Memory as a Wasting Asset: Pricing Flash Endurance for Embodied Agents, and the Limits of Doing So
arXiv:2606.18144v1 [cs.AI] 16 Jun 2026
Josef Chen KAIKAKU [email protected]
June 2026
Abstract A robot’s flash endurance is a non-renewable stock: every persisted write spends one of a few thousand program/erase cycles and never refills, yet no fielded robot memory system prices which memories are worth an erase cycle. We treat embodied memory as depreciating capital and price that stock with a single endurance shadow price η, which makes cost-minimizing placement across a RAM / on-board NVM / cloud hierarchy a threshold in a wear-augmented per-byte index. The index is cost-optimal whatever the sign of the value–write association χ; only when χ > 0 does the optimum turn non-monotone, sending a robot’s most valuable memories off its flash. The pivot is thus empirical, and we measure χ on real robot logs at a pre-specified gate: its sign is a property of the deployment regime—positive on recurrent long-horizon manipulation (χ̂ = +1.0 × 10−3 , replicated at full power), null on a shorter-horizon suite, and negative on non-recurrent teleoperation. Two boundaries scope the result. The endurance budget is dormant on premium 3,000-P/E TLC at datasheet prices and binding on the commodity QLC/eMMC (∼ 1,000 P/E) that cheaper edge robots run. And where it binds, a learned wear-aware controller only ties price-based routing on task value, because realized value is tier-invariant across RAM, NVM, and cloud: the rent governs device lifetime and cost, not task performance. Whether wear-aware placement improves task value remains open—χ is measured against a value proxy, and the non-monotone optimum, while proven, is not yet observed in data.
1
Introduction
A robot ships with a finite quantity of flash. Every block of its on-board NAND tolerates a fixed number of program/erase cycles (roughly 3,000 for the TLC parts that dominate edge platforms [31, 59]), after which it wears out and is gone. On-board memory is therefore not free storage: each persisted write spends a fraction of a stock that does not refill, so the right object to reason about is memory as a depreciating capital asset carrying a per-period user cost [23], not a scratchpad of unlimited capacity (fig. 1). No fielded embodied-memory system prices this. They decide what to keep; none decide which kept memory is worth an erase cycle, in which physical tier it should sit, or what spending the cycle costs in joules and device lifetime. Any embodied memory system must answer three questions: when to write a memory, where that memory should physically live, and what it is worth to keep it there. The first is the subject of our predecessor AURA [9], a learned write gate; this paper supplies the open pair. The program is 1
A robot's memory is a wasting asset Every memory written to on-board flash spends one of ~3,000 program/erase cycles that never come back.
So every kept memory faces a price:
RAM renewable — refills every period
keep in RAM
fast, volatile — pay capacity rent
persist to flash
spend an erase cycle: cash wear + rent η
resets each step
offload to cloud
On-board NAND endurance
rent someone else's endurance, pay latency
non-renewable — ~3,000 erases, then the cell is gone
forget
free now — risk paying to re-acquire later device life writes draw it down — no refill
η — the price of one erase cycle No fielded robot prices this. This paper builds that price.
Figure 1: The core idea in one picture. On-board NAND endurance is a non-renewable stock: each persisted write spends one of a few thousand program/erase cycles and is gone, whereas RAM capacity refills every period. Every retained memory therefore faces a priced choice—keep in RAM, persist to flash (spending an erase cycle), offload to cloud, or forget—governed by a single endurance rent η. No fielded embodied-memory system prices this; we build that price.
a three-stage arc, WHEN → WHERE → WORTH (fig. 2): AURA gates the write; this paper places the retained item across a RAM / on-board NVM / cloud hierarchy; and an economic layer prices the scarce resource that placement consumes: the erase cycle. WHEN. Embodied-memory systems are uniformly what-to-remember machinery [21, 53, 57]: they gate, retrieve, or consolidate content to maximize task success. AURA is the author’s own when-to-write gate on a single constant-size store. None of these decide the physical tier an item occupies, its joule/erase cost, or its economic worth. WHERE. Datacenter wear-aware caching already exhibits the core qualitative behavior we study (keeping write-heavy objects off endurance-limited flash [15, 38, 62]), and we do not claim the phenomenon. What is unoccupied is the embodied object: in a robot, placement is coupled simultaneously to energy, to task-conditioned depreciating value, and to a cloud-offload tier that trades a transmit-energy-plus-latency penalty for a saved erase. We port wear-aware admission into that regime rather than reinvent it, and stress-test where the port survives. WORTH. The binding constraint is an exhaustible stock, so the consumed erase cycle carries a present-value scarcity rent η [24]; memory becomes depreciating capital with a user cost [12, 23, 65]. We make η the operative economic object: it fixes the placement boundary, signs how placement reacts to the 2025–26 memory-price supercycle [55], and, because spending an erase cycle consumes device lifetime, doubles as a fleet e-waste / embodied-carbon lever [6, 58]. A measured, not assumed, antecedent. The wear-augmented index and its rent η are the optimal policy form however value and write-intensity covary; η binds whenever the endurance
2
WHEN
WHERE
WORTH
AURA
THIS PAPER
ECONOMICS
the learned write gate
wear-aware placement
the endurance rent η
(arXiv:2606.02775)
RAM / on-board NVM / cloud
fleet P&L · device lifetime
the research-program arc: the placement layer between writing and pricing memory
Figure 2: Research-program-arc banner. AURA decides when to write; this paper decides where memory lives; the economic layer prices what it is worth (the endurance rent η).
stock is scarce, and scarcity is a regime, not a given—dormant on premium TLC at datasheet prices, binding on the commodity QLC/eMMC cheaper edge robots run (section 5.3). The non-monotone refinement (Proposition 2) needs one further primitive, a positive value–write association χ > 0, which we treat as a falsifiable antecedent and measure on real robot logs at a pre-specified $25 gate, with a published kill criterion, before any controller is trained (Assumption A5). The headline empirical finding is that χ’s sign is a property of the deployment regime, not a universal law: positive on recurrent long-horizon manipulation with a small backbone, null on a shorter-horizon suite, and negative on non-recurrent teleoperation. The coupling tracks long-horizon recurrence—re-observation of valuable scenes couples write-intensity with value, whereas value-agnostic teleoperation churn decouples it—and it is real but small. We report it only where a pre-specified cross-backbone agreement floor is met: a larger OpenVLA-7B backbone places items on a near-orthogonal value axis and is uninterpretable against the headline rather than a disconfirmation. Full estimates, clustering, and corrections are in sections 5.1 and 5.2. Contributions. 1. Measurement: the value–write coupling’s sign is regime-dependent (sections 5.1 and 5.2). On real robot logs at a pre-specified $25 gate, χ is positive on recurrent long-horizon manipulation (LIBERO-Long, SmolVLA-0.5B; Holm-reject, CI excluding zero), null on a shorterhorizon suite, and negative on non-recurrent teleoperation (DROID; post-hoc), with a recurrence dose-response that replicates at full power (ρ = 0.94, p < 10−4 ). The coupling tracks longhorizon recurrence, not a dataset. We pair it with a cross-backbone agreement floor (pre-specified Spearman ≥ 0.6), below which cross-model sign claims are uninterpretable, as our OpenVLA-7B arm shows (ρs = 0.05). 2. Theory: a wear-augmented placement index and a conditional non-monotone optimum (section 3). Cost-minimizing placement across RAM/NVM/cloud is a threshold in a per-byte index set by a single endurance shadow price η, optimal regardless of the sign of χ. On this sign-agnostic spine, a proven strictly-non-monotone-in-value optimum (Proposition 2) holds on the χ > 0 branch—with the antecedent measured, not assumed (Assumption A5). 3. Boundary: when the pricing layer is live, and when it is not (sections 5.3 and 5.4). At datasheet prices the endurance budget is dormant on premium 3,000-P/E TLC but binding on the commodity QLC/eMMC (∼ 1,000 P/E) cheaper edge robots run. Where it binds, a 3.15M-parameter learned controller is genuinely endurance-aware—it strictly beats the naive all-NVM strategy—but only ties the strongest cost-matched baseline on a task-value proxy. The tie is structural: with the cloud repriced at its slow value and connectivity swept, the wear-aware 3
Pricing the Erase Cycle — what it establishes, and what it doesn't Four tiers of evidence: proven as theory, measured on a value proxy, gated by hardware regime, and one negative result.
The theory holds, as math. PROVEN
Cost-optimal placement is a threshold in a wear-augmented index, priced by one number — the endurance rent η — whatever the sign of the value–write coupling. Under χ > 0 the optimum is strictly non-monotone in value. (Props. 1–4.)
The coupling's sign depends on the regime. MEASURED
REGIMEGATED
NOT YET SHOWN
Positive on recurrent long-horizon manipulation (χ ̂ = +1.0 × 10−3); the doseresponse replicates at full power (ρ = 0.94, p < 10−4). Null on a shorter-horizon suite; negative on teleoperation (DROID). Measured on a value proxy, not task success.
Dormant on premium flash, binding on commodity flash.
On 3,000-P/E TLC the endurance budget does not bind (η = 0). On the ≈ 1,000-P/E QLC/eMMC cheaper edge robots run, the same write demand binds it at datasheet prices, with device life of 0.2–1.8 years; the down-crossing lies outside support.
The payoff is priced, not demonstrated.
The non-monotone curve is proven, not observed — its down-crossing lies beyond the measured value support. Even where the budget binds, the wear-aware policy ties pricebased routing on task value, because realized value is tier-invariant.
Figure 3: Every headline claim, by epistemic tier. What is proven (the wear-augmented index and the conditional non-monotone optimum, as theorems); measured on a value proxy (the regime-dependent sign of χ, replicated but small); regime-gated (the budget does not bind on premium 3,000-P/E TLC but binds at datasheet prices on commodity QLC/eMMC ; the down-crossing lies outside support); and a negative result (even where the budget binds, the wear-aware policy ties price-based routing on task value—realized value is tier-invariant).
advantage stays zero because LIBERO write-intensity is nearly constant (CV(w) = 0.13%), collapsing the index to value-ranking; a synthetic control recovers the advantage only once CV(w) is large (fig. 11). On today’s hardware and workloads simple price-based routing suffices, and whether wear-awareness improves task value is unresolved. 4. Economics: a calibrated capital model of wasting memory (sections 3.6 and 3.7). Signed price comparative statics over an oligopoly band (Proposition 4) confirm three of four predicted signs. Re-solving across the 2025–26 NAND supercycle cuts the equilibrium rent ηsim by ≈ 39% ⋆ = 0.91 holds fixed—the shock hits the wear margin, not the while the break-even durability vBE placement boundary—and a bounded corollary links cost-optimal forgetting to device-lifetime extension (Corollary 1). Every headline claim carries an epistemic tier, sorted in fig. 3: what is proven as theory, what is measured on a value proxy, what is regime-gated by the hardware, and the one negative result. (Each number maps to its run and data file in the reproducibility appendix.) The lead empirical figure is fig. 7, the measured backbone×regime χ matrix; the model-derived ⋆ wear phase diagram (fig. 4) and its interior down-crossing vDC are a theory illustration, deferred to section 3.
2
Related Work
Our regime sits at the intersection of six literatures. Two of its ingredients are old: datacenter storage already keeps high-value, write-heavy items off endurance-limited flash, and already solves 4
an endurance-budgeted admission knapsack. What is new is the joint embodied object—three-tier RAM/NVM/cloud placement under a simultaneous energy and non-renewable endurance budget, with task-conditioned depreciating value and a priced exhaustible-stock shadow price η—which no single prior literature spans. Each subsection names the closest prior art and what it leaves open.
