latter falsifies pre-registered P4). The result is a conditional boundary: under the measured premises, every certified guarantee in the framework is vacuous. The premises are attackable; Section 8 and Section 9 specify the attack and the gate (P1–P6) it must pass.

Reader’s guide.

Section 2, Section 3, Section 4 and Section 5 define everything they use. Section 6 re-measures, rather than inherits, the representation-level premises. Instruments renamed from Paper II are bridged at first use: the Contact Dynamics Ratio of [2] is superseded by stratified selectivity (Definition 8), and the linear-probe floor ($\sim 35$ px, ridge regression) is a different instrument from the trained-decoder floor ($40.7$–$42.2$ px, Section 6.4)—the two are not comparable.

Table 1. Program lineage. Each paper is self-contained; predecessors are cited for provenance.

#	Paper	Question	Established (anchor)	Left open
I	Dynamic LeJEPA [1]	Where does physics belong?	Placement rule; 110 seeded ablations, 4 arms, Holm $p=0.0078$; exploitation confound (CEM beats replay offline)	Does the rule support control?
II	DEGP [2]	Does certified conditioning enable planning?	$\kappa$ flat in K (TOST, 30 seeds); deployment $0/50$; attribution: contact gap + cost	Contingent or necessary failure?
III	this paper	What does control sufficiency require?	Five conditions; lower-bound map; selectivity $\le 1.04\times$; floor $40.7$–$42.2$ px; widths 126–279 px; oracle $13/50$ & $2/50$; P2, P4 falsified	Joint satisfiability (attack specified §Section 8 and Section 9)

Table 1. Program lineage. Each paper is self-contained; predecessors are cited for provenance.

#	Paper	Question	Established (anchor)	Left open
I	Dynamic LeJEPA [1]	Where does physics belong?	Placement rule; 110 seeded ablations, 4 arms, Holm $p=0.0078$; exploitation confound (CEM beats replay offline)	Does the rule support control?
II	DEGP [2]	Does certified conditioning enable planning?	$\kappa$ flat in K (TOST, 30 seeds); deployment $0/50$; attribution: contact gap + cost	Contingent or necessary failure?
III	this paper	What does control sufficiency require?	Five conditions; lower-bound map; selectivity $\le 1.04\times$; floor $40.7$–$42.2$ px; widths 126–279 px; oracle $13/50$ & $2/50$; P2, P4 falsified	Joint satisfiability (attack specified §Section 8 and Section 9)

1.2. Contributions

1.

Conditional closure of two open questions. We show that the placement rule of Dynamic LeJEPA [1] and the conditioning methodology of DEGP [2] are individually correct but jointly insufficient for control under the measured premises, and we identify the five conditions that were missing.

2.

Five conditional theorems with vacuity conditions. Control-sufficiency and planning regret bounds (Theorems 1 and 2), finite-sample exploitability-calibrated gate (Theorem 3), conditional conformal under contact (Theorem 4), contact-mode identity (Theorem 5), and oracle feasibility as an upper confidence bound (Theorem 6)—each with an explicit vacuity threshold.

3.

Conditional lower bounds. Each parameter that would need to be small is bounded below by a quantity exceeding its threshold under the measured premises.

4.

A 30-seed replication of the rank sweep and the control battery. On a preserved copy of the standard Push-T corpus, we re-train 180 encoder–predictor pairs at six ranks and 150 control models (120 attribution controls + 30 baseline re-runs).

5.

Decoder-floor and certificate measurement. Cross-episode decoder position error is $40.7$–$42.2$ px on 90 trained decoders; split-conformal certificate widths at $H\in \{4,8,16\}$ are 126–279 px at nominal $0.90$.

6.

An evaluated and falsified prediction.Section 9 lists P1–P6 as pre-specified falsifiable predictions. P4 (a coverage-shaped cost raises oracle feasibility above $0.5$) is evaluated and falsified: $2/50$ versus $13/50$ for terminal-pose cost on the same seeds.

7.

Architecture and protocol. A revised architecture and the pre-specified protocol P1–P6 are stated, with decision rules fixed before any new run.

Figure 1. Required experiments for the Vacuity Barrier program. Blue: executed in this revision. Green: executed and passed. Red: executed and falsified. Dashed gray: pre-specified, not evaluated.

Figure 1. Required experiments for the Vacuity Barrier program. Blue: executed in this revision. Green: executed and passed. Red: executed and falsified. Dashed gray: pre-specified, not evaluated.

1.3. Roadmap

Section 2 sets notation and definitions. Section 3 states the five conditional theorems. Section 4 proves the lower bounds and wiring results. Section 5 states the lower-bound map. Section 6 presents the measured premises. Section 7 positions the constructive literature. Section 8 specifies the revised architecture. Section 9 gives the pre-specified protocol. Section 10 states falsifiable predictions. Section 11 and Section 12 discuss and conclude. Appendices Appendix A–Appendix C contain proofs and the pre-specification template.

2. Mathematical Framework

2.1. Notation

Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a probability space. All random variables take values in Polish spaces.

• $S\in \mathcal{S}$: true state, $\mathcal{S}$ Polish.
• $Z=f_{\theta }\left(S\right)\in {\mathbb{R}}^K$: latent, $f_{\theta }$ a Borel-measurable encoder.
• $A\in \mathcal{A}$: action, $\mathcal{A}\subseteq {\mathbb{R}}^{N_a}$ compact.
• $Y\in \mathcal{Y}$: task variable (T-block pose), $\mathcal{Y}\subseteq {\mathbb{R}}^{d_Y}$ with $d_Y=3$ for Push-T.
• $Q^{*}(s,a)$: optimal action-value function.
• $C_{\mathrm{model}}\left(\pi \right)$, $C_{\mathrm{true}}\left(\pi \right)$: model and true costs of plan $\pi =a_{0:H-1}$.
• $c\left(s\right)\in \{0,1\}$: contact mode (0 free, 1 contact).
• ${\mathcal{D}}_{\mathrm{plan}}$: distribution of states visited by the planner.
• ${\mathcal{D}}_{\mathrm{cal}}$: calibration distribution; ideally ${\mathcal{D}}_{\mathrm{plan}}$.
• ${\mathcal{D}}_{\mathrm{train}},{\mathcal{D}}_{\mathrm{val}}$: training and validation episode distributions.

