Mathematical proof of the Fisher-Escolà Q statistical distribution in quantum consciousness modeling

May combine two claims.

Type I errors appeared in 2-5 % of right-tailed tests at α = 0.05 but approached zero as α decreased.

“Type I errors appeared in 2–5 % of right-tailed tests at α = 0.05 but approached zero as α decreased.”

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Type II errors occurred in left-tailed tests at 1-4 % at α = 0.05 but diminished with stricter significance levels.

“Type II errors occurred in left-tailed tests (1–4 % at α = 0.05) but also diminished with stricter significance levels.”

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The Q of Fisher-Escolà distribution follows a beta distribution with parameters a, b, loc, and scale (QFisher-Escolà ∼ beta(a, b, loc, scale)), as confirmed by maximum likelihood estimation.

“Through maximum likelihood estimation, we mathematically confirmed that QFisher-Escolà ∼ beta(a, b, loc, scale).”

May combine two claims. The quote only partly states this claim.

The EM1 circuit, with 10 qubits, Hadamard and CNOT gates, induced nonlocal correlations and generated a maximally entangled system that systematically violates Mermin's inequality.

“These CNOT gates induced nonlocal correlations between qubit states, allowing for the generation of a maximally entangled system that systematically violates Mermin’s inequality.”

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The calculations of CQ and IQ based on the 150 density matrices yielded CQ ≈ 0.9999 and IQ ≈ 0.5001.

“The calculations of CQ (30) and IQ (31) based on these 150 density matrices yielded ideal values for a highly entangled system (CQ ≈ 0.9999; IQ ≈ 0.5001).”

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The analytical solution to the QFI quantum integral successfully stabilizes decoherence, ensuring balanced information levels-neither excessively sensitive to contextual perturbations nor overly restrictive-thereby preserving quantum information processing across the 150 systems analyzed through 150 quantum and empirical density matrices.

“The analytical solution to the QFI quantum integral successfully stabilizes decoherence, ensuring balanced information levels—neither excessively sensitive to contextual perturbations (which induce decoherence) nor overly restrictive, thereby preserving quantum information processing across the 150 systems analyzed through 150 quantum and empirical density matrices.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

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The rates of false positives and false negatives were negligible, supporting its classification as a high-power distribution with potential external validity, particularly when used under probabilistic sampling conditions.

“The rates of false positives and false negatives were negligible, supporting its classification as a high-power distribution with potential external validity, particularly when used under probabilistic sampling conditions.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

May combine two claims.

The ideal value for near-perfect entanglement is CQ ∼ 1, consistent with the values obtained from our circuit.

“The ideal value for near-perfect entanglement is CQ ∼ 1, consistent with the values obtained from our circuit.”

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All density matrices in our circuit clearly violated Mermin's inequality, confirming that the qubits were entangled.

“all density matrices in our circuit clearly violated Mermin’s inequality, confirming that the qubits were entangled.”

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The models compared to the QFisher-Escolà distribution had p-values: normal = 3.4542e-17, gamma = 0.2872, beta = 0.9192, log-normal = 0.2274.

“The models compared to the distribution in this graph were the normal distribution (p-value = 3.4542e-17), gamma distribution (p-value = 0.2872), beta distribution (p-value = 0.9192), and log-normal distribution (p-value = 0.2274).”

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The EM1 system-circuit demonstrated consistent effectiveness across 150 distinct density matrices, meeting the required quantum properties.

“it demonstrated consistent effectiveness across 150 distinct density matrices, meeting the required quantum properties—a notable result that supports its adoption by other researchers.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

May combine two claims.

The relationship ∑λk/∑λresidual follows an F-distribution, F(d1,d2), making it a form of Fisher's statistic.

“By considering λresidual and λk, we can define their mathematical relationship in a way that yields a statistic expected to follow an F-distribution, F(d₁, d₂), making it a form of Fisher’s statistic (see Eq. (33)):”

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Using experimental values N = 106, J = 2, K = 38 from Escolà-Gascón [12], Vk ∼ F(68, 37) and β ∼ F(104, 1).

