Integrating Out, Twice:The Open-System Case That Neural-Network Ensemble Theory Is Missing

Jin Lei

arxiv: 2606.09950 · v1 · pith:MDNBLAUZnew · submitted 2026-06-08 · 💻 cs.LG · nucl-th· physics.comp-ph· physics.data-an

Integrating Out, Twice:The Open-System Case That Neural-Network Ensemble Theory Is Missing

Jin Lei This is my paper

Pith reviewed 2026-06-27 17:11 UTC · model grok-4.3

classification 💻 cs.LG nucl-thphysics.comp-phphysics.data-an

keywords neural network ensemblesopen systemsSchur complementnon-Hermitian operatorsensemble theoryattention mechanismsexpert routingGaussian marginalization

0 comments

The pith

Neural-network ensemble averaging produces only closed-system covariances and misses the open case that conserves flux into a continuous spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Averaging a network over random parameters equals marginalizing a Gaussian sector via the Schur complement. When the eliminated sector is closed, the result is a covariance and its inverse—the full output of current ensemble methods. The open case, drawn from scattering theory, projects onto channels while the rest carries probability irreversibly into a continuum, yielding a non-Hermitian generator that tracks exactly what is lost. Tests on a truncated attention map, token-level transfer operator, and sparse expert router find this distinctive open content absent. The structural reason is that mainstream neural objects supply only finite or dissipative sectors, not the required continuous-spectrum wave dynamics.

Core claim

The closed case of neural ensemble averaging is the Schur complement of a Gaussian block that returns a covariance and its inverse; this maps the neural tangent kernel to the Fisher sensitivity kernel, the infinite-width limit to the Gaussian-process emulator, and the lazy-to-feature transition to the validity boundary of a reduced-basis emulator. The open case requires an eliminated sector with continuous spectrum and wave-like dynamics to produce a non-Hermitian effective generator that itemizes conserved flux, as in the nuclear optical model. The three tested partitions lack this sector, so the open ledger is either absent, an artifact of the partition, or pinned near a floor by the train

What carries the argument

The Schur complement of the eliminated block, which returns covariance and inverse when the sector is closed and a non-Hermitian generator that conserves lost probability when the sector is open with continuous spectrum.

If this is right

The neural tangent kernel is the Fisher sensitivity kernel under the closed-case dictionary.
The infinite-width Gaussian limit is the Gaussian-process emulator.
The lazy-to-feature transition marks the validity boundary of a reduced-basis emulator.
The conserved flux ledger appears only where openness is genuinely present with a continuous-spectrum sector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Architectures whose internal partitions naturally eliminate a continuum sector could make the open ledger usable for uncertainty that current ensembles treat as epistemic.
Training objectives that penalize flux loss may systematically suppress the open-sector signature even when the partition geometry allows it.
Replacing relaxational layers with wave-propagating ones in selected blocks would provide a direct test of whether the missing dynamics can be engineered inside existing networks.

Load-bearing premise

That the three tested neural objects constitute representative instances of genuine openness with an eliminated continuous-spectrum sector.

What would settle it

Observation of a neural-network partition whose eliminated sector exhibits continuous spectrum and produces wave-like rather than relaxational dynamics, yielding a non-Hermitian generator whose flux ledger matches the open-system predictions.