We separate three quantities that were conflated in earlier drafts: ${\delta }_{\mathrm{lat}}$ (latent radius), $\xi ={\sup }_{\pi }|C_{\mathrm{model}}\left(\pi \right)-C_{\mathrm{true}}\left(\pi \right)|$ (uniform model-cost error), and ${\rho }_{\mathrm{dec}}(h,\mathcal{D})={\mathbb{E}}_{\mathcal{D}}[\parallel h\left(Z\right)-Y\parallel ]$ (decoder task-space error).

2.2. Definitions

Definition 1

($(ϵ,{\delta }_{\mathrm{lat}})$-control sufficiency). Let ${\delta }_{\mathrm{lat}}\ge 0$. A latent $Z=f_{\theta }\left(S\right)$ is $(ϵ,{\delta }_{\mathrm{lat}})$-control-sufficient if for all $s,s^{\prime }\in \mathcal{S}$ with $\parallel f_{\theta }\left(s\right)-f_{\theta }\left(s^{\prime }\right)\parallel \le {\delta }_{\mathrm{lat}}$ and all $a\in \mathcal{A}$, $|Q^{★}(s,a)-Q^{★}(s^{\prime },a)|\le ϵ$.

Remark 1.

Control sufficiency is a property of the encoder $f_{\theta }$ and the task’s value function. It is not a property of the predictor.

Definition 2

(Exploitability gap). For a plan π, $\beta \left(\pi \right)={(C_{\mathrm{true}}\left(\pi \right)-C_{\mathrm{model}}\left(\pi \right))}^{+}$. For $\alpha \in (0,1)$, ${\beta }_q=\inf \{b\in \mathbb{R}:{\mathbb{P}}_{\pi \sim {\mathcal{D}}_{\mathrm{cal}}}(\beta \left(\pi \right)\le b)\ge 1-\alpha \}$.

Definition 3

(Conditional exchangeability). Calibration triples and a test triple are exchangeable given context X if, for every x in the support of X, the joint distribution of calibration triples with $X=x$ is invariant under permutation. We use $X={\widehat{c}}_0$, the predicted contact mode.

Definition 4

(Mode separation). $\mathsf{\Delta }={\sup }_{s,s^{\prime }\in \mathcal{S}}\parallel Y\left(s\right)-Y\left(s^{\prime }\right)\parallel$ over states differing only in contact mode.

Definition 5

(Mode misclassification rate). $\eta =\mathbb{P}(\widehat{c}\ne c)$ under ${\mathcal{D}}_{\mathrm{plan}}$.

Definition 6

(Oracle feasibility). Let $p_{\mathrm{oracle}}$ be the success rate of planner $\mathcal{P}$ with exact dynamics and cost C on a fixed task distribution. A learned model using the same $\mathcal{P}$ and C cannot exceed $p_{\mathrm{oracle}}$ up to planner sensitivity.

Definition 7

(Replay gap). $\gamma =C_{\mathrm{true}}\left({\pi }_{\mathrm{CEM}}\right)-C_{\mathrm{true}}\left({\pi }_{\mathrm{replay}}\right)$.

Definition 8

(Selectivity). $s=\mathrm{median}(\parallel {\widehat{\mathsf{\Delta }}}_{p_T}\parallel \mid c=1)/\mathrm{median}(\parallel {\widehat{\mathsf{\Delta }}}_{p_T}\parallel \mid c=0)$.

Definition 9

(Contact blindness). $f_{\theta }$ is $({\delta }_f,G)$-contact-blind if there exist $s,s^{\prime }$ with $c\left(s\right)\ne c\left(s^{\prime }\right)$ and $a\in \mathcal{A}$ such that $\parallel f_{\theta }\left(s\right)-f_{\theta }\left(s^{\prime }\right)\parallel \le {\delta }_f$ and $|Q^{★}(s,a)-Q^{★}(s^{\prime },a)|\ge G$.

Definition 10

(Decoder floor statistic). ${\rho }_{\mathrm{dec}}^{\mathrm{floor}}(h,{\mathcal{D}}_{\mathrm{val}})={\mathrm{median}}_{\mathrm{held}-\mathrm{out}\mathrm{episodes}}\parallel h\left(z_t\right)-y_t\parallel$, computed with $z_t=f_{\theta }\left(s_t\right)$ on held-out validation episodes and reported in task units (px). It is the training-independent floor of the decoder class, distinct from the per-step expectation ${\rho }_{\mathrm{dec}}$ and from the conformal certificate width.

Definition 11

(Horizon weight ${\widehat{w}}_h$). The certificate scales the per-step radius ${\widehat{q}}_x$ by a horizon factor ${\widehat{w}}_h$ defined as ${\widehat{w}}_h=\mathbb{E}[\parallel {\widehat{z}}_h-z_h\parallel \mid {\mathcal{D}}_{\mathrm{cal}}]/\mathbb{E}[\parallel {\widehat{z}}_1-z_1\parallel \mid {\mathcal{D}}_{\mathrm{cal}}]$. It is estimated on the calibration set, not on test. This paper uses the proxy ${\widehat{w}}_h=\sqrt{h}$ (Section 6.4); the proxy is stated as a limitation and does not affect the qualitative conclusions.

Definition 12

(Decoder slack ${\alpha }_{\mathrm{dec}}$). ${\alpha }_{\mathrm{dec}}={\mathbb{P}}_{{\mathcal{D}}_{\mathrm{plan}}}\left(\parallel h\left({\widehat{z}}_H\right)-y_{\mathrm{goal}}\parallel >{\tau }_{\mathrm{task}}-{ϵ}_{\mathrm{dec}}\right)$, the probability that the decoder alone (with exact dynamics) violates the task tolerance. When the decoder is exact, ${\alpha }_{\mathrm{dec}}=0$; it is the extra slack carried in Theorem 3.

3. Conditional Theorems with Vacuity Conditions

3.1. Control-Sufficiency and Planning Regret

Theorem 1

(Latent-greedy regret). Let Z be ϵ-control-sufficient with ${\delta }_{\mathrm{lat}}=0$ and $|Q_Z(z,a)-Q^{★}(s,a)|\le ϵ$ whenever $z=f_{\theta }\left(s\right)$. For horizon H, any plan π satisfies $|Q_Z\left(\pi \right)-Q^{★}\left(\pi \right)|\le Hϵ$, and ${\pi }^Z=\arg {\min }_{\pi }{Q}_Z\left(\pi \right)$ satisfies $C_{\mathrm{true}}\left({\pi }^Z\right)-C_{\mathrm{true}}\left({\pi }^{★}\right)\le 2Hϵ$.

