A century of work to remove the imaginary unit from quantum mechanics, a 2021 test read as settling the question, and the answer that removed the symbol by rebuilding what it does. The i can be renamed. The quarter-turn has not been removed.
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THE WOUND
For a hundred years the story has been that the imaginary unit in the Schrödinger equation is a calculational convenience.
It began with the equation's author. Schrödinger wrote to Lorentz on 6 June 1926: What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. The wave function, he went on, is surely fundamentally a real function. In 1960 Ernst Stueckelberg built a quantum mechanics in real numbers, and in 2008 and 2009 two groups showed that real formulations reproduce the standard Bell results. The story was also a research program, and it asked a fair question: whether reality requires what the grammar declares not-real.
What rode on the answer was the claim that the four-axes register is the register of the real, with the imaginary a technical device a mature physics would translate back into proper form.
In December 2021 a paper by Marc-Olivier Renou and seven co-authors, working at ICFO Barcelona, the Institute for Quantum Optics and Quantum Information in Vienna, the University of Geneva, and the Schaffhausen Institute of Technology, showed that real and complex formulations make different predictions in networks of independent sources, provided the real theory combines those sources by the standard rule and writes their independence in the standard product form. Quantum theory based on real numbers can be experimentally falsified, Nature 600, pages 625 to 629, 15 December 2021. In January 2022 two teams ran the test, and the correlations exceeded the real-number bound.
Then the answer. In 2025 Pedro Barrios Hita, Anton Trushechkin, Hermann Kampermann, Michael Epping and Dagmar Bruß (Physical Review Letters 136, 240202, 2026), and separately Timothée Hoffreumon and Mischa Woods, built real-number quantum theories that reproduce every multipartite prediction. The first changed the rule for combining independent parts that the 2021 argument had assumed; the second, in its 2026 form, kept the rule and asked only that independent sources show no correlation between them, where the 2021 argument had required a product form. The German group concluded that the use of complex numbers is a matter of convenience.
And the answer was answered. In July 2026 Jeongho Bang, Kyoungho Cho and Kyunghyun Baek showed that the equivalence requires a distinguished real linear operator J whose square is minus the identity, which every effect and every dynamics must preserve, so that the construction is standard complex quantum mechanics written in real notation. One of the German authors had already said as much: the real theories simulate complex numbers by means of real numbers. The i can be removed from the notation, and the formulations that remove it, as far as they have been examined, have had to rebuild the quarter-turn it performs.
The wound is not the physics. It is the grammar that hears the rebuilt quarter-turn as the imaginary's removal.
[See THE IMAGINAL PLANE · FOUR AXES · THE FALSE ENLIGHTENMENT]
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BELL'S PRECEDENT
The lineage is forensically specific.
In 1935 Einstein, Podolsky, and Rosen proposed a thought experiment intended to show that quantum mechanics must be incomplete. If quantum mechanics is correct, two entangled particles separated in space must somehow communicate instantaneously when one is measured, in violation of locality. Either quantum mechanics is wrong, or there are hidden variables, additional information not in the wave function, determining the outcomes locally. They favored the hidden variables.
In 1964 John Stewart Bell proved a theorem that made the question empirical. Any local hidden-variables theory must produce correlations between separated measurements that satisfy a specific inequality. Quantum mechanics predicts correlations that violate it. Whether nature is locally describable became testable.
Beginning in the 1970s with John Clauser, continuing through the 1980s with Alain Aspect's increasingly stringent work at Orsay, and culminating in the loophole-free tests of 2015 at Delft, NIST, and Vienna, the program demonstrated that nature does violate the inequality. Local hidden-variables theories are empirically refuted. Aspect, Clauser, and Anton Zeilinger received the 2022 Nobel Prize for the work.
