Neural Admissibility Program

• The Neural Admissibility Program (NAP) is a framework for evaluating neural mechanisms of language according to whether they maintain the formal invariants of linguistic computation.
• NAP distinguishes different levels of admissibility: weak, operational, and causal, focusing on preserving hierarchical grouping, commutativity, recursive closure, access to substructures, and structured workspace transitions.
• Applications and relevance of NAP include understanding syntactic computations, analyzing neural development, and investigating structured workspace transitions in cognitive processes.

The Neural Admissibility Program (NAP) is a research framework for evaluating neural mechanisms of language according to whether they preserve the formal invariants of linguistic computation. Proposed in “Formal Properties of Language as Constraints on Neural Dynamics” (Murphy, 13 Sep 2026), NAP shifts the criterion for neural implementation from predictive correlation or decoding accuracy to mechanistic admissibility: a neural system must preserve hierarchical grouping, commutativity, recursive closure, access to substructures, and structured workspace transitions, while discarding distinctions that the formal computation treats as irrelevant.

1. Conceptual foundations

NAP treats linguistic theories as specifying formal objects, operations, and invariants. Let a formal theory provide a space of computational objects $S$, operations $M_i$, and invariants $I_j$. A neural implementation supplies an encoding $\mathcal N:S\to X$ into neural state space $X$ and physiological operations $G_i$. The implementation is admissible only if the formal and neural operations approximately commute:

$$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$

and if the relevant invariants remain recoverable and usable within the neural system.

For binary composition, the core condition is

$$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$

Here, $G$ denotes the physiological operation, or coordinated family of operations, that realizes formal composition. The framework does not require a single brain region, frequency band, or neural population to implement Merge. Instead, it requires that the distributed mechanism as a whole preserve the computational properties of Merge.

The paper distinguishes three levels of admissibility:

• Weak admissibility: neural activity is statistically sensitive to a formal property.
• Operational admissibility: the activity occurs at the appropriate time and scale, dissociates from lexical and task confounds, and is exploited downstream.
• Causal admissibility: perturbing the activity selectively disrupts the operation while sparing control processes.

This distinction separates neural sensitivity to syntax from the actual implementation of syntactic computation. A neural signal can predict a linguistic variable without being the state used by downstream composition, and therefore without being operationally admissible.

NAP consequently treats decoding as a necessary but insufficient evaluation method. The central comparison is not which model decodes a linguistic label most accurately, but which model preserves every distinction required by the formal computation while discarding distinctions that the computation identifies as irrelevant.

2. Algebraic structure of syntactic composition

The proposed target is a free commutative non-associative magma generated by a finite set of lexical items and syntactic features $\mathrm{SO}_0$:

$M_i$0

where

$M_i$1

Merge is commutative,

$M_i$2

but non-associative:

$M_i$3

It is also recursively closed: the output of Merge is itself a syntactic object that can participate in further Merge operations.

For three terminals, the structures

$M_i$4

are distinct despite containing the same lexical multiset. A neural implementation must therefore preserve bracketing history. Vector addition is inadmissible for this role because it is associative:

$M_i$5

A bag-of-words representation is commutative but similarly loses grouping. Conversely, an ordered role-filler code may preserve too much information by retaining daughter order, which abstract Merge treats as irrelevant.

The distinction between abstract syntax and surface order is central. Let $M_i$6 be planar ordered binary trees and $M_i$7 non-planar trees. A forgetful map

$M_i$8

removes planar embedding. A language-specific externalization map may then select a surface ordering through a section

$M_i$9

The parser may use temporal order, but the abstract composite passed to subsequent recursive composition must not retain daughter order unless that order is independently required by the grammar or interface.

3. Core neural admissibility invariants

Non-associative hierarchical grouping

A neural mechanism must distinguish alternative bracketings:

$I_j$0

This requirement follows from the role of constituency in syntactic interpretation, including scope, binding, selection, reconstruction, and phrase-level meaning. For example, the structures

$I_j$1

and

$I_j$2

contain the same lexical items but differ in grouping.

An explicit symbolic tree is not required. Grouping may instead be preserved in neural trajectories, population geometry, phase relations, or transient state configurations. However, a representation based only on sentence length, lexical surprisal, total activation, or another scalar summary cannot establish this invariant.