2.1
Wear-aware flash caching and storage
The closest prior work is Flashield [15]: its learned admission filter uses DRAM to keep writeamplifying objects off flash under a write-rate cap, already exhibiting the qualitative phenomenon we analyze—persistence is not monotone in an object’s worth. What we add is its driver and formalization: a proven down-crossing driven by a priced non-renewable endurance stock and value depreciation δ, in a three-tier energy-budgeted embodied loop with a cloud-offload action Flashield lacks. CacheSack [62] solves a per-category admission knapsack that cuts Google datacenter flash wearout by 17.8%—our placement-index skeleton at cloud scale, but with endurance as a soft cost term rather than a hard finite stock, and no energy, depreciating value, or multi-tier action. Kangaroo [38] supplies the lifetime-bounded write-cost-threshold admission rule we adopt as a cost-matched baseline (table 2). Managed-Retention Memory [32] calls for an endurance budget plus retention-aware placement but supplies no controller, depreciating-value model, or shadow-price theorem; DPRO [35] learns per-content retention with a soft P/E cost but keys on content popularity, a single tier, and no energy or capital layer. The learned write-avoidance line [56, 63, 64] makes steering writes around flash standard practice, so our controller is a necessity, not a novelty claim.
2.2
Embodied / robot memory and VLA models
Embodied-memory systems decide what to remember; none decide where a retained item lives or what it costs in joules and erase cycles. Surprise-gating [21] produces a value proxy we can place under a budget, but not a placement decision; MemER [53] bounds context cost by keeping ≤ 8 keyframes and lists discarding them as future work, naming the eviction-under-budget gap we close; ReMEmbR [3] and KARMA [57] build and prune memory stores for recall and prompt relevance, not for a joule or erase budget across tiers. Our own AURA [9] is the launch point—a when-to-write gate on a single constant-size store—which leaves the where-and-worth layer open and serves as the single-tier baseline our controller must dominate (table 2). MemGPT’s OS-style token paging [44] moves text between fast and slow stores to relieve capacity, abstracting away the hardware-wear and dollar economics that are our subject. The backbones we evaluate, SmolVLA [51] and OpenVLA [30], carry no persistent-memory mechanism; their edge-deployability is what makes RAM/NVM/cloud placement economically live.
2.3
Economic and decision-theory foundations
The economics is assembled from mature toolkits, each applied to a new object. The Hall–Jorgenson user cost of capital [23, 26] supplies “memory as a depreciating asset” with per-period rent = holding cost + depreciation δ; Hotelling’s exhaustible-resource theory [24] supplies the erase cycle as a unit of a non-renewable stock carrying a scarcity rent η (demoted to a bounded caveat, section 3.5, since the closed-form price path fails under stochastic demand). That η decouples the per-item program follows from Lagrangian relaxation—Whittle’s restless-bandit subsidy [60], the Gittins index [20], weakly-coupled-MDP relaxation [1], and constrained-MDP duality [2]—with the new ingredient that our coupling constraint is an intertemporal stock, not a per-period one, which yields the value down-crossing. Pricing agent memory as depreciating capital is itself not new in the token-budget setting: Token Economics [12] and the Marginal-Token-Allocator [65] give cache-as-inventory shadow 5
prices, but for token and context budgets with no physical endurance stock, RAM/NVM/cloud placement, energy term, or non-monotone optimum—so our claim is re-scoped to the physical P/E stock. Omri et al. [43] profile stateful agent-memory cost without a shadow price or capital model, and inference-aware deployment economics [49] motivates pricing write, hold, and retrieve jointly over the horizon.
2.4
Edge/cloud offloading, robot hardware, and cost anchors
Robot computation offloading is a learned when-to-offload decision: Chinchali et al. [13] solve perception offload under stochastic networks with deep RL; we reuse that machinery for the persistversus-offload decision and use their policy as the “this is just offloading” rebuttal baseline (it offloads compute, not a persistent store, with no endurance stock). Neurosurgeon [27] and the split-computing survey [37] supply the bandwidth-and-energy cost terms, but partition the compute graph, not memory state. We correct the cloud-tier dollar term for concurrency using Patil’s utility-aware methodology [45] (a naive per-token estimate is off by 1/U , a 2.5–24× penalty), and anchor edge-decode energy/latency to our own batch-1 measurements [10]. Endurance and energy constants are datasheet-pinned to the Jetson Thor and Orin platforms [41, 42].
2.5
Caching theory, learned policies, and RL-for-systems
The offline optimum our hindsight solver relaxes is Belady’s clairvoyant replacement [7], with cost-aware competitive vocabulary from weighted-paging primal-dual analysis [5] (whose fetch cost is renewable, not a consumable wear stock). The learned-caching line gives our recipe and baselines: LRB [52] regresses to a relaxed Belady boundary and Parrot [34] imitates the oracle, validating the behavior-cloning warm-start we use—except our oracle solves a knapsack-over-time under an endurance budget. Baleen [61] is the nearest write-cost-aware learned cache, but admits under a write-rate constraint with no non-renewable stock, depreciation, energy, or cloud tier. Our PPO-on-placement follows RL-for-systems precedent [25, 36, 39]; the novelty is the resource (memory tiers with consumable endurance), not the verb. Because learned caches underperform heuristics under abundant cache or distribution shift [11, 48], we report cost-matched baselines and bound the worst case to a tuned heuristic.
2.6
Memory market band and cs.CY / policy context
The 2025–26 memory supercycle hands the price-statics layer a dated, citable grid: enterprise TLC NAND roughly $0.06–0.22/GB across the Low/Base/High band, anchored by TrendForce ASP tracking [55], Counterpoint server-DRAM analysis [14], and Epoch AI’s component cost model [17]. The cs.CY claim—endurance-aware placement extends device life and defers fleet embodied carbon— rests on Weppe et al.’s ≈ 22 kg CO2 e/TB for 3D NAND [58], within the ACT carbon-modeling framework [22] and AI-hardware LCA [46, 50]. Its bounds: no regulation sets a numeric P/E floor, lifetime extension can overstate real savings [6], and superlinear new-hardware efficiency can justify replacement over retention [54]. The EU circular-economy regime (Right-to-Repair [19], Ecodesign-for-Sustainable-Products [18]) brings storage products in scope, so endurance-aware placement complements, and is not mandated by, the policy frontier.
6
3
Model
This section states the formal core: primitives and assumptions A1–A6, the placement index, the conditional non-monotonicity result (Prop. 2), the RAM-survival result (Prop. 3), the Hotelling caveat with the cs.CY corollary, and the price comparative statics P5a–d. Proofs are deferred to section A; fig. 1 shows the priced loop and the box below states its logic in plain English. The economic logic in plain English A robot’s flash ships with a fixed stock of erase cycles. The cost-minimizing planner prices that stock with one number, the endurance rent η—what one erase cycle is worth in the best feasible plan. Each candidate memory then faces a capital-budgeting test: persist on-board only if being local covers storage, energy, and the full user cost of wear (cwear + η) wi ; else rent the cloud’s endurance (paying latency) or forget it and risk re-acquisition. When memory prices move, the budget still binds and the rent re-clears—re-pricing the marginal memory while the persist/evict boundary barely moves (section 3.7).
3.1
Primitives and assumptions
Discrete time t = 0, . . . , T − 1, finite horizon T , discount γ ∈ (0, 1). At each t an item stream arrives (admitted upstream by AURA; admission is exogenous here; we price placement). Item i has type θi = (vi , δi , λi , si , wi , κi ) drawn i.i.d. from a joint law F . Assumption A1 (value / depreciation / retrieval). Base value vi ≥ 0 (monetized task-success gain); geometric staleness δi ∈ (0, 1] (value at age a is vi e−δi a ); Poisson retrieval rate λi > 0. Assumption A2 (size / write-intensity / recompute). Size si > 0; NVM erase ops per period if resident wi ≥ 0; recompute cost κi ≥ 0 if discarded and later needed. v and w are distinct coordinates of θ: nothing in the model forces them to co-move. Assumption A3 (tiers). k ∈ {R, N, C} = {RAM, NVM, cloud}. Per-byte holding rents P P pR , pN , pC ≥ 0; RAM hard capacity i∈Rt si ≤ CR (multiplier µR ≥ 0); power cap i powi,t ≤ P (multiplier µP ≥ 0). Access cost ≈ 0 (RAM), I/O energy eN (NVM), transmit energy + latency eC + ℓπ (cloud). Assumption A4 (non-renewable endurance: the load-bearing asymmetry). NVM residency consumes erase cycles from a fixed stock TX −1 X
wi ≤ Eend = (erase cycles/block) × (blocks).
t=0 i∈Nt
This is the only constraint integrated over time; RAM and power are flow constraints that reset each period. Let η ≥ 0 be the single multiplier on this budget: the shadow price of one erase cycle. Assumption A5 (value–write association: empirically testable antecedent). Let w̄(v) := E[w | v] d and define the association coefficient χ := dv w̄(v) (globally sign χ = sign CovF (w, v) when w̄ monotone). Assumption A5 is not assumed: it is estimated on real robot logs at a pre-specified $25 go/no-go gate with a published kill: if χ ≤ 0 the non-monotone headline (Proposition 2) is withdrawn for the monotone index (Proposition 1). Assumption A6 (recompute sub-linearity). κ̄(v) := E[κ | v] has κ̄(v)/v non-increasing: high-value items are not proportionally more expensive to regenerate. Testable at the gate. 7
The program. The agent chooses xi,t ∈ {R, N, C, ∅}, earns ρi,k (t) = (depreciated value) − (access cost) − (recompute if discarded and needed), and solves P Pi∈Rt si ≤ CR hX X i t max E γ ρi,xi,t (t) − pxi,t si − cwear wi 1[xi,t = N ] − migr s.t. pow ≤ P Pi P i,t {xi,t } t i w ≤E t
i∈Nt
i
∀t, ∀t,
end .
(1) The Lagrangian carries flow multipliers µR (t), µP (t) and the single stock multiplier η.
3.2
The placement index and the wear-augmented index
Renewable limit Eend → ∞ ⇒ η = 0. Given (µR , µP ) the per-period problem decouples across items. With discounted locality return Vi =
X
γ a λi e−δi a vi =
a≥0
λ i vi , 1 − γe−δi
(2)
place i in the fastest tier whose marginal per-byte rent its index clears, the cutoff statistic being the per-byte index λi (vi + κi ) Ii = (3) si (1 − γe−δi ) strictly increasing in vi (higher value ⇒ faster tier: monotone), the “obvious” Belady/Gittins-withrent regime, not the contribution. Proposition 1 (monotone renewable benchmark). With η = 0 and the energy ordering eN < eC +ℓπ (on-board access cheaper than cloud), 1[xi = N ] is weakly increasing in vi , ceteris paribus. (Proof in section A. Tested: endurance-relaxed arm, expect monotone fit.) Binding endurance η > 0: the wear-augmented index. Now Eend binds. NVM residency carries an extra per-period cost (cwear + η)wi : cash wear plus the scarcity value of the consumed erase cycle. With the RAM multiplier µR carried explicitly, the four tier-specific net returns are fast ΠR − (µR + pR )si , i = Vi
fast ΠN − pN si − (cwear + η)wi , i = Vi
(4)
slow ΠC − pC si , i = Vi
Π∅ i = − Pr[needed] κi ,
(5)
∅ R C with Vislow = Vifast − λi ℓπ/(1 − γe−δi ). Item i → NVM iff ΠN i ≥ max(Πi , Πi , Πi ). The NVM-vscloud break-even: λi ℓπ + (pC − pN )si ≥ (cwear + η) wi (BE-NC) 1−γe−δi
|
{z
}
Bi :=value of locality
and the NVM-vs-RAM break-even (the fallback v1 dropped): (µR + pR )si − pN si ≥ (cwear + η) wi
(BE-NR)
i.e. NVM beats RAM iff its wear cost is below RAM’s capacity-rent premium. The endurance price P P η is fixed by the budget binding, t i∈Nt (η) wi = Eend , a one-dimensional monotone root-find (Nt (η) shrinks as η ↑). 8
Each tier return Πxi reads as a per-item income statement: revenue is the discounted value of recalls (Vifast , Vislow ), against RAM rent, NVM occupancy pN si , cash-wear depreciation cwear wi , the cloud’s storage-plus-latency charge, or the write-off cost of forgetting. Placement is then capital budgeting, with η the hurdle rate the scarce flash imposes. Only the NVM line carries depreciation plus scarcity rent η wi —that single extra term is what makes the optimum non-monotone (section 3.3).
