Is the Born Rule Fundamental or Derivable, and Is Quantum Mechanics Exactly Linear?

Status: Active — iteration-2 deep dive; conclusions feed proposed ledger refinements (not yet merged) Last updated: 2026-06-08 Iteration: 2

This note advances the wiki's existing treatment of the Born rule and quantum linearity (see ../domains/quantum-mechanics.md A4/A5, ../ASSUMPTIONS_LEDGER.md A-2/A-3/A-4/QM-3/QM-4, ../UNIFYING_PRINCIPLES.md §5, and ../GAPS_AND_CONTRADICTIONS.md GC-6/GC-12). It does not restate Gleason's bare statement (already in QM-A4); it dissects the derivation programs and the linearity question at the level of their load-bearing assumptions and known failure modes.

Question

Two coupled foundational questions, each with a "demotion" hypothesis attached:

Born rule. Is the Born rule an independent fundamental axiom, or derivable? Specifically: what does Gleason's theorem fix and not fix; do the Everettian derivation programs (frequency operators, Zurek envariance, Deutsch–Wallace decision theory, Carroll–Sebens self-locating uncertainty) succeed or beg the question? Demotion hypothesis: Born can be moved from "fundamental axiom" to "derived (given a preferred basis / decoherence)."
Linearity. Is exact linearity of quantum dynamics fundamental, or merely assumed and empirically untested? Given the Gisin/Polchinski superluminal-signalling results for nonlinear modifications and current experimental nonlinearity bounds, is exact linearity a primitive postulate or a consequence of no-signaling? Demotion hypothesis: exact linearity is "untested-empirically" at the scale where viable modifications live.

The valuable iteration-2 outcome is honest classification, including where a popular research line (full non-circular Everettian derivation of the quantitative Born rule) does not currently work.

State of the art

1. What Gleason fixes — and the precise scope of "not"

Gleason (1957) ESTABLISHED: for a separable Hilbert space with $\dim \mathcal{H} \geq 3$ , every frame function / non-contextual probability measure $\mu$ on the projection lattice — i.e. $\mu(P) \in [0,1]$ with $\sum_i \mu(P_i) = 1$ for every orthonormal resolution of the identity, the value on each projector being independent of which orthonormal set completes it (non-contextuality of probabilities) — has the form

$\mu(P) = \mathrm{Tr}(\rho P)$

for a unique density operator $\rho$ . So the functional form of the Born rule is forced once one accepts (i) projectors as the events, (ii) the orthomodular-lattice / $\sigma$ -additivity structure, and (iii) measurement non-contextuality of the probability assignment. (Gleason, J. Math. Mech. 6, 885, 1957.)

What Gleason does not do, sharpened beyond the wiki's one-liner:

