Decision Coherence Law

• Decision Coherence Law is a family of constraints linking actions and internal representations to ensure correctness at the decision moment.
• In systems, it mandates a single authoritative state snapshot and temporal bounds to evaluate interacting irreversible actions consistently.
• In quantum and language-model contexts, it emphasizes aligned sector coherence and deductive chain support to preserve trainability and factual reasoning.

Searching arXiv for the requested topic and related papers to ground the article in current research. Decision Coherence Law denotes a family of law-like constraints that couple admissible action, inference, or trainability to a coherent internal representation of the relevant state, beliefs, or dynamical modes. In current arXiv usage, the term appears both as an explicit systems principle—“for agents taking irreversible actions whose effects interact, operating constructively requires that interacting decisions be evaluated against a coherent representation of reality at the moment they are made”—and as an interpretive label for domain-specific coherence laws in quantum learning, probabilistic decision making, and language-model reasoning (Jiang, 15 Jan 2026). In the most developed quantum-trainability formulation, the decisive quantity is readout-visible sector coherence in a noisy $U(1)$-equivariant quantum neural network, where gradient survival is governed by an accumulated coherence-loss variable rather than by global channel diagnostics (Ugail et al., 28 Jun 2026). A plausible implication is that “Decision Coherence Law” is best treated as a recurrent scientific pattern: correctness is preserved only when the specific representation that a system actually uses for action remains coherent at the decision point.

1. Systems law: coherent reality at decision time

The explicit formulation of Decision Coherence Law in systems research is given in “Context Lake: A System Class Defined by Decision Coherence” (Jiang, 15 Jan 2026). The law is stated as a normative correctness requirement for collective agent systems: for agents taking irreversible actions whose effects interact, operating constructively requires that interacting decisions be evaluated against a coherent representation of reality at the moment they are made. The paper operationalizes this requirement through three derived conditions: semantic operations as native capabilities, transactional consistency over all decision-relevant state, and operational envelopes bounding staleness and degradation under load.

Within that framework, a coherent representation of reality is a single authoritative state snapshot against which decisions are evaluated. The paper formalizes this through a transactional consistency invariant: at any moment, there exists exactly one authoritative representation of state against which decisions are evaluated. It further imposes a temporal envelope $\Delta$, written as

$$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$

and a concurrency envelope $C$, under which transactional consistency and the temporal bound are maintained for all decisions under sustained concurrent operation.

The systems consequence is an impossibility result. The Composition Impossibility Theorem states that decision coherence under continuous mutation cannot be achieved by composing independently advancing systems while preserving the independently advancing system class, unless a single system already enforces decision coherence as a non-bypassable property. The corollary, Authority Localization, states that under continuous mutation, decision coherence can be enforced only within a single system boundary. The proposed system class, Context Lake, is therefore defined by enforcement of a single authoritative logical representation of reality for decisions, timely visibility of all decision-relevant mutations, preservation of guarantees under sustained concurrency, and native semantic interpretation and retrieval.

This systems usage is the most literal form of the term. It is concerned neither with epistemic coherence in the abstract nor with post hoc reconciliation, but with correctness at decision time. The paper’s central contrast is with eventual consistency, batch reconciliation, and siloed semantic systems: guarantees that apply after the decision window do not prevent conflicts once irreversible actions have already occurred.

2. Training-law formulation in noisy equivariant quantum neural networks

In “A Coherence Law for Trainability in Noisy Equivariant Quantum Neural Networks,” the decisive question is which physical quantity determines whether gradients survive decoherence in noisy $U(1)$-equivariant brickwork circuits with conserved charge $\hat N = \sum_j \frac{I-Z_j}{2}$ (Ugail et al., 28 Jun 2026). The answer is a compact training law that can be interpreted as a Decision Coherence Law: trainable gradients survive according to readout-visible sector coherence, quantified by an aligned coherence rate entering the gradient as an essentially linear decay in the product of noise strength and depth.