3.3
Headline: clean non-monotonicity
Parameterize a value stratum by v, holding λ, δ, s at conditional means; within it w has conditional mean w̄(v) and locality B(v). Two structural facts drive everything: (1) locality is O(v 0 ): with λ, δ, s frozen at their conditional means, B(v) is flat in v to first order (B ′ (v) ≈ 0). Freezing λ here is a modeling choice, and a load-bearing one: if retrieval rate co-varies positively with value, as the very recurrence mechanism that yields χ > 0 would suggest (valuable scenes are re-observed more often), then B(v) rises in v and the down-crossing weakens. We therefore carry |B ′ (v)| < (cwear + η)|χ| as an explicit clause of the existence condition (7) rather than assume it away, and flag dλ/dv as an unmeasured primitive that bounds the result. (2) wear cost is strictly increasing in v iff χ > 0: the (BE-NC) RHS has v-derivative (cwear + η)χ. Definition 1 (endurance threshold). Let vmax be the largest v with w̄(v) > 0 and set η̄ :=
B(vmax ) − cwear w̄(vmax )
(6)
the smallest erase-cycle price at which the wear term overtakes locality at the top of the value support (assumed > 0; else the headline is vacuous and the Phase-0 gate kills it). Proposition 2 (non-monotone-in-value optimum: clean conditions). Assume Assumptions A1 to A4 and A6, the RAM-slack bound of Proposition 3, and the empirically-verified antecedent Assumption A5 with χ > 0. Then for every endurance price η > η̄ (equivalently, Eend below the level that induces η̄), the optimal local-persistence probability Pr[x = N | v]—a probability rather than a hard indicator, because within each value stratum the remaining primitives (λ, δ, s) vary and smooth the per-item threshold into a curve—is strictly non-monotone in v: it rises on a low⋆ value interval and strictly falls on a high-value interval. The interior down-crossing vDC solves ⋆ ⋆ ⋆ ⋆ B(vDC ) = (cwear + η)w̄(vDC ) and satisfies ∂vDC /∂η < 0 and ∂vDC /∂χ < 0. The down-crossing exists iff d E[w | v] > 0, with |B ′ (v)| < (cwear + η)|χ| on the high-v interval. (7) (i) η > η̄ AND (ii) χ = dv
Neither condition alone suffices: (i) without (ii) is the monotone index (Proposition 1); (ii) without (i) gives η = 0, wear bounded by cwear w̄(v), so for small cwear no down-crossing: the falsifiable knife-edge the Phase-0 gate decides. (Proof in section A. Tested: value-stratified persist regression, high-v negative coefficient at η > 0, vanishing at Eend → ∞.)
3.4
The RAM-capacity multiplier survival bound
Proposition 3 (down-crossing exists ∀µR ; location shifts left via endogenous η). The item ejected ⋆ from NVM at vDC lands in RAM iff its wear cost exceeds RAM’s capacity-rent premium, (cwear + ⋆ η)w̄(vDC ) > (µR + pR − pN )s; if µR is high enough that the reverse holds for all v, the ejected 9
Wear phase diagram (illustrative; model-derived, not a measured curve)
local-persist probability Pr[x = N ∣ v]
η = 0 monotone reference
1.0 η > η̄ persist probability (rise-then-fall, Prop. 3)
⋆ interior down-crossing vDC lies BEYOND measured support at the measured χ ̂
0.8
0.6
0.4 price-statics break-even ⋆ vBE = 0.91 (inside support)
0.2 measured value support
extrapolation
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
item value v (monetized task-success gain, normalized)
Figure 4: Wear phase diagram (illustrative; model-derived). Headline subject: the η > η̄ persist⋆ probability curve (rise-then-fall in value v), whose theory interior down-crossing vDC lies beyond the normalized value support at the measured point estimate (section 3.8); the faint η = 0 monotone step (Proposition 1) is ⋆ the contrast reference. The annotated marker is the price-statics break-even vBE = 0.91 (the persist/evict crossover inside support [0.056, 1.034]); it and the “do-NOT-persist-locally” band boundary are read from the real price-statics overlay; the rise-then-fall persist-probability curve is the closed-form shape of Proposition 2 and is model-derived, not a measured empirical curve. A measured value-stratified persist regression is not available: the offline H1 test (the non-monotone-persistence hypothesis) was vacuous at the non-binding S0 regime (today’s base prices), and the binding-regime closed-loop eval (section 5.3) returned a flat high-value persistence slope (H1 p = 1.0 in the primary cell), so it does not support the curve. We therefore present the curve as an illustration of the theory, not an empirical claim.
item routes to cloud ΠC or recompute Π∅ (cheap by Assumption A6). The down-crossing exists for every µR ≥ 0; µR controls only the destination of the spared endurance, with RAM-slack bound ⋆ )/s − (p − p ). Because η is endogenous, raising µ pushes µR < µmax := (cwear + η)w̄(vDC R N R R ⋆ /∂η < 0 RAM-bound items into NVM (BE-NR), raising erase demand and hence η, and since ∂vDC ⋆ (µ ) moves left as µ ↑. the crossing vDC R R ⋆ (Proof in section A. Tested: RAM-pressure sweep; down-crossing persists, vDC shifts left, destination RAM→cloud as CR ↓.) Figure 4 is the visual statement of Props. 1–3: the rise-then-fall persist curve with the down⋆ crossing vDC marked, the faint η = 0 monotone reference, and the placement-region backdrop.
3.5
Hotelling: a bounded caveat
The endurance constraint is formally an exhaustible-stock problem [24], but that scaffold justifies one fact only: an erase cycle spent today is unavailable tomorrow, so its user cost carries a present-value scarcity rent η—and this rent, not the cash wear cwear , makes placement multi-period. We do not claim the path ηt = η0 (1 + r)t (with a positive cash term only the rent grows at r; endurance is partially recoverable via wear-leveling; non-stationary retrieval makes η non-closed-form ex ante). The operative object is the dual of a finite-horizon constrained (PO)MDP, estimated as ηb. As noted in section 2.3, pricing memory as depreciating capital is not new in the token setting; what is new 10
Table 1: Unified price scenario band, used identically in the model, the calibration, and the pre-specified experiment plan. Anchors: Counterpoint Research [14], Elinfor (relaying TrendForce) [16], TrendForce [55]. Scenario S0 ≡ Base (today) S1 ≡ High (DRAM cycle) S2 ≡ High (NAND shock) Low (floor)
DDR5 $/GB NAND TLC $/GB egress $/GB basis ∼9 ∼ 16 ∼ 12 3
∼ 0.13 ∼ 0.18 ∼ 0.22 0.06
∼ 0.09 ∼ 0.09 ∼ 0.09 0.09
current spot [14] DRAM-led tightening [55] NAND-led tightening [16] low-price boundary
is that η here is the dual of a physical non-renewable P/E stock (Assumption A4) under multi-tier energy-budgeted placement.
3.6
cs.CY corollary: endurance rent → device lifetime → fleet e-waste
The rent η is also a device-lifetime price. Spending an erase cycle consumes a fixed fraction 1/Eend of NAND life; at the budget binding the policy’s cumulative erases map one-to-one to calendar lifetime. Corollary 1 (bounded lifetime lever). Lowering fleet-wide NVM-erase demand by a fraction q (the controller’s saving vs. an endurance-blind baseline) extends flash-limited device life by ≈ q to first order, deferring replacement embodied carbon at the ≈ 22 kg CO2 e/TB NAND anchor [58]. This is a directional, not magnitude-certified lever: replacement is multi-causal; SSDs are the dominant and growing component of device carbon [22, 50], end-of-use ̸= end-of-life, and lifetime-extension can overstate real savings [6, 54]. So η wires the core model to a falsifiable cs.CY claim, cost-optimal forgetting is also lifetime-extending, without a precise carbon number (reported as a TCO/carbon implication, not an experimental target).
3.7
Price comparative statics
Placement is a function of the cloud-tier price vector p = (pHBM , pDDR , pNAND , pegress , penergy ): RAM rent pR tracks DDR/LPDDR $/GB; cwear , pN track enterprise NAND $/GB and $/P-E-cycle; pC and ℓπ + eC track egress $/GB and energy $/kWh; η is endogenous. We sign the statics over a three-point band (table 1), never a point estimate, since memory prices are oligopoly/contract-set [14, 55]. The same Low/Base/High band is used everywhere a price enters the paper. Proposition 4 (signed statics P5a–d). Writing σN for the population NVM share and Θ for the NVM-persist value-threshold, each sign follows from (BE-NC)/ (BE-NR) plus the budget identity fixing η: P5a pNAND ↑: persist region shrinks, ∂σN /∂pNAND < 0 and ∂η/∂pNAND < 0 (NVM↓, η ↓; clean). P5b pDDR ↑: conditional on an interior RAM allocation, (BE-NR) tilts toward NVM and ∂η/∂pDDR ≥ 0. Under binding RAM the static is (near-)zero: the RAM shadow price µR absorbs the entire pDDR shock one-for-one, so it does not propagate to the endurance margin (dη ≈ 0). Empirically this is the operative regime (section 5), so P5b returns an inconclusive null: a clean identification of the boundary condition under which the proven sign is observable, not a contradiction.
11
P5c pegress , penergy ↑: cloud dearer, B(v) ↑, more onboard, η ↑ (σN ↑ first order; threshold move directional, not magnitude-signed). P5d Eend ↑: ∂η/∂Eend < 0; as η → 0 we recover Proposition 1 and non-monotonicity vanishes (clean boundary; a small-endurance phenomenon). (Proof in section A. Tested: price-band sweep, re-solve η(p), σN (p) at S0/S1/S2.) A falsifiable conjecture (cross-partial). Bandwidth b enters only B(v), so extra bandwidth diverts writes to cloud, relaxes endurance, lowers η, and lifts the value of remaining onboard items. This predicts ∂ 2 (fleet task value) > 0, (8) ∂b ∂(1/Eend ) i.e. the marginal task-value of bandwidth rises with endurance tightness—for memory-bound fleets on cheap NAND, buying radio is partly buying flash lifetime. We state this as a model-predicted conjecture (H6, table 3), not a theorem: it is not derived in Proposition 4, and its mechanism has two opposing channels. The empirical sweep (section 5.5) returns +0.50 with a bootstrap CI that straddles zero ([−0.34, +1.25]), so we report it as directional, not confirmed.