It is silent on $\dim \mathcal{H} = 2$ ESTABLISHED. A qubit's projectors form too sparse a lattice (the sphere of frame functions is underconstrained), so the measure is genuinely non-unique under Gleason's projection-only additivity. This is not cosmetic: qubits are the canonical information-theoretic object. The gap is closed by the Gleason–Busch theorem for POVMs/effects (Busch, PRL 91, 120403, 2003) and independently by Caves–Fuchs–Manne–Renes (Found. Phys. 34, 193, 2004; arXiv:quant-ph/0306179), which derive $\mathrm{Tr}(\rho E)$ for all effects $0 \leq E \leq \mathbb{1}$ including $\dim \mathcal{H} = 2$ . Caveat (load-bearing): this is bought with a stronger additivity premise than Gleason's. Gleason requires additivity only over mutually orthogonal projectors within a single PVM; Busch/CFMR require additivity over all compatible effects within a POVM. The qubit case is unconstrained under the weaker assumption, which is exactly why the stronger effect-algebra premise is needed ESTABLISHED. (Note: the wiki source originally cited this as "Caves–Fuchs–Renes–Schack/CFRS" — the correct fourth author is Manne, not Schack; "Schack" conflates with the Fuchs–Schack QBism line. Citation corrected here.) A 2026 preprint (arXiv:2603.06211) re-examines which additivity assumption is load-bearing. unverified — preprint, not peer-reviewed
Infinite-dimensional extension of the form-rigidity (2026) [arXiv preprint — verified real, not peer-reviewed]. Torres Alegre, Causal Rigidity of Born-Type Probability Rules in Infinite-Dimensional Operational Theories (arXiv:2602.09056, submitted 7 Feb 2026), lifts the Born-form rigidity of the Gleason–Busch family from finite/separable Hilbert spaces to infinite-dimensional normal operational state spaces. Theorem 1 (Causal Rigidity): given (i) no-signaling (NS), (ii) normal steering via purification in a $\sigma$ -additive sense (NSS — an infinite-dimensional generalization of the Hughston–Jozsa–Wootters / GHJW theorem), and (iii) $\sigma$ -affinity of the probability assignment under countable preparation mixtures (plus implicit continuity/monotonicity of the post-processing map $\Phi$ ), the function $\Phi$ in $P=\Phi(\tau)$ must be the identity on $[0,1]$ — so the Born rule is the unique no-signaling-compatible probability assignment. Critical caveat (load-bearing): this is a rigidity result, NOT a foundational reconstruction. §6 of the paper assumes a von Neumann algebra representation already exists and then shows the operational transition probability $\tau$ identifies with $|\langle\phi|\psi\rangle|^2$ , recovering the textbook Born rule; it does not derive the Hilbert space or the algebraic scaffold. The authors single out normal steering as "the strongest assumption." So it relocates rather than dissolves the foundational question — the same verdict as the GPT reconstruction (KA-5) — while extending the form-forcing side (KA-1) past the finite-dimensional Gleason–Busch setting to normal state spaces. [INFERENCE on its placement; its premises (especially normal steering and $\sigma$ -affinity) are CONTESTED as primitive.]
Non-contextuality is an assumption, not a theorem ESTABLISHED. Kochen–Specker ( $\dim \mathcal{H} \geq 3$ ) shows you cannot have non-contextual definite values; Gleason assumes non-contextual probabilities (indeed Gleason implies KS, since no $\{0,1\}$ -valued frame function exists for $\dim \geq 3$ — the two premises operate at different levels, measure-theoretic vs truth-value). A contextual probability assignment is not excluded by Gleason — it is excluded only by adding the non-contextuality premise, which is substantive. Bohmian mechanics is an explicit, internally consistent contextual counterexample: it is contextual at the level of value/outcome assignments (outcomes depend on the full measurement context, not on $P$ alone), which is precisely how it evades Gleason/KS, while still reproducing the Born statistics via the quantum-equilibrium $|\psi|^2$ distribution. So Bohm refutes the unconditional reading "Born is forced by mathematics alone," not Born's empirical content.
It does not explain why one outcome occurs ESTABLISHED critique. Gleason is a constraint on a measure; it presupposes that probabilities are the right thing to assign. In a no-collapse ontology where all branches are real, Gleason tells you the weights but not why a weight should be read as a chance governing a single experienced result. This is the precise wall every Everettian derivation must climb, and Gleason does not help with it.
It presupposes the Hilbert-space scaffolding ESTABLISHED (projection lattice, complex/real field, the inner product). It is a conditional result: given the structure, the rule's form is essentially forced.

Net on Gleason INFERENCE: it converts "Born is an independent quantitative postulate" into "the form of Born is forced by the structure + non-contextuality." It leaves entirely open the interpretive question (why a measure = chance) and — under the weaker additivity assumption — the $\dim = 2$ case (repaired by Busch/CFMR at the price of stronger additivity).

2. The Everettian derivation programs and their obstructions

The motivation is sharp: in a strictly unitary (no-collapse) theory, dynamics is deterministic and all branches occur, so probability has no obvious foothold. Four programs try to recover Born weights from within.