“Considering the experimental values reported by Escolà-Gascón [12], we have N = 106, J = 2, and K = 38. Thus, we obtain Vk ∼ F(68, 37) and β ∼ F(104, 1).”

• cites [12] Escolà-Gascón, Á. (2025). Evidence of Quantum-Entangled Higher States of Consciousness. Computational and Structural Biotechnology Journal. Article Accepted for Publication (Ref. CSBJ-D-25-00108). PMC11960655 40171221 sourceNot checked yet

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Classified as probably not a substantial claim.

The results from Table 3, Table 4, Table 5 are consistent with the theoretical deductions outlined at the beginning of the subsection.

“The results from Table 3, Table 4, Table 5 are consistent with the theoretical deductions outlined at the beginning of this subsection.”

May combine two claims.

For a right-tailed test, the critical QFisher-Escolà values are: α=0.05: 0.3596; α=0.01: 0.3920; α=0.001: 0.4286; α=0.0001: 0.4586.

“| 0.0500 | 0.3596 | 0.2162 | 0.2044 | 0.3745 | | 0.0100 | 0.3920 | 0.1911 | 0.1825 | 0.4040 | | 0.0010 | 0.4286 | 0.1655 | 0.1592 | 0.4382 | | 0.0001 | 0.4586 | 0.1464 | 0.1415 | 0.4667 |”

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After solving the integral and normalizing, the final form is fQ(Q)≈cQ^{a−1}(1−Q)^{b−1}, confirming that QFisher-Escolà fits a beta distribution under the given assumptions.

“After solving the integral and normalizing, we obtain the final form, shown as Eq. (71):(71)fQ(Q)≈cQa−1(1−Q)b−1where c is a normalization constant ensuring that the total probability integrates to 1. This confirms that QFisher-Escolà fits a beta distribution under the given assumptions.”

• AssumptionRests on an assumption: The components Vk, β, CQ, and IQ are independent, so their joint density is the product of their individual densities.

The quote only partly states this claim.

The empirical data from 150 quantum circuits on IBM's supercomputers provide the first real-world statistical validation of a quantum-classical probabilistic model in neuroscience.

“These empirical data provide the first real-world statistical validation of a quantum-classical probabilistic model in neuroscience, bridging the gap between theoretical speculation and empirical analysis.”

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The β parameter in the Q statistic served as an indirect and ad hoc measure of Q, making reliable statistical inferences about quantum transfer impossible.

“The main limitation of this parameter was that it served as an indirect and ad hoc measure of the Q statistic. This limitation made it impossible to conduct reliable statistical inferences about whether any quantum effect was actually being transferred via the latent patterns.”

• cites [12] Escolà-Gascón, Á. (2025). Evidence of Quantum-Entangled Higher States of Consciousness. Computational and Structural Biotechnology Journal. Article Accepted for Publication (Ref. CSBJ-D-25-00108). PMC11960655 40171221 sourceNot checked yet

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Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

In order for quantum effects to become observable outside their native domain, a certain degree of realism must be introduced.

“The core of the paradox is this: in order for quantum effects to become observable outside their native domain, a certain degree of realism must be introduced.”

Probably not a substantial claim: may not pass S4 (based).

If Mermin's inequality were not violated, certain hidden variables could predict phase shifts in qubit states, contradicting the intended function of the Q statistic.

“Recall that if Mermin’s inequality were not violated, certain hidden variables could predict phase shifts in qubit states, contradicting the intended function of the Q statistic.”

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Only the experimental group yielded significant findings, providing evidence in favor of a quantum transition effect.

“The results showed that only the experimental group yielded significant findings, providing evidence in favor of a quantum transition effect.”

• cites [12] Escolà-Gascón, Á. (2025). Evidence of Quantum-Entangled Higher States of Consciousness. Computational and Structural Biotechnology Journal. Article Accepted for Publication (Ref. CSBJ-D-25-00108). PMC11960655 40171221 sourceNot checked yet

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Probably not a substantial claim: may not pass S4 (based).

Q was the first statistic to merge quantum and classical sources of variation.

“Since Q was the first statistic to merge two ontologically distinct sources of variation—quantum and classical—there were valid reasons to question whether its statistical behavior should conform to the predictions of normalized models.”