3.8
Calibration: measured primitives in the model
We plug the measured primitives (δ̂, m̂, χ̂) and the datasheet cost constants into eqs. (2) to (8). Scope (binds every number in this subsection): the canonical χ̂ = +1.016×10−3 , 95% physicalscene-clustered CI [+3.81 × 10−4 , +1.65 × 10−3 ] (n=3,032, 379 clusters), is the LIBERO-LONG, SmolVLA-0.5B headline; on a non-recurrent teleoperation distribution DROID measures a significantly negative slope at high power (χ̂ = −8.95 × 10−3 , CI [−1.61 × 10−2 , −4.09 × 10−3 ]), a measured opposite-sign regime, so every calibrated association magnitude is recurrent-regime-scoped. δ̂ = 0.032/step (half-life 21.7 steps, CI [16.2, 29.6], R2 =0.997) is the DROID recurrence-kernel anchor, used only as a value-decay timescale. We mix a DROID-anchored δ̂ with the LIBERO-scoped χ̂ and flag the mix as such: we do not claim δ is regime-robust. The LIBERO state-kernel δ ≈ 0.0004 measures state persistence, not value depreciation, so it cannot serve; the LIBERO value-decay δ ≈ 0.054 (sparse, R2 =0.15) is the same-regime quantity but is ∼69% larger than the DROID 0.032 and, used instead, would give M ≈ 16× (half-life ≈ 13 steps) rather than 24.3×. We headline the DROID-anchored value M = 24.3× for its far tighter fit (R2 =0.997 vs. 0.15) but flag that this is a fit-quality choice, not evidence of cross-regime stability of the value-decay timescale; the LIBERO value-decay M ≈ 16× is the same-regime robustness alternative. m̂ = 0.0503 is the normalized endurance scarcity markup solved on the pooled Phase-0 logs under a capped budget (binds at ≤ 0.75× measured write demand, flat across binding budgets); at uncapped datasheet conditions the budget does not bind and the markup is zero. Two duals, two unit systems: m̂ is the Phase-0 normalized scarcity-markup ratio (+5.0% on the cash wear of one erase; Jorgenson user cost = cwear (1 + m̂), a unit-free indicator that endurance binds), while ηsim ≈ 2.4 × 10−4 in the price-statics results (section 5.5) is a separately-solved equilibrium dual in the simulator’s objective PP units. They are the same kind of object—the dual on w ≤ Eend —in two different objectives, so we do not compare them numerically; the bare symbol η is reserved for the model-theoretic shadow price. Depreciation and the locality multiplier. With γ = 0.99, D = 1 − γe−δ̂ = 0.0412 (CI [0.0329, 0.0516]), so a persisted item is worth M = 1/D = 24.3× its per-step value (CI [19.4, 30.4]).
12
Wear cost and the endurance user cost. The datasheet anchors (TLC 3,000 P/E, WAF 3, 128 GB module) give TBW=128 TB and a base cash wear of $0.13 per TB-written, i.e. 5.55 × 10−6 $/(block P/E). The Jorgenson user cost of one erase, cwear (1 + m̂), is 5.83 × 10−6 $/(block P/E) at base NAND (band [2.69, 9.86] × 10−6 over the Low/Base/High NAND price band, table 1); the rent, not the cash term, makes placement multi-period. Calibrated break-even (BE-NR). (BE-NR) with slack RAM (µR =0) and the base DDR– NAND gap ($8.87/GB capacity-rent premium) yields: ≈ $4.8 × 10−5 of task value per GB-day justifies NVM persistence at base NAND ($0.13/TB cash wear + 5% rent). The RAM premium exceeds wear cost by ∼ 5 orders of magnitude: at measured intensities capacity rent, not wear, evicts from RAM; wear bites only once endurance binds (η > 0), the regime we isolate. ⋆ : outside support. Our data do not pin the locality level B Calibrated down-crossing vDC ⋆ ⋆ = χ /χ̂ ≈ 1.62 is independently, so we do not headline a number for vDC (the point estimate vDC hi an anchoring identity, not a B-grounded measurement). The anchoring-invariant conclusion is that at the measured χ̂ the interior down-crossing lies beyond the measured value support, reaching the support edge only at the upper end of the χ-CI. The non-monotone optimum is therefore sign-correct but quantitatively dormant at measured intensities, becoming empirically live ⋆ =0.91 only under tighter endurance. It is a distinct object from the price-statics break-even vBE (section 5.5). This is the headline calibration caveat.
Calibrated cs.CY lever (bounded). The measured write-intensity spread (≈ 9.5% of mean) bounds the divertible erases: device-life extension under the index policy vs. LRU (Corollary 1) ⋆ ranges from ≈ 0% at the measured χ̂ (since vDC is beyond support) up to a ≤ 9.5% ceiling when ⋆ endurance is tight enough to pull vDC interior, deferring ≤ 2.82 kg CO2 e per 128 GB module life at the Weppe anchor [58]: directional, not magnitude-certified. Calibrated cross-partial (H6). The conjecture of (8) calibrates to a directional, CI-inconclusive estimate: flipping one item from NVM to cloud spares w̄ ≈ 1.05 P/E-cycles (≈ 2.9 × 10−7 $ of endurance relief at base NAND), and the swept price-grid estimate is +0.50 with CI [−0.34, +1.25] (section 5.5). We report it as directional—for memory-bound fleets on cheap NAND, buying radio is partly buying flash lifetime—not confirmed; full CI propagation ships in the artifact (section B).
4
Experimental Design
The design is pre-specified as a five-phase pipeline with published kill criteria (fig. 6); the frozen plan ships in the released artifact (section B). One scope note on what “pre-specified” means here: the plan is a frozen, version-controlled document committed before the corresponding runs, with per-phase gates and kill criteria fixed in advance, but it is internally timestamped rather than lodged with an external registry (e.g. OSF), a weaker guarantee that we state plainly. Headline backbone: SmolVLA-0.5B [51]; OpenVLA-7B [30] is a single scale-stress arm (the pre-specified crossbackbone confirmation requires Spearman ≥ 0.6, which it does not meet; see section 5). Datasets: LIBERO-LONG [33], DROID, a 100-ep Phase-1 sample enlarged for a high-power cross-dataset re-test, analyzed on the pre-specified 1,200-new-only subset (n = 9,598 frames, 1,200 physical-scene clusters; lerobot/droid_1.0.1) [28]. Primary metric: task-success-per-joule-per-erase.
13
Calibrated economic primitives — persisted item worth M = 24.3 × its per-step value (DROID-anchored; 16 × under the LIBERO value-decay alt.)
Break-even task value to justify NVM (BE-NR)
User cost of one erase = cash wear + scarcity rent
1e−5
1e−5
8
≈ $4.8 × 10 /GB-day at base ($0.13/TB cash wear + 5% rent)
6 4.78e-05
4
2.21e-05
2
0
Jorgenson user cost ($/block P/E)
wear+rent break-even ($/GB-day)
cash wear cwear 8.09e-05 −5
1.0
9.86e-06
endurance markup m̂ (+5.0%)
0.8
5.83e-06
0.6
0.4 2.69e-06
0.2
0.0 Low $0.06/GB
Base $0.13/GB
Low $0.06/GB
High $0.22/GB
Base $0.13/GB
High $0.22/GB
Figure 5: Calibrated economic primitives. Left: the wear+rent break-even ($/GB-day) at Low/Base/High NAND (≈ $4.8 × 10−5 at base). Right: the Jorgenson user cost of one erase, decomposed into datasheet cash wear plus the +5.0% endurance markup (m̂). The persisted item is worth M = 24.3× its per-step value (half-life 21.7 steps).
Phase 0 value–write gate χ̂, ρs on real robot logs (≤ $25)
Phase 1 value labeling SmolVLA-0.5B counterfactuals + value model
kill: ρs ∈ [−0.1, +0.1] ∧ recompute < 10% (passed)
floor: cross-backbone ρs ≥ 0.6 (not met: 7B uninterpretable)
Phase 2 placement controller set transformer, 3.15M BC warmstart + PPO
Phase 3 offline eval battery cost-matched ladder McNemar + Holm
Closed loop / VLA-inthe-loop binding-regime rollouts + causal gate
cap: ≤ 50M params ≥ 3 seeds (met)
H1–H6 kill criteria; H2 negative
causal gate ≥ +8 pp, measured +0.0 pp (abort)
Figure 6: Experiment-architecture pipeline. Five pre-specified stages, each with a published kill criterion (red): the $25 Phase-0 gate decides whether the non-monotone branch is admissible before any training spend; Phase 1 labels item value via action counterfactuals; Phase 2 trains the 3.15M-parameter placement controller; Phase 3 runs the cost-matched McNemar/Holm battery; the final stage closes the loop in the binding endurance regime, and the VLA-in-the-loop causal gate aborted at +0.0 pp.
4.1
Phase 0: value–write go/no-go gate (≤ $25, pre-specified)
Measures whether write-intensity wi is associated with item value vi through a channel that is not the shared retrieval-frequency process. The value proxy v̂i is counterfactual task-outcome attribution on an independent sample via batched SmolVLA masked-vs-unmasked passes, with a forward-value regression confirmation; wi is measured on the complement sample. We pre-specify both estimators: d Spearman ρs (w, v̂) (rank screen) and the local-slope χ̂ = dv E[w | v] (kernel/local-linear, percentilebootstrap 95% CI); χ̂ is decision-relevant. Kill criterion: pivot to the monotone-index design iff (proxy spot-check passed) AND χ̂ ≤ 0 with CI excluding χ > 0 AND ρs ∈ [−0.1, +0.1] AND cheap-recompute fraction < 10%, on both LIBERO and DROID. The gate is deliberately conjunctive across datasets; section 5 reports that DROID alone later met these thresholds while LIBERO did not, so the AND-gate correctly did not fire and the non-monotone branch is retained but scoped to the recurrent regime.
14
Table 2: Cost-matched baseline ladder. Each baseline is constrained to the identical energy + erase + $ budget as ours. Tag
Baseline
Role
all-cloud / all-RAM / all-NVM extremes LRU / LFU / size-based Flashield-style [15] CacheSack-style [62] Chinchali offload [13] surprise-gated [21] AURA single-tier [9] random / hindsight oracle
4.2
bound frontier; all-NVM hits the cliff classic caching monotone-in-score floor ML admission, cost-matched pre-empt rebuttal knapsack admission, cost-matched A3 rebuttal learned offload, cost-matched “just offloading” rebuttal embodied-memory heuristic SOTA heuristic arm write-if-gated, no tier choice must strictly dominate floor / ceiling frontier bounds
Phase 1: value labeling
P1a: batched counterfactual labeling of a stratified item subset with frozen SmolVLA-0.5B (importance-weighted to the population; exact #forward-passes pre-specified). P1b: a ≤ 50M supervised value model (v̂i , λ̂i , δ̂i ) generalizing sparse labels. P1c: OpenVLA-7B single scale-stress arm (pre-specified cross-backbone Spearman ≥ 0.6 target), not a per-step counterfactual sweep.
4.3
Phase 2: placement controller (≤ 50M; BC warm-start + PPO; ≥ 3 seeds)
A small transformer over the item set + a system-state token. Reward = success-proxy −λ·energy −ν·erase −ζcloud ·cloud-$ (the cloud weight is ζcloud ; ρ is reserved for the Phase-0 Spearman, χ for the value–write slope; no symbol does double duty: in particular the two value crossings carry distinct ⋆ denotes the price-statics break-even on support (= 0.91, section 5.5) and symbols throughout: vBE ⋆ vDC the theory interior down-crossing (beyond support at the measured χ̂, Proposition 2); the bare ⋆ ). Hindsight-oracle warm start, then PPO over stochastic word “down-crossing” always refers to vDC network/energy/endurance regimes; split by scene (70/15/15).
4.4
Phase 3: baseline ladder, δ fitting, sensitivity
Table 2 is the cost-matched baseline ladder. δ is fitted from retrieval-recurrence decay (reported fitted-vs-assumed). A 4-axis stitched-episode sensitivity battery (stitch-boundary incl. adversarial, depreciation, stream-length, scope-sensitivity) is the headline robustness claim.
4.5
Pre-specified hypotheses
Eval N ≥ 200 held-out episodes/seed (≥ 600 paired at the 3-seed floor; 5 seeds for headline H1/H3/H4). Paired McNemar vs. each cost-matched baseline, Holm–Bonferroni over the family {H1, H1b, H2, H3, H4, H4b, H5, H6}. Three of these (H4, H4b, H5) require a swept-box / CR -sweep run that did not land, so the realized corrected family is {H1, H1b, H2, H3, H6}; we flag the unrun members rather than quietly drop them. Every value–write test pre-specifies both ρs (screen) and χ̂ (decision-relevant). Statistics methodology follows Chen [8]. Table 3 lists each falsifiable statement and its kill criterion.
