Battery-Powered Tableaux

• Battery-powered tableaux are dual-purpose constructs that serve as interactive visualizations linking BEB transit and grid metrics, and as combinatorial objects (standard Young tableaux on battery shapes).
• They employ precise data encodings and linked interactive views to support optimization tasks such as charge scheduling, voltage compliance, and cost minimization.
• The concept also extends to battery-aware computation on small devices, where modular brick decomposition drives energy-efficient multimodal inference.

Searching arXiv for the cited papers to ground the article in the current record. Battery-powered tableaux denotes two distinct technical constructs in recent arXiv literature. In power–transit systems, it is an interactive, multi-layer spatio-temporal visualization that fuses battery–electric bus transit operations with distribution-grid status for monitoring, exploration, and cost-effective, constraint-aware charging (Bagherinezhad et al., 2020). In enumerative combinatorics, it denotes standard Young tableaux of battery shapes, a family of truncated skew shapes motivated by chess batteries and enumerated through generalized hypergeometric functions (Shoan, 2022). An adjacent battery-aware systems literature, although not using the tableau terminology, develops battery-powered multimodal inference frameworks on heterogeneous SoCs and supplies a complementary perspective on scheduling, memory coordination, and energy-aware operation under tight resource constraints (Li et al., 25 Sep 2025).

1. Terminological scope and disciplinary usage

In the systems literature, a battery-powered tableaux is a visualization artifact: a web-based tableau that links battery-electric bus operations, charging-station behavior, market prices, and feeder-level grid states. Its role is operational situational awareness, with linked spatial and temporal views exposing how BEB charging load appears as a flexible load at mapped distribution nodes (Bagherinezhad et al., 2020). In the combinatorics literature, the same phrase refers to standard Young tableaux on battery shapes, where “battery” names an added vertical chain attached above a chosen column of a rectangular Young diagram, reflecting a chess battery motif rather than electrochemical storage (Shoan, 2022).

An adjacent usage appears in battery-powered on-device AI systems. NANOMIND is not itself a tableau, but it exemplifies a battery-aware systems methodology in which model pipelines are decomposed into modular “bricks,” mapped to heterogeneous accelerators, and coordinated through unified-memory scheduling on a small device (Li et al., 25 Sep 2025). This suggests that, across otherwise unrelated domains, “battery-powered” tends to index resource-constrained operation and the need for explicit coordination among coupled subsystems.

Usage	Object	Core technical elements
Power–transit operations	Interactive multi-layer spatio-temporal visualization	BEB SOC, charging events, voltages, currents, power flows, TOU, LMP
Enumerative combinatorics	SYT of battery shapes	Truncated skew shapes, hook-length ratios, hypergeometric series
Adjacent battery-aware systems	On-device multimodal inference framework	Brick decomposition, dynamic offloading, TABM, low-bit kernels

A common misconception is that the phrase has a single established meaning. The literature instead supports a domain-specific polysemy: visualization and tableaux in the operator-interface sense on one side, and tableaux in the Young-tableau sense on the other.

2. Battery-powered tableaux in interdependent bus–grid operations

The power-systems meaning arises in "Spatio-Temporal Visualization of Interdependent Battery Bus Transit and Power Distribution Systems" (Bagherinezhad et al., 2020). The tableau is designed to illustrate the interdependency of battery bus transit and power distribution systems and the associated components. It monitors bus locations, state of charge of batteries, charging power demand, station active/reactive power, node voltage magnitude, node active/reactive load, line current, active/reactive power flow, locational marginal price, and time-of-use tariff.

The architecture is explicitly two-layered. The transit layer includes bus AVL/GPS and planned timetable data, BEB battery telemetry, and station information such as identifier, geographic layout, and number of BEBs at station. The power distribution layer includes SCADA-like node and line metrics. Interdependency is encoded by mapping each on-route charging station to a distribution node so that BEB charging load appears as a flexible load at the corresponding node. The underlying data model is a graph representation $G=(N,L)$ across the transit and distribution layers, with gray connectors visualizing station-to-node membership.

The demonstrated testbed uses Park City, UT transit with 9 routes, inter-station distances 3–15.5 miles, 45 BEBs, 7 on-route charging stations, and a 33-node radial distribution network in which node 1 is the substation. The temporal horizon is 24 hours, from 5:00 AM to 5:00 AM the next day, with 5-minute granularity for a total of 288 time steps. The prototype streams precomputed, time-indexed data to the front-end as static files such as JSON or CSV, while the operational direction is REST/WebSockets ingestion from AVL, charger telemetry, and SCADA/AMI feeds.