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This optimization from latent patterns is statistically significant and stable according to previous biological findings.

“Although modest, this optimization is statistically significant and stable according to previous biological findings [12], [15], [42], [58].”

• cites [12] Escolà-Gascón, Á. (2025). Evidence of Quantum-Entangled Higher States of Consciousness. Computational and Structural Biotechnology Journal. Article Accepted for Publication (Ref. CSBJ-D-25-00108). PMC11960655 40171221 sourceNot checked yet

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• cites [15] Escolà-Gascón Á. Our brains sense the future through a new quantum-like implicit learning mechanism Brain Res Bull 216 2024 111048 10.1016/j.brainresbull.2024.111048 39128676 sourceNot checked yet

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• cites [42] Lufityanto G. Donkin C. Pearson J. Measuring intuition Psychol Sci 27 5 2016 622 634 10.1177/0956797616629403 27052557 sourceNot checked yet

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Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Introducing variability to enhance system information increases QFI values, making the system more sensitive to perturbations.

“However, introducing variability to enhance system information increases QFI values, making the system more sensitive to perturbations.”

Probably not a substantial claim: may not pass S4 (based).

The Q of Fisher-Escolà distribution is the first statistical model to integrate quantum and classical probabilities.

“This study introduces and formalizes the Q of Fisher-Escolà distribution, the first statistical model to integrate quantum and classical probabilities, enabling robust inferential analysis in neuroscience and consciousness studies.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The Fisher-Escolà paradox was formally noted by Tegmark [65] and later discussed by other researchers [66].

“This paradox was formally noted by Tegmark [65] and later discussed by other researchers [66].”

• cites [65] Tegmark M. Importance of quantum decoherence in Brain Processes Phys Rev E 61 4 2000 4194 4206 10.1103/physreve.61.4194 11088215 sourceNot checked yet

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• cites [66] Tegmark M. Consciousness as a state of matter Chaos, Solitons Fractals 76 2015 238 270 10.1016/j.chaos.2015.03.014 sourceNot checked yet

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Multipartite systems do not have fixed thresholds or strictly bounded metrics.

“whereas Bell’s inequality sets an experimental threshold of 2√2 and a theoretical threshold of √16 [56], [57], multipartite systems do not have fixed thresholds or strictly bounded metrics [32].”

• cites [32] Kaniewski J. Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities Phys Rev Lett 117 7 2016 10.1103/physrevlett.117.070402 27563939 sourceNot checked yet

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Probably not a substantial claim: may not pass S2 (checkable), S4 (based). May combine two claims.

The proposed solution involves an integral-based formalism that allows for the stabilization of quantum decoherence: introducing just enough realism to make quantum information accessible, without entirely eliminating its quantum characteristics.

“The proposed solution involves an integral-based formalism that allows for the stabilization of quantum decoherence: introducing just enough realism to make quantum information accessible, without entirely eliminating its quantum characteristics.”

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The theoretical distribution of Q remained unknown and had to be constructed based on new logical rules.

“However, because Q integrates quantum sources of variation and there are no prior references for this type of metrological fusion, its theoretical distribution remained unknown and had to be constructed based on new logical rules that would allow for its mathematical deduction.”

Probably not a substantial claim: may not pass S4 (based).

Adding realism to quantum measurements means introducing a level of decoherence, which inevitably reduces or modifies some of the system's quantum properties.

“Adding realism to quantum measurements means introducing a level of decoherence, which inevitably reduces or modifies some of the system’s quantum properties.”

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Classical variances are highly sensitive and susceptible to fluctuations, and when combined with exceedingly small quantum variances, this can truncate the Q calculation, rendering it non-interpretable and non-functional.

“The mathematical paradox in optimizing the original Q statistic arises from the fact that classical variances exhibit high sensitivity, making them susceptible to fluctuations. When combined with quantum variances, which are exceedingly small, this can truncate the Q calculation, rendering it non-interpretable and non-functional.”

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The Fisher-Escolà Q distribution integrates quantum and classical probabilities and facilitates hypothesis testing in consciousness research.