15
Table 3: Pre-specified hypotheses, tests, and kill criteria. ID
Falsifiable statement
Test / metric
Kill criterion
H1
Optimal local-persist is non-monotone in v when η > 0 (negative high-v coeff).
persist regression; sign + p (Holm).
H1b
Persist set monotone-shrinking as endurance tightens (Prop. 1). Controller beats every cost-matched baseline on success-per-joule-per-erase.
∂(persist frac)/∂(1/Eend ); bootstrap CI. McNemar, Holm.
Controller strictly dominates the AURA single-tier arm. Non-monotonicity occupies ≥ 10% of the datasheet-plausible box. ⋆ vDC exists ∀µR ; location shifts with η (Prop. 3). Boundary sensitive to Eend (vanishes as Eend → ∞). Marginal task-value of bandwidth increases in endurance tightness.
McNemar per-joule-pererase. fraction of swept box with down-crossing. down-crossing per µR ⋆ cell; ∂vDC /∂η < 0. ∂Θ/∂Eend < 0.
no negative high-v coeff at p < 0.05 in any η > 0 regime. not decreasing in 1/Eend at p < 0.05. fails strongest costmatched baseline at p < 0.05. no strict dominance at p < 0.05. < 10% of the box.
H2
H3 H4 H4b H5 H6
5
∂ 2 (value)/∂b ∂(1/Eend ) > 0.
⋆ vDC absent in a cell OR slope not negative. boundary insensitive to Eend . flat in tightness (nonfatal).
Results
All χ are computed with the verbatim Phase-0 local-linear-slope estimator under physical-scene clustering (resampling physical scene_id, not seed-reshuffle pseudo-clusters) with 1,000-resample 95% scene-clustered bootstrap CIs; the canonical χ table shipped in the artifact (chi_canonical_table.json, section B) is the paper’s source of truth. Controller and price-statics results are reported exactly as the data say them; a negative or null result is reported as a finding, not a failure.
5.1
Phase-0 gate and the backbone×regime χ matrix
Table 4 is the canonical value→write matrix. On LIBERO-Long with the SmolVLA-0.5B backbone, χ̂ = +1.016 × 10−3 (CI [+3.81 × 10−4 , +1.65 × 10−3 ], ρs = +0.10, cheap-recompute fraction < 10%): it excludes zero positive and survives Holm correction across the three-arm χ family (rank 2, α = 0.025, reject). The cheap forward-value proxy tracks the full SmolVLA counterfactual at held-out rcf = 0.92 (pre-specified floor 0.40)—an internal proxy-consistency check, not a validation of v̂ against realized task success, which the +0.0pp causal gate (section 6.1) leaves open; every χ here is therefore a coupling to the value proxy. The rejection does not depend on the family’s composition—it holds as a single test, in a two-arm pre-specified family, and at rank 2 of the three-arm family—so the inclusion of the post-hoc DROID arm, whose unadjusted significance we read only as exploratory, does not carry the headline. The effect is real but small: the χ-implied w-swing across the 5th–95th value percentile is 4.92 × 10−4 , ≈ 5.6% of w’s dynamic range, so it supports the conditional branch of Proposition 2, not a universal monotone law. The effect is LIBERO-Long-specific: a second, shorter-horizon LIBERO suite (goal/object/spatial) is null (χ̂ = −1.58 × 10−3 , CI [−3.84 × 10−3 , +6.5 × 10−4 ] straddles zero, n=1,520 items / 190 clusters, ρs ≈ 0; second-suite row of the canonical χ table), so the coupling tracks long-horizon recurrence, not LIBERO-as-a-dataset. The enlarged DROID arm is significantly negative (χ̂ = −8.95 × 10−3 , CI [−1.61 × 10−2 , −4.09 × 10−3 ], Holm rank 1, reject) but post-hoc / exploratory: the enlargement was launched after a negative, underpowered 100-scene pilot (a forking path), so we report it on the pre-specified 1,200-new-only subset (excluding the motivating episodes) as a suggestive 16
Table 4: Canonical value→write matrix (physical-scene clustering, 1,000-resample bootstrap, Holm across the three-arm χ family). Source: chi_canonical_table.json. Backbone SmolVLA-0.5B SmolVLA-0.5B SmolVLA-0.5B OpenVLA-7B
χ̂ (raw)
Regime / suite LIBERO-Long (recurrent) LIBERO goal/obj/spatial DROID (enlarged, post-hoc) LIBERO-Long (scale-stress)
βstd −3
+1.016×10 −1.58×10−3 −8.95×10−3 −2.44×10−4
ρs Verdict
+0.118 +0.10 CI> 0; Holm-reject −0.064 ≈ 0 straddles 0; null −0.084 −0.077 CI< 0; Holm-reject −0.094 −0.006 straddles 0; uninterp.
The value → write coupling's sign is a property of the regime value & write DECOUPLE (χ < 0)
COUPLE (χ > 0) χ ̂ = +1.02e − 03
SmolVLA-0.5B · LIBERO-Long (recurrent)
positive · Holm-reject · n=3,032/379
χ ̂ = −1.58e − 03
SmolVLA-0.5B · LIBERO goal/object/spatial
null · CI straddles 0 · n=1,520/190
χ ̂ = −8.95e − 03
SmolVLA-0.5B · DROID (enlarged, post-hoc)
negative · Holm-reject · n=9,598/1200
χ ̂ = −2.44e − 04
OpenVLA-7B · LIBERO-Long (scale-stress)
uninterpretable · ρs =0.05≪0.6 · n=3,032/379
−10−2
−10−3
−10−4
0
10−4
10−3
value → write local slope χ ̂ (95% scene-clustered bootstrap CI)
Figure 7: The value→write coupling’s sign is regime-dependent. Canonical χ̂ with 95% sceneclustered CIs: positive and CI-excluding-zero on recurrent LIBERO-Long (SmolVLA-0.5B), significantly negative on non-recurrent DROID (post-hoc), and CI-straddling on the OpenVLA-7B scale-stress arm, whose cross-backbone agreement (ρs = 0.05 ≪ 0.6) makes its sign uninterpretable.
regime-difference signal, not a confirmation. The OpenVLA-7B arm is uninterpretable, not a disconfirmation. The pre-specified confirmation criterion is a cross-backbone Spearman ≥ 0.6 between the two backbones’ per-item value rankings. Measured on the identical 3,032 LIBERO frames it is only ρs = 0.05 (CI [0.016, 0.086]), far below the floor: OpenVLA-7B and SmolVLA place items on near-orthogonal value axes, so a χ sign difference between them is two incomparable measurements, not a contradiction of the headline mechanism. Independently, under physical-scene clustering OpenVLA’s own χ straddles zero (−2.44 × 10−4 , p = 0.18). We therefore withdraw any cross-backbone “sign reversal” claim and report the arm as null/uninterpretable (fig. 7). Gate decision. The pre-specified kill is conjunctive across datasets: DROID independently met the kill thresholds (χ̂ ≤ 0 with CI excluding χ > 0, ρs = −0.077 ∈ [−0.1, +0.1]), but LIBERO-Long did not, so the AND-gate did not fire. We proceed on the non-monotone branch, scoped to the recurrent regime per the measured boundary.
17
density (log)
Write-intensity dispersion 10
2
10
1
LIBERO: std=0.0013 range [1.003,1.011] recurrence-driven
LIBERO (recurrence) DROID (churn)
DROID: std=0.024 range [1.0006,1.63] churn-driven
100
10−1
10−2 1.0
1.1
1.2
1.3
1.4
1.5
1.6
episode length (recurrence proxy, steps)
Recurrence couples value ↔ write (LIBERO, χ > 0); control churn decouples it (DROID, χ < 0)
Recurrence structure LIBERO: 10 long-horizon tasks, heavily repeated DROID: 909 unique instr / 1,300 scenes, 35 shared by >1 scene
800
600
400
200
0
write-intensity w
LIBERO (repeated scenes)
DROID (909 unique instr.)
Figure 8: Recurrence-driven vs. churn-driven write-intensity: the mechanism behind the sign. Left: w is tight on LIBERO-Long (std 0.0013, dominated by re-observation of recurrent scenes) and wide on DROID (std 0.024, range 1.0006–1.63, driven by teleoperation churn). Right: episode-length dispersion, a recurrence proxy (268 ± 57 steps LIBERO vs. 303 ± 230 DROID; 909 unique instructions over the 1,300-scene full DROID pool. The χ estimate is reported on the pre-specified 1,200-new-only analyzed subset: 1,300 is the full pool, 1,200 the analyzed subset; the difference is the 100-scene motivating pilot excluded to avoid the forking path).
5.2
Recurrence dose-response (mechanism test)
To test the recurrence mechanism directly, we interpolate episode mixes blending the non-recurrent DROID regime (χ < 0) with the recurrent LIBERO-Long regime (χ > 0), with dose = fraction of recurrent episodes (fig. 9). The pre-specified design is a twelve-level, three-seed sweep across the sign-flip band. At its first run the DROID frame loader drew from a 100-episode sample, so the realized recurrence range was compressed and the χ̂-trend was underpowered (Spearman +0.35, p = 0.12); a Fisher combination with an earlier seven-level sweep reached p = 0.039, but that combine assumed an independence the two sweeps only partially have and sat one rounding step from failing. Rather than lean on it, we re-ran the same pre-specified twelve-level grid at full power, pointing the loader at the full lerobot/droid_1.0.1 (600 distinct DROID scenes vs. 100): the only change is the dataset, not the design or the estimator. The dose-response replicates decisively. χ̂ rises—rank-monotone, with minor level-to-level wobble but no trend reversal—from −5.2 × 10−3 at pure churn (dose 0) to +1.67 × 10−2 at the recurrent end (dose 0.5). Across the twelve level means the trend is Spearman ρ = 0.94 (level-clustered permutation p < 10−4 ); the scale-free rank-association trend is ρs -vs-dose = 0.97 (p < 10−4 ); and every other test agrees (Kendall τ = 0.82; OLS R2 = 0.83; per-seed level-block p < 10−4 ; low-vs-high-dose sign flip, split at 0.25, Mann–Whitney p = 10−5 ). The properly-powered replication thus supersedes both the underpowered single sweeps and the fragile Fisher combine: the recurrence mechanism is confirmed, not merely suggested. The full-power run cost $1.12 on one L40S. Two bounds remain. First, this is a dose-response in the value proxy-to-write coupling χ̂; whether the proxy itself tracks realized task success is the separate, unvalidated question of section 6.1. Second, the design and grid were pre-specified, but this full-DROID re-run was executed after
18
0.025 0.020
Recurrence dose-response replicates at full power (600 DROID scenes) old 100-scene means (underpowered) 600-scene per-seed (12×3) 600-scene level means
value → write slope χ ̂
trend (OLS, R 2=0.83)
0.015 0.010 0.005 0.000 −0.005
pre-specified 12-level grid, 3 seeds, full DROID (600 scenes vs old 100): level-mean Spearman ρ = 0.94, perm p < 10−4; scale-free ρs ρ = 0.97 per-seed level-block p < 10−4; Kendall τ = 0.82; sign-flip MW p = 10−5 χ :̂ −5.2 × 10−3 (pure churn) → +16.7 × 10−3 (recurrent)
−0.010 0.0
0.1
0.2
0.3
0.4
0.5
recurrent fraction of the episode mix (design dose: LIBERO-Long share)
Figure 9: Recurrence dose-response replicates at full power. χ̂ vs. the recurrent (LIBERO-Long) fraction of the episode mix, on the pre-specified twelve-level grid (three seeds) re-run with the full droid_1.0.1 (600 DROID scenes; light dots are seeds, filled circles level means, orange the OLS trend, R2 = 0.83). The loader-capped 100-scene means (hollow grey) cluster near zero; at full power χ̂ rises (rank-monotone) from −5.2 × 10−3 (pure churn) to +1.67 × 10−2 (recurrent), Spearman ρ = 0.94, permutation p < 10−4 across all trend tests. This supersedes the underpowered sweeps and the retired Fisher combine.
review (the natural fix to the disclosed loader limitation), so we report it as a pre-specified-design replication at proper power, not as the original frozen primary.