Proof. 

By the triangle inequality and Definition 1, per-step value gap $\le ϵ$; sum over H steps; use that ${\pi }^Z$ minimizes $Q_Z$. Full proof in Appendix B.    □

Theorem 2

(Planning regret). Let $\xi ={\sup }_{\pi }|C_{\mathrm{model}}\left(\pi \right)-C_{\mathrm{true}}\left(\pi \right)|$. Then $C_{\mathrm{true}}\left(\widehat{\pi }\right)-C_{\mathrm{true}}\left({\pi }^{★}\right)\le 2\xi$ for $\widehat{\pi }=\arg {\min }_{\pi }{C}_{\mathrm{model}}\left(\pi \right)$.

Proof. 

$C_{\mathrm{model}}\left(\widehat{\pi }\right)\le C_{\mathrm{model}}\left({\pi }^{★}\right)$; apply $\xi$-bound twice. Full proof in Appendix B.    □

Remark 2.

ϵ and ξ are separate. No model-error bound follows from control sufficiency alone.

Vacuity.

Informative only if $\xi <(C_{\max }-C_{\min })/2$ and $Hϵ<(C_{\max }-C_{\min })/2$.

3.2. Exploitability-Calibrated Gate

Theorem 3

(Finite-sample exploitability-calibrated gate). Under exchangeability with X trivial, $\mathbb{P}(G=1\wedge C_{\mathrm{true}}\left(\pi \right)>\tau +\widehat{q})\le \alpha +TV({\mathcal{D}}_{\mathrm{cal}},{\mathcal{D}}_{\mathrm{test}})$. With decoder error, $\mathbb{P}(\mathrm{execute}\wedge \mathrm{violation})\le \alpha +{\alpha }_{\mathrm{dec}}+TV$.

Proof. 

If $G=1$, $C_{\mathrm{model}}\left(\pi \right)\le \tau$. Violation requires $\beta >\widehat{q}$; apply finite-sample conformal; union bound for the decoder term. Full proof in Appendix B.    □

Vacuity.

$\tau +\widehat{q}+TV>{\tau }_{\mathrm{task}}$.

3.3. Conditional Conformal under Contact

Theorem 4

(Mondrian conformal under contact). Under within-stratum exchangeability with $X={\widehat{c}}_0$, the conformal set ${\mathcal{C}}_{\alpha }=\{z_{1:H}:{\max }_h\parallel {\widehat{z}}_h-z_h\parallel \le {\widehat{q}}_x{\widehat{w}}_h\}$ satisfies $\mathbb{P}(z_{1:H}^{★}\in {\mathcal{C}}_{\alpha }\mid X=x)\ge 1-\alpha$.

Proof. 

Split-conformal argument within each stratum [10]. Full proof in Appendix B.    □

Vacuity.

Vacuous when $n_x+1<1/\alpha$, within-stratum exchangeability fails, or X is unobservable.

3.4. Contact-Mode Identity

Theorem 5

(Contact-mode identity). With ${\mathsf{\Delta }}_{\mathrm{bar}}=\mathbb{E}[{\ell }_{\widehat{c}}-{\ell }_c\mid \widehat{c}\ne c]$, $E=\mathbb{E}\left[{\ell }_c\right]+\eta {\mathsf{\Delta }}_{\mathrm{bar}}$. If ${\ell }_{\widehat{c}}-{\ell }_c\le \mathsf{\Delta }$ a.s., then $E\le \max (E_0,E_1)+\eta \mathsf{\Delta }$.

Proof. 

Decompose over $\widehat{c}=c$ and $\widehat{c}\ne c$. Full proof in Appendix B.    □

3.5. Oracle Feasibility as an Upper Confidence Bound

Theorem 6

(Oracle feasibility). Any learned model using the same planner and cost has success rate at most $p_{\mathrm{oracle}}+{ϵ}_{\mathrm{planner}}$. With $0/50$, the one-sided $95\%$ upper bound is $p_U=1-0.{05}^{1/50}\approx 0.058$.

Proof. 

The learned model only changes cost evaluation and dynamics prediction; apply Clopper–Pearson. Full proof in Appendix B.    □

Remark 3

(Attribution of the $0.058$). Theorem 6 instantiates $p_U$ at the $0/50$ measurement reported inDEGP [2]—the deployment study this paper builds on—not at this paper’s oracle runs. This paper’s own oracle runs (Table 8) instantiate the same Clopper–Pearson formula at different sample counts: $13/50$ yields $p_U\approx 0.38$ and $2/50$ yields $p_U\approx 0.12$, both reported as one-sided $95\%$ upper bounds in that table. The three values are the same bound under three different sample counts; they are not competing estimates.

4. Lower Bounds on Vacuity Parameters

4.1. Lower Bound on $ϵ$

Theorem 7

(Latent-Lipschitz lower bound). If $f_{\theta }$ is $({\delta }_f,G)$-contact-blind and $Q_Z$ is L-Lipschitz, then $ϵ\ge (G-L{\delta }_f)/2$.

Proof. 

Triangle inequality: $G\le 2ϵ+L{\delta }_f$. Full proof in Appendix A.    □

4.2. Lower Bound on Decoder Error

Theorem 8

(Shannon lower bound). For a source $Y\in {\mathbb{R}}^{d_Y}$ with differential entropy $h\left(Y\right)$ (in bits) decoded from a representation Z with mutual information $I(Z;Y)$ (in bits), the per-dimension mean-squared decoder error obeys

$${\rho }_{\mathrm{MSE}}(h,\mathcal{D})\ge {\displaystyle \frac{1}{2\pi e}}{2}^{2(h\left(Y\right)-I(Z;Y))/d_Y}.$$

(1)

Proof. 

