Physical computation requires six boundaries

Scope

“Energy per operation” is meaningful only after both operation and energy boundary are fixed. A logical erasure bound, terminal energy of one device, energy of a clocked circuit, wall-plug energy of a workload, facility energy, and fabrication-to-retirement burden are related measurements, but they are not substitutes. Crossing those boundaries without changing the claim is the main category error this chapter prevents (C-1100, C-1151).

The working contract has six layers:

The evidence base is the information thermodynamics and physical computation audit, normalized as C-1100 through C-1151. The maintained quantitative definitions are in the boundary-qualified mathematical contract, and the twelve hostile comparisons are in Fixture F-010. The result composes current candidates; it adds no principle or candidate.

flowchart TB
    task["Sealed useful-task contract<br/>quality · latency · risk · throughput · retention horizon"] --> fundamental
    subgraph boundaries["Six non-substitutable accounting boundaries"]
        direction TB
        fundamental["1 · Fundamental information operation<br/>initial distribution · Hamiltonian · temperature<br/>retained side information · final error · protocol time"]
        device["2 · Device transition<br/>terminal work/heat · waveform · parasitics<br/>temperature · duration · transition error · stability"]
        circuit["3 · Circuit and control<br/>clock · interconnect · converters · leakage<br/>ancillae · sensing · correction · reset"]
        workload["4 · Workload and data movement<br/>software · precision · hierarchy · routing<br/>utilization · retries · accepted outcomes"]
        facility["5 · Facility and cooling<br/>wall power · power conversion · cooling<br/>network/storage allocation · interval · site"]
        lifecycle["6 · Embodied lifecycle<br/>fabrication yield · package · deployment<br/>maintenance · replacement · end of life"]
        fundamental -->|"physical implementation"| device
        device -->|"composed and controlled"| circuit
        circuit -->|"executes declared workload"| workload
        workload -->|"served by facility"| facility
        facility -->|"amortized with hardware history"| lifecycle
    end
    finite["Joint time–error–stability ledger<br/>finite-time excess · error consequence<br/>retention · correction · throughput capacity"] --> boundaries
    information["Joint information ledger<br/>measurement · controller · feedback · memory<br/>fluctuation ensemble · valid TUR scope"] --> boundaries
    uncertainty["Versioned evidence<br/>meter/calibration lineage · coverage interval<br/>model support · allocation sensitivity"] --> boundaries
    lifecycle --> compare{"Matched-budget comparison<br/>same useful-task acceptance<br/>held-out regimes · ablations · complete null stack"}
    nulls["Mature nulls<br/>compression · caching · compiler locality · DVFS/gating<br/>ECC/retry · reversible/adiabatic controls · matched accelerators"] --> compare
    compare -->|"Pareto gain survives all boundaries"| retain["Retain support only for existing candidate scope"]
    compare -->|"gain moves boundary or fails support"| retire["Hard retirement<br/>no principle or candidate promotion"]

Editable source: boundary-qualified-physical-computation.mmd.

The useful outcome is the firewall. Every lower-level saving must survive the next boundary without lowering quality, raising unacceptable risk, missing the latency/throughput contract, or exporting work.

Biological observation

Biochemical sensing and adaptation make the boundary problem concrete. In the audited models, copy number, integration time, receptor statistics, energy supply, precision, and response speed remain distinct resources (C-1135, C-1136). A result for one biochemical network does not become a universal information price, and it does not set accelerator energy. Its useful contribution is a measurement discipline: name the physical states, dynamics, observation interval, error variable, and supplied work.

Feedback experiments add a second lesson. A controlled subsystem can extract work or cool while the sensor, memory, controller, actuator, or coupled demon dissipates energy. The joint boundary restores the balance (C-1130, C-1131, C-1132, C-1133). Continuous information flow can be assigned to parts only when the joint transition structure supports that decomposition (C-1134).

Three observations transfer:

1. sensing, state retention, response, and reset are physical parts of the same loop;
2. precision, speed, stability, and energy form a frontier, not one scalar; and
3. a subsystem benefit is provisional until the coupled system closes.

The transfer stops there. The audited biological models do not establish a digital-training lower bound. Predictive-information and stochastic-learning connections remain plausible rather than established for deployed AI (C-1137, C-1138).