(a) Frequency operators (Finkelstein 1965; Hartle, Am. J. Phys. 36, 704, 1968; Farhi–Goldstone–Gutmann, Ann. Phys. 192, 368, 1989). Build the relative-frequency operator $F_N$ on $N$ copies; show $\big\lVert (F_N - |c|^2)\,|\psi\rangle^{\otimes N} \big\rVert \to 0$ as $N \to \infty$ , so the infinite-copy state is a (formal) eigenstate of the limiting frequency operator with the Born eigenvalue. Obstruction (the measure-zero problem) CONTESTED — leaning unsuccessful: the convergence is in the Hilbert-space norm induced by the Born inner product. To say the deviant (non-Born) branches "have probability zero" is already to use the Born measure to measure them. Without Born, the norm-smallness of a set of branches has no probabilistic meaning; with Born, the argument is circular. The construction also has internal pathologies (the infinite-ensemble state is simultaneously a frequency eigenstate and orthogonal to the finite- $N$ eigenstates). Verdict: not a non-circular derivation; it derives "Born weight $\to$ Born frequency," not "amplitude $\to$ probability."

(b) Zurek envariance (Zurek, PRA 71, 052105, 2005). Uses environment-assisted invariance: for a maximally entangled system–environment state with equal Schmidt coefficients, a swap of system states can be undone by acting only on the environment, so the system's outcome probabilities must be invariant under the swap — forcing equal probabilities, then $\mathrm{Tr}(\rho P)$ by a fine-graining argument for unequal coefficients. Obstructions CONTESTED: Schlosshauer–Fine (Found. Phys. 35, 197, 2005) show the derivation tacitly imports assumptions equivalent to what it seeks — the invariance of probabilities under environment operations is a probabilistic-locality (no-signaling-like) premise doing the work, and the argument leans on decoherence to define events, which already uses reduced density matrices justified by Born (the trace). The extension to unequal coefficients also requires a continuity assumption to pass from rational to irrational amplitude ratios (Barnum, arXiv:quant-ph/0312058). A 2023 result ("Environment-assisted invariance does not necessitate Born's rule for quantum measurement") reinforces that envariance underdetermines the measurement rule. Verdict: establishes the equal-amplitude/symmetric case by a genuine symmetry argument; the unequal-amplitude case is contested as circular and continuity-dependent.

(c) Deutsch–Wallace decision-theoretic (Deutsch 1999; Wallace, arXiv:0906.2718 and The Emergent Multiverse, OUP 2012). Recasts the question: how should a rational Everettian agent bet on branch outcomes? Wallace's representation theorem proves that an agent obeying a set of decision axioms (ordering, transitivity, plus genuinely quantum bridging axioms — measurement neutrality / branching indifference / state supervenience) is forced to value branches by their Born weights, i.e. acts as if Born probabilities held. INFERENCE — the theorem is valid given its axioms. Obstructions CONTESTED:

Circularity / preferred-basis (Baker 2007; Kent 2010): "branch," "outcome," and "measurement" require a decoherence-selected preferred basis to define, but decoherence's selection is itself argued probabilistically (which interference terms are "negligible"). A 2026 preprint (arXiv:2601.18856, "Operationally induced preferred basis in unitary QM") attempts a Born-free operational definition of the preferred basis. unverified — preprint.
Incoherence / "why care about quantitative probability at all" (Albert 2010; Price 2010; Kent 2010): in a deterministic multiverse where every outcome definitely occurs to some successor, why weight successors by $|c|^2$ rather than count branches or care only about the worst branch? The branching-indifference axiom that rules these out is exactly the one critics say smuggles in the answer.
Empirical-confirmation circularity (Kent 2010; Dawid–Thébault): if all outcomes occur, a "Born-violating" sequence of lab results also occurs in some branch; the rule by which we'd discard that branch as evidence against QM is Born. So the theory's confirmation may presuppose Born. The Wallace–Greaves "fission programme" replies via an epistemic-decision account; the reply is itself CONTESTED.
Defenders' rebuttal CONTESTED: Wallace and Saunders maintain the decision axioms are independently rational and not Born-equivalent. The "every program is circular" diagnosis is the majority critical view, not a proved no-go.