The visual design is tightly specified. The left panel shows the distribution layer, where purple rectangles encode nodes and blue polylines encode lines; node color intensity represents node active load, line width represents current, and line color intensity represents active power flow. The right panel shows the transit layer, where charging stations are circles in a bird’s-eye geographic layout, circle radius encodes the number of BEBs currently at the station, and green shade encodes station active power. Gray interdependency connectors link a station to its grid node, and selected or hovered links highlight in dashed orange. The interface also includes panning and zooming, a time slider, a data panel, a scrollable BEBs table with diverging bars for SOC and charging power magnitude, and a chart-view dropdown.

The point of the tableau is not merely display density. It is a linked-view system in which hovering a station highlights its mapped grid node and vice versa, while hovering a BEB flashes the most-recent station in red and the next-destination station in green. The paper characterizes this as a “show, then tell” design: high-level cues first, then detailed charts and contextual values. A second misconception follows from the richness of the interface: the tableau itself does not run power flow; it visualizes provided metrics and is designed to be coupled with co-optimization rather than replacing it.

3. Data encodings, interaction logic, and optimization coupling

The monitored variables are encoded so that operational linkages remain visually salient (Bagherinezhad et al., 2020). BEB location is categorical, distinguishing on-road versus at station states. Velocity appears as a table column and reflects schedule-driven constant speed for each segment. SOC and charging power demand are shown as diverging bars and as time-series charts on demand. Station utilization is conveyed through circle radius, while station active/reactive power is conveyed through shading and time series. Price data are split into an LMP green line and a TOU red line, with peak hours indicated from 6 PM to 12 AM.

These encodings are not independent glyph choices; they are designed to support specific analytical tasks. Operators can visually align station active power traces with low-tariff windows, inspect how concentrated charging shifts line flows and voltages, and compare wholesale procurement exposure with retail charging cost exposure. Multi-selection adds several components to the same chart, and nearest-line hover highlighting with opacity dimming improves line discrimination. This design supports exploratory diagnosis without collapsing transit and grid phenomena into a single aggregate index.

The mathematical models that the tableau is designed to surface are standard charge-scheduling and radial-feeder constraints. Battery SOC evolves as

$$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$

Optional operational-reserve bounds are given as $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$. Availability and dwell-time are enforced through

$$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$

and charger-port capacity through

$\sum_b u_{s,b,t} \le N^{\text{port}}_s.$

The charging-cost objective can be written either in TOU form,

$$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$

or in procurement-cost form,

$$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$

On the grid side, the tableau is meant to interface with DistFlow constraints for radial feeders:

$$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$

$$V_j^2 = V_i^2 - 2\big(r_{ij}P_{ij} + x_{ij}Q_{ij}\big) + \big(r_{ij}^2 + x_{ij}^2\big)\,\ell_{ij}, \quad \ell_{ij} = \frac{P_{ij}^2 + Q_{ij}^2}{V_j^2}.$$

Voltage and thermal limits are

$$V^{\min} \le V_{j,t} \le V^{\max}, \quad I_{ij,t} \le I^{\max}_{ij},$$

and the nodal-load composition is

$$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$0

A feeder-capacity constraint is also given:

$$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$1

The tableau therefore occupies a boundary position between visualization and optimization. It focuses on monitoring and exploration, yet its variables and interactions are already organized around optimization primitives: TOU/LMP-aware cost minimization, route/dwell-time feasibility, voltage compliance, thermal limits, and feeder capacity. Recommended extensions include alerting for voltage violations, line overloads, station over-capacity, and SOC below reserve; uncertainty display with dashed lines or lighter opacity; and predictive features such as SOC forecasting, arrival-time estimation, and LMP forecasting.

4. Battery-powered tableaux as standard Young tableaux of battery shapes

In combinatorics, the phrase is formal rather than metaphorical. "Enumeration of Tableaux of Unusual Shapes" defines battery shapes and studies the number of standard Young tableaux on them (Shoan, 2022). A partition $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$2 is a weakly decreasing sequence of positive integers summing to $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$3, and its Young diagram $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$4 is the left-justified array of cells whose $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$5th row has length $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$6. A standard Young tableau of shape $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$7 is a bijection $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$8 with entries strictly increasing left-to-right in each row and top-to-bottom in each column. For skew shapes $$\text{SOC}_{b,t+1} = \text{SOC}_{b,t} + \eta_c \frac{P^{\text{ch}}_{b,t}\,\Delta t}{E_{\text{bat},b}} - \frac{1}{\eta_d}\frac{P^{\text{dis}}_{b,t}\,\Delta t}{E_{\text{bat},b}}, \quad 0 \le \text{SOC}_{b,t} \le 1.$$9, the definition is analogous on $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$0.