The setup fixes the input in a single charge sector,

$$\mathcal H = \bigoplus_r \mathcal H_r,\qquad \rho_0=P_r\rho_0P_r,$$

with local sector-projected readout $O_B=P_r Z_0 P_r$, noisy output

$$f_\theta^\gamma=\operatorname{Tr}\!\bigl[O_B\,\Phi_\theta^\gamma(\rho_0)\bigr],$$

and trainability measure

$$M(\theta_i;\gamma)=\mathbb E_\theta[(\partial_{\theta_i}f_\theta^\gamma)^2],\qquad D(\gamma)=\log\frac{M(\theta_i;0)}{M(\theta_i;\gamma)}.$$

The paper identifies two distinct mechanisms. Causality determines where gradients can live: because the circuit is local brickwork, the noiseless gradient vanishes whenever $\Delta$0. Coherence determines whether those active gradients survive noise: the derivative is carried by off-diagonal elements within the active charge sector, so symmetry-preserving dephasing can preserve sector populations while still destroying trainability by contracting the relevant intra-sector off-diagonals.

The key quantity is the readout-visible aligned coherence rate,

$\Delta$1

where $\Delta$2. This is a Rayleigh quotient restricted to the gradient-carrying direction rather than a worst-case rate over the full operator space. The perturbative coherence-loss law is

$\Delta$3

which in the restricted-isotropic family collapses to linear decay in $\Delta$4.

The finite-noise numerical law is reported in compact form as

$\Delta$5

with coefficient of determination $\Delta$6. In the main sweep, noise depth alone explains only $\Delta$7, whereas adding the coherence rate lifts the fit to $\Delta$8; the partial $\Delta$9 of the coherence rate after controlling for depth is approximately $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$0. The sharpest test is a correlated-dephasing channel with large worst-case sector coherence rate $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$1 but near-zero aligned rate $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$2; the law predicts no measurable gradient degradation, and numerically $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$3 is observed at all noise levels.

Two structural conclusions follow. First, the light-cone reduction theorem gives a size-independent lower bound on the noiseless gradient second moment for active parameters, so adding qubits outside the light cone does not dilute active gradients at fixed depth and sector. Second, trainability is governed neither by average gate infidelity nor by unitarity loss nor by a global off-diagonal diagnostic, but by the contraction of the specific off-diagonal mode that the readout can observe. In this sense, the “decision” is the update step itself: if aligned sector coherence is retained, the circuit remains trainable; if it is not, gradient-based learning fails.

3. Coherent factuality and reasoning-path admissibility in LLMs

In “Conformal LLM Reasoning with Coherent Factuality,” the relevant analogue of Decision Coherence Law is coherent factuality: an ordered output $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$4 of distinct claims is coherently factual if each $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$5 is deducible from $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$6 and $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$7 (Rubin-Toles et al., 21 May 2025). The paper’s central objection is that independent factuality of isolated claims is not sufficient for reasoning tasks, because a reasoning step may be true yet unsubstantiated or ill-placed in the chain.

The structural device is the deducibility graph, ideally a DAG whose descendants of $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$8 are precisely substantiated claims. Since exact oracle graphs are unavailable, the paper works with approximate deducibility graphs satisfying two properties: ancestor-connected subgraphs that admit coherent orderings must have every topological ordering coherently factual, and ancestor-connected subgraphs that admit no coherent ordering cannot be extended to a larger coherent subgraph. Candidate outputs are generated by thresholding nodewise risk scores and pruning nodes missing ancestors, producing a family of ancestor-connected subgraphs $$\text{decision\_time} - \text{retrieval\_time} < \Delta,$$9.

The resulting conformal nonconformity score is defined on subgraphs rather than claims. With split conformal calibration, the paper proves the guarantee

$C$0

and, under the approximate-graph assumptions, an upper bound of

$C$1

The operational rule is to output the highest-risk ancestor-connected subgraph below the calibrated limit.

Empirically, on MATH and FELM, the method “consistently produces correct and substantiated orderings of claims, achieving coherent factuality across target coverage levels.” At target $C$2, the descendant-weighted variant achieves approximately $C$3 factuality while retaining approximately $C$4 of claims; at $C$5 factuality it retains nearly $C$6 of claims. The paper further reports that post-filtering coherent outputs improves bootstrapping substantially: for $C$7, zero-shot error is $C$8, coherent post-filter error is $C$9, and independent post-filter error is $U(1)$0.