“Building on this foundation, we introduce the Fisher-Escolà Q distribution, a novel statistical framework that integrates quantum and classical probabilities, facilitating hypothesis testing in consciousness research.”

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The Q of Fisher-Escolà distribution enables hypothesis testing and predicting subjective experiences, with applications in neuroscience and computational automation.

“It enables hypothesis testing and predicting subjective experiences, with applications in neuroscience and computational automation.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

We have provided solid mathematical and statistical evidence supporting the idea that certain aspects of consciousness may align with quantum mechanisms.

“However, we have provided solid mathematical and statistical evidence supporting the idea that certain aspects of consciousness may align with quantum mechanisms.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

Probably not a substantial claim: may not pass S3 (load-bearing).

Constructing a new theoretical distribution that allows computing probabilities for explained variance-integrating quantum and non-quantum effects in learning sequences-does not imply or demonstrate that conscious experience has a quantum origin.

“constructing a new theoretical distribution that allows us to compute probabilities for explained variance—integrating quantum and non-quantum effects in learning sequences—does not imply or demonstrate that conscious experience has a quantum origin.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

This new distribution opens new avenues for testing consciousness theories, particularly those suggesting that quantum entanglement-based effects play a role in the emergence of conscious experience.

“This new distribution opens new avenues for testing consciousness theories, particularly those suggesting that quantum entanglement-based effects play a role in the emergence of conscious experience.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

Probably not a substantial claim: may not pass S3 (load-bearing).

Sample size impacts pattern detection, following the Central Limit Theorem.

“Sample size impacts pattern detection, following the Central Limit Theorem [10].”

• cites [10] Durrett R. Probability: Theory and Examples 2019 Cambridge University Press 10.1017/9781108591034 sourceNot checked yet

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The previous equations for MLE of beta parameters do not have a closed-form analytical solution, so iterative numerical methods based on Newton-Raphson are employed.

“At this stage, it is essential to recognize that the previous equations do not have a closed-form analytical solution. Therefore, we employ iterative numerical methods based on the Newton-Raphson algorithm, also known as gradient-based numerical methods.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The quantile function varies depending on the type of hypothesis test (right-tailed, left-tailed, or two-tailed).

“It is important to note that the quantile function varies depending on the type of hypothesis test (right-tailed test, left-tailed test, or two-tailed test).”

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An IQ value around 0.5 represents an optimal balance, because excessively high IQ indicates high sensitivity to random perturbations and IQ near 0 indicates hermetic information.

“One might assume that an IQ ∼ 1 would be ideal, but this interpretation is incorrect. An excessively high IQ would indicate that the system is highly sensitive to random perturbations, potentially disrupting quantum entanglement properties. Conversely, an IQ ∼ 0 would also be undesirable, as it would suggest that the density matrix contains hermetic or non-extractable information, making it inacce”

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For concurrence, an F(149, 9) model was applied, where concurrence values closest to 0.5 were the most probable, accounting for possible system decoherence induced by perturbations.

“For concurrence, we applied an F(149, 9) model, where concurrence values closest to 0.5 were the most probable, accounting for possible system decoherence induced by perturbations.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

These effects play a role in the phenomenology of learning processes, but they neither confirm nor disprove that consciousness itself operates on quantum principles.

“While these effects play a role in the phenomenology of learning processes, they neither confirm nor disprove that consciousness itself operates on quantum principles.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

May combine two claims.

Eq. (32) preserves the properties of the original Q statistic, while its new formulation enables Q to be recalculated for quantum systems with up to 10 qubits, ensuring the metric remains within the 0-1 range.

“Eq. (32) preserves the properties of the original Q statistic, while its new formulation enables Q to be recalculated for quantum systems with up to 10 qubits, ensuring the metric remains within the 0–1 range.”

May combine two claims.

Without violating local statistical limits, quantum entanglement would be mathematically impossible, and the phasic behavior of qubits would not transition in a way that affects their collapse.

“Without violating local statistical limits, quantum entanglement would be mathematically impossible, and the phasic behavior of qubits would not transition in a way that affects their collapse.”