5.3
When the budget binds: commodity edge storage, not premium TLC
The dormancy we report—η = 0 at datasheet prices—is a property of the premium storage we pinned to (3,000-P/E TLC on a 128-GB module, TBW= 128 TB), not of the embodied-memory problem. That configuration gives a device write-lifetime of ≈ 5.2 years at the measured fleet write demand (24.4 TB/robot/yr, section 6.2)—right at the edge of a 3–5-year deployment, so the budget only just fails to bind. But cheap edge robots run denser, cheaper NAND—commodity QLC and eMMC, whose endurance is ∼ 1,000 P/E (a few hundred for the cheapest parts) rather than 3,000 [40, 47]. There the same write demand exhausts the stock within a deployment: a 128-GB QLC part lasts ≈ 1.8 years, a 64-GB QLC part ≈ 0.7, a 32-GB eMMC part ≈ 0.3 (fig. 10). On the storage commodity edge robots actually use, the endurance budget binds at datasheet prices (η > 0) and the wear-pricing layer is live. The dormancy is a knife-edge artifact of the premium-TLC pin, flipping to firmly-binding under exactly the cheaper-NAND regime the regime map (fig. 15) anticipates. What binding buys: cost and lifetime, not task value. We ran the placement ladder in two binding regimes, and both tie endurance-blind routing on task value. (i) Under an artificial S2 cap (RAM-scarce, cloud expensive, endurance capped at 0.4× write demand; 5 seeds × 5 cells, binary success proxy) the controller strictly beats the naive all-NVM AURA strategy (McNemar b = 200, c = 0) but only ties trivial cloud-routing (H2 p = 1.0): routing to cloud avoids the endurance wall 19
Dormancy is a property of premium TLC — on commodity QLC/eMMC the budget binds at datasheet prices 6
device write-lifetime (years)
5
write demand 24.4 TB/robot/yr (paper's own figure); commodity QLC ∼ 1,000 P/E (a few hundred for the cheapest), premium TLC 3,000. The budget binds when device life < deployment.
5.25 yr typical deployment horizon (3–5 yr)
4
3 BINDS
2
1.75 yr BINDS
1
0
0.66 yr
128GB TLC
128GB QLC
64GB QLC
BINDS
BINDS
0.26 yr
0.20 yr
32GB eMMC QLC
64GB QLC (300 P/E)
Figure 10: The endurance budget binds on commodity storage. Device write-lifetime at the measured fleet write demand (24.4 TB/robot/yr) across storage classes: premium 128-GB TLC lasts ≈ 5.2 years (dormant, at the edge of deployment), but commodity QLC/eMMC (∼ 1,000 P/E [40, 47]) wears out in 0.2–1.8 years, well inside a 3–5-year deployment—so the budget binds at datasheet prices and the wear-pricing lever is live.
for free, so the binary proxy saturates. (ii) Under the realistic commodity-QLC binding regime with a graded net-realized-value metric, the wear-augmented index—clairvoyant optimum and deployable η-routing alike—again ties endurance-blind routing (LRU, size-based; advantage 0.00, 95% CI [0, 0] over scene-clustered resamples). The reason is structural: realized value is tier-invariant—RAM, NVM, and cloud all serve the item—so the endurance rent reshapes costs (erases, energy, dollars, device lifetime), not task value, and a simple rule that routes off flash captures the same value. A task-value payoff would need both a regime where flash is forced and scarce and a value signal validated against realized task success (section 6.1); that is future work. Why the index ties: write-intensity has no dispersion to exploit. The tie is not an artifact of treating the cloud as a free escape hatch. Re-pricing the cloud tier at the model’s slow value V slow (eq. (4)) and sweeping connectivity from connected to fully disconnected leaves the wear-aware advantage at 0.00 in every regime. The cause is the value–write joint distribution: on LIBERO, write-intensity is nearly constant (CV(w) = 0.13%, w ∈ [1.003, 1.011]) while value varies roughly two orders of magnitude more (CV(v) ≈ 24% on the same items). The wear-augmented index ranks placement by surplus per erase; dividing by an almost-constant w leaves the ranking unchanged, so the index collapses to value-ranking and the two policies coincide. Figure 11 makes the boundary quantitative: on a synthetic control with χ > 0 held fixed and CV(w) swept, the advantage is flat at zero until CV(w) ≳ 0.2 and reaches ≈ 4% at CV(w) = 0.5, while LIBERO sits two orders of magnitude below, at the floor. Wear-aware placement thus pays off precisely when high-value memories are disproportionately rewritten—a property of the workload, not the policy—which LIBERO-class manipulation does not exhibit. The recurrence–dispersion tension: why the win regime is empty. The dispersion the lever needs (CV(w) ≳ 20%) is not merely unmet on LIBERO—it is structurally out of reach, because the
20
Wear-aware placement matters only when write-intensity varies; LIBERO's is nearly constant
wear-aware advantage over endurance-blind routing (% net value)
synthetic control (χ > 0 fixed; CV(w) swept)
4 wear-aware lever bites
3
LIBERO (measured): CV(w) = 0.13%, no advantage (index ≡ value-ranking)
2
1
0 10−3
10−2
10−1
write-intensity dispersion CV(w) = σw/w̄ (log scale)
Figure 11: When wear-aware placement beats endurance-blind routing. The advantage in net realized (modeled) value is flat at zero until write-intensity dispersion CV(w) is substantial (synthetic control, χ > 0 fixed, CV(w) swept; cloud priced at the model’s V slow in the binding commodity-QLC regime). LIBERO’s measured CV(w) = 0.13% sits at the floor, where the wear-augmented index collapses to value-ranking. Value is the proxy v̂, not realized task success.
two ingredients a wear-aware win requires (a positive coupling χ > 0 and high write-dispersion) are anti-correlated across every way real robots generate writes (fig. 12). Recurrence (manipulation) makes χ positive—re-observing valuable scenes couples value with writes—but re-observing the same scenes homogenizes the stream (CV(w) ≤ 0.4% on recurrent LIBERO and on a real SO-101 and a sim ALOHA arm). Churn (teleoperation) spreads write-intensity but decouples it from value, driving χ < 0. And navigation—the escape hatch we predicted, where landmarks recur unequally—does have the highest dispersion of any workload (CV(w) of 7–10% on the Berkeley GNM datasets), but its frequently-traversed places are low-value transit while distinctive landmarks are seen rarely, so value and writes anti-correlate (χ: ρs = −0.13, −0.22; proxy validity ρ ≥ 0.92). We then surveyed write-dispersion across ≈ 20 Open-X / LeRobot workloads and ran the χ pipeline on the highest-dispersion ones. Across thirteen workloads with measured χ—spanning the ecosystem, multiple embodiments, and all three write-generating mechanisms, plus the full-power recurrence dose-response (section 5.2; χ rises monotonically with recurrence, ρ = 0.94, which lifts χ exactly as it homogenizes the writes)—the win quadrant (χ > 0 and CV(w) > 20%) is empty, and empty by mechanism, not by sampling. The single closest approach is a bimanual xArm dataset (ρs = +0.31, χ̂ = +0.34 at CV(w) = 16%, proxy 0.88)—the lone positive coupling at high dispersion—but it is underpowered (n = 70 episodes, χ CI [−0.11, +0.78] straddling zero) and still short of the 20% threshold, so we report it as a suggestive frontier, not a counterexample: coordination-rich manipulation is the regime a future positive result should target. (All external embodiments use LeRobot v3.0, video-decoded through the same image-counterfactual χ pipeline with action/state z-scored per dataset so CV(w) is comparable; held-out proxy validity ρ ≥ 0.76 throughout.) A wear-aware placement win would require a workload where high-value memories are frequently and unequally rewritten with positive coupling; no natural regime we measured supplies all three, and
21
The recurrence--dispersion tension: the wear-aware win regime is empty in real robot data xArm bimanual: the lone χ > 0 at high CV --- but CI straddles 0, n = 70, and 16% < 20% threshold (suggestive only)
value → write coupling χ (Spearman ρs)
0.4 0.3 ALOHA ⇒ χ > 0, recurrence but writes homogenize
0.2
wear-aware win possible (χ > 0 AND CV(w) > 20%)
LIBERO-Long
EMPTY in all real data
UR5
0.1 LIBERO-Goal/Obj/Spatial
0.0 GNM-recon (nav)
−0.1
DROID-1200
DROID-enlarged
SO-101
−0.2
churn / navigation ⇒ high CV(w), GNM-cory (nav) but χ < 0
−0.3 −0.4
external robot embodiments
10−1
100
101
write-intensity dispersion CV(w), % (log scale)
Figure 12: The recurrence–dispersion tension. A wear-aware win needs both χ > 0 and high writedispersion CV(w), but they are anti-correlated across all three write-generating mechanisms: recurrence (manipulation) gives χ > 0 at low dispersion; churn (teleoperation) and navigation give high dispersion but χ < 0. Four in-house workloads (round/plus/square/diamond) and nine external robot workloads (stars) from an ≈ 20-workload Open-X / LeRobot survey trace an anti-correlated band; the win quadrant (χ > 0, CV(w) > 20%, threshold from fig. 11) is empty. The lone positive coupling at high dispersion is a bimanual xArm dataset (ρs = + 0.31 at CV(w)=16%), flagged suggestive because its χ CI straddles zero (n=70) and it remains below the 20% threshold—a frontier for future work, not a counterexample. The full-power recurrence dose-response is reported separately (section 5.2).
we state this as the precise, falsifiable boundary rather than engineer a workload to cross it.
5.4
Placement controller: a negative result (H1, H2, H3)
The 3.15M-parameter controller (BC warm-start + PPO, 5 seeds) returns a negative result. H1 (non-monotone deny-NVM-to-high-value persistence slope) is not rejected on any seed (p ≥ 0.40; the slope is flat at 0.0). H2 (beat the strongest cost-matched baseline) ties on every seed (p = 1.0). H3 (controller ̸= AURA single-tier) is seed-dependent, and where it rejects it is an energy-metric-gaming artifact: two seeds reject AURA by emitting MIGRATE actions that deflate the joule denominator, driving a controller/oracle ratio to 2.11—impossible against a clairvoyant oracle (fig. 13). The one non-gaming seed converges to AURA-identical behavior (McNemar b = c = 0). The verdict is invariant across all four stream-stitch families and every price regime, confirming it is a fixed policy property, not a wear effect; PPO does not beat the BC warm-start. The central cause: at datasheet S0-connected prices the endurance budget never binds (the solved dual is zero with 196-KB LIBERO frames), making H1/H3 partly vacuous—itself a boundary result, which the binding-regime tests (section 5.3) address.