Proposed AI translation

Begin with an accepted useful outcome

For requested outcome $j$, define a preregistered acceptance indicator

where $A_j\in\{0,1\}$ is acceptance [dimensionless], $Q_j$ and $q_j$ are measured and required task-quality vectors in the same task-native units, $L_j$ and $L_j^{\max}$ are measured and maximum latency [s], $\rho_j$ and $\rho_j^{\max}$ are measured and maximum risk vectors [failure/request], and $\Theta$ and $\Theta_j$ are delivered and required throughput [accepted outcome/s]. Rejection, abstention, timeout, retry, silent corruption, and blocked side effects remain in the request and resource ledgers. $\mathbf 1[\cdot]$ is the dimensionless indicator; $\succeq$ and $\preceq$ mean that every registered vector component passes in its declared direction.

Let

where $N_{\mathrm{req}}$ is requested outcomes [request] and $N_{\mathrm{acc}}$ is accepted outcomes [accepted outcome]. Every energy intensity in this chapter uses that denominator. If $N_{\mathrm{acc}}=0$, the intensity is undefined and the arm fails; it is not zero.

Carry six typed records

Each run produces six linked records rather than one “energy” field:

Every record carries immutable hardware/software/calibration versions, timestamps, uncertainty, validity support, missingness, and the parent/child identity needed to trace replacement or recalibration. This is the physical- energy specialization of the project's versioned observation contract.

Make boundary escalation explicit

A result can support only its measured level:

1. theorem or ideal protocol result;
2. isolated device result;
3. closed circuit result;
4. accepted workload result;
5. allocated facility result; or
6. amortized lifecycle result.

Promotion from one level to the next requires a new measurement, not a larger claim. This keeps the chapter aligned with reliability under mission profiles, where device population and history are part of the evidence, and with operator-qualified sensing, where physical observations remain tied to their operator, calibration, and support.

Efficiency mechanism

One vector, not one number

For a sealed run, report

where the components are respectively theorem-qualified fundamental lower bound, device-terminal energy, complete circuit/control energy, workload IT energy, allocated facility energy, and allocated embodied energy [J]. The vector is not a sum: in many measurements $E^{\mathrm{dev}}\subset E^{\mathrm{circ}}\subset E^{\mathrm{IT}}\subset E^{\mathrm{fac}}$. For boundary $b$, useful intensity is

where $b\in\{\mathrm{dev,circ,IT,fac,emb,life}\}$ and $N_{\mathrm{acc}}$ is accepted outcomes [accepted outcome]. Distance between $E^{\mathrm{fund}}$ and any measured component is descriptive only after their operations and denominators match; it is not a system ranking (C-1151).

Boundary 1 — generalized erasure

For physical microstate $z\in\mathcal Z$, probability $p(z)$ [dimensionless], Hamiltonian $\mathcal H(z)$ [J], bath temperature $T$ [K], and Boltzmann constant $k_B=1.380649\times10^{-23}$ J/K, define nonequilibrium free energy

For an isothermal transformation under the selected theorem's assumptions, expected work on the system obeys

where $p_0,p_1$ are initial and final microstate distributions, $\mathcal H_0,\mathcal H_1$ are initial and final Hamiltonians [J], and $W_{\mathrm{on}}$ is work on the system [J]. Logical entropy alone is insufficient for nondegenerate or nonequilibrium memories (C-1103). Side information, correlation, and a finite reservoir change the accounting (C-1105, C-1107).

The familiar binary special case is

where $\epsilon\in[0,1/2]$ is symmetric reset-error probability [error/transition] and $h(\epsilon)$ is binary entropy [nat]. At $\epsilon=0$, a uniformly distributed degenerate bit yields $k_BT\ln2$ (C-1101). A biased state instead follows its entropy (C-1102). The system must still charge the consequence of allowed errors (C-1104).

Finite duration adds a separate coordinate. For protocol $\pi$ of duration $\tau_\pi$ [s], define excess work

Compare it only for matched initial/final state, error, bath, and allowed controls. Finite-time excess, finite error, and stability are distinct (C-1118, C-1119, C-1122). Finite-time quantum erasure has additional model-specific cost and fluctuation structure; classical quasistatic expressions cannot simply be relabeled (C-1120).

Information, fluctuations, and theorem scope

Individual trajectories below a mean bound are compatible with fluctuation relations (C-1106, C-1127, C-1128, C-1129). The implementation therefore stores the full work distribution, sample-selection rule, reverse protocol, rare-event coverage, and estimator uncertainty rather than only a mean or minimum.

For feedback, close the physical loop:

where the six terms are energy crossing the plant, sensor, record memory, controller, actuator, and reset boundaries [J]. Extracted plant work is signed; it cannot cancel an unmeasured controller.