This formulation is not about global truth conditions alone. It imposes a law on admissible reasoning paths: a claim may appear only if it is supported by its predecessors and the underlying graph structure. A plausible implication is that, for reasoning systems, decision coherence is naturally expressed as path coherence rather than pointwise factuality.

4. Belief–decision coherence in decision theory, risk, and choice

In probabilistic decision making with LLMs, “Do LLMs Act Like Rational Agents? Measuring Belief Coherence in Probabilistic Decision Making” studies whether elicited probabilities can be the true beliefs of some rational utility maximizer whose choices match observed actions (Yamin et al., 6 Feb 2026). The base model is

$U(1)$1

extended to a random utility model and a prospect-theoretic random utility model. The principal falsifiable condition is conditional independence: $U(1)$2 If elicited beliefs $U(1)$3 are truthful, then $U(1)$4 must hold. The paper reports that all model–dataset pairs have conditional mutual information significantly greater than $U(1)$5, so the null is rejected everywhere. It also proves that, under RUM or PT-RUM with IIA, pairwise choice probabilities must be monotone in the elicited belief $U(1)$6. Empirically, monotonicity is usually respected for Yes-versus-No choices, but violations are more common for pairs involving Defer. This yields a precise belief–decision coherence criterion: the reported probability must be decision-sufficient.

A different decision-theoretic use of coherence appears in “Coherence and elicitability,” where coherence refers to coherent, law-invariant risk measures in the Artzner sense, and decision coherence enters through elicitability (Ziegel, 2013). The paper shows that law-invariant spectral risk measures such as Expected Shortfall are not elicitable unless they reduce to minus the expected value, while the class of elicitable law-invariant coherent risk measures consists of certain expectiles. In the paper’s own interpretation, coherence in risk measurement and coherence in forecast evaluation are jointly compatible only for negative expectiles $U(1)$7 with $U(1)$8.

At a more general level of choice under uncertainty, “Interpreting, axiomatising and representing coherent choice functions in terms of desirability” develops a direct coherence theory for choice functions and rejection functions (Bock et al., 2019). The primitive object is a set $U(1)$9 of desirable option sets, interpreted as finite option sets that contain at least one desirable option. Coherence is axiomatized by $\hat N = \sum_j \frac{I-Z_j}{2}$0–$\hat N = \sum_j \frac{I-Z_j}{2}$1: removal of zero, exclusion of $\hat N = \sum_j \frac{I-Z_j}{2}$2, inclusion of positive singletons, closure under positive combinations, and monotonicity under supersets. The central representation theorem states that a set of desirable option sets $\hat N = \sum_j \frac{I-Z_j}{2}$3 is coherent iff there exists a non-empty family of coherent sets of desirable options $\hat N = \sum_j \frac{I-Z_j}{2}$4 such that

$\hat N = \sum_j \frac{I-Z_j}{2}$5

With additional properties such as totality, the mixing property, and Archimedeanity, the representations sharpen to sets of strict total orders, lexicographic probability systems, coherent lower previsions, or linear previsions.

Across these papers, decision coherence is not a single axiom but a compatibility requirement between what is stated, what is preferred, and what is chosen. In one case the incompatibility is between elicited beliefs and observed actions; in another it is between coherent capital-allocation behavior and coherent forecast comparison; in a third it is between menu-dependent choices and an underlying desirability structure.

5. Related quantum coherence laws

Several quantum papers articulate adjacent coherence laws that illuminate the broader term. In “Born’s rule as a quantum extension of Bayesian coherence,” the Born rule is presented as an additional Dutch-book coherence constraint for agents who believe they are gambling on sufficiently quantum-like systems (DeBrota et al., 2020). In SIC form, the rule is written as

$\hat N = \sum_j \frac{I-Z_j}{2}$6

The claim is explicitly normative: an agent who accepts the structural assumptions of the argument but refuses this Born-rule relation is vulnerable to a Dutch book. Here coherence is not trainability or systems correctness, but rational consistency among probability assignments across different experiments.