Commutativity

At the level of the abstract composite,

$I_j$3

Commutativity does not imply that the parser is insensitive to input order. It requires that daughter order not determine the object supplied to the next recursive composition.

An ordered role-filler code such as

$I_j$4

may decode bracketing highly accurately while also preserving which daughter occupied which role. Such a code is inadmissible for abstract Merge unless the order-bearing information is removed before the composite is reused.

Recursive closure

Merge must be closed over its outputs:

$I_j$5

Neurally, the composite must be reusable as a constituent:

$I_j$6

Transient synchrony between two daughter populations is insufficient if it does not produce a new state that can serve as input to later composition. A possible implementation uses a workspace code $I_j$7 and a binding-population code $I_j$8:

$I_j$9

The write-back operation must preserve the properties required for later extraction and composition.

Access to substructures

A syntactic object is not merely a completed sentence vector. Its internal constituents must remain accessible for movement, reanalysis, interpretation, scope, agreement, and reconstruction.

Workspaces may be represented as forests:

$\mathcal N:S\to X$0

The paper uses a Hopf-algebraic coproduct to formalize decomposition into accessible subforests and residual quotients:

$\mathcal N:S\to X$1

Here, $\mathcal N:S\to X$2 is an extracted subforest and $\mathcal N:S\to X$3 is the residual object. A Merge transition can be schematized as

$\mathcal N:S\to X$4

where $\mathcal N:S\to X$5 extracts candidate substructures, $\mathcal N:S\to X$6 selects a pair, $\mathcal N:S\to X$7 binds them, and $\mathcal N:S\to X$8 reassembles the workspace.

A wh-dependency, for example, may require extracting which book from an already constructed object, retaining a quotient containing the gap, and re-grafting the extracted constituent elsewhere. A global embedding that supports sentence-level classification but cannot retrieve a constituent-specific representation fails this invariant.

Structured workspace transitions

Parsing is modeled as a trajectory through a structured workspace state space rather than as an undifferentiated increase in neural activation. The transition operator is

$\mathcal N:S\to X$9

Its vertices are workspaces and its edges are admissible Merge transitions. If $X$0 is irreducible, a Perron–Frobenius eigenfunction $X$1 yields a maximal-entropy Doob transform:

$X$2

A Gibbs deformation incorporates transition costs:

$X$3

The resulting neural predictions concern opening possible attachment sites, filtering inadmissible operations, selecting a transition, closing a constituent, and retrieving an accessible substructure. Candidate dynamical implementations include continuous-time neural networks, winnerless competition, and stable heteroclinic channels. These systems become NAP candidates only when their states and transitions are equivariant with the formal workspace graph.

4. Proposed neural architecture

NAP does not identify Merge with a single circuit. It factorizes composition into extraction, selection, binding, reassembly, and sealing.

Content-addressable workspace memory

Content-addressable memory is proposed as a substrate for storing partial and completed constituents, correcting noisy or partial cues, retrieving inactive substructures, and supporting coproduct extraction and product reassembly. Bipartite expander Hopfield networks are discussed as candidate stores because they can provide robust attractors and error correction.

Higher-order or simplicial extensions may be needed for structured, typed objects rather than unstructured patterns. The limitations of this proposal include the equilibrium character of attractor models, uncertainty about cortical implementation and developmental learning, and the fact that attractor cleanup alone does not implement extraction, movement, or recursive Merge.

Graph-structured transient dynamics

Continuous-time neural networks and stable heteroclinic channels are proposed as schedulers over workspaces. Continuous-time neural networks provide graph-to-attractor control and sequential attractors. Stable heteroclinic channels provide reproducible but perturbable sequences of metastable states, dwell times, and sensitivity to new input.

In a heteroclinic realization, a workspace $X$4 maps to a metastable state $X$5, and each admissible transition $X$6 maps to a heteroclinic connection

$X$7

The unstable directions of $X$8 provide possible next operations. The local branching structure may be reflected in

$X$9

Such dynamics do not automatically implement syntax. They require a mapping between neural states, workspace configurations, and formally admissible transitions.