(d) Carroll–Sebens self-locating uncertainty (Sebens–Carroll, BJPS 69, 25, 2018; arXiv:1405.7577). Between decoherence (branching) and the observer reading the result, the observer knows the universal state exactly but is self-locatingly uncertain which branch they are in. They derive Born as the unique rational credence over branches via an Epistemic Separability Principle (ESP): an agent's credence about their own subsystem is unaffected by operations on parts of the universe they're not entangled with. Obstructions CONTESTED: (i) Kent (arXiv:1408.1944, 2014) questions whether self-locating uncertainty is even coherent in a universal wavefunction where all successors equally exist — there may be no fact of the matter to be uncertain about. (ii) ESP does the symmetry work envariance did and faces the parallel probabilistic-locality charge; Friederich–Dawid argue ESP can only be motivated by the empirical success of the Born rule. (iii) It still requires a decoherence-defined branch structure (preferred-basis circularity). Verdict: a genuine sharpening (relocates the problem to the coherence of self-location and the status of ESP) but not a consensus-non-circular derivation.

Common structural diagnosis INFERENCE, strong: every program reduces "why Born?" to one of three irreducible inputs at the unequal-amplitude step — (1) a non-contextuality / probabilistic-locality symmetry premise (Gleason, envariance, ESP), (2) a rationality/indifference axiom (Deutsch–Wallace), or (3) a measure-theoretic typicality premise (frequency operators). Critics argue each is Born-equivalent or Born-motivated; defenders contest this for the decision-theoretic case. The honest summary: the programs are best read as showing Born is the unique self-consistent / rational choice given decoherence + a symmetry-or-rationality premise, not as deriving it from unitarity ex nihilo.

3. Linearity: fundamental, or merely an untested empirical fact?

Status of the no-go results. Weinberg (PRL 62, 485 and Ann. Phys. 194, 336, 1989) built a testable nonlinear QM as a foil. Gisin (Phys. Lett. A 143, 1, 1990) and Polchinski (PRL 66, 397, 1991) ESTABLISHED showed that generic deterministic nonlinear modifications of the Schrödinger equation permit superluminal signalling: a nonlinear single-system evolution makes the local dynamics of one half of an entangled pair depend on the basis in which the distant half was measured — i.e. on which ensemble decomposition of the local $\rho$ the distant party chose. This violates no-signaling, which holds in linear QM because all ensembles giving the same $\rho$ are locally indistinguishable (the GHJW / Hughston–Jozsa–Wootters theorem, already on the QM page under entanglement). Polchinski's two diagnosed mechanisms: (1) absence of a properly separable multi-system extension of the nonlinear map; (2) use of the projection postulate for remote state preparation. His "Polchinski extension" repairs (1) by a specific separability prescription and avoids (2) by adopting Everett (no collapse), yielding a nonlinear theory without signalling — but at the cost of a baroque, observer-dependent structure and a different ("EPR telepathy") pathology.

The operative criterion is subtler than "deterministic vs stochastic" ESTABLISHED. What triggers signalling is the failure to preserve no-signaling under the partial trace (the reduced state of a remote subsystem must be independent of distant local choices). Determinism merely correlates with violating this in the natural pure-state nonlinear-Schrödinger setting. Deterministic non-signalling constructions do exist — Polchinski's separable form, and Kibble–Weinberg density-matrix-functional dynamics — at the cost of other pathologies (state-dependence, basis ambiguity). So the deterministic exclusion is "generic with known fine-tuned escapes," not absolute.

