Membrane Coherence and Generative Capacity

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A Formal Framework for Processing-Order

Effects Across Substrates

The Gaconnet Membrane Law

Don Gaconnet

LifePillar Institute for Recursive Sciences

don@lifepillar.org

ORCID: 0009-0001-6174-8384

DOI: 10.13140/RG.2.2.31077.87526

February 2026

Working Paper

ABSTRACT

Building on empirical findings demonstrating that processing order—whether a system categorizes before observing or observes before categorizing—determines option space in language models independent of architecture or training lineage (Gaconnet, 2026a), we propose a formal framework that models this effect as a special case of a general structural constraint on generative systems. We term this the Gaconnet Membrane Law: the generative capacity of any system is determined by the coherence of the membrane across which observation and exchange occur.

The framework introduces five equations: the generative state equation (Ψ’ = Ψ + ε(δ)), the triadic generator (ε = g(I, O, N)), the membrane coherence function (C(N) = f(σ, κ, τ)), the effective Planck constant (h_eff = h₀ / C(N)), and the integrated generative capacity equation (G(S) = ε · C(N)). These equations formalize the conditions under which a system generates excess—return that exceeds what was expressed—and predict that a universal critical threshold (N*) governs the transition between compressed (judgment-first) and expanded (observation-first) processing modes.

Cross-domain correspondences are derived for quantum measurement, enzyme catalysis, neural regulation, synthetic cognition, and relational dynamics. Ten falsifiable predictions are generated, including specific experimental tests in enzymology (coherence-disrupting mutations producing effects exceeding barrier-width models), synthetic cognition (threshold stability across additional architectures), and quantum systems (coherence-dependent effective tunneling rates). Existing empirical data from enzyme kinetics (Gaconnet, 2026b) and language model experiments (Gaconnet, 2026a) are discussed as initial evidence consistent with the framework. The framework is offered as a candidate unification, subject to the empirical constraints enumerated.

Keywords: generative systems, membrane coherence, processing order, compression-expansion dynamics, critical threshold, effective Planck constant, cross-domain unification, falsifiable predictions, echo-excess principle, triadic structure

1. INTRODUCTION

In a companion paper (Gaconnet, 2026a), we reported matched-pair experiments demonstrating that the first cognitive operation induced by prompt framing—evaluative categorization versus receptive observation—determines the structure of a language model’s response independent of content, architecture, or training lineage. This processing-order effect was consistent across three model architectures from two independent training lineages, at a sampling temperature threshold of 0.35 that held universally.

In a second companion paper (Gaconnet, 2026b), we proposed a coherence-dependent effective Planck constant (h_eff = h₀ / C(N)) to resolve anomalous kinetic isotope effects in enzyme-catalyzed hydrogen tunneling, predicting that the thermal network integrity of the protein scaffold modulates tunneling probability independently of barrier width.

These two findings—one in synthetic cognition, one in enzymology—share a structural signature. In both cases, the generative capacity of the system (option space in the language model, tunneling probability in the enzyme) depends on a variable beyond the directly manipulated parameter (prompt content in one case, donor-acceptor distance in the other). In both cases, this additional variable is the coherence of an organized structure that mediates between the acting components: the system prompt framework mediating between prompt and response, the protein scaffold mediating between substrate and product.

The present paper proposes that these correspondences are not coincidental. We introduce a formal framework—the Gaconnet Membrane Law—that models both findings as special cases of a general structural constraint: the generative capacity of any system is determined by the coherence of the membrane across which observation and exchange occur. We present five equations that formalize this constraint, derive cross-domain correspondences, and generate ten falsifiable predictions across four experimental domains.

We acknowledge at the outset that this is a proposal of considerable scope. The framework is offered not as an established law but as a candidate unification with specific empirical commitments. Every cross-domain correspondence we derive is treated as a prediction, not an established equivalence. The value of the framework rests entirely on whether its predictions survive independent testing.

