Defensive publication · energy-native compute

A Thermodynamic Bound on Reproducibility in Energy-Native Computation, and the Crossings-Tax Spectrum

Collapse the crossings between source and compute state and efficiency rises but noise floods in. Reproducibility is bought back not with a conversion crossing but with a dissipation-built basin, at a price with a lower bound.

David Jean Charlot, PhD  ·  Michael Hiner

The Charlot Lab & The Hiner Lab, Institute for Physical AI @ JBI

Correspondence: contact@physicalai-bmi.org · physicalai-bmi.org · Interactive companion: physicalai-bmi.org/research/hiner-lab#topic-energy

1. What does a crossing cost?

Let a computing system map an energy source to a computational state variable through a chain of physical domains. Write $K$ for the number of domain crossings that survive between the two. Each crossing $c$ carries a conversion inefficiency $\lambda_c\in(0,1)$ and freezes an interface, so the end-to-end efficiency is multiplicative,

and is monotonically improved by deleting crossings. The spectrum orders mechanisms by crossings remaining, high to low: S0 separated (source · supply · regulated rail · digital compute); S1 harvested-separated; S2 conditioning-collapsed (bursty supply drives asynchronous logic, no clean rail); S3 storage-collapsed (the store is the state); S4 co-integrated (source and compute share a substrate, distinct mechanisms); S5 identity (the source variable is the compute-state variable, $K=0$). S5 is the asymptote and the object of interest.

2. How do carriers and rungs combine?

Crossed with the physical carrier, the spectrum has forbidden and empty cells that locate the unbuilt work. The S5-identity mechanism and its status per carrier:

The empty-but-implementable cells (ionic and thermal) are the flagged prior art: in each, two literatures name one mechanism twice (harvesting versus computing), and this review did not locate a published device that makes one coordinate serve both. Searched: Web of Science, arXiv (cond-mat.mes-hall, physics.app-ph), IEEE Xplore and Google Scholar, terms {iontronic memristor, Soret computing, thermodiffusive logic, ionic thermoelectric memristor, thermal-gradient in-memory compute}, through July 2026; an unfound device and a nonexistent device are different things and only one is checkable. Nuclear is the only carrier forbidden to identity, which is precisely why a betavoltaic constant-power floor is the deterministic reference: pure power, no compute coupling, no noise to inject.

3. Why is the diagonal a design law?

The spectrum has a diagonal. Every crossing deleted to remove the conversion tax is a crossing at which the state could have been re-clocked, re-quantized, or restored to enforce reproducibility. Efficiency and bit-invariance are therefore bought with the same coin, the retained interface: S0 is maximally reproducible and maximally wasteful; S5 is maximally efficient and inherits the raw physics (including the noise) of its carrier. The naive repair, converting and rounding at an added boundary, reintroduces exactly the crossing that was deleted. The remaining sections disclose an invariance that costs no crossing.

4. What makes a basin label invariant?

Let the substrate be dissipative, descending a free-energy landscape $V$ with discrete attractor basins $\{B_\ell\}$. Define the committed observable as the basin label rather than the microstate,

Three mechanisms make the label invariant at no domain crossing, because each is intrinsic to the substrate. (i) Basin coarse-graining: a perturbation below the barrier is rejected as the dynamics roll back down; the settling is the error correction, and the landscape does the rounding that a converter would otherwise do. (ii) Physical redundancy: many noisy elements summed on a shared electrode average by the law of large numbers, and the summation is a Kirchhoff junction (a wire), so independent noise never moves the committed label; only a common-mode excursion can. (iii) Hysteresis as free quantization: a hysteretic device is a physical Schmitt trigger, rejecting noise inside the loop width without an analog-to-digital conversion. Attestation then pays exactly one crossing, and it is asymptotically free: signing the label costs $O(\log|\{B_\ell\}|)$ bits of decision entropy, not the joules-heavy analog trajectory.

5. Is there a Landauer bound for reproducibility?

This bound is the same one derived in the companion report, where it is interactive: drag the failure probability and the supply noise and watch the required barrier move against the Landauer erasure floor. It is not reproduced as a second widget here on purpose.