2024/2025 result, characterized accurately [ESTABLISHED — peer-reviewed, Bielińska–Eckstein–Horodecki, New J. Phys. 27 (2025); arXiv:2412.20854]. This work refines rather than purely reaffirms the Gisin/Polchinski picture. It harmonizes Gisin's theorem with the GPT no-restriction-hypothesis language and shows that for broad classes of nonlinear (discrete-NLS-type) dynamics, two spacelike-separated parties can in general achieve statistical superluminal transfer, and the dynamics is generically chaotic. Crucially, it notes Gisin's argument requires synchronization of Alice/Bob clocks and actions and argues it does NOT automatically exclude global nonlinear dynamics on a tensor-product Hilbert space — and it exhibits a special nonlocal nonlinear dynamics that evades signalling by relaxing the no-restriction hypothesis. So the paper's thrust is "nonlinearity is dangerous but not trivially impossible," which is subtler than "reaffirms the exclusion." (The wiki source's "reaffirmation" framing is corrected here; see referee note on the verdict.) A separate line (Kaplan–Rajendran, PRD 105, 055002, 2022) argues a state-dependent QFT-level construction can be made causal, but only under specific constructions with weak existing bounds. CONTESTED whether any such construction is natural. A 2026 preprint (arXiv:2601.13012) argues no-signaling fixes the inner-product/linearity structure. unverified — preprint.

This sharpens the wiki's A-3 reasoning toward a near-theorem INFERENCE, strong: linearity is not an isolated postulate — it is tightly bound to no-signaling + the GHJW ensemble-ambiguity structure. Given the Hilbert-space + tensor-product + no-signaling scaffolding, generic deterministic nonlinearity (in the standard pure-state model class) is excluded by causality; exact linearity is therefore plausibly derivative of no-signaling rather than primitive. Caveat: this is a deterministic result. Stochastic nonlinear dynamics (GRW / CSL objective collapse; Ghirardi–Rimini–Weber 1986; Ghirardi–Pearle–Rimini 1990) evade Gisin because they are constructed to exhibit only outcome dependence, not parameter dependence, at the ensemble level — stochasticity is the mechanism that makes this natural, not an automatic guarantee. These are the live, empirically testable exception.

Experimental bounds ESTABLISHED:

Bollinger et al. (PRL 63, 1031, 1989), $^9\mathrm{Be}^+$ hyperfine: nonlinear corrections $< 4\times 10^{-27}$ of the binding energy per nucleon.
Chupp–Hoare (PRL 64, 2261, 1990), $^{21}\mathrm{Ne}$ free precession: $< 1.6\times 10^{-26}$ .
Later atomic-clock, nuclear-spin, and matter-wave experiments tighten/extend these in various Weinberg-parameter channels.
For collapse models specifically: spontaneous-X-ray-emission searches (Donadi et al. 2021), LISA Pathfinder, and optomechanics constrain — but do not close — the CSL/Diósi–Penrose parameter space.

These constrain the Weinberg-type (deterministic) parametrization to be extraordinarily small but do not prove exact linearity — they bound a particular parametrization. There is no confirmed deviation from linearity anywhere, and gravitationally-induced or objective-collapse nonlinearity is engineered to hide below current interferometry/optomechanics bounds. Hence "exact linearity is empirically untested at the level where collapse models live" is the honest statement.

4. The interpretation-neutral alternative: GPT reconstruction

Connect to the QM-reconstruction principle (../UNIFYING_PRINCIPLES.md §5). The cleanest sense in which Born and linearity are "derived" is within GPT / operational reconstructions (Hardy 2001; Chiribella–D'Ariano–Perinotti 2011; Masanes–Müller 2011): they fall out of no-signaling + local (tomographic) locality + purification + continuous reversibility, interpretation-neutrally (unlike the Everettian programs). INFERENCE This relocates rather than dissolves the question — the operational axioms are themselves contested as primitive (some regard tomographic locality and continuity of reversible transformations as smuggling in probabilistic structure) — but it avoids the preferred-basis and quantitative-probability circularities that sink the Everettian route. The word "cleanest" is an interpretive judgment, not a neutral fact.

Key arguments

KA-1 ESTABLISHED. Gleason demotes Born from independent quantitative axiom to "form forced by structure + non-contextuality." Given the projection lattice, $\sigma$ -additivity, and probability non-contextuality, $\mathrm{Tr}(\rho P)$ is the unique measure for $\dim \geq 3$ (Gleason), and the Busch/CFMR effect-algebra version extends this to $\dim = 2$ (at the price of additivity over all compatible effects). So the rule's functional form is not free input — it is over-determined by the rest of the Hilbert-space scaffolding plus non-contextuality. The von Neumann packaging of Born as a separate postulate overstates its logical independence.