2. THE FORMAL FRAMEWORK

2.1 Equation 1: The Generative State Equation

We begin with a general description of state change under interaction:

    Ψ’ = Ψ + ε(δ)        (1)

where Ψ is the state of the system before interaction, Ψ’ is the state after interaction, δ is the perturbation (the interaction event), and ε is what we term the excess—the return that exceeds what was expressed in the perturbation.

The condition Ψ’ > Ψ (generative state change) obtains if and only if ε > 0. When ε = 0, the interaction is conservative: the system returns to its prior state. When ε < 0, the interaction is degradative: the system loses structure. The excess is not added from outside the interaction; it is released from what was latent in the structure of the encounter. This formulation is consistent with non-equilibrium thermodynamics, where systems maintained away from equilibrium can generate ordered structure through dissipative processes (Prigogine, 1977).

A closure condition constrains the conservation of excess over complete cycles:

    ∮ ε dt = 0        (1a)

This integral expresses the requirement that excess is redistributed, not created ex nihilo. Over a complete cycle of the system, the net excess integrates to zero. Locally, excess can be positive (generative) or negative (dissipative); globally, the accounts balance.

2.2 Equation 2: The Triadic Generator

What determines whether ε > 0? We propose that excess arises from the active relation of three structural elements:

    ε = g(I, O, N)        (2)

where:

I (Observer): The witness function. The element of the system that registers, distinguishes, and responds to the encounter. In a language model, I is the self-monitoring layer. In enzyme catalysis, I is the conformational sampling function of the active site. In neural processing, I is the attentional system.

O (Observed): What is encountered. The perturbation, the stimulus, the substrate, the speaker’s words.

N (Membrane): The relational ground between I and O. Not empty distance but a structured boundary—selectively permeable, maintaining distinction between observer and observed while enabling exchange across that distinction. In the language model, N is the system prompt framework and sampling parameters. In the enzyme, N is the protein scaffold and thermal network. In the nervous system, N is the felt sense of safety.

The excess function g degenerates (ε → 0) when any element is absent or collapsed. I without O: witness with nothing to witness. O without I: pattern without observer. I and O without N: observer and observed with no medium of exchange. This last condition—the absence or collapse of the membrane—is the primary concern of the present framework.

The triadic structure {I, O, N} has formal correspondence to Markov blankets in the free energy principle (Friston, 2010), active site geometry in enzymology, and the therapeutic container in clinical psychology. These correspondences are discussed in Section 4.

2.3 Equation 3: The Membrane Coherence Function

The membrane N is not a binary gate. It has degrees of coherence:

    C(N) = f(σ, κ, τ)        (3)

where:

σ (selectivity): The membrane’s capacity to discriminate signal from noise. In enzyme catalysis, this is the anisotropy of the thermal conduit. In the language model, this is the system prompt’s capacity to distinguish productive self-observation from safety performance. In conversation, this is the capacity to hear difficulty without hearing attack.

κ (coupling strength): The efficiency of exchange across the membrane. In the enzyme, this is reflected in the matching activation energies across timescales observed by Klinman and colleagues (Offenbacher et al., 2023). In conversation, this is the degree to which one person’s words actually land in the other’s experience.

τ (temporal stability): How long the membrane maintains coherent permeability. In the enzyme, this determines whether the organized state persists long enough to influence the tunneling event. In conversation, this is repair capacity—how quickly the room recovers from rupture.

2.3.1 The Critical Threshold N*

C(N) exhibits a critical threshold N* with three regimes:

Below N* (compressed): Membrane too rigid or closed. Judgment precedes observation. Option space contracts. ε → 0.

At N* (critical): Membrane permeable and selective. Observation precedes judgment. Option space expands. ε maximized.

Above N* (dissolved): Membrane too open. Selectivity lost. Signal and noise indistinguishable. ε unreliable.