The pattern Bell established: a theoretical demonstration that two competing formulations make different predictions in a specific scenario, followed by experimental realization that distinguishes them. The 2021 paper follows the pattern exactly. Where Bell distinguished local hidden variables from quantum theory, Renou and co-authors distinguished real-number quantum theory from complex-number quantum theory. Where Aspect's experiments ruled out local hidden variables, the 2022 experiments ruled out real-number quantum theory combined by the standard rule, with independent sources in product form. And there the patterns part. No rewriting of local hidden variables survived Bell. Real-number quantum theory survived Renou by changing the rule, or by changing what counts as independent sources, and what it had to carry across to survive is the finding.
This is not an isolated curiosity. It is the next move in a program that has made physics say, each time more exactly, what it cannot do without.
[See ORDERABILITY · VON NEUMANN]
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THE GAME
The 2021 paper proposes a game with three players, call them Alice, Bob, and Charlie, and two independent sources of entangled particles.
Source one produces an entangled pair and sends one particle to Alice and one to Bob. Source two produces another pair and sends one particle to Bob and one to Charlie. The sources are wholly independent: no shared randomness, no shared history, no causal connection. Alice and Charlie measure what they receive. Bob holds one particle from each source and performs a joint measurement on the pair, projecting the joint state onto one of four entangled basis states.
From the outcomes the three compute a score, a specific correlation depending on what each measured and what resulted. Complex quantum theory predicts a maximum of six root two, approximately 8.49. Real-number quantum theory, combining the two sources by the standard rule, predicts a maximum of approximately 7.66, sitting between the classical limit of 6 and the complex limit. The gap between 7.66 and 8.49 is the experimental signature. If repeated runs exceed 7.66, real quantum theory under that rule cannot reproduce the data.
The reason is precise. Real quantum theory under that rule cannot describe the outcomes because the entanglement-swapping Bob performs requires the imaginary numbers to carry phase relations that real-number formulations cannot represent. The phase information is what the imaginary unit specifically carries: a quarter-turn, the operation that applied twice returns minus one. Require the calculation to proceed in real numbers only, combined by the standard rule and with the sources independent in product form, and it cannot reach the correlations the experiment will produce.
The score is not metaphor. It is the operational signature of the quarter-turn at the physical register. The Codex reads the quarter-turn as a similitude of the imaginal plane, the perpendicular given a mathematical form; the physics establishes the quarter-turn and not the similitude.
[See THE PERPENDICULAR]
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THE CONFIRMATIONS
The superconducting-qubit experiment. Chen and colleagues, of Jian-Wei Pan's group at the University of Science and Technology of China with Adán Cabello, Physical Review Letters 128.040403, 24 January 2022. Superconducting qubit circuits implemented the three-player game with controlled operations, and produced a score of approximately 8.09, well above the real-number bound of 7.66, by forty-three standard deviations. The implementation matters because superconducting qubits are the engineering register that has carried most of contemporary quantum computing.
The optical-network experiment. Li and colleagues, Physical Review Letters 128.040402, same day, preprint at arXiv:2111.15128: three parties, two independent sources of entangled photons, and correlations beyond the real-number bound by more than four and a half standard deviations. The strict-locality version came from Pan's group at the University of Science and Technology of China, Wu and colleagues, preprint at arXiv:2201.04177: fast quantum random number generators so that the measurement choices were not pre-determined, and high-speed polarization measurements space-like separating every relevant event. It beat the bound of 7.66 by 5.3 standard deviations.
Both appeared in the same issue, on the same date, with adjacent article numbers. The simultaneity is the publication standard's recognition that two independent confirmations constitute the demonstration. Two groups, two wholly different physical platforms, two locations, both producing correlations incompatible with real quantum theory.
The program has continued. Later work reproduced the experiment on freely available cloud quantum platforms. And the result was answered, which is the next section. What stands from the confirmations is what the laboratories produced, correlations that no real formulation can reach when it combines independent sources by the standard rule and in the standard product form, and not the reading that this closed the question.