Phase-coupled sealing

The paper proposes that a local composition is incomplete until it is sealed. A schematic sealing operation suppresses active daughter carriers:

$G_i$0

while writing the composite to a phase-addressed register:

$G_i$1

Here, $G_i$2 encodes structural depth and $G_i$3 represents the composite category or content. Sealing suppresses live daughter activity without deleting the daughters; their structures remain available in associative memory and can later be retrieved.

The predicted closure signature is reduced variability, including reduced Fano factor and reduced noise dimensionality, together with stabilization of the composite population state. Reopening a constituent should produce a transient expansion in variability and effective dimensionality.

A phase-address implementation assigns depth to phase:

$G_i$4

The number of non-aliased depth addresses is bounded by the carrier ratio:

$G_i$5

For $G_i$6 Hz and $G_i$7 Hz, the predicted capacity is approximately four depth addresses. The paper emphasizes that phase alone is insufficient for arbitrary tree identity; full grouping requires the combination of phase address, composite identity, and sealing history.

5. Candidate binding operations

Corrected entropy gate

The Marcolli–Berwick entropy gate combines two functions through

$G_i$8

For second-order Rényi entropy,

$G_i$9

The original gate accumulates an entropy offset at each application. If

$$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$0

then

$$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$1

The offset is depth- and tree-shape-dependent and diverges as $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$2, which is implausible for a bounded neural population or phase code.

The corrected gate retains only the optimized mixture:

$$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$3

This removes the unbounded recursive offset and preserves scale under nesting. It also introduces limitations: it is idempotent,

$$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$4

collapses self-Merge, approaches fixed mean pooling as $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$5, and does not inherit the original generic-injectivity theorem.

In simulations, the corrected gate preserved three-terminal bracketing most accurately at an intermediate commitment level. Its matched-terminal rotation decoding at $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$6 dB was approximately $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$7 for $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$8, $$\mathcal N\circ M_i \approx G_i\circ(\mathcal N\times\mathcal N),$$9 for $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$0, $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$1 for $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$2, and $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$3 for $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$4. It also outperformed fixed mean pooling on a balanced four-leaf contrast, where mean pooling was near chance because both bracketings reduce to

$$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$5

Meld

The paper proposes Meld as a neural binding operation:

$$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$6

where $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$7 and $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$8 are constituent population states, $$\mathcal N\circ\mathcal M \approx G\circ(\mathcal N\times\mathcal N).$$9 is a shared weight matrix, $G$0 is an output gain, and $G$1 is obtained from the corrected entropy optimization.

Meld has four principal components:

• Shared synapses enforce exchange symmetry:

$G$2

• Content-dependent sublinear integration makes the composite sensitive to disagreement between constituents.
• Saturation prevents unbounded amplitude growth under recursion.
• Non-associativity preserves grouping while allowing the output to be reused as a constituent.

Unlike the corrected mixture alone, Meld is not idempotent:

$G$3

which need not equal $G$4.

The paper argues that Meld satisfies the binding-level NAP invariants: it is commutative, non-associative, recursively closed, bounded, disagreement-sensitive, and order-blind in the composite consumed by subsequent composition. It does not implement the complete Merge architecture; extraction, selection, workspace storage, reassembly, and sealing must be provided by other mechanisms.

6. Simulations, evidence, and limitations

The simulations used 128-point Gabor-wavelet lexical atoms, randomized Gaussian envelopes, carrier frequencies from 6–26 cycles, and normalization to zero mean and unit variance. Candidate composition laws included additive composition, fixed mean pooling, ordered role-filler coding, recurrent composition, saturating mean pooling, tied-weight units, Meld, and unsaturated Meld.

Bracketing contrasts were height-matched rotations of

$G$5

with terminal identity, terminal order, height, node count, and leaf-depth multiset controlled. Order contrasts compared $G$6 with $G$7. The simulations used 60 independent lexicons per condition, isotropic Gaussian noise from $G$8 to $G$9 dB, and nearest-centroid decoding.