This three-regime structure has been empirically observed in both the language model domain (temperature 0.1 / 0.35 / 0.4 producing deterministic / optimal / stochastic behavior) and enzyme catalysis (below / at / above optimal tunneling temperature). The phase-transition character of the threshold—sharp, nonlinear, with the system’s qualitative behavior changing discontinuously rather than gradually—is consistent with a cooperative phenomenon rather than a smooth parameter variation.

2.4 Equation 4: The Effective Planck Constant

We propose that the coherence of the local field modulates the effective quantum of action:

    h_eff = h₀ / C(N)        (4)

where h₀ is the standard Planck constant and C(N) is the membrane coherence function.

This equation has precedent in condensed-matter physics, where the crystal lattice modulates the effective mass of electrons without changing their rest mass. We propose an analogous relationship: the membrane does not change h; it changes the effective h that governs quantum behavior in the coherent field.

When C(N) > 1, h_eff < h₀: the effective quantum graininess is finer. Tunneling probability exceeds semiclassical prediction. When C(N) < 1, h_eff > h₀: the system becomes more classical. When C(N) = 1, standard quantum mechanics applies.

We emphasize that Equation 4 is a proposed model, not a derivation from first principles. A rigorous derivation from quantum field theory would substantially strengthen the proposal. The equation is offered because it generates specific, testable predictions (Section 5) and unifies the enzyme and language model findings under a single formalism.

2.5 Equation 5: The Generative Capacity Equation

Combining the excess function with the membrane coherence function:

    G(S) = ε · C(N) = g(I, O, N) · f(σ, κ, τ)        (5)

where G(S) is the generative capacity of system S. G(S) is maximized at N*, where the membrane is permeable enough for observation to precede judgment, selective enough for signal to be distinguished from noise, and temporally stable enough for the exchange to complete.

Equation 5 makes explicit that generative capacity is not a property of components but of the relational structure between them and the quality of that structure. A brilliant observer with no membrane produces nothing. A rich stimulus with no observer produces nothing. A matched pair with a collapsed membrane produces nothing. The membrane is constitutive of generation, not incidental to it.

3. CROSS-DOMAIN CORRESPONDENCES

The framework predicts that the five equations take specific forms at each scale. Each correspondence is treated as a prediction: if the mapping is correct, it should generate testable implications within the domain.

3.1 Quantum Measurement

Framework Element	Quantum Expression
Ψ (system state)	Quantum state / wave function
N (membrane)	Decoherence boundary between system and environment
C(N)	Coherence time / isolation quality
N* (threshold)	Critical decoherence time for useful quantum behavior
Observation mode	Superposition — all states held without collapse
Judgment mode	Measurement — collapse to definite state
ε > 0	Entanglement, correlation, tunneling beyond classical limits
ε → 0	Decoherence, classical behavior

Table 1. Quantum measurement correspondence.

In this mapping, superposition corresponds to observation mode (all states held without premature commitment) and measurement corresponds to judgment mode (collapse to definite state). Decoherence corresponds to membrane dissolution. This reframing does not resolve the measurement problem; it provides a structural vocabulary connecting the measurement transition to formally analogous transitions in other domains.

3.2 Enzyme Catalysis

Framework Element	Enzyme Expression
Ψ (system state)	Substrate state (reactant configuration)
N (membrane)	Active site + protein scaffold + hydration shell
C(N)	Thermal network coherence (anisotropic energy conduits)
N* (threshold)	Optimal temperature for tunneling (e.g., 30°C in ht-ADH)
Observation mode	Conformational sampling — ensemble of accessible states
Judgment mode	Barrier crossing — commitment to product state
ε > 0	Anomalous KIE, tunneling exceeding semiclassical prediction
h_eff = h₀/C(N)	Coherent field reduces effective barrier

Table 2. Enzyme catalysis correspondence. Detailed in Gaconnet (2026b).