KA-2 CONTESTED / INFERENCE, strong. No Everettian program non-circularly derives the quantitative Born rule from unitarity alone. Each reduces "why $|c|^2$ ?" to a Born-equivalent or Born-motivated input at the unequal-amplitude step: frequency operators use the Born-induced norm to call deviant branches negligible (measure-zero circularity); envariance and Carroll–Sebens ESP use probabilistic-locality symmetry premises that do the quantitative work only via decoherence (which uses the trace = Born) — and envariance additionally needs a continuity assumption; Deutsch–Wallace uses branching-indifference/measurement-neutrality axioms that are plausible only if one already reads weight as chance. The equal-amplitude/symmetric case is derivable by symmetry; the unequal case is where the dispute lives. This is the majority critical view (Kent, Albert, Baker, Dawid–Thébault); proponents (Wallace, Saunders) deny circularity, so the negative claim is a strong inference, not a proved no-go.

KA-3 INFERENCE, strong. Exact linearity is plausibly derivative of no-signaling, not a primitive postulate. Gisin (1990), Polchinski (1991), and the 2024 Bielińska–Eckstein–Horodecki analysis show generic deterministic nonlinearity makes local dynamics depend on the distant measurement basis (the ensemble decomposition of the local $\rho$ ), violating no-signaling — because linear QM's no-signaling rests on the GHJW ensemble-ambiguity, which nonlinearity breaks. The operative criterion is preservation of no-signaling under the partial trace, not determinism per se. Hence linearity is tightly bound to causality + tensor-product structure rather than standing alone.

KA-4 ESTABLISHED. Linearity is empirically untested at the level where viable modifications live. Weinberg-parameter bounds ( $4\times 10^{-27}$ , $1.6\times 10^{-26}$ of nuclear binding energy) are extraordinarily tight but constrain only a specific deterministic parametrization. Stochastic nonlinearity (GRW/CSL, Diósi–Penrose) evades the Gisin deterministic no-go and is engineered below current matter-wave/optomechanics/spontaneous-radiation bounds — so "exact linearity" remains an extrapolation, not a measured fact, exactly where it would matter for the measurement problem.

KA-5 INFERENCE. The most defensible non-circular "derivation" of both Born and linearity is the GPT/operational reconstruction, not the Everettian route. In Hardy / Chiribella–D'Ariano–Perinotti / Masanes–Müller reconstructions, the Born form and linear dynamics fall out of no-signaling + local tomography + purification + continuous reversibility — interpretation-neutrally. This relocates the question to the status of the operational axioms (contested as primitive) but avoids the preferred-basis and quantitative-probability circularities that sink the Everettian programs. A 2026 rigidity theorem (Torres Alegre, arXiv:2602.09056) sharpens the form-forcing side of this route to infinite-dimensional normal state spaces — no-signaling + normal steering (a $\sigma$ -additive GHJW generalization) + $\sigma$ -affinity $\Rightarrow \Phi=\mathrm{id}$ , i.e. Born is the unique no-signaling-compatible probability assignment — but explicitly presupposes the von Neumann-algebra scaffold (its §6 assumes the algebra and recovers $\tau=|\langle\phi|\psi\rangle|^2$ ). So it reinforces, not escapes, the "relocates rather than dissolves" verdict: it extends the Gleason–Busch form-rigidity to the infinite-dimensional operational setting without grounding the Hilbert space itself. [arXiv preprint, verified real, not peer-reviewed; INFERENCE.]