The results showed that additive composition was at chance for bracketing because it is associative. Ordered role-filler and recurrent laws decoded bracketing strongly but retained daughter order. Fixed mean pooling was order-blind but failed the balanced four-leaf contrast. Meld remained order-blind and preserved grouping across tested depths and temperatures. Its rotation decoding across depths one through five was approximately

$\mathrm{SO}_0$0

At depth five, its performance was close to that of the ordered role-filler law while remaining at chance for daughter-order decoding. Meld decoded the balanced four-leaf contrast at approximately $\mathrm{SO}_0$1, with $\mathrm{SO}_0$2, and showed no evident temperature window across $\mathrm{SO}_0$3.

Phase-address simulations found categorical depth aliasing when the carrier ratio was insufficient. At a carrier ratio of four, depths three and four were decoded at ceiling while depths separated by four aliased. At a ratio of seven, all six tested depths were decoded at ceiling. A containment code instead showed monotonic degradation:

$\mathrm{SO}_0$4

These results are simulations rather than neural recordings. The simulated signal-to-noise ratio is an oracle quantity and is not a calibrated neural recording measure.

The paper distinguishes demonstrated results from theoretical proposals. The simulations demonstrate properties of the candidate composition laws, including the associativity of additive composition, the order sensitivity of role-filler codes, the commitment-band behavior of the corrected entropy gate, and the grouping and order-blindness of Meld. They do not establish that cortical networks implement content-addressable workspaces, continuous-time graph schedulers, phase-coupled sealing, or Meld-like binding.

Important limitations include the absence of a proof of generic injectivity for Meld, the non-faithfulness of the corrected gate on the diagonal and in its mean-pooling limit, the reliance on oracle encodings, the unspecified relationship between pointwise soft minima and cortical dynamics, the assumption of shared weights, and the lack of an integrated biological model combining workspace, scheduler, binding, and sealing mechanisms. The phase code is capacity-limited and depends on frequency ratios, while phase unwrapping remains a substantive assumption. Full compatibility between Meld and the proposed extraction and reassembly operations also remains to be shown.

7. Empirical predictions and significance

NAP generates predictions at multiple levels of neural organization. At the composition level, matched-terminal composites should decode bracketing while daughter order remains at chance in the constituency-preserving subspace. Composite states should depend on daughter disagreement at fixed summed input, show sublinear or min-like integration, saturate, and remain reusable in later composition.

At the workspace level, previously built constituents should be reactivated during movement, agreement, scope, reconstruction, and reanalysis. Reopening a sealed constituent should be slower and more variable than reading an active constituent. Directed inter-areal interactions should distinguish extraction, selection, binding, reassembly, and reanalysis.

At the level of closure, constituent completion should produce a burst or phase transition, suppress daughter carriers, stabilize a composite label, and reduce Fano factor and noise dimensionality. Reopening should produce an opposite-signed expansion in variability and dimensionality.

At the level of hierarchy, phase-address codes should show categorical depth aliasing at

$\mathrm{SO}_0$5

while containment codes should show monotonic degradation with depth. Relevant phase-locking should be measured at $\mathrm{SO}_0$6 relations rather than only at $\mathrm{SO}_0$7:

$\mathrm{SO}_0$8

The framework also predicts nonzero interactions between phrase-structure and theta-role violations:

$\mathrm{SO}_0$9

A further prediction concerns representational geometry: the subspace that best decodes bracketing should have lower mutual coherence than a semantic-similarity subspace,

$M_i$00

Selective perturbation of phase-addressing or sealing dynamics should impair structure-sensitive operations while sparing associative bracket-counting strategies. Disrupting interfaces between workspace, scheduler, binding, and sealing populations should reduce downstream usability even when local decodability remains intact.

The central significance of the Neural Admissibility Program is methodological. It establishes that neural model comparison must be constrained by the algebraic and computational properties of the target operation before predictive performance is interpreted. A mechanism that decodes bracketing accurately may be inadmissible if it retains daughter order, fails recursive closure, or cannot provide access to internal constituents. Conversely, a mechanism with lower raw decoding accuracy may be more faithful if it preserves the required invariants and discards irrelevant distinctions.

NAP therefore proposes the following criterion for neural theories of language: a mechanism is acceptable not merely when it predicts linguistic variables, but when its dynamics preserve the formal structure of linguistic computation, support its recursive use, expose its internal substructures, and implement the appropriate transitions between computational workspaces.