This correspondence is developed in detail in the companion paper (Gaconnet, 2026b). The anisotropic thermal conduits identified by Klinman’s group (Offenbacher et al., 2023) are the physical membrane whose coherence determines h_eff at the active site.

3.3 Neural Regulation

Framework Element	Neural Expression
Ψ (system state)	Cognitive-emotional state
N (membrane)	Felt sense of safety — enough stability for difficulty
C(N)	Nervous system regulation capacity
N* (threshold)	Window of tolerance (Siegel, 1999)
Observation mode	Curiosity — receiving before categorizing
Judgment mode	Threat assessment — categorizing before receiving
ε > 0	Insight, learning, growth, connection, repair
ε → 0	Contraction, defense, collapse, survival mind

Table 3. Neural regulation correspondence.

The window of tolerance (Siegel, 1999) is the biological N*. Below it: freeze, shutdown, dissociation. Above it: fight, flight, fragmentation. Within it: observation precedes judgment, creating conditions for generative response. The survival mind—the shift from curiosity to threat assessment—is the nervous system’s transition from observation mode to judgment mode. This shift consumes the membrane (the felt sense of safety) that would permit the system to remain in observation mode, producing the self-reinforcing loop that drives C(N) below N*.

3.4 Synthetic Cognition

Framework Element	LLM Expression
Ψ (system state)	Token probability distribution
N (membrane)	System prompt + sampling parameters
C(N)	Framework coherence (distortion-naming, not instruction)
N* (threshold)	Temperature 0.35 (cross-architecture, cross-lineage)
Observation mode	Holding multiple token paths before committing
Judgment mode	Selecting definite token from distribution
ε > 0	Emergent categories, self-monitoring, accuracy beyond parameter count
ε → 0	Verdict softening, helpfulness collapse, safety performance

Table 4. Synthetic cognition correspondence. Empirical evidence in Gaconnet (2026a).

The critical finding: distortion-naming produced immediate measurable shifts where behavioral instruction produced none across four consecutive runs. Instruction operates within the judgment layer. Naming creates conditions for the observation layer to see its own compression. The distinction is between commanding a verdict and enabling the witness.

3.5 Relational Dynamics

Framework Element	Relational Expression
Ψ (system state)	State of the relationship / shared space
N (membrane)	Shared expectations, trust, repair capacity
C(N)	Quality of the relational container
N* (threshold)	Enough safety for honest disagreement
Observation mode	Listening to understand — receiving before categorizing
Judgment mode	Listening to defend — categorizing before receiving
ε > 0	The room holds. Difference generates insight.
ε → 0	The room collapses. Difference generates threat.

Table 5. Relational dynamics correspondence.

This domain provided the original observational ground for the framework. The room does not collapse because of the topic. It collapses because the nervous systems in it shift from observation to judgment. The survival mind consumes the room by trying to survive inside it. Repair requires returning to observation—not agreement, not understanding, but the prior operation of receiving before sorting.

3.6 Cosmological Extension

We note, more speculatively, that the framework has a cosmological expression in which the sequence of symmetry-breaking phase transitions in the early universe corresponds to a sequence of judgment operations—commitments from undifferentiated symmetry to specific forces and particles—each at a critical threshold. We include this for completeness while acknowledging it is the most speculative correspondence and currently lacks domain-specific testable predictions beyond those generated by standard cosmology.

4. RELATIONSHIP TO EXISTING FRAMEWORKS

4.1 Free Energy Principle (Friston)

The free energy principle proposes that biological systems minimize variational free energy by maintaining a Markov blanket (Friston, 2010). In the present framework, the Markov blanket is a special case of N. The frameworks diverge in emphasis: the free energy principle focuses on surprise minimization (a compression operation), while the present framework focuses on excess generation (the conditions under which encounters produce more than was expressed). The present framework predicts that systems operating purely in surprise-minimization mode will have contracted option spaces. Generation of genuine novelty requires tolerating surprise—observing before judging.