For a stationary continuous-time Markov jump process and registered integrated current $J_t$ over time $t$ [s], the original steady-state thermodynamic uncertainty relation has the scoped form

where $\Sigma_t$ is expected entropy production in units of $k_B$ [dimensionless] and $J_t$ is measured in its registered integrated-current unit. The process, current, stationarity, Markov property, time-reversal convention, observation support, and entropy-production estimator must be established first (C-1139). Other finite-time, initial-state, non-Markovian, deterministic, or quantum settings do not inherit this formula unchanged (C-1140, C-1141).

Boundaries 2 and 3 — device, circuit, and real crossover

For device transition $k$ over $[t_k^0,t_k^1]$, terminal energy is

where $C_k$ is supplied channels [channel], $V_{k,c}$ is calibrated voltage [V], $i_{k,c}$ is signed current [A], and $t$ is time [s]. Heat requires a calibrated thermal balance; terminal electrical energy is not relabeled as heat.

Logical reversibility can avoid compulsory erasure at intermediate steps (C-1112), but history, ancillae, output copy, communication, uncomputation, retention, and final reset remain physical (C-1113, C-1114). Its strongest comparison is therefore against reversible pebbling, checkpoint/recompute, compiler elimination, and an optimized irreversible circuit at equal service.

For an idealized adiabatic RC path, the crossover model is

where $R$ is resistance [ohm], $C$ is capacitance [F], $\tau$ is transition duration [s], $V$ is voltage [V], $\gamma$ is a waveform coefficient [dimensionless], $P_{\mathrm{leak}}$ is leakage power [W], and the remaining terms are clock, control, I/O, and reset energy [J]. Resistive loss may decrease approximately with $RC/\tau$ in its slow-ramp regime (C-1115); leakage and power-clock costs can create a finite optimum (C-1116). A fabricated energy-recovery processor establishes feasibility in its measured range, not zero energy or universal superiority (C-1117).

The crossover is real only when $E^{\mathrm{adiabatic}}<E^{\mathrm{ordinary}}$ at matched quality, error, throughput, capacity, layout/process, temperature, and cyclic closure. Here, $E^{\mathrm{ordinary}}$ is complete energy of the conventional comparison [J]. The ordinary $CV^2$ switching-loss expression is a circuit model, not Landauer erasure (C-1147).

Retention and correction

For memory tier $m$, charge

where $N_m^{\mathrm w}$, $N_m^{\mathrm r}$, and $N_m^{\mathrm{ref}}$ are write, read, and refresh counts [operation]; the corresponding $e$ terms are measured energy [J/operation]; and the remaining terms are correction, scrub, movement, and idle energy [J]. Report retention distribution, raw and post-correction errors, miscorrection, silent loss, endurance, and accepted retrievals.

Retention depends on barrier, temperature, time, and loss probability in the activated bistable null (C-1121). Analog storage adds thermal/device noise, finite usable precision, drift, calibration, and conversion (C-1123, C-1124). Noise-assisted or stochastic advantage is plausible only for selected workloads against matched deterministic and pseudorandom nulls (C-1125).

Boundary 4 — workload and locality

For hierarchy link $\ell\in\mathcal L$, let $B_\ell$ be transferred bytes [byte] and $\widehat e_\ell$ be calibrated energy [J/byte] at the registered process, voltage, precision, distance, rate, and utilization. Movement energy is

Measure register, local memory, cache, on-chip network, off-package memory, host, storage, and network separately. Sparse or modular execution also pays indices, routing, load imbalance, synchronization, conversion, cache misses, and idle capacity. Data movement can dominate arithmetic on measured accelerators (C-1145); a hierarchy-aware model's transfer to another system remains plausible until wall-plug validation (C-1146).

Modularity can save locality while losing accessible correlation or adding reset and communication cost (C-1142). A physical process optimized for one input prior can add mismatch dissipation under drift (C-1143), and circuit topology can change thermodynamic cost for the same logical function (C-1144). These are direct constraints on sparse predictive computation and the working architecture.

Boundaries 5 and 6 — facility and lifecycle

For a synchronized facility interval $r$,

where $E_r^{\mathrm{fac}}$ is total facility energy [J] and $E_r^{\mathrm{IT}}$ is IT-equipment energy [J] under the declared ISO/IEC 30134-2 boundary and measurement category. PUE is neither task energy nor carbon intensity (C-1148). Workload attribution still needs synchronized meters, accepted outcomes, network/storage shares, weather, and sensitivity to cooling-overhead allocation.

For hardware cohort $h$, lifecycle energy is

where the terms are fabrication, packaging, transport, deployment, operation including facility share, maintenance, replacement, and end-of-life primary energy [J]. Fabrication and packaging can change a use-phase ranking (C-1149). Specialized hardware is superior only when saved accepted-service energy outweighs new fabrication, low utilization, support life, maintenance, and replacement across uncertainty (C-1150).