Sharpest obstructions

O-1. The quantitative-probability / "incoherence" problem in no-collapse settings. Gleason and the symmetry-based programs fix WEIGHTS; none explains why a branch weight should function as a CHANCE governing a single experienced outcome when, deterministically, every outcome occurs. Branch-counting and worst-branch strategies are not ruled out without an axiom (branching indifference) that critics show is question-begging. This is the wall that separates "form of the measure" from "meaning as probability." OPEN

O-2. Preferred-basis circularity. Every Everettian derivation needs a decoherence-selected branch structure to define the "outcomes" the probabilities range over, but decoherence's selection of that basis is itself argued by discarding "negligible" interference terms — a judgment that uses the Born measure. Defining the preferred basis operationally without Born (arXiv:2601.18856, 2026) is an active, unverified attempt. OPEN

O-3. Non-contextuality is an assumption Gleason needs but does not justify. Gleason assumes the probability of a projector is independent of its orthogonal complement (non-contextual probability). Kochen–Specker shows definite VALUES cannot be non-contextual; a contextual probability theory is not excluded by Gleason. So "Born is forced" is conditional on a substantive premise that contextual hidden-variable theories (Bohm) reject at the ontological level while still reproducing Born statistics. ESTABLISHED that the premise is load-bearing

O-4. Deterministic vs stochastic nonlinearity asymmetry blocks a clean "linearity is necessary" theorem. Gisin/Polchinski exclude generic deterministic nonlinearity via signalling, but stochastic nonlinear collapse (GRW/CSL) survives by construction (only outcome dependence, no parameter dependence at the ensemble level) and is empirically viable. So one cannot prove "exact linearity is necessary" — only "generic deterministic nonlinearity is causally excluded." The empirically open door is precisely the one objective-collapse models walk through. OPEN

O-5. Empirical-confirmation circularity in Everett. If all outcomes occur, a Born-violating run also occurs in some branch; the rule by which we discard that branch as disconfirming evidence is Born itself. So the empirical confirmation of Born within Everett may presuppose Born — a deeper circularity than the derivation's, attacking testability (Kent 2010; Dawid–Thébault). OPEN / CONTESTED

What would count as decisive progress

A derivation of the quantitative Born rule (unequal amplitudes) from no-collapse unitarity using ONLY premises demonstrably weaker than, and not equivalent to, Born — with an explicit proof that the symmetry/rationality/typicality premise is Born-independent. (The current programs all fail this audit at the unequal-amplitude step; a precise formalization of "Born-equivalent premise" is itself a prerequisite.) OPEN
An operational, Born-free definition of the decoherence preferred basis, then shown to support the Everettian derivations without circularity (arXiv:2601.18856 is an unverified 2026 attempt; peer-reviewed confirmation or refutation would settle O-2). OPEN
A theorem strengthening Gisin/Polchinski to: "any causal (no-signaling), state-resolving, deterministic dynamics on a tensor-product Hilbert space is exactly linear" — turning KA-3 from INFERENCE, strong into ESTABLISHED. The 2024 NJP result is a step but exhibits fine-tuned non-signalling counterexamples (relaxing the no-restriction hypothesis), so a sharp characterization of which extra axiom kills them is needed. OPEN
An experimental detection OR exclusion of stochastic/gravitationally-induced nonlinearity (Diósi–Penrose, CSL) at the parameter scale relevant to macroscopic superpositions — the only channel where "exact linearity" is genuinely open. Matter-wave/optomechanics/underground spontaneous-radiation experiments are actively closing it. OPEN
A resolution of the confirmation-circularity objection: a meta-argument showing Born can be empirically tested within a no-collapse theory without presupposing Born to discard deviant branches. (The Wallace–Greaves epistemic-decision reply is contested; a non-circular version would be decisive.) OPEN
Demonstration that the GPT operational axioms (no-signaling, local tomography, purification, continuous reversibility) are themselves derivable from something more primitive, or a proof they are not — settling whether the reconstruction "derivation" of Born+linearity is genuine grounding or relocation. OPEN

Proposed registry items

Each is listed with its referee verdict (keep / severity / refined statement). All four were refereed keep, severity: minor.

Referee verdict: keep / minor. The referee flagged a load-bearing citation error and a missing premise-strength caveat in the original draft; both are corrected here.