• cites [25] Hensen B. Bernien H. Dréau A.E. Reiserer A. Kalb N. Blok M.S. Ruitenberg J. Vermeulen R.F. Schouten R.N. Abellán C. Amaya W. Pruneri V. Mitchell M.W. Markham M. Twitchen D.J. Elkouss D. Wehner S. Taminiau T.H. Hanson R. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres Nature 526 7575 2015 682 686 10.1038/nature15759 26503041 sourceNot checked yet

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If each group in an experimental research design is exposed to different conditions, then distinct qubit circuit systems should be used.

“If each group in an experimental research design is exposed to different conditions, then, for ontological reasons, distinct qubit circuit systems should be used.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

In biological and behavioral sciences, no real applications of circuits exceeding 10 qubits have been found.

“In biological and behavioral sciences, we have found no real applications of circuits exceeding 10 qubits (although some theoretical circuits propose models with more than 10 qubits, their applicability in neuroscience, biology, and behavioral sciences remains uncertain).”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Concurrence (CQ) measures the degree of entanglement in a system, where CQ > 0.5 indicates a partially entangled system.

“Concurrence (CQ) measures the degree of entanglement in a system, where CQ > 0.5 indicates a partially entangled system [73].”

• cites [73] Wootters W.K. Entanglement of formation of an arbitrary state of two qubits Phys Rev Lett 80 10 1998 2245 2248 10.1103/physrevlett.80.2245 sourceNot checked yet

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Stabilizing the integral of the QFI by dividing it by the maximum density function value serves as a functional correction but does not constitute a structural solution to the decoherence problem.

“stabilizing the integral of the QFI by dividing it by the maximum density function value serves as a functional correction but does not constitute a structural solution to the problem.”

• AssumptionRests on an assumption: The Q and QFisher-Escolà statistics provide probabilities or proportions of quantum variability (violating Bell’s or Mermin’s inequality) and variability related to latent factors, which could explain patterns in the response matrix structures.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

This interpretative criterion suggests that IQ follows Fisher's transformation, which is adjustable to a normal distribution N(μ, σ).

“This interpretative criterion suggests that IQ follows Fisher’s transformation, which is adjustable to a normal distribution N(μ, σ).”

May combine two claims.

The paper mathematically demonstrated that QFisher-Escolà follows a beta distribution by applying transformations, integrating the joint density function, and verifying the final functional form.

“Therefore, we have mathematically demonstrated the property that ensures QFisher-Escolà follows a beta distribution by applying transformations, integrating the joint density function, and verifying the final functional form.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Global Quantum Concurrence (CQ) as defined in Eq. (30) is widely used in this type of circuit and provides an approximation of the entanglement levels of qubit states for each execution and density matrix.

“Eq. (30) is widely used in this type of circuit and provides an approximation of the entanglement levels of qubit states for each execution and density matrix [47], [52].”

• cites [47] Mintert F. Carvalho A. Kus M. Buchleitner A. Measures and dynamics of entangled states Phys Rep 415 4 2005 207 259 10.1016/j.physrep.2005.04.006 sourceNot checked yet

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• cites [52] Rungta P. Bužek V. Caves C.M. Hillery M. Milburn G.J. Universal state inversion and concurrence in arbitrary dimensions Phys Rev A 64 4 2001 10.1103/physreva.64.042315 sourceNot checked yet

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No theory can hold scientific credibility without empirical-statistical validation.

“no theory can hold scientific credibility without empirical-statistical validation [36].”

• cites [36] Lenharo M. Consciousness theory slammed as ‘pseudoscience’ — sparking uproar Nature 2023 10.1038/d41586-023-02971-1 37730789Not checked yet

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The weight of β has been previously discussed in Escolà-Gascón, where several estimation methods were proposed.

“The weight of β has been previously discussed in Escolà-Gascón [12], where several estimation methods were proposed.”

• cites [12] Escolà-Gascón, Á. (2025). Evidence of Quantum-Entangled Higher States of Consciousness. Computational and Structural Biotechnology Journal. Article Accepted for Publication (Ref. CSBJ-D-25-00108). PMC11960655 40171221 sourceNot checked yet

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