Evidence status

The ledger contains exactly 52 claims:

• 46 established within their stated theorem, experiment, device, circuit, workload, facility, or lifecycle boundary;
• 5 plausible transfers that still require target-system evidence; and
• 1 disputed system-level inference.

The status distribution is not a confidence score for one architecture. It describes separate claims with separate support:

Four platforms experimentally approach or test Landauer-scale erasure: a colloidal memory (C-1108), a feedback trap (C-1109), a nanomagnetic bit (C-1110), and a cryogenic molecular nanomagnet (C-1111). They establish the physical principle in their declared protocols. They do not measure complete computers. The same evidence discipline applies upward:

1. a theorem establishes a bound only for its model;
2. a device experiment establishes its controlled physical transition;
3. a processor or accelerator establishes its measured circuit/workload range;
4. facility metering establishes its interval and allocation; and
5. lifecycle assessment establishes its functional unit and inventory cases.

The unresolved scientific object is not another universal constant. It is the measured crossover between implementations at equal accepted service. Fixture F-010 preserves that question without promoting its evaluation contract into a new P- bundle.

Speculative extensions

Only the five plausible claims license active extension work.

Physical stochasticity for matched workloads

Physical noise could be useful when the task already requires sampling, probabilistic search, or exploration. The comparison must match stationary distribution or target posterior, bias, mixing, tail coverage, latency, task quality, device/circuit energy, and lifecycle cost against high-quality digital pseudorandom sampling (C-1125). A noisy device does not earn credit merely for producing variation.

Predictive retention rather than historical retention

If a continual system stores only state that improves prediction, it may avoid updates whose information is nonpredictive. The surviving question is physical: does the predictive objective reduce writes, movement, correction, and retained capacity after ordinary predictive compression, caching, event-triggered updates, and recomputation are matched (C-1137)? This joins memory and consolidation with Candidate 017 and Candidate 018.

Thermodynamic learning efficiency on actual hardware

Toy stochastic-learning results can generate hypotheses about which updates carry useful information, but they do not bind digital gradient training (C-1138). A valid transfer would jointly measure optimizer, arithmetic, activation/gradient memory, communication, data loading, checkpointing, accepted validation outcomes, and hardware/facility energy under the same learning contract.

Hierarchy-aware conditional execution

A route-energy model could decide whether skipping arithmetic saves more than its indices, movement, arbitration, synchronization, imbalance, and idle capacity cost (C-1146). It must transfer across held-out model, sequence/graph length, sparsity pattern, cache-fit, precision, topology, and utilization regimes. This is the energy test for Candidate 001, not a new routing candidate.

Lifecycle-qualified physical compilation

Specialized hardware may move recurring computation into a lower-energy substrate, but the gain becomes real only after yield, package, converters, calibration, utilization, service life, software support, maintenance, replacement, and displaced-hardware assumptions are propagated (C-1150). Candidate 006 already owns that experiment. The governing quantity is accepted lifetime service, not peak component efficiency.

Failure modes

The broader energy model should consume these typed records, while reliability under mission profiles supplies the device population, temperature, wear, correction, repair, and retirement state. Neither chapter can replace the other's denominator.

Measurable predictions

Fixture F-010 converts the chapter into twelve equal-budget experiments:

All tracks use the same accepted-task contract and preserve failed devices, rejected requests, timeouts, retries, uncorrectable errors, silent corruption, abstention, idle capacity, maintenance, and replacement in their denominators. Confirmation groups withhold physical devices, fabrication cohorts, waveform and duration regimes, target errors, temperatures, retention horizons, workload families, hierarchy patterns, controller versions, sites, seasons, lifecycle cases, and future time.

The common result vector is

where $f_{\mathrm{acc}}$ is accepted fraction [dimensionless], $Q$ is task quality [task-native unit], $L_{0.50}$ and $L_{0.99}$ are median and 99th percentile latency [s], $\rho$ is risk [failure/request], the three $e$ terms are energy intensity [J/accepted outcome], $E^{\mathrm{err}}$ is error-consequence energy [J], $C^{\mathrm{cap}}$ is provisioned capacity [device s], $G$ is greenhouse-gas inventory [kg CO$_2$e], $W$ is water inventory [m$^3$], and $M$ is a material/labor vector in declared native units.

A physical-efficiency claim survives only when its simultaneous uncertainty region is no worse on every hard-gated coordinate and strictly better on at least one preregistered primary coordinate against the strongest compatible null across required sensitivity cases. Passing supports only the existing candidate scope named in F-010. Failure identifies the boundary that produced the apparent saving and retires the wider claim.