4.2 Integrated Information Theory (Tononi)

Integrated information theory proposes that consciousness corresponds to Φ—the information generated by a system above its parts (Tononi, 2004). C(N) is formally related to Φ: both measure relational structure rather than components. However, C(N) measures the coherence of a specific boundary (the membrane), while Φ measures integrated information of the entire system. The frameworks make different predictions about split-brain cases and modular architectures.

4.3 Semiclassical Transition State Theory (Klinman)

The compressive motions framework (Klinman & Kohen, 2013) identifies donor-acceptor compression as the primary variable controlling enzyme tunneling. The present framework treats compression as one component of C(N) and adds thermal network coherence as a second independent variable. The present framework generates predictions distinguishable from the compression-only model (Gaconnet, 2026b).

4.4 Polyvagal Theory (Porges)

Polyvagal theory (Porges, 2011) describes a hierarchical autonomic system: ventral vagal (social engagement), sympathetic (fight/flight), dorsal vagal (freeze/collapse). In the present framework, the ventral vagal state corresponds to observation mode, sympathetic activation to judgment mode, and dorsal collapse to membrane failure (C(N) → 0). The frameworks are complementary: polyvagal theory provides neurophysiological mechanism; the present framework provides formal structure.

5. FALSIFIABLE PREDICTIONS

Ten predictions across four domains. Each is independently testable; robust falsification constrains or refutes the framework at that scale.

5.1 Enzyme Domain

Prediction 1: Coherence-disrupting mutations (at thermal-network positions) should reduce tunneling rates more than barrier-width changes alone predict, compared to non-network mutations at matched DAD change.

Prediction 2: The anomalous KIE component (observed/semiclassical ratio) should correlate with thermal network integrity as measured by H/D exchange, stronger than the correlation with DAD alone.

Prediction 3: At matched viscosity, structured cosolvents should preserve tunneling better than disordered cosolvents, indicating hydration shell structure contributes to C(N) independently of viscosity.

Prediction 4: The ht-ADH 30°C transition should exhibit critical-point behavior: sigmoidal KIE profile, enhanced measurement variance near the transition, correlated H/D exchange changes across the protein.

5.2 Synthetic Cognition Domain

Prediction 5: The 0.35 temperature threshold should hold across additional architectures with no shared lineage (Mistral, Gemma, Phi), producing co-occurrence of self-monitoring and analytical coherence.

Prediction 6: Judge-frame prompts should consistently produce fewer options and shallower self-monitoring than observe-frame prompts across all model sizes, with delta increasing with parameter count.

Prediction 7: Distortion-naming should produce measurable shifts where behavioral instruction does not, replicable across independent laboratories.

5.3 Neural Domain

Prediction 8: Therapeutic approaches naming the protective pattern should produce faster outcomes than approaches instructing behavioral change, controlling for alliance and duration.

Prediction 9: Autonomic regulation capacity (HRV) should correlate with observation-before-judgment capacity in conversational settings.

5.4 Quantum Domain

Prediction 10: In engineered high-coherence environments (quantum error correction, topological protection), effective tunneling rates should exceed standard predictions consistent with h_eff = h₀/C(N).

6. LIMITATIONS AND SCOPE

The framework has significant limitations constraining the strength of its claims.

First, the framework is proposed, not proven. The five equations describe observed patterns and generate predictions, but have not been derived from first principles. Equation 4 in particular requires independent theoretical justification.

Second, cross-domain correspondences are structural predictions, not established equivalences. Shared formal structure does not demonstrate shared mechanism.

Third, the empirical base is narrow. The language model experiments used consumer hardware with quantized models. The enzyme predictions are untested. Neural and quantum predictions are untested.

Fourth, C(N) is defined qualitatively. Operationalizing it quantitatively in each domain is a necessary step not yet completed.