The Shannon lower bound for squared-error distortion follows from the entropy-power inequality. For a $d_Y$-dimensional source with differential entropy h (bits) and rate $R=I(Z;Y)$ (bits), the minimum achievable MSE per dimension is $D\ge {\displaystyle \frac{1}{2\pi e}}{2}^{2(h-R)/d_Y}$. The constant $2\pi e$ comes from the Gaussian’s differential entropy in bits, $h_{\mathcal{N}}={\displaystyle \frac{d_Y}{2}}{\log }_2\left(2\pi e{\sigma }^2\right)$, inserted into the rate–distortion function $R\left(D\right)={\displaystyle \frac{d_Y}{2}}{\log }_2({\sigma }^2/D)$ and inverted for D. Both h and I are measured in the same units (bits); using nats replaces $2\pi e$ by $2\pi$ and the base-2 exponent by the natural exponent. Full derivation in Appendix A.    □

4.3. Lower Bound on ${\beta }_q$

Lemma 1

(Replay gap bounds the exploitability gap). If $C_{\mathrm{model}}\left({\pi }_{\mathrm{CEM}}\right)\le C_{\mathrm{model}}\left({\pi }_{\mathrm{replay}}\right)$, then $\beta \left({\pi }_{\mathrm{CEM}}\right)\ge \gamma -\xi$.

Proof. 

$\beta \left({\pi }_{\mathrm{CEM}}\right)\ge C_{\mathrm{true}}\left({\pi }_{\mathrm{CEM}}\right)-C_{\mathrm{true}}\left({\pi }_{\mathrm{replay}}\right)-\xi =\gamma -\xi$. Full proof in Appendix A.    □

Theorem 9

(Lower bound on ${\beta }_q$). Under Lemma 1, if $\gamma >\xi$, then ${\beta }_q\ge \gamma -\xi >0$.

4.4. Lower Bound on $\eta$

Theorem 10

(Prior-corrected Bayes bound). $\eta \ge \min ({\pi }_0,{\pi }_1)(1-TV(P_0,P_1))$. For Push-T the majority-class baseline is $0.364$.

Proof. 

Bayes error identity for binary classification with priors. Full proof in Appendix A.    □

4.5. Planner Impossibility

Theorem 11

(Planner impossibility). Under Theorem 6, $p_{\mathrm{oracle}}\le p_U\approx 0.058$ at $95\%$ confidence when the deployment measurement is the DEGP $0/50$ null.

Proof. 

Clopper–Pearson for $0/50$. Full proof in Appendix A.    □

4.6. Decoder Transfer Bound

Lemma 2

(Transfer gap). $|{\rho }_{\mathrm{dec}}(h,{\mathcal{D}}_{\mathrm{val}})-{\rho }_{\mathrm{dec}}(h,{\mathcal{D}}_{\mathrm{train}})|\le TV({\mathcal{D}}_{\mathrm{train}},{\mathcal{D}}_{\mathrm{val}})·diam\left(\mathcal{Y}\right)$.

Corollary 1

(Transfer-gap bound). ${\rho }_{\mathrm{dec}}(h,{\mathcal{D}}_{\mathrm{val}})\ge {\rho }^{★}\left({\mathcal{D}}_{\mathrm{val}}\right)-TV·diam\left(\mathcal{Y}\right)$.

4.7. Exchangeability Failure

Theorem 12

(Coverage gap under shift). $|\mathbb{P}(Y\in {\mathcal{C}}_{\alpha })-(1-\alpha )|\le TV({\mathcal{D}}_{\mathrm{cal}},{\mathcal{D}}_{\mathrm{test}})$.

Proof. 

Standard conformal coverage bound under covariate shift [8]. Full proof in Appendix A.    □

4.8. Gate Impossibility

Theorem 13

(Gate abstention). If $\tau +\widehat{q}+TV·diam\left(\mathcal{Y}\right)>{\tau }_{\mathrm{task}}$, every plan with $C_{\mathrm{model}}\left(\pi \right)\le \tau$ abstains or has certified bound exceeding ${\tau }_{\mathrm{task}}$.

4.9. Wiring Results

Proposition 1

(Decoder-floor measured decomposition). ${\rho }_{\mathrm{dec}}{(h,{\mathcal{D}}_{\mathrm{held}})}^2={\rho }_{\mathrm{dec}}{(h,{\mathcal{D}}_{\mathrm{fit}})}^2+{\mathsf{\Delta }}_{\mathrm{held}}^2+{\mathsf{\Delta }}_{\mathrm{overfit}}^2$. Measured finding: $40.7$–$42.2$ px.

Proposition 2

(Attribution control). Shuffled target satisfies $I(g_{\mathrm{perm}};z_{t+1}\mid z_t,a_t)=0$. Measured finding: coordinate channel and both controls indistinguishable from baseline at $r\in \{16,64\}$.

Corollary 2

(Lipschitz bridge diagnostic). $L_h\in [791,5388]$ px per latent unit for the ReLU MLP decoder. With ${\tau }_{\mathrm{task}}=100$ px, non-vacuity requires ${\widehat{q}}_{\mathrm{lat}}{\widehat{w}}_H\le 100/L_h$. Measured task-space widths: 126–279 px.

5. The Lower-Bound Map

Reading the $ϵ$ row.

The Table’s $ϵ$ row is a proxy, not a numerical evaluation of the bound $(G-L{\delta }_f)/2$. The logic is: flat selectivity at every coupled rank ($\le 1.04\times$) means that contact and free strata are not separated in predicted task-space movement; combined with the evidence that the same latents do decode contact at $5.3$–$6.0\times$ (Paper II attribution), this gives cross-mode state pairs at matched latent distance (small ${\delta }_f$) with large value gap G—the contact-blindness premise of Definition 9 and Theorem 7. The bound therefore binds. Neither G nor ${\delta }_f$ is instantiated numerically in this paper; the selectivity measurement is the evidence that the premise is met, and instantiating $G,{\delta }_f$ is left to Paper IV.

Table 2. Lower-bound map with 30-seed measured premises.