read the original abstract

Averaging a neural network over its random parameters and marginalizing a Gaussian sector are the same operation, the Schur complement of the eliminated block, and when that block is closed it returns a covariance and its inverse. That is all a network ensemble produces, the closed case. The open case is missing, and nuclear reaction theory has it worked out. Projecting a scattering problem onto a chosen set of channels, with the rest carrying probability irreversibly to a continuum, leaves a non-Hermitian effective generator that conserves and itemizes exactly what it loses: the nuclear optical model and its generalized optical theorem. I set the two cases side by side using only the moments of a distribution, the algebra of Gaussians, and block inversion, no field theory, and give the closed-case dictionary in full: the neural tangent kernel is the Fisher sensitivity kernel, the infinite-width Gaussian limit is the Gaussian-process emulator, and the lazy-to-feature transition is the validity boundary of a reduced-basis emulator. I then test the open export on a truncated attention map, a token-level transfer operator, and a sparse expert router, and report a mostly negative result. The conserved flux ledger ports wherever openness is genuinely present, but its distinctive content is absent, an artifact of the chosen partition, or pinned near a floor by the training objective, and the operationally useful uncertainty turns out to be epistemic, living in the closed half of the correspondence, not the open one. The negative has a structural reason this note makes precise: the open case needs an eliminated sector with a continuous spectrum and wave-like, not relaxational, dynamics, which mainstream learning's finite or dissipative objects do not supply. This is a note, not a result; its main finding is that negative one, and its value is the map that locates it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note correctly shows that standard NN ensemble objects only deliver the closed Gaussian case via Schur complement and cannot produce the non-Hermitian open-system generator with conserved flux from nuclear theory, but the negative rests on three partitions that may simply lack the required continuous-spectrum sector.

read the letter

The punchline is that neural-network ensembles, as currently built, only ever marginalize closed sectors and therefore cannot export the open-system optical model with its itemized loss of probability to a continuum. The paper walks through the closed case with plain block inversion and Gaussian moments, recovering the usual NTK-Fisher and infinite-width GP equivalences plus the lazy-to-feature boundary as a reduced-basis limit. That part is clean and uses no extra machinery.

It does the mapping honestly and without overclaim. The three tests (truncated attention, token transfer operator, sparse router) return mostly negative on the flux ledger and related signatures, which aligns with the claim that those objects eliminate only finite or dissipative pieces rather than a wave-like continuum.

The soft spot is exactly the one the stress-test flags: if the chosen partitions never project onto a continuous spectrum carrying irreversible flux, the absence of open signatures is expected from the algebra and does not yet prove that no NN construction could ever supply one. The paper treats this as structural and frames the work as a note rather than a positive result, which is fair, but it leaves the representativeness of the examples unexamined.

This is for readers already following physics-to-ML analogies who want a precise boundary condition on when those analogies stop. It will not change how people train networks or derive kernels. The algebra is solid and the negative is stated plainly, but the scope is narrow and the central claim is partly definitional on the partitions tested. I would not send it to referees; it reads as an internal clarification rather than something that needs external validation.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that neural-network ensemble theory realizes only the closed case of marginalization over random parameters, which via the Schur complement on a Gaussian sector yields a covariance and its inverse; the open case from nuclear reaction theory—producing a non-Hermitian effective generator that conserves and itemizes flux to a continuous-spectrum eliminated sector—is absent. It supplies a closed-case dictionary (NTK as Fisher sensitivity kernel, infinite-width limit as GP emulator, lazy-to-feature transition as reduced-basis validity boundary) using only moments and block inversion, then tests three NN objects (truncated attention map, token-level transfer operator, sparse expert router) and reports mostly negative results for open-case signatures such as the conserved flux ledger. The structural negative is attributed to the lack of an eliminated sector with continuous spectrum and wave-like (non-relaxational) dynamics in mainstream finite or dissipative NN objects.

Significance. If the structural diagnosis is correct, the work supplies a precise diagnostic map locating why open-system features have not appeared in ensemble theory and unifies several existing NN equivalences in a parameter-free manner via moments and block inversion. The closed-case dictionary is a clear contribution; the negative result on the three objects, while preliminary, identifies a concrete obstruction that future constructions would need to overcome.

major comments (2)