22
Placement controller: a negative result — the apparent wins are metric-gaming, not wear-aware placement
Invariant to adversarial stitching (192 cells)
ratio > 1.0 exceeds the clairvoyant oracle 2.110 → the apparent win is a metric-gaming artifact
2.0
1.5 1.132 0.996
1.0
0.5
0.0
seed 42 H3 reject (gaming)
seed 137 AURA-identical (b=c=0)
seed 2024 H3 reject (gaming)
controller / oracle ratio (per cell)
controller / oracle primary-metric ratio
Per-seed H3: 2 of 3 are gaming artifacts
mean 0.846 H3 32/48
mean 0.843 H3 32/48
mean 0.853 H3 32/48
mean 0.843 H3 32/48
2.0
1.5
1.0
0.5 ratio flat across stitch types (spread 0.010) → a fixed policy property, not a wear-economics effect
0.0
random
similarity ordered
adversarial
drift injected
Figure 13: Controller robustness and the metric-gaming finding. Left: per-seed H3 outcome (controller/oracle primary-metric ratio); the non-gaming seed is AURA-identical (ratio 0.996), the rejecting seeds exceed the oracle (ratios 1.13, 2.11, the impossibility signature of gaming). Right: the H3 verdict is invariant across all four stitch families (spread 0.010; adversarial ≡ random), confirming the difference is a fixed policy property, not a wear-economics effect.
5.5
Price statics and the cross-partial (P5a–d, H5, H6)
The placement simulator on the real LIBERO-labeled items with the fitted δ̂ = 0.032 and datasheet constants, swept across the S0/S1/S2 price grid returns three of four pre-specified signs confirmed and one inconclusive null. P5a (pNAND ↑⇒ η ↓), P5c (pegress ↑⇒ η ↑), and P5d (Eend ↑⇒ η ↓) confirm with tight bootstrap CIs and 100% sign-stability. P5b (pDDR ↑⇒ η ↑) is an inconclusive null (+7.2 × 10−8 ≈ 0, CI [−4.6 × 10−7 , +4.6 × 10−7 ], 25% sign-stable): under binding RAM the shadow price µR absorbs the DDR shock one-for-one, so it never reaches the endurance margin, a clean identification of the boundary condition, exactly as rescoped in Proposition 4. The NVM-share σN partials are inconclusive by construction (σN is budget-pinned at ≈ Eend /w̄ once η > 0, so the price signal lives in η). The cross-partial (H6) point estimate is +0.50 (87% sign-stable) but its bootstrap CI [−0.34, +1.25] straddles zero, so it is directionally consistent, CI-inconclusive: H6’s non-fatal kill is not triggered, but the sign is not CI-confirmed either. Quantitative headline. The equilibrium rent declines monotonically across the price band (ηsim : LOW 3.16 → S0 2.42 → S1 1.90 → S2 1.47, all ×10−4 , simulator units; see section 3.8 for why these ⋆ = 0.91 at base prices. The 2025–26 are not on the m̂ scale); the break-even item value is vBE NAND supercycle (S0→S2) cuts ηsim by ≈ 39% (2.42 → 1.47 × 10−4 ) while leaving σN (+0.7 ⋆ = 0.91 unchanged: the price shock is absorbed by the wear margin, not pp) and the break-even vBE the placement boundary (fig. 14). Not yet run. The phase-diagram-measure and RAM-pressure hypotheses H4/H4b (Proposition 3) require a dedicated swept-box / CR -sweep run that has not landed; we report them as outstanding rather than fill them from an unrelated artifact.
23
Price comparative statics: the supercycle re-prices the marginal item, not the boundary
Rent falls across the price band
0.400
NVM share budget-pinned ( ≈ 0.29–0.30)
3.5 0.375 S0 → S2 NAND supercycle: ηsim cut ≈ 39% (2.42 → 1.47)
3.0 2.5 2.0 1.5 1.0
population NVM share σN
equilibrium rent ηsim (×10−4, simulator units)
Computed in the budget-binding regime (endurance capped below write demand). At today's measured write intensities the budget does not bind and η = 0 — this panel prices the regime where it does.
0.5
⋆ break-even vBE = 0.91 stable
0.350 0.325 0.300 0.275 0.250 0.225 0.200
0.0 Low (floor)
S0=Base (today's $)
S1=High (DRAM)
Low (floor)
S2=High (NAND)
S0=Base (today's $)
S1=High (DRAM)
S2=High (NAND)
Figure 14: Price comparative-statics fan over the unified Low/Base/High band (real re-solved outputs): the equilibrium rent ηsim declines monotonically across the band (left) while the population NVM share σN stays ⋆ budget-pinned at ≈ 0.29–0.30 (right), with the break-even vBE = 0.91 annotated.
6
Discussion
What the priced model buys. The central object is the endurance shadow price η: it fixes the persist/evict boundary, signs how placement reacts to the memory-price supercycle (section 3.7), and doubles as a device-lifetime price. The learned controller adds little on top: in both binding regimes (section 5.3) it ties price-based routing on task value, because realized value is tier-invariant across RAM/NVM/cloud—so once η and the wear-augmented index are in hand, simple price-based routing suffices on today’s hardware. The genuinely open question is therefore narrower than “does the controller help”: whether a regime exists where flash is forced and scarce (so the tier choice is not free) and a value signal validated against realized task success (section 6.1) makes placement causally move performance. That is the next experiment.
6.1
Limitations: what this work does not establish
1. The association χ > 0 is regime- and backbone-conditional, not universal: positive only for SmolVLA-0.5B on recurrent long-horizon data, null on a second LIBERO suite, negative on teleoperation (section 5.1). The non-monotone branch (Proposition 2) is claimed only in that regime; the monotone index and rent η are sign-agnostic and hold in every measured cell. 2. The DROID negative is post-hoc, not pre-specified: the enlargement followed a negative, underpowered pilot. We label it exploratory, report it on the pre-specified 1,200-new-only subset, and treat it as a regime-difference signal pending replication. 3. Cross-backbone comparison is uninterpretable by a pre-specified criterion: OpenVLA7B and SmolVLA agree on item value at only ρs = 0.05 (floor 0.6), so we do not read the 7B arm as a disconfirmation. A cross-backbone agreement floor is a necessary validity check before any cross-model sign claim. 4. A pre-specified third backbone (pi0-3.5B) was loaded but deferred after a checkpoint state-dict mismatch left its vision tower random-initialized, which would invalidate its value proxy; we report the two-backbone scale axis (SmolVLA-0.5B, OpenVLA-7B) instead, with details in section B. 24
When does wear-aware placement switch on? recurrencedriven
DORMANT (today)
η = 0 — the budget does not bind, so wear-pricing is moot regardless of the sign of χ
deployment recurrence →
LIBERO-Long (recurrent) χ ̂ = +1.0 × 10−3 — positive, but dormant at today's endurance
χ ̂ = −9 × 10−3 (opposite sign)
SWITCHES ON
η > 0 AND χ > 0: non-monotone optimum is live
the regime this paper prices in advance
tight, but churn-driven (η > 0, χ ≤ 0: wear-aware index, but monotone)
DROID (teleop churn) endurance budget starts binding
churndriven abundant (today's NAND)
tight (denser, cheaper NAND; heavier writes)
endurance tightness (write demand ÷ flash endurance budget) →
Figure 15: When wear-aware placement switches on. The lever needs both a binding endurance budget (η > 0, the right of the map) and a positive value–write coupling (χ > 0, the top): only the upper-right cell activates the non-monotone optimum. Our measured datasets sit in the endurance-abundant band as run (LIBERO-Long recurrent χ > 0; DROID churn-driven χ < 0), but the binding (right) column is live today on the commodity QLC/eMMC cheaper edge robots use (section 5.3), not merely future. What remains priced in advance is the upper-right corner alone—binding endurance and a recurrent, χ > 0 regime together.
5. The controller win is not demonstrated; one H3 rejection was a metric-gaming artifact (a seed exceeding a clairvoyant oracle by deflating the energy metric; section 5.4). At datasheet prices the budget never binds, so the offline H1/H3 tests are partly vacuous. 6. The placement→task-success causal chain is not yet demonstrated with a VLA in the loop. We built and ran the project’s first true VLA-in-the-loop arm: a LIBERO-finetuned SmolVLA-0.5B acting in real LIBERO-Long physics (not labeling static frames): 18 episodes, 3,960 environment steps, 0 errors. The pre-specified causal-room gate required oracle-memory minus no-memory ≥ +8 pp and measured +0.0 pp: the 0.5B backbone solves none of the three hardest LIBERO-Long tasks at a 220-step budget, so there is no success signal for placement to perturb, and the minimal training-free memory channel does not reproduce the published end-to-end-trained memory effect. We aborted the full campaign per the frozen kill criterion rather than engineering a favorable coupling, and we explicitly scope placement→task-success causal validation, which would require a trained memory-augmented backbone, to future work. A direct consequence bears on the headline: every χ we report is a coupling between write-intensity and a counterfactual value proxy v̂, internally validated against the full SmolVLA counterfactual (rcf = 0.92) but not validated end-to-end against realized task success, since the causal gate found no task-success signal to anchor it. The measured regime-dependence of sign χ stands as a property of v̂; tying it to realized task value awaits a backbone that can solve the tasks. 7. Welfare: we minimize the operator’s private expected cost: no claim on social welfare or Pareto-efficiency. 25
8. Market equilibrium: p is an exogenous shadow/contract price; section 3.7 is partial-equilibrium, operator-side. 9. Closed-form η path: the (1 + r)t path is not claimed; η is a learned dual under non-stationary demand. 10. Separability failure modes: the index/threshold form is exact only under item separability; under retrieval complements, write-amplification, non-stationary Ft , and lumpy items, the direction of non-monotonicity survives but the closed-form threshold does not: the case for a learned controller, theory as scaffold.
6.2
cs.CY implications
By Corollary 1, cost-optimal forgetting is also lifetime-extending: a fleet-wide erase saving q defers replacement embodied carbon at the ≈ 22 kg CO2 e/TB anchor [58] (which prices NAND manufactured, not bytes written). For a 1,000-robot fleet the solver puts device write-lifetime at ≈ 5.2 years, mapping to ≈ 24.4 TB of replacement NAND per year (fleet capacity ÷ write-lifetime, not TB-written) and fleet embodied carbon of order ≈ 540 kg CO2 e/yr; the supercycle shortens device life ≈ 2.4%, second-order relative to the 39% swing in ηsim . These are order-of-magnitude, relative figures, not forecasts. Lifetime extension complements rather than substitutes for fleet-refresh economics [4, 54] and circular-economy policy [18, 19]; edge data-residency rules add a privacy rationale for local persistence [29]. Author continuity. AURA [9] is the front-end write gate and a baseline arm; Chen [10] supplies edge-decode cost anchors; Chen [8] is the statistics-methodology precedent.
7
Conclusion
Under a binding non-renewable write-endurance budget, the cost-minimizing memory placement rule is a threshold in a wear-augmented per-byte index governed by an endurance shadow price η, a rule whose form does not depend on the sign of the value–write association. On this sign-agnostic spine, the optimum becomes strictly non-monotone in item value under one further condition, a positive association χ > 0 that we measure at a pre-specified gate rather than assume. Our central empirical finding is that the antecedent’s sign is a property of the deployment regime: positive on recurrent long-horizon data with a small backbone, null on a second suite, negative on non-recurrent teleoperation, and uninterpretable across backbones that fail a cross-backbone agreement floor. The measured boundary of relevance matters as much, and it cuts both ways: on premium 3,000-P/E TLC the endurance budget does not bind, the confirmed coupling is small, the calibrated down-crossing lies outside the measured value support, and our learned controller does not yet beat a simple price-based routing rule under a binary success proxy—but on the commodity QLC/eMMC (∼ 1,000 P/E) that cheaper edge robots actually run, the same measured write demand exhausts the endurance stock within a deployment, so the budget binds at datasheet prices and the pricing layer is economically live (section 5.3). Yet even there, the wear-aware policy ties simple endurance-blind routing on task value: realized value is tier-invariant across RAM/NVM/cloud, so the rent reshapes cost and device lifetime, not performance—and the placement gain stays zero because the wear-aware index has no write-dispersion to exploit. That last point is structural: the recurrence that makes χ positive homogenizes the write stream, so χ > 0 and high dispersion are anti-correlated and the win regime sits empty in every workload we measure (fig. 12). Re-solving the calibrated model across the 2025–26 memory-price supercycle cuts the equilibrium rent by 26
≈ 39% while leaving the persist/evict boundary fixed, and the rent doubles as a device-lifetime price. Wear-aware placement is thus economically live today on commodity edge storage (fig. 15), but its task-value payoff awaits a regime where flash is forced and scarce and a value signal validated against realized task success. That is the next step.