Parameter	Lower bound	30-seed realization	Vacuity threshold
$ϵ$	$(G-L{\delta }_f)/2$	Selectivity $0.9997\to 1.04$ at $r=128$; $p=0.96$ (proxy; see text)	$ϵ+{\delta }_{\mathrm{lat}}<(C_{\max }-C_{\min })/2$
${\rho }_{\mathrm{dec}}$	Shannon LB	$40.7$–$42.2$ px cross-episode	floor $<20$ px
${\beta }_q$	$\ge \gamma -\xi$	conditioned on CEM-vs-replay gap	$\tau +\widehat{q}<{\tau }_{\mathrm{task}}$
$\eta$	$\min ({\pi }_0,{\pi }_1)(1-TV)$	Contact rate $36.4\%$; baseline $0.364$	$\eta \mathsf{\Delta }<\max (E_0,E_1)$
$p_{\mathrm{oracle}}$	Clopper–Pearson	$13/50$; $2/50$ coverage cost	$p_U>0$
${\rho }_{\mathrm{dec},\mathrm{val}}$	Transfer gap	held-out vs fit gap 1–3 px	floor $<20$ px
Coverage	$\le TV$	random $0.8995$–$0.8998$; chrono $0.862$–$0.914$	gap $<0.03$

Table 2. Lower-bound map with 30-seed measured premises.

Parameter	Lower bound	30-seed realization	Vacuity threshold
$ϵ$	$(G-L{\delta }_f)/2$	Selectivity $0.9997\to 1.04$ at $r=128$; $p=0.96$ (proxy; see text)	$ϵ+{\delta }_{\mathrm{lat}}<(C_{\max }-C_{\min })/2$
${\rho }_{\mathrm{dec}}$	Shannon LB	$40.7$–$42.2$ px cross-episode	floor $<20$ px
${\beta }_q$	$\ge \gamma -\xi$	conditioned on CEM-vs-replay gap	$\tau +\widehat{q}<{\tau }_{\mathrm{task}}$
$\eta$	$\min ({\pi }_0,{\pi }_1)(1-TV)$	Contact rate $36.4\%$; baseline $0.364$	$\eta \mathsf{\Delta }<\max (E_0,E_1)$
$p_{\mathrm{oracle}}$	Clopper–Pearson	$13/50$; $2/50$ coverage cost	$p_U>0$
${\rho }_{\mathrm{dec},\mathrm{val}}$	Transfer gap	held-out vs fit gap 1–3 px	floor $<20$ px
Coverage	$\le TV$	random $0.8995$–$0.8998$; chrono $0.862$–$0.914$	gap $<0.03$

6. Measured Premises

6.1. Setup

All experiments use the preserved Push-T corpus of DEGP [2]: gym-pusht with pymunk 6.11.1; 250 episodes × 200 steps $=50,000$ transitions; $96\times 96\times 3$ observations plus 2-D agent position; episode-level split 200 train / 50 validation; success is coverage $\ge 0.95$. Latent dimension $K=128$. $36.4\%$ contact, $8.0\%$ block-moving.

Table 3. Premise provenance. What is inherited, what is re-measured, and what is new in this paper.

Quantity	Value	Status in this paper	Source
Push-T corpus ($250\times 200$)	$50,000$ transitions	Inherited; deterministic regeneration (see Table 4)	[2]
Contact / block-moving rates	$36.4\%$ / $8.0\%$	Inherited, measured	[2]
Coupled-encoder selectivity	$\le 1.04\times$ ($p=0.96$)	Re-measured (supersedes $1.20\times$)	§Section 6.2
Attribution controls	Indistinguishable (Holm $p=1$)	Re-measured (120 controls)	§Section 6.3
Decoder floor	$40.7$–$42.2$ px	New (90 decoders)	§Section 6.4
Certificate widths	126–279 px @ $0.90$	New	§Section 6.4
Oracle feasibility	$13/50$; $2/50$	New (P4 falsified)	§Section 6.5

Table 3. Premise provenance. What is inherited, what is re-measured, and what is new in this paper.

Quantity	Value	Status in this paper	Source
Push-T corpus ($250\times 200$)	$50,000$ transitions	Inherited; deterministic regeneration (see Table 4)	[2]
Contact / block-moving rates	$36.4\%$ / $8.0\%$	Inherited, measured	[2]
Coupled-encoder selectivity	$\le 1.04\times$ ($p=0.96$)	Re-measured (supersedes $1.20\times$)	§Section 6.2
Attribution controls	Indistinguishable (Holm $p=1$)	Re-measured (120 controls)	§Section 6.3
Decoder floor	$40.7$–$42.2$ px	New (90 decoders)	§Section 6.4
Certificate widths	126–279 px @ $0.90$	New	§Section 6.4
Oracle feasibility	$13/50$; $2/50$	New (P4 falsified)	§Section 6.5

Figure 2. Code-module dependency graph. The pipeline flows compactly from left to right: data ingestion, training and control script execution, checkpoint generation, evaluation, and final output artifacts.

Figure 2. Code-module dependency graph. The pipeline flows compactly from left to right: data ingestion, training and control script execution, checkpoint generation, evaluation, and final output artifacts.

Corpus provenance.

The Push-T corpus is the only inherited artifact in this paper. Its release [2] includes a seeded collection script—fixed RNG seed, pymunk 6.11.1, gym-pusht, and the exact $250\times 200$ protocol above—that is documented to regenerate the corpus deterministically. For this paper we used a preserved copy of that release rather than re-running the collection script, and we did not perform a bit-identity check against a regenerated corpus. Table 4 records the SHA-256 hash of the preserved corpus file actually used for every measurement reported below, alongside the reserved slot for the regenerated file. Two consequences follow. (i) Every quantity in this paper is conditioned on the preserved file; any drift from a regenerated corpus is unmeasured here. (ii) Closing the chain is a single-command check—run the released script, hash the output, populate the right-hand column—and we list this as an explicit deliverable to Paper IV. The provenance chain is thus closed on seed, environment, and collection protocol, and open on bit-level identity; the paper reports it as such rather than claiming a closure that was not performed.

Table 4. Corpus provenance. The preserved file is what every measurement in this paper conditions on. The regenerated slot is reserved for the one-command bit-identity check deliverable to Paper IV.

Artifact	SHA-256 (preserved file, this paper)	SHA-256 (regenerated)
push_t_corpus_250x200.npz	TODO: sha256sum	TODO: run released script and fill
Collection script commit	TODO: fill from [2] repository	—
Environment	pymunk 6.11.1, gym-pusht	pymunk 6.11.1, gym-pusht
RNG seed	fixed (released)	fixed (released)

Table 4. Corpus provenance. The preserved file is what every measurement in this paper conditions on. The regenerated slot is reserved for the one-command bit-identity check deliverable to Paper IV.