In “Experimental verification of a coherence factorization law for quantum states,” the law concerns the dynamics of a coherence measure $\hat N = \sum_j \frac{I-Z_j}{2}$7 under genuinely incoherent operations (Zheng et al., 2023). The stronger element-wise factorization theorem is

$\hat N = \sum_j \frac{I-Z_j}{2}$8

and the induced coherence factorization law is

$\hat N = \sum_j \frac{I-Z_j}{2}$9

The paper experimentally verifies this relation for qubits and qutrits and reports a limited ququad test consistent with the law. This is a multiplicative law of coherence decay rather than a decision law, but it shares the same structural theme: a global quantity becomes controllable through a compact invariant.

In “Coherence manipulation in asymmetry and thermodynamics,” coherence is treated as time-translation asymmetry, with covariant operations and catalytic assistance (Kondra et al., 2023). The central structural criterion is a set-inclusion law on reachable coherence gaps: $$\mathcal H = \bigoplus_r \mathcal H_r,\qquad \rho_0=P_r\rho_0P_r,$$0 The main theorem states that if this holds, then there exists an approximately catalytic covariant transformation from $$\mathcal H = \bigoplus_r \mathcal H_r,\qquad \rho_0=P_r\rho_0P_r,$$1 to $$\mathcal H = \bigoplus_r \mathcal H_r,\qquad \rho_0=P_r\rho_0P_r,$$2. The paper interprets this as a major step toward a fully general law of quantum thermodynamics in which free-energy monotonicity handles populations while $$\mathcal H = \bigoplus_r \mathcal H_r,\qquad \rho_0=P_r\rho_0P_r,$$3-inclusion handles coherence. A plausible implication is that the trainability law of noisy equivariant QNNs belongs to a broader family of quantum coherence laws in which specific symmetry-restricted modes, rather than full-state diagnostics, determine what transformations remain possible.

6. Scope, misconceptions, and unifying themes

The literature does not support a single, domain-independent theorem called Decision Coherence Law. Rather, it supports a family of structurally similar laws. In systems, the law is explicit and concerns interacting irreversible actions evaluated against a coherent, timely, authoritative state (Jiang, 15 Jan 2026). In noisy equivariant QNNs, the corresponding law states that trainable gradients are governed by readout-visible aligned sector coherence, not by generic noise metrics (Ugail et al., 28 Jun 2026). In reasoning with LLMs, coherent factuality replaces independent factuality as the relevant target for ordered derivations (Rubin-Toles et al., 21 May 2025). In decision theory, coherence can mean compatibility of beliefs with actions, of coherent risk measures with elicitable scoring, or of choice functions with an underlying desirability representation (Yamin et al., 6 Feb 2026).

Several recurrent misconceptions are directly addressed by these works. One is that global or worst-case diagnostics suffice. The quantum trainability paper shows that a large worst-case sector coherence rate does not determine gradient loss when the aligned rate along the visible gradient mode is near zero (Ugail et al., 28 Jun 2026). Another is that local truth suffices for reasoning correctness. The coherent-factuality paper shows that independent factuality is inadequate when the validity of a claim depends on its place in a deductive chain (Rubin-Toles et al., 21 May 2025). A third is that eventual consistency or later reconciliation is enough for multi-agent correctness; the systems law rejects this because post-decision agreement cannot undo interacting irreversible actions (Jiang, 15 Jan 2026). A fourth is that coherent risk measurement automatically supports coherent forecast comparison; the elicitability results show that this fails for spectral risk measures such as Expected Shortfall (Ziegel, 2013).

The unifying theme is selective coherence. None of these laws demand that every internal degree of freedom remain globally coherent. They instead isolate the coherence that is decision-relevant: the backward-light-cone sector mode that carries a gradient, the ancestor-connected subgraph that supports a claim, the state snapshot against which interacting agents act, the elicited belief that should suffice for action choice, or the structural conditions under which a scoring rule can compare forecasts. This suggests a common encyclopedic characterization: a Decision Coherence Law is a law-like criterion identifying the particular representation whose coherence is necessary for a system’s outputs to remain admissible, trainable, rational, or correct.