A
Proofs
Proof of Proposition 1. With η = 0 the three persistent returns share an identical slope in v. Writing Vi = λi vi /(1 − γe−δi ), both Vifast (entering ΠR , ΠN ) and Vislow = Vifast − λi ℓπ/(1 − γe−δi ) (entering ΠC ) are affine in vi with the same coefficient λi /(1 − γe−δi ); the tier-specific terms (µR +pR , pN si , pC si , and at η = 0 the bounded cash wear cwear wi ) are v-independent intercepts. Hence ΠR , ΠN , ΠC are parallel lines in v: the choice among the persistent tiers is fixed by their intercepts and does not vary with v: there is no unique “largest v-coefficient” tier. The only v-dependent margin is persistvs-discard: Π∅ = − Pr[needed]κi has the smallest v-slope (zero, or sub-linear by Assumption A6), so max(ΠR , ΠN , ΠC ) − Π∅ is strictly increasing in v. Single-crossing of this margin gives a threshold v † above which some persistent tier dominates discard. Above v † , when the v-independent intercepts make NVM the best persistent tier (i.e. ΠN ≥ max(ΠR , ΠC ), the condition eN < eC + ℓπ on the energy/latency terms), 1[xi = N ] is weakly increasing in vi , a step up driven by the persist-vs-discard margin, not by any tier owning the steepest v-slope. Proof of Proposition 2. Define the slack g(v) := B(v) − (cwear + η)w̄(v). (i) Low v: V small, even locality below the discard/cloud option: Pr[x = N | v] low, rising as V (v) clears the cloud margin. (ii) Moderate v: g(v) > 0 and V (v) beats the cloud option: persist. (iii) High v: B ′ (v) ≈ 0 (fact 1) while (cwear + η)χ > 0 (fact 2), so g ′ (v) < 0; for η > η̄ the slack crosses zero from above ⋆ , after which (BE-NC) fails and the item routes to cloud (or cheap recompute, at interior vDC ⋆ : rise-then-fall. By the implicit function Assumption A6). Hence Pr[x = N | v] strictly falls past vDC ⋆ ′ ⋆ ⋆ /∂η = −g /g , where theorem on g(vDC ; η, χ) = 0 with gv = B (vDC ) − (cwear + η)χ < 0: ∂vDC η v ⋆ ) < 0, so ∂v ⋆ /∂η = −w̄(v ⋆ )/[(c ′ ⋆ gη = −w̄(vDC wear + η)χ − B (vDC )] < 0. Writing w̄(v) = w0 + χv DC DC ⋆ ⋆ /∂χ = −g /g < 0. The χ > 0 requirement locally, gχ = −(cwear + η)vDC < 0, giving ∂vDC χ v ′ ′ is necessary: at χ = 0, g (v) = B (v) ≈ 0 and no interior down-crossing exists (monotone, Proposition 1). Proof of Proposition 3. The down-crossing is fixed by (BE-NC), the NVM-vs-(cloud/recompute) margin, whose RHS rises in v via χ > 0 independent of µR . µR enters only (BE-NR), i.e. which non-NVM tier wins. Both ΠC = V slow − pC s and Π∅ = − Pr[needed]κ are µR -free and available ⋆ (recompute cheap for high v by Assumption A6); so Pr[x = N | v] falls past vDC for any µR ≥ 0, R C ∅ with arg max(Π , Π , Π ) re-labeling the receiver. The location result follows from the budget ⋆ /∂η < 0. identity (raising µR raises NVM demand, hence η) and ∂vDC Proof of Proposition 4 (sketch). Each sign differentiates the break-even conditions PP (BE-NC)/(BE-NR) and the budget identity w = Eend implicitly defining η(p). P5a/P5d are clean (single-signed channels); P5b/P5c carry a directional label where the second-order η-feedback opposes the first-order channel, as noted.
27
B
Artifact and Reproducibility
All figures and statistics are regenerable from the released artifact bundle (wamp_reproducibility/, distributed as supplementary material with this paper): the code/ tree (simulator, eval harness, cost model, χ estimators, price-statics and calibration scripts, tests), the frozen pre-specified plan (experiments/experiment_plan.md), the canonical χ re-analysis table (experiments/chi_reanalysis/chi_canonical_table.json, the source of truth for every χ reported here), every per-phase analysis output, and the run and cost registries (runs/, 46 billable rows summing to $18.3764). A MANIFEST.md at the artifact root maps each paper claim to its regeneration script and data file. The decision log (D-001–D-016) and the multi-round governance audit trail are referenced by identifier throughout and are included in the complete repository release alongside this code/data core. The only exclusions are the raw PPO training-log directories (≈146 MB, regenerable from the pinned seeds and configs) and the external datasets and checkpoints, which are not redistributed but are pinned by HuggingFace slug and revision in the dataset card. Total project compute spend was ≈ $18.38 (the original pre-specified campaign spent ≈ $17.26—its kill criteria, not the budget, stopped it—and the subsequent full-power replication and commodity-storage binding analyses added ≈ $1.12). Software versions (pinned). Throughout this appendix, internal campaign codes are used for provenance: W1/W2 denote the first and second χ batteries (W2 is the dose-response sweep, “W2 top-up” its pre-specified replication), W3a the closed-loop diagnostic folded into section 5.3, and D-nnn entries reference the shipped decision log. The campaign used three coexisting Python stacks (D-003): a main train/eval env (transformers 4.55.4) for the Phase-2 controller, W3a diagnostic, P3 statics, and calibration; the Modal χ-estimation containers (transformers 4.51.3 with lerobot[smolvla] 0.3.3) for every SmolVLA arm (Phase-0 gate, labeling, DROID re-test, all χ batteries, and the deferred pi0-3.5B probe); and an isolated OpenVLA env (transformers 4.40.1 + timm 0.9.10) for the 7B arm. The headline and regime χ results depend on the 4.51.3 stack and the OpenVLA arm on 4.40.1; no χ estimate depends on the main env. Full version pins ship in code/pyproject.toml and the artifact. Random seeds. Controller training/eval seeds Modal {42, 137, 2024} and Lambda {7, 99}; bootstrap seeds {42, 137, 2024, 7, 99, 2718} (primary = 137); W2 top-up seeds {1001, 1002, 1003}. All χ CIs use 1,000-resample physical-scene-clustered bootstraps. Hardware and per-phase cost. Modal (L40S, T4) and a single Lambda H100-SXM lane (terminated and verified). Key per-phase spend from runs/cost_registry.csv: The table is exhaustive over the 46 billable rows of runs/cost_registry.csv (REFERENCE-only rows excluded); rows sum exactly to the $18.3764 cumulative total. The original pre-specified campaign spent $17.2582 (its kill criteria, not the budget, stopped it); the subsequent full-power replication and commodity-storage binding analyses added $1.12. Data and checkpoint slugs (HuggingFace). Datasets: lerobot/droid_1.0.1 (full DROID; analyzed 1,200-new-only subset), lerobot/droid_100 (rev 87301a2), and the LIBERO suites via openvla/modified_libero_rlds (rev 6ce6aaa). Checkpoints: lerobot/smolvla_base, HuggingFaceVLA/smolvla_libero, moojink/openvla-7b-oft-finetuned-libero-10, and lerobot/pi0 (deferred). Revisions are pinned in the dataset card where determinable from experiments/p1_staging_manifest.json; otherwise stated as latest as of 2026-06-12. Full
28
Phase / arm
Hardware
Infra & staging smoke Phase-0 full gate Phase-1 full labeling Phase-2 placement controller (≥ 3 seeds) DROID re-test (enlarged) Lambda lane (seeds 7/99 + OpenVLA arm) χ re-analysis (OpenVLA relabel) Phase-3 price comparative-statics W1+W2 χ suites + dose-response pi0-3.5B probes (deferred, D-013) W3a closed-loop diagnostic W2 top-up dose-response VLA-loop smoke (HV0/HV1, parallel)
Modal L40S Modal L40S Modal L40S Modal L40S/T4 Modal L40S Lambda H100-SXM Modal L40S CPU (local) Modal L40S Lambda H100-SXM Modal L40S Modal L40S Modal L40S
Cost (USD) 0.0230 0.2748 0.9736 2.4542 0.5540 7.2600 0.8500 0.0000 1.1931 0.0570 0.0800 0.8004 2.7381
Original campaign subtotal Full-power 600-scene replication (section 5.2) Modal L40S Commodity-storage binding + ladder analyses (section 5.3) CPU (local)
17.2582 1.1182 0.0000
Cumulative project total
18.3764
per-arm episode/scene counts, preprocessing, and license notes are in the dataset card pointer: experiments/DATASET_CARD.md. Controller hyperparameters. Placement controller (code/configs/phase2_controller.yaml): a 3.15M-parameter transformer (dmodel =256, 8 heads, 4 layers, FF mult 4, 5 actions {keep-RAM, write-NVM, offload-cloud, discard, migrate}, max 512 items; ≤ 50M cap). Training: behaviorcloning warm-start (5 epochs, lr 10−3 , hindsight-oracle targets) then clipped-surrogate PPO (rollout length 2048, lr 3 × 10−4 , γ=0.95, GAE λ=0.95, clip ϵ=0.2, 4 PPO epochs, minibatch 256, 200,000 total steps), ≥ 3 seeds; scene split 70/15/15; reward = success-proxy −λE energy −ν erase −ρC cloud-$ − migration-cost 0.05. Script → claim map. Headline χ̂ + matrix: phase0_gate.py, w1w2_chi.py (→ chi_ canonical_table.json); OpenVLA scale-stress arm: openvla_confirmation_arm.py; DROID enlarged (post-hoc): droid_retest.py; W2 dose-response + top-up: w2_topup.py, power_ analysis{,2,3}.py; controller (negative result): phase2_train.py/full.py, w3a_rollout.py (→ GATE_VERDICT.json); χ canonical re-analysis: chi_estimators.py (a verbatim port of phase0_gate._local_linear_slope). Run-ID provenance. The run identifiers behind each result (relocated here from inline tags for the camera-ready build) are listed in table 5.
References [1] Daniel Adelman and Adam J. Mersereau. Relaxations of weakly coupled stochastic dynamic programs. Operations Research, 56(3):712–727, 2008. doi: 10.1287/opre.1070.0445. [2] Eitan Altman. Constrained Markov Decision Processes. Chapman and Hall/CRC, 1999. URL https://www-sop.inria.fr/members/Eitan.Altman/TEMP/h.pdf.
29
Table 5: Run-ID provenance for the main results. Result
Run ID(s)
Phase-0 value–write gate Canonical χ matrix / re-analysis Phase-1 labeling / regime mechanism Recurrence dose-response Controller (H1/H2/H3) Closed-loop W3a Price statics / cross-partial Calibration (δ-anchor) VLA-loop (aborted)
phase0-full-20260612 (GATE-cov) chi-reanalysis-20260612 (D-012); W1+W2 battery phase1-full-20260612; droid-retest-20260612 W2 battery; W2-topup P2 eval battery (64 cells/seed, Modal/Lambda lanes) w3a-diagnostic-5seed-20260612 p3-pricestat-20260612 econ-calibration-20260612; droid-retest-20260612 vla-loop-smoke-parallel-20260612
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