Artifact	SHA-256 (preserved file, this paper)	SHA-256 (regenerated)
push_t_corpus_250x200.npz	TODO: sha256sum	TODO: run released script and fill
Collection script commit	TODO: fill from [2] repository	—
Environment	pymunk 6.11.1, gym-pusht	pymunk 6.11.1, gym-pusht
RNG seed	fixed (released)	fixed (released)

6.2. 30-Seed Rank Sweep

Findings.

(1) Selectivity is flat; the $2.0\times$ target is not met at any rank. (2) Contact skill peaks at $0.0119$ at $r=128$. (3) Free skill is negative at every coupled rank. (4) The ${ϵ}_{\mathrm{eff}}T\le 0.5$ budget holds at every rank. (5) Median $\kappa$ stays below $1.5$. (6) Decodability degrades mildly with coupling.

Table 5. 30-seed rank sweep. Mean over 30 paired seeds; p = Holm-adjusted paired Wilcoxon vs. the $r=0$ reference column. The $r=0$ column is the baseline (all other columns test against it); p-value rows therefore have five entries, one per non-reference column.

	$\mathit{r}=0$ (ref.)	$\mathit{r}=8$	$\mathit{r}=16$	$\mathit{r}=32$	$\mathit{r}=64$	$\mathit{r}=128$
Selectivity	$0.9997$	$0.9994$	$1.0004$	$1.009$	$1.018$	$1.040$
		$p=1$	$p=1$	$p=0.48$	$p=0.33$	$p=0.96$
Skill, contact	$0.00043$	$0.00041$	$0.00041$	$0.0013$	$0.0053$	$0.0119$
		$p=1$	$p=1$	$p=5.5\times {10}^{-5}$	$p=3.7\times {10}^{-8}$	$p=9.3\times {10}^{-9}$
Skill, free	$-0.0031$	$-0.0034$	$-0.0036$	$-0.0044$	$-0.0058$	$-0.0031$
$\overline{m}$ (leaked moment)	$1.91$	$1.83$	$1.77$	$1.77$	$1.76$	$2.01$
${ϵ}_{\mathrm{eff}}T$ (mean)	$0.00033$	$0.00077$	$0.0012$	$0.0223$	$0.0789$	$0.1964$
$\kappa$ (median)	$1.0007$	$1.0016$	$1.0024$	$1.045$	$1.161$	$1.484$
$R^2$ (held-out pos.)	$0.8845$	$0.8844$	$0.8845$	$0.8845$	$0.8795$	$0.8757$
		$p=1$	$p=1$	$p=1$	$p=0.002$	$p=4.6\times {10}^{-5}$

Table 5. 30-seed rank sweep. Mean over 30 paired seeds; p = Holm-adjusted paired Wilcoxon vs. the $r=0$ reference column. The $r=0$ column is the baseline (all other columns test against it); p-value rows therefore have five entries, one per non-reference column.

	$\mathit{r}=0$ (ref.)	$\mathit{r}=8$	$\mathit{r}=16$	$\mathit{r}=32$	$\mathit{r}=64$	$\mathit{r}=128$
Selectivity	$0.9997$	$0.9994$	$1.0004$	$1.009$	$1.018$	$1.040$
		$p=1$	$p=1$	$p=0.48$	$p=0.33$	$p=0.96$
Skill, contact	$0.00043$	$0.00041$	$0.00041$	$0.0013$	$0.0053$	$0.0119$
		$p=1$	$p=1$	$p=5.5\times {10}^{-5}$	$p=3.7\times {10}^{-8}$	$p=9.3\times {10}^{-9}$
Skill, free	$-0.0031$	$-0.0034$	$-0.0036$	$-0.0044$	$-0.0058$	$-0.0031$
$\overline{m}$ (leaked moment)	$1.91$	$1.83$	$1.77$	$1.77$	$1.76$	$2.01$
${ϵ}_{\mathrm{eff}}T$ (mean)	$0.00033$	$0.00077$	$0.0012$	$0.0223$	$0.0789$	$0.1964$
$\kappa$ (median)	$1.0007$	$1.0016$	$1.0024$	$1.045$	$1.161$	$1.484$
$R^2$ (held-out pos.)	$0.8845$	$0.8844$	$0.8845$	$0.8845$	$0.8795$	$0.8757$
		$p=1$	$p=1$	$p=1$	$p=0.002$	$p=4.6\times {10}^{-5}$

Reading the $ϵ$ row of Table 2.

The flat selectivity in this Table is the proxy for the vacuity of the control-sufficiency bound: it is the empirical signature that the contact-blindness premise of Definition 9 holds at every coupled rank, hence that the lower bound $ϵ\ge (G-L{\delta }_f)/2$ binds without G and ${\delta }_f$ being numerically instantiated (see the paragraph after Table 2).

6.3. Attribution Controls

Findings.

No control produces a selectivity gain.

Table 6. 150 control models (120 attribution controls + 30 baseline re-runs). Holm-adjusted $p=1$ for every condition on every metric.

	baseline $\mathit{r}=0$	shuffle $\mathit{r}=16$	shuffle $\mathit{r}=64$	random-II $\mathit{r}=16$	random-II $\mathit{r}=64$
Selectivity	$0.9998$	$0.9961$	$0.9996$	$1.0035$	$0.9983$
Skill, contact	$0.0018$	$-0.0005$	$0.0036$	$0.0029$	$-0.0041$
Skill, free	$0.0001$	$-0.0087$	$-0.0051$	$0.0015$	$-0.0028$
${ϵ}_{\mathrm{eff}}T$ (sup)	$115.3$	552	618	133	114
$\kappa$ (median)	$1.0000$	$1.0000$	$1.0000$	$1.0008$	$1.0009$

Table 6. 150 control models (120 attribution controls + 30 baseline re-runs). Holm-adjusted $p=1$ for every condition on every metric.

	baseline $\mathit{r}=0$	shuffle $\mathit{r}=16$	shuffle $\mathit{r}=64$	random-II $\mathit{r}=16$	random-II $\mathit{r}=64$
Selectivity	$0.9998$	$0.9961$	$0.9996$	$1.0035$	$0.9983$
Skill, contact	$0.0018$	$-0.0005$	$0.0036$	$0.0029$	$-0.0041$
Skill, free	$0.0001$	$-0.0087$	$-0.0051$	$0.0015$	$-0.0028$
${ϵ}_{\mathrm{eff}}T$ (sup)	$115.3$	552	618	133	114
$\kappa$ (median)	$1.0000$	$1.0000$	$1.0000$	$1.0008$	$1.0009$

6.4. Decoder Floor and Certificate Widths

Table 7. Decoder floor and split-conformal certificate widths at $r\in \{0,64,128\}$, 30 seeds per rank.