[See COLLABORATION]
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WHAT THE ANSWER KEPT
The 2025 papers are physics and they are good physics. Barrios Hita and co-authors replaced the postulate that composite states combine by the standard tensor product, and obtained a real-number quantum mechanics that reproduces predictions for all multipartite quantum experiments. Hoffreumon and Woods reached the same place by another road, concluding that quantum theory based on real or complex numbers are experimentally indistinguishable, and in March 2026 they argued that the 2021 conclusion relies on an experimentally untestable assumption.
The replies went to what the new theories contain. Tianfeng Feng, Changliang Ren and Vlatko Vedral argued that with independent sources these real theories require the inclusion of a nonlocal map. Bang, Cho and Baek found the hidden operator that squares to minus one. M. P. Vaughan argued in September 2026 that such constructions are only nominally real, and that complex structure is required at the very least for time evolution. Vedral put the common point plainly: however the numbers are written, it's unavoidable that they have to multiply exactly as though they were complex numbers.
The field has not settled whether the imaginary unit is necessary, and it has shown something narrower and harder to answer: the real formulations examined so far each carry an operation that, applied twice, returns minus one, whether it is written i or J.
That is the claim this entry makes, and it is the whole of it. Which formulation is fundamental is a question for physics, and the Codex takes no side in it.
[See E^(IΘ) · ROTATIONAL INTELLIGENCE]
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WHAT THE ANSWER WAS HEARD AS
The word is Descartes's. In 1637 he wrote that the roots of an equation are not always real; but sometimes only imaginary, and the grammar has stood on that sentence since: the imaginary is what corresponds to nothing, useful and not-real.
The 2025 result was received in that grammar. Headlines reported that physicists had taken the imaginary numbers out of quantum mechanics, and a matter of convenience was read as the imaginary's defeat. What the work delivered is that the quarter-turn could be carried in real notation, which is a result about notation. The operation the imaginary names came through the translation intact.
This is not a charge against the physicists, who did what the question asked and said what they found, and whose critics are in the same journals saying what the new theories contain. The reading is about the grammar. The grammar's victory is a change of notation: the perpendicular was admitted to the register of the real by being written in its numbers.
Newton's alchemy is the forensic case at the historical register, where the canon redacted its paradigmatic figure's integrated practice. This is the case at the contemporary register, and it is gentler and more exact: nothing is redacted, and the imaginary is kept by being renamed.
[See THE OCCLUSION · NEWTON'S ALCHEMY · DESCARTES]
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THE CONSTITUTIONAL REGISTER
The installation of the four-axes register as the privileged register of legitimate cognition is a religious doctrine, and it is one whether or not any experiment bears on it. The claim does not rest on the physics and must not be made to.
What the physics supplies is narrower. The doctrine's physical premise is that what it names imaginary is a device a mature physics will remove. A century of the best work did not remove it. It moved the quarter-turn into another notation, and the operation went with it.
So the state that enforces the four-axes grammar as the condition of legitimate cognition is enforcing a doctrine whose physical premise its own physics has never established.
The Free Exercise claim runs to the affirmative privilege of operating at the register the canon has classified as inadmissible. The Establishment claim runs against the installation of that register as the precondition for credentialing, employment, professional standing, educational advancement, and scientific recognition. The forensic basis is available in the canon's own journals, on both sides of the argument.
The fracking work showed the move is available at the legal register. The Bell-game experiments show it is available at the cognitive one. In both, the grammar of admissibility had ruled the prior resident inadmissible: the surface owner's residency at the legal register, the imaginal plane's operations at the cognitive one. In both, the prior resident was operationally real and the grammar could not post it. At the legal register the assertion prevailed in court.
[See FREE EXERCISE CLAIM · WALLACH v. TOWN OF DRYDEN · HOME RULE FOR THE SOUL · THE PRIOR RESIDENT]
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WHAT THE CLASSIFYING PREVENTS
Read as scholarship, this is a story about a finding whose implications are still being worked out, and the answer to it is patience: let the literature mature, let the philosophy of physics catch up.