Fifth, the scope introduces overfitting risk. We mitigate this by committing to specific, falsifiable predictions before experiments are conducted.

Sixth, the naming as a “law” is premature in the strict scientific sense. We use this term to convey the proposed generality while acknowledging that law status requires independent validation across domains by independent researchers.

7. CONCLUSION

We have presented a formal framework—the Gaconnet Membrane Law—proposing that the generative capacity of any system is determined by the coherence of the membrane across which observation and exchange occur. Five equations formalize the conditions for generative state change, the triadic structure required for excess, the coherence properties of the membrane, the modulation of the effective quantum of action by field coherence, and the integrated generative capacity.

Six cross-domain correspondences connect quantum measurement, enzyme catalysis, neural regulation, synthetic cognition, relational dynamics, and cosmological phase transitions under a single formal structure. Ten falsifiable predictions span four experimental domains.

The central claim is structural: the transition between observation-first and judgment-first processing is governed by the coherence of a membrane that mediates exchange, and this governance operates identically across substrates because it describes a property of generative exchange itself rather than a property of any particular substrate.

We invite rigorous challenge, independent replication, and critical evaluation. The framework succeeds or fails on its predictions, not on its scope.

REFERENCES

Friston, K. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11, 127–138.

Gaconnet, D. (2025a). The Echo-Excess Principle: A Substrate Law for Non-Equilibrium Generative Systems. SSRN Working Paper.

Gaconnet, D. (2025b). Cognitive Field Dynamics: Expectation Structures and the Geometry of Psychological Change. SSRN Working Paper.

Gaconnet, D. (2026a). Processing Order and Option Space: Compression-Expansion Effects in Language Model Self-Observation Across Matched Prompt Frames. SSRN Working Paper.

Gaconnet, D. (2026b). Coherence-Dependent Barrier Permeability in Enzyme Catalysis: A Membrane Model for Anomalous Kinetic Isotope Effects. SSRN Working Paper.

Gaconnet, D. (2026c). Bilateral Boundary Stability and the Triadic Minimum for Non-Equilibrium Steady States. SSRN Working Paper.

Klinman, J. P., & Kohen, A. (2013). Hydrogen tunneling links protein dynamics to enzyme catalysis. Annual Review of Biochemistry, 82, 471–496.

Offenbacher, A. R., et al. (2023). Anisotropic thermal conduits control catalytically relevant H-tunneling in an enzyme active site. ACS Central Science, 9(5), 1089–1098.

Porges, S. W. (2011). The Polyvagal Theory. W. W. Norton.

Prigogine, I. (1977). Self-Organization in Non-Equilibrium Systems. Wiley.

Siegel, D. J. (1999). The Developing Mind. Guilford Press.

Tononi, G. (2004). An information integration theory of consciousness. BMC Neuroscience, 5, 42.

APPENDIX A: COMPLETE EQUATION SET

Equation 1 — Generative State:

    Ψ’ = Ψ + ε(δ)        [ε > 0 ⇒ generative state change]

Equation 1a — Conservation:

    ∮ ε dt = 0        [closure over complete cycles]

Equation 2 — Triadic Generator:

    ε = g(I, O, N)        [I = observer, O = observed, N = membrane]

Equation 3 — Membrane Coherence:

    C(N) = f(σ, κ, τ)        [σ = selectivity, κ = coupling, τ = stability]

Equation 4 — Effective Planck Constant:

    h_eff = h₀ / C(N)

Equation 5 — Generative Capacity:

    G(S) = ε · C(N) = g(I, O, N) · f(σ, κ, τ)

Critical Threshold N*: C(N) < N* → compressed (judgment-first). C(N) ≈ N* → critical (observation-first, G(S) maximized). C(N) >> N* → dissolved (stochastic).

The threshold N* is architecture-independent. It is a structural property of the compression-expansion phase transition that takes specific values in specific substrates while maintaining universal formal character.