	$\mathit{r}=0$	$\mathit{r}=64$	$\mathit{r}=128$	Paper earlier draft
Decoder floor, val median px	$42.24$	$40.91$	$40.67$	$\approx 50$
Cert H=4 width px	$125.9$	$129.3$	$139.4$	83–115
Cert H=8 width px	$178.0$	$182.8$	$197.1$	97–135
Cert H=16 width px	$251.7$	$258.5$	$278.7$	109–164
Coverage, random split	$0.8995$	$0.8998$	$0.8997$	$0.895$–$0.900$
Coverage, chronological split	$0.862$	$0.897$	$0.914$	$0.79$–$0.84$

Table 7. Decoder floor and split-conformal certificate widths at $r\in \{0,64,128\}$, 30 seeds per rank.

	$\mathit{r}=0$	$\mathit{r}=64$	$\mathit{r}=128$	Paper earlier draft
Decoder floor, val median px	$42.24$	$40.91$	$40.67$	$\approx 50$
Cert H=4 width px	$125.9$	$129.3$	$139.4$	83–115
Cert H=8 width px	$178.0$	$182.8$	$197.1$	97–135
Cert H=16 width px	$251.7$	$258.5$	$278.7$	109–164
Coverage, random split	$0.8995$	$0.8998$	$0.8997$	$0.895$–$0.900$
Coverage, chronological split	$0.862$	$0.897$	$0.914$	$0.79$–$0.84$

6.5. Terminal-Cost and Coverage-Cost Oracles

Table 8. Oracle with exact dynamics. P4 prediction: coverage-shaped cost raises $p_{\mathrm{oracle}}\ge 0.5$. Result: falsified. One-sided $95\%$ Clopper–Pearson upper bounds on the success rate: $13/50\Rightarrow p_U\approx 0.38$; $2/50\Rightarrow p_U\approx 0.12$; $0/100\Rightarrow p_U\approx 0.037$.

Cost	Seeds	n	Success	95% CP upper ${\mathit{p}}_{\mathit{U}}$
Terminal-pose	7000–7049	50	13 ($26.0\%$)	$\approx 0.38$
Terminal-pose	5000–5049	50	16 ($32.0\%$)	$\approx 0.44$
Coverage-shaped	7000–7049	50	2 ($4.0\%$)	$\approx 0.12$
Random actions	both	100	0 ($0\%$)	$\approx 0.037$

Table 8. Oracle with exact dynamics. P4 prediction: coverage-shaped cost raises $p_{\mathrm{oracle}}\ge 0.5$. Result: falsified. One-sided $95\%$ Clopper–Pearson upper bounds on the success rate: $13/50\Rightarrow p_U\approx 0.38$; $2/50\Rightarrow p_U\approx 0.12$; $0/100\Rightarrow p_U\approx 0.037$.

Cost	Seeds	n	Success	95% CP upper ${\mathit{p}}_{\mathit{U}}$
Terminal-pose	7000–7049	50	13 ($26.0\%$)	$\approx 0.38$
Terminal-pose	5000–5049	50	16 ($32.0\%$)	$\approx 0.44$
Coverage-shaped	7000–7049	50	2 ($4.0\%$)	$\approx 0.12$
Random actions	both	100	0 ($0\%$)	$\approx 0.037$

7. What the Constructive Literature Achieves

The table’s structure is the point. Each of the five papers climbs exactly one wall—Bounou control sufficiency, Destrade the planner cost signal, Parthasarathy exploitability under shift, Wang predictor conditioning, Mirzaeedodangeh certificate transfer—and none climbs two. The two entries that touch two conditions at once do so because those conditions are already coupled in the cited mechanism; they do not close the conjunction of all five. Section 8’s architecture is therefore the first attempt in this program to combine all five under one stack, and the P1–P6 gate of Section 10 is what it must pass. Two of the five walls correspond to conditions whose vacuity this paper measures directly ($ϵ$ and $p_{\mathrm{oracle}}$); the remaining three correspond to conditions whose vacuity is inherited from DEGP [2] and instantiated in Table 2.

Table 9. Constructive literature placement: five papers, five walls, no conjunction. The “Condition” column maps each paper to the five conditions formalized in Section 3 where a clean mapping exists.

Paper	Wall climbed	Mechanism	Regime	Condition
Bounou et al. [3]	Control sufficiency	Differentiable LQR	Smooth dynamics	$ϵ$
Destrade et al. [4]	Cost / planner signal	IQL value alignment	2D navigation	${\beta }_q$
Parthasarathy et al. [5]	Exploitability under shift	Online/adversarial FT	PushT, maze	${\beta }_q$, exchangeability
Wang et al. [6]	Predictor conditioning	Curvature regularizer	PushT, maze	$ϵ$, $\delta$ (predictor side)
Mirzaeedodangeh et al. [7]	Certificate transfer under shift	ARCP + shift budget	Car-pedestrian	exchangeability, coverage

Table 9. Constructive literature placement: five papers, five walls, no conjunction. The “Condition” column maps each paper to the five conditions formalized in Section 3 where a clean mapping exists.

Paper	Wall climbed	Mechanism	Regime	Condition
Bounou et al. [3]	Control sufficiency	Differentiable LQR	Smooth dynamics	$ϵ$
Destrade et al. [4]	Cost / planner signal	IQL value alignment	2D navigation	${\beta }_q$
Parthasarathy et al. [5]	Exploitability under shift	Online/adversarial FT	PushT, maze	${\beta }_q$, exchangeability
Wang et al. [6]	Predictor conditioning	Curvature regularizer	PushT, maze	$ϵ$, $\delta$ (predictor side)
Mirzaeedodangeh et al. [7]	Certificate transfer under shift	ARCP + shift budget	Car-pedestrian	exchangeability, coverage

8. Architecture (Specified, Not Evaluated)

1.

Encoder $f_{\theta }$. Entropic variance–covariance (unchanged from DEGP) or frozen pretrained features.

2.

Contact-conditional predictor $g_{\varphi }$. Gated hybrid, target selectivity $>2.0\times$.

3.