A truncated shape is a line-convex diagram obtained by deleting cells from the northeast corner of a Young diagram, possibly skew. Battery shapes are special truncated skew shapes obtained by extending a prescribed column upward. Fix a partition $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$1 and integers $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$2 and $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$3. The battery shape $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$4 is obtained by adjoining a vertical chain of length $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$5 directly above the upper cell of the $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$6th column of $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$7. In the thesis, the main base shape is the rectangle $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$8, that is, $$\text{SOC}_{\min} \le \text{SOC}_{b,t} \le \text{SOC}_{\max}$$9 rows of equal length $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$0. The parameterization $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$1 therefore means a base rectangle with $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$2 columns and $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$3 rows, an attached upward column of length $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$4 above the top of column $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$5, and total size $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$6.

Within this literature, “battery-powered tableaux” simply means SYT of battery shapes. The word “battery” refers to the added column and to the motivating chess battery, not to energy storage. This distinction matters because it avoids a category error that would otherwise arise from importing the engineering sense of “battery-powered” into a purely enumerative setting.

The classical baseline is the hook-length formula for straight shapes:

$$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$7

For rectangles $$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$8, this becomes

$$0 \le P^{\text{ch}}_{b,t} \le P^{\max}_{s}\,u_{s,b,t}, \quad u_{s,b,t} \in \{0,1\} \text{ only during dwell windows at station } s,$$9

The thesis then introduces generalized hypergeometric functions,

$\sum_b u_{s,b,t} \le N^{\text{port}}_s.$0

where $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$1 is the Pochhammer symbol,

$\sum_b u_{s,b,t} \le N^{\text{port}}_s.$2

These functions provide the principal language in which battery-shape enumeration is expressed.

5. Hypergeometric enumeration, higher batteries, and chess motivation

The central exact formula in the thesis concerns attachment above column $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$3 (Shoan, 2022). For the battery shape $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$4, Theorem 4.1 states that the number of SYT equals

$\sum_b u_{s,b,t} \le N^{\text{port}}_s.$5

Because $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$6 is a nonpositive integer, the $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$7 series terminates. The derivation uses a pivot-cell method. The new cell immediately above the top of column 2 is marked as the pivot, one sums over all possible pivot entries, smaller-than-pivot entries form a constrained “bullet” subtableau, larger-than-pivot entries are transformed by a rotate-complement operation, and the resulting ratio of hook-length products simplifies to rising factorials that match the hypergeometric term.

The thesis also provides finite-sum reduction machinery. Theorem 3.7 gives the integer-parameter Gauss-type identity

$\sum_b u_{s,b,t} \le N^{\text{port}}_s.$8

for integers $\sum_b u_{s,b,t} \le N^{\text{port}}_s.$9 with $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$0 and $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$1. Theorem 3.9 gives a contiguous relation for $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$2 with integer parameters. Iterating that recursion reduces the original $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$3 to a $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$4, and for small $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$5 yields explicit rational forms. One explicit closed form stated in the thesis is for $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$6:

$$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$7

Specializations in fixed $$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$8 are also explicit. For example,

$$\min \sum_t \pi^{\text{TOU}}_t \sum_b P^{\text{ch}}_{b,t}\,\Delta t,$$9

and

$$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$0

For higher batteries, the bullet subtableaux may occupy multiple columns beneath the pivot, and the enumeration becomes multi-parameter. The thesis introduces a multiple hypergeometric function to package these sums. Theorems 5.5, 5.6, 5.9, and 5.11 express the counts for $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$1 respectively as

• $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$2: $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$3,
• $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$4: $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$5,
• $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$6: $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$7,
• $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$8: $$\min \sum_t \pi^{\text{LMP}}_t\,P^{\text{grid}}_t\,\Delta t.$$9.

The chess motivation is structurally important rather than anecdotal. Enumerative chess problems often ask for the number of feasible move orders when the multiset of moves is fixed. A chess battery induces precedence constraints, and these can be encoded as a poset. For Ferrers posets, linear extensions are in bijection with SYT, so the number of feasible move orderings in the chess battery problem equals the number of SYT of the associated battery shape. Buchanan’s problem maps to $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$0, and the thesis evaluates the corresponding quintuple multiple hypergeometric series to obtain an exact integer, thereby resolving the enumeration.

A further misconception is that battery shapes are merely ad hoc examples. The thesis instead positions them as a systematic family of truncated skew shapes for which specialized hypergeometric representations are available, complementing rather than replacing general tools such as the hook-length formula for straight shapes and Naruse-type approaches for skew shapes.