[Testing section (three NN objects)] Testing section on the three NN objects: the partitions applied to the truncated attention map, token-level transfer operator, and sparse expert router must be shown explicitly to eliminate a sector whose spectrum is continuous and whose dynamics are wave-like rather than relaxational; absent such verification, the reported absence of conserved-flux signatures follows by construction from the Schur-complement algebra and does not establish that no NN ensemble can realize the required open sector.
[Abstract and conclusion] Abstract and concluding paragraph: the claim that 'mainstream learning's finite or dissipative objects do not supply' the eliminated continuous-spectrum sector is load-bearing for the central negative; the three chosen examples are treated as representative, yet the manuscript provides no general argument that every possible partition of an NN object must eliminate only finite or dissipative sectors.

minor comments (1)

The manuscript would benefit from an explicit side-by-side equation block comparing the closed-case Schur complement (covariance) with the open-case non-Hermitian generator and its optical theorem, to make the dictionary immediately usable.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify points where additional clarification would strengthen the manuscript. We respond to each major comment below.

read point-by-point responses

Referee: Testing section (three NN objects): the partitions applied to the truncated attention map, token-level transfer operator, and sparse expert router must be shown explicitly to eliminate a sector whose spectrum is continuous and whose dynamics are wave-like rather than relaxational; absent such verification, the reported absence of conserved-flux signatures follows by construction from the Schur-complement algebra and does not establish that no NN ensemble can realize the required open sector.

Authors: We agree that the partitions should be characterized explicitly to confirm the nature of the eliminated sectors. In the revised manuscript we will add explicit descriptions of the partitions for the truncated attention map, token-level transfer operator, and sparse expert router, together with arguments based on their finite dimensionality or dissipative character showing why the eliminated sectors lack continuous spectra and wave-like dynamics. This will make clear that the reported absence of open-case signatures follows from the structure of the tested objects rather than from the algebra alone. revision: yes
Referee: Abstract and conclusion: the claim that 'mainstream learning's finite or dissipative objects do not supply' the eliminated continuous-spectrum sector is load-bearing for the central negative; the three chosen examples are treated as representative, yet the manuscript provides no general argument that every possible partition of an NN object must eliminate only finite or dissipative sectors.

Authors: The three examples are presented as representative of mainstream finite or dissipative NN objects. We acknowledge that the manuscript supplies no general proof that every conceivable partition of every NN object must eliminate only finite or dissipative sectors; such a proof would require a classification of all possible architectures and partitions and lies outside the scope of this note. We will revise the abstract and conclusion to qualify the claim accordingly while preserving the structural diagnosis and the negative result for the tested cases. revision: partial

standing simulated objections not resolved

A general argument establishing that no possible partition of any neural-network object can eliminate a continuous-spectrum sector with wave-like dynamics.

Circularity Check

0 steps flagged

Derivation self-contained; no load-bearing reductions to inputs or self-citations

full rationale

Closed-case results follow directly from moments, Gaussian algebra, and block inversion (Schur complement) with no fitted parameters or self-citations invoked as justification. Open-case negative result is anchored in external nuclear-reaction-theory results whose authors do not overlap with the present paper, so the central claim does not reduce to any quantity defined inside the paper's own objects or fits. The three tested partitions are presented as representative instances rather than tautological choices that force the negative outcome by construction. No step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard linear-algebra facts (Schur complement, block inversion) and the algebraic properties of Gaussians; no free parameters are introduced, no new entities are postulated, and the open-system construction is imported from existing nuclear-reaction literature rather than invented here.

axioms (2)

standard math Block inversion and Schur complement yield the marginal covariance and its inverse for a partitioned Gaussian
Invoked to equate parameter averaging with marginalization; stated in the abstract as the common operation for both closed cases.
domain assumption An eliminated sector with continuous spectrum produces a non-Hermitian effective generator obeying a generalized optical theorem
Taken from nuclear reaction theory; used to define the missing open case that neural objects are claimed to lack.

pith-pipeline@v0.9.1-grok · 5869 in / 1624 out tokens · 19850 ms · 2026-06-27T17:11:51.741838+00:00 · methodology

discussion (0)

Reference graph

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