Transfer-oriented decoder $h_{\psi }$. Target held-out floor $<20$ px.

4.

Planner. CEM with a hybrid cost: coverage shaping plus a pose anchor,

$$C_{\mathrm{plan}}\left({\widehat{z}}_H\right)=-\mathrm{coverage}\left({\widehat{z}}_H\right)+{\lambda }_{\mathrm{pose}}{∥\widehat{h}\left({\widehat{z}}_H\right)-y_{\mathrm{goal}}∥}^2,$$

(2)

matching Paper IV’s Eq. 3.

5.

Certificate head $c_{\chi }$. Trained with pinball loss.

6.

Gate. Execute iff width $\le {\tau }_{\mathrm{task}}$ and health $=1$.

7.

Exploitability diagnostic. Real-action replay floor plus action-space distance.

Why the hybrid. Step 0 therefore tests a repaired cost family: the pure coverage-shaped cost was falsified at the oracle level ($2/50$, P4), and terminal-pose alone reached only $13/50$; the hybrid combines coverage shaping with a pose anchor. This is exactly the “planner repair” Step 0 names, and it removes the ambiguity a naive re-use of the pure coverage-shaped cost would inherit into Paper IV’s pre-registration.

9. Experimental Protocol (Pre-Specified)

9.1. Step 0: Planner Repair on Exact Dynamics

Test the hybrid cost family of Eq. 2 against the two falsified/degenerate extremes: pure coverage-shaped ($2/50$, P4 falsified) and pure terminal-pose ($13/50$). Gate: $p_{\mathrm{oracle}}\ge 0.5$ on the hybrid; else stop.

9.2. Step 1: Decoder Transfer

Floor $<20$ px; else decoder-centric premise fails.

9.3. Step 2: Contact-Conditional Predictor

Selectivity $>2.0\times$; else encoder lacks selectivity.

9.4. Step 3: B-PLAN Conformal Calibration

Overall $\ge 0.87$, contact $\ge 0.80$.

9.5. Step 4: Closed-Loop Evaluation

Success $\ge 10/50$ under the gate; violations $\le \alpha +{\alpha }_{\mathrm{dec}}$.

9.6. Step 5: Exploitability Diagnostic

CEM does not beat real-action replay.

10. Falsifiable Predictions

Table 10. Predictions as premise tests.

ID	Premise	Threshold	Status / If fails
P1	Decoder transfer	Floor $<20$ px	not evaluated
P2	Contact selectivity	$>2.0\times$	falsified (max $1.04\times$)
P3	Conditional exchangeability	Overall $\ge 0.87$	not evaluated
P4	Oracle feasibility	$p_{\mathrm{oracle}}\ge 0.5$	falsified ($2/50$ coverage, $13/50$ terminal)
P5	Gate safety	Violations $\le \alpha +{\alpha }_{\mathrm{dec}}$	not evaluated
P6	No exploitation	CEM ≤ replay	not evaluated

Table 10. Predictions as premise tests.

ID	Premise	Threshold	Status / If fails
P1	Decoder transfer	Floor $<20$ px	not evaluated
P2	Contact selectivity	$>2.0\times$	falsified (max $1.04\times$)
P3	Conditional exchangeability	Overall $\ge 0.87$	not evaluated
P4	Oracle feasibility	$p_{\mathrm{oracle}}\ge 0.5$	falsified ($2/50$ coverage, $13/50$ terminal)
P5	Gate safety	Violations $\le \alpha +{\alpha }_{\mathrm{dec}}$	not evaluated
P6	No exploitation	CEM ≤ replay	not evaluated

11. Discussion

11.1. What a Failure Would Mean

Each failure is a field, not a verdict. P1 (decoder floor $40.7$–$42.2$ px vs. $<20$ px) opens transfer-oriented decoder design. P2 (selectivity flat $\le 1.04\times$ vs. $>2.0\times$) opens contact-conditional prediction and encoder–predictor coupling. P3 (conditional exchangeability, overall $\ge 0.87$) opens stratified conformal calibration under contact. P4 (oracle feasibility, falsified) opens the planner-cost family of Section 8. P5 (gate safety, violations $\le \alpha +{\alpha }_{\mathrm{dec}}$) opens certified decision rules under decoder slack. P6 (no exploitation, CEM ≤ replay) opens the real-action-replay diagnostic as a standard offline-planning check. The failures are not verdicts on the framework; they are the specifications of what Paper IV must build.

11.2. What the 30-Seed Run Establishes

Selectivity is flat; free-stratum skill is negative; controls are indistinguishable from baseline; conditioning holds; decoder floor is $40.7$–$42.2$ px; oracle ceiling $13/50$ terminal, $2/50$ coverage.

11.3. The Evaluated P4 Prediction

$2/50$ versus $13/50$ for terminal-pose cost. P4 is falsified.

Figure 3. Summary of measured premises from Parts 2–4.

Figure 3. Summary of measured premises from Parts 2–4.

11.4. Limitations

Theorems are conditional. One benchmark. Certificate widths use a proxy $H^{1/2}$ scaling (Definition 11). Corpus provenance is closed on seed, environment, and collection protocol, and open on bit-level identity (Table 4); closing it is a single-command check reserved as a deliverable to Paper IV. Non-significant differences at 30 seeds not read as equivalence. Architecture is specified but not evaluated.

12. Conclusions

Dynamic LeJEPA asked where physics belongs; DEGP asked whether certified conditioning enables planning. Both receive conditional responses here. A 30-seed replication sharpens the barrier: selectivity is flat, free-stratum skill is negative, controls are indistinguishable from baseline. Decoder floor: $40.7$–$42.2$ px. Certificate widths: 126–279 px at nominal $0.90$. P4 is evaluated and falsified. The result is a boundary paper: a map of the exact assumptions a control-sufficient latent world model requires.

Program status.

After these three papers, the field knows four things it did not know before: (1) physics-informed world models fail for an identified reason—wrong placement (Paper I [1]); (2) offline planning metrics cannot certify control readiness—the exploitation confound (Paper I); (3) certified conditioning does not rescue deployment—the best-certified instance fails $0/50$, localized to contact prediction (Paper II [2]); (4) that failure is quantified and conditional: five requirements, each violated by a measured margin, each attackable in principle (this paper). What remains open is joint satisfiability—the fourth paper’s pre-registered question.