Escape from a basin under supply noise of intensity $D_\sigma$ (the noise variance density, an effective temperature) follows Kramers' rate[4],

Holding the label over a commit window $\tau$ with failure probability at most $\varepsilon$ requires $1-e^{-r\tau}\le\varepsilon$, hence a minimum barrier, and (since each rejected perturbation dissipates at least the barrier it climbs back down) a minimum energy per invariant commit:

This is a thermodynamic lower bound on the energy required to make an output invariant against supply noise of a given amplitude, and it is distinct from Landauer's $k_BT\ln 2$ per erased bit. At one worked operating point (a thermally limited substrate, $D_\sigma=k_BT$ with $T=300$ K, and a commit failure probability $\varepsilon=10^{-9}$) $\ln(1/\varepsilon)=20.7$, so $\Delta E_b^{\min}\approx 20.7\,k_BT=8.6\times10^{-20}$ J per commit, against $k_BT\ln 2=2.9\times10^{-21}$ J for erasure at the same temperature: a factor of about 30. Inputs: $k_B=1.381\times10^{-23}$ J K$^{-1}$; $T=300$ K (assumption: room-temperature substrate, replaceable); $\varepsilon=10^{-9}$ (assumption: one broken commit per $10^9$, replaceable); prefactor set to unity, which is the loosest case.[1]: Landauer prices erasure, this prices reproducibility. Spend less than $\Delta E_b^{\min}$ and the basins are too shallow to reject the noise; the invariance is not available to purchase at any lower price. Whether this bound is tight, or can be beaten by a coarse-graining cleverer than $k_BT\ln(\text{states})$, is stated as the open theoretical question, not asserted as settled.

6. Why does reversibility invert here?

The bound inverts a premise the low-power field treats as settled. Reversible and adiabatic computing minimize energy per operation by not dissipating[2]; but a non-dissipative substrate has no attractor basins, and so preserves supply noise instead of coarse-graining it. Dissipation is what builds the basins that reject the noise for free. The substrate that buys boundary-invariance without a crossing is therefore minimally dissipative but deliberately not reversible, sitting above the Landauer floor by exactly the barrier height it needs,

so that reproducibility and reversibility pull against each other at S5. The attractor-network lineage of this construction is classical[3]; what is disclosed is its use as a crossing-free invariance primitive priced by the bound above.

7. What does it cost in representational density?

Free invariance is paid in bits. Deep, well-separated basins occupy the state range at a coarser spacing $w(\Delta E_b)$, and the two are tied only once the carrier's achievable curvature is capped: for a periodic landscape $V(x)=-\tfrac{\Delta E_b}{2}\cos(2\pi x/w)$ with $|V''|\le\kappa_{\max}$, the period obeys $w\ge 2\pi\sqrt{\Delta E_b/2\kappa_{\max}}$, so $\text{bits/device}=\log_2(R/w)$ falls as $-\tfrac12\log_2\Delta E_b$. Assumption a reader can replace: $\kappa_{\max}$ is a per-carrier material property and must be named row by row in the §2 matrix, with no cap on curvature the barrier can be raised at fixed spacing and no bits are lost,

This is the true Pareto surface of energy-native computation: representational density traded against noise margin, a landscape-geometry tradeoff in which barrier height replaces the guardband of a regulated design.

8. Claim

The disclosed object is the following, taken together and priced thermodynamically rather than in raw energy: (a) the crossings-tax ordering S0–S5 and the carrier matrix that locates the empty-but-implementable identity cells; (b) the basin-label invariance construct, in which the committed and attested observable is the attractor-basin label of a dissipative substrate and the three intrinsic mechanisms of §4 supply reproducibility at no domain crossing; and (c) the bound $E_{\text{inv}}\ge D_\sigma\ln\!\big(\omega_0\omega_b\tau/2\pi\gamma\varepsilon\big)$ of §5, which collapses to the quoted form $D_\sigma\ln(1/\varepsilon)$ only when the attempt-rate prefactor satisfies $\omega_0\omega_b\tau/2\pi\gamma\ge 1$; with its two corollaries, the reversibility inversion and the density–margin Pareto. All constituent relations (the multiplicative efficiency, Kramers' rate, the Landauer floor, the Kirchhoff sum, the Nernst/Soret and photonic identity mechanisms) are public physics; no term is enclosable.

Where does this stand?

The binding constraint differs by carrier, and §7 names it for each. The thermal carrier is the clearest case and it is materials: the in-domain thermopower must be large enough that the gradient both drives the state change and separates basins by more than the minimum barrier of §5. That is a coefficient a candidate material either reaches or does not, which makes the assessment a lookup rather than an argument: name the coefficient the material reaches, and recompute the bound.

What would settle it. For each carrier in the matrix, the measured coefficient against the threshold the bound implies, on a named material. Several rows of that matrix were not located by this review and are marked nascent rather than ruled out, which is the position this review takes: a row with no measurement is a row not yet filled, and the distance between those two readings is exactly what a published coefficient closes.

References

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