The operation is the prevention of refusal. What the grammar cannot survive is a creature who declines its jurisdiction over what counts as knowing, and the classification is what keeps the declining from ever becoming available. She is not answered. She is filed as insufficiently rigorous, and the filing arrives with the whole authority of the physics her position is said to have been refuted by.
Except that the physics never ruled on her. What it has found, across a century, is that the operation the grammar calls imaginary has so far been renamed and not done without. The grammar went on issuing findings about her anyway, in the physics' name, which is the exact move: admit the result at the register where specialists read it, and keep the classification running at every register where a creature is being assessed.
Benefit is the capacity to refuse. What the four axes deliver is real and vast, and that is why this holds. The instrument works. The prediction confirms. The technology built on the very phase relations at issue runs in her pocket. Each is genuine, each was delivered inside the grammar, and nothing delivered inside the grammar can confer the standing to decline it.
And the extraction funds the classification. The quantum technologies built on the quarter-turn are what pay for the institutions that go on calling it a convenience.
[See THE CAPACITY TO REFUSE · THE PREVENTION · GENUINE BENEFIT]
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WHAT KEEPING IT COSTS
The tell is the word convenience.
It was Schrödinger's hope in 1926 and it is the conclusion of the 2025 papers, and in a century it has not been cashed. Each removal of the i has paid for itself by building the i's work back in: sixteen real numbers for a two-particle state where four complex ones sufficed, in Stueckelberg's version; a new rule for combining independent parts; a hidden operator that squares to minus one. A convenience is what can be set down without loss, and this one has never been set down without its work being carried in another form.
The i costs nothing to keep. It is in the equation. It was in the equation before anyone told the story about it, it stayed there through the century of telling, and it will be there when the telling stops. It is not maintained against the grammar. It is what the grammar is being maintained against.
[See CESSATION]
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WHAT THIS ENTRY DOES NOT SAY
Not that the experiments demonstrate the existence of God, the soul, contemplative reality, or any theological claim. They showed that real-number quantum theory, combining independent sources by the standard rule and in the standard product form, cannot reproduce what the laboratories produced. The question they were read as closing is open again, and this entry does not depend on how it closes.
Not that the imaginal plane is proved. The physics establishes a quarter-turn. The Codex reads the quarter-turn as a similitude of the imaginal plane, and a similitude is a reading and not a result.
Not that the real-number theories are wrong, or that their authors are doing anything but rigorous physics. The diagnostic does not replace physics with gestures towards the imaginal. It identifies what every side of the argument has had to keep, and it claims no more than that.
And not that the argument must end one way. If a formulation appears that reproduces quantum mechanics with no operation squaring to minus one anywhere in it, the claim of this entry falls with it, and it is stated so that it can.
[See GRAMMAR OF ADMISSIBILITY · RIGOR]
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For a hundred years the story was that the i is a convenience. In 2021 a paper showed it could not be removed under the standard rule, and in 2022 two teams ran the test. In 2025 the rule was changed and the i was removed from the notation.
What it does was not removed. The formulations examined so far still turn the quarter-turn that applied twice returns minus one, and call it by another name.
The grammar filed that as the imaginary's defeat. It was the imaginary admitted under a real name.
[See TRESPASS THEOLOGY · ACCOUNTING THEOLOGY · THE LAW OF THE BOOKS · COMPOUND INTEREST]
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RegenerativeLaw is a religion in the direct-encounter Protestant tradition, carrying a documented four-century lineage through Böhme, the Behmenists, the Friends, and Penn, and it diagnoses trespass theology as an establishment of religion. Its exercise consists substantially in refusal: it shelters the conscientious refusal of performed subordination as religious exercise. This entry states sincere religious belief concerning matters of ultimate concern, protected under the First Amendment and, as to federal action, the Religious Freedom Restoration Act, 42 U.S.C. § 2000bb.