6. Battery-aware computation on small devices: an adjacent systems tradition

A third, adjacent strand of literature concerns battery-powered computation itself rather than tableaux. "Tiny but Mighty: A Software-Hardware Co-Design Approach for Efficient Multimodal Inference on Battery-Powered Small Devices" presents NANOMIND, a hardware–software co-design inference framework for Large Multimodal Models that breaks large models into modular bricks and maps each to its ideal accelerator on a unified-memory SoC (Li et al., 25 Sep 2025). The paper is relevant here because it makes battery-awareness explicit at the scheduler, memory, and kernel levels, and therefore provides a systems analogue to the resource-coupling logic seen in the bus–grid tableau.

The decomposition is by modality and compute pattern: vision encoder and projector, audio encoder, multimodal embedding, and LLM decoder. Vision encoder plus projector are mapped to the NPU because they use high-throughput tensor operations and static shapes; LLM decode is mapped to the GPU because it is streaming, has varying sequence lengths, and is dominated by GEMM and attention; Whisper.cpp and Piper TTS run on CPU, with optional ASR offload to NPU if supported. The prototype uses a Rockchip RK3566 SoC with quad Cortex-A55 up to 1.6 GHz, Mali G52-2EE GPU, and integrated NPU, together with four LPDDR4x modules in parallel and shared UMA for CPU/GPU/NPU.

The scheduling layer is driven by a lightweight CPU scheduler that orchestrates brick placement, applies battery-aware execution modes, and coordinates producer–consumer flow through the Token-Aware Buffer Manager. TABM manages a ring buffer in unified memory with state transitions

$$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$1

The NPU writes embeddings directly into the ring buffer, the GPU binds the same memory for LLM input with zero-copy, and pool reuse reduces redundant allocations and synchronizations. At low battery, the system enters event-driven “On-Demand Cascade Inference” following “load → execute → release,” which minimizes peak memory and energy.

Battery awareness is formalized through thresholds $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$2 and $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$3 on battery level $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$4. If $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$5, the system maximizes parallelism. If $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$6, proportional throttling uses

$$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$7

to control camera FPS and memory read/write rates. If $$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$8, cascade mode is used. The energy definition is

$$P_{ij} = \sum_{k \in \mathcal{D}(j)} P_{jk} + p_j + r_{ij}\,\ell_{ij}, \quad Q_{ij} = \sum_{k \in \mathcal{D}(j)} Q_{jk} + q_j + x_{ij}\,\ell_{ij},$$9

over bricks $$V_j^2 = V_i^2 - 2\big(r_{ij}P_{ij} + x_{ij}Q_{ij}\big) + \big(r_{ij}^2 + x_{ij}^2\big)\,\ell_{ij}, \quad \ell_{ij} = \frac{P_{ij}^2 + Q_{ij}^2}{V_j^2}.$$0, and a possible scheduling objective is to minimize energy subject to a latency constraint $$V_j^2 = V_i^2 - 2\big(r_{ij}P_{ij} + x_{ij}Q_{ij}\big) + \big(r_{ij}^2 + x_{ij}^2\big)\,\ell_{ij}, \quad \ell_{ij} = \frac{P_{ij}^2 + Q_{ij}^2}{V_j^2}.$$1.

The kernel layer uses fused dequant–GEMM for W4A16 on GPU, a linear attention kernel for streaming decode, and RKNN paths for SigLip/CLIP/Whisper with FP16/INT8 on NPU. Standard affine quantization is written as

$$V_j^2 = V_i^2 - 2\big(r_{ij}P_{ij} + x_{ij}Q_{ij}\big) + \big(r_{ij}^2 + x_{ij}^2\big)\,\ell_{ij}, \quad \ell_{ij} = \frac{P_{ij}^2 + Q_{ij}^2}{V_j^2}.$$2

The quantitative results reported are a 42.3\% reduction in energy consumption, an 11.2\% reduction in GPU memory usage, throughput comparable to Jetson Nano running NanoVLM at approximately 35.7 tokens/s for Qwen2-VL-2B 4-bit, and a 36.2\% reduction in end-to-end latency compared to Orange Pi 5 Ultra running rkllm. With a standard 2000 mAh commercially available power bank, low-power cascade mode sustains up to approximately 20.8 hours at an average approximately 0.375 W draw, while LLaVA-OneVision with a camera runs for nearly half a day depending on workload duty cycle and throttling.

This literature does not redefine battery-powered tableaux. A plausible implication is instead methodological: battery-aware visualization, combinatorial enumeration, and on-device multimodal inference all expose situations in which structure becomes operationally useful only after decomposition into explicit components—stations and nodes, columns and pivot cells, or modality-specific bricks—and after the introduction of coordination rules that respect resource or precedence constraints.