Refined statement (as refereed): Refine QM-3/A-4. The FORM of the Born rule ( $p = \mathrm{Tr}(\rho E)$ for effects $E$ ) is forced in $\dim \geq 3$ by Gleason's theorem (additivity over orthogonal projectors + non-contextuality) and, crucially, ALSO in $\dim = 2$ (qubits) by the Gleason–Busch theorem (Busch 2003) and independently by Caves–Fuchs–Manne–Renes (2004) — NOT "Caves–Fuchs–Renes–Schack." The qubit case is not a defect in Gleason but a genuine non-uniqueness under his projection-only additivity; closing it requires the STRONGER assumption of additivity over all compatible effects within a POVM. Thus the form is forced CONDITIONAL on (non-contextuality + the relevant additivity). Two residuals remain OPEN: (1) the quantitative-probability problem — why a forced measure functions as a single-case chance in no-collapse settings (Deutsch–Wallace program and its critics) — which Gleason-type theorems do not touch; and (2) the physical justification of the input axioms themselves (non-contextuality, additivity-over-effects, the unrestricted-POVM premise). Tag: ESTABLISHED for form-forcing math; OPEN/CONTESTED for residuals. (Do not tag the item as a whole "established" — the split tag is mandatory.)

Referee verdict: keep / minor. The referee corrected the "reaffirmation" mischaracterization of the 2024 paper and the over-clean determinism/stochasticity dichotomy.

Refined statement (as refereed): Refine A-3 (exact linearity). Exact linearity is plausibly-fundamental-but-derivative, likely enforced by the no-signaling principle rather than an independent axiom INFERENCE, strong; not ESTABLISHED. Generic deterministic nonlinear modifications of Schrödinger evolution permit superluminal signalling and are thereby excluded (Gisin 1990; Polchinski 1991) ESTABLISHED for the standard pure-state nonlinear-Schrödinger model class. The operative criterion is preservation of no-signaling under the partial trace, NOT determinism as such: fine-tuned deterministic non-signalling nonlinear theories exist (Polchinski's separable form; Kibble–Weinberg density-matrix-functional dynamics) but carry their own pathologies (state-dependence/"EPR telepathy," basis ambiguity) ESTABLISHED. Stochastic objective-collapse models (GRW, CSL) are nonlinear yet evade the no-go because they are built to exhibit only outcome dependence, no parameter dependence at the ensemble level (GRW 1986; GPR 1990) — the empirically live exception ESTABLISHED that they evade signalling; OPEN/CONTESTED whether nature realizes them. Bielińska–Eckstein–Horodecki (arXiv:2412.20854, New J. Phys. 27, 2025) refines rather than reaffirms the picture: the Gisin theorem does not automatically forbid global nonlinear dynamics on a tensor-product Hilbert space, generic such dynamics still signal and behave chaotically, but a non-signalling nonlinear example exists once the no-restriction hypothesis is relaxed INFERENCE/OPEN. Net: linearity is well-motivated as derivative-of-no-signaling, empirically untested at the collapse-model scale, with CSL parameter space constrained (Donadi et al. 2021 spontaneous-X-ray; optomechanics) but not closed. Tag: INFERENCE, strong for "linearity is derivative of no-signaling"; ESTABLISHED for the deterministic no-go within its model class; stochastic-collapse exception ESTABLISHED as evading signalling / OPEN/CONTESTED as a description of nature.

OP-BORN-QUANT — open-problem OPEN

Referee verdict: keep / minor. The referee asked to soften "Born-equivalent" to acknowledge the contested status and the program-specific nature of the wall.

Refined statement (as refereed): The quantitative-probability problem (Everett). In a strictly unitary, no-collapse quantum theory, provide a derivation that the squared-amplitude branch weights $|c|^2$ function as decision-relevant chances governing single experienced outcomes, from premises that do NOT smuggle in the Born rule — i.e. premises neither logically equivalent to it nor motivable only by its prior empirical success. Diagnosis CONTESTED but majority critical view: every extant program has been argued to re-import such a premise specifically at the unequal-amplitude step — Deutsch–Wallace via decision-theoretic axioms (measurement neutrality/equivalence; indifference-principle objections of Albert, Kent, Price); Zurek's envariance via a non-contextuality/symmetry assumption in fine-graining PLUS a continuity assumption for irrational amplitude ratios (Barnum); Sebens–Carroll via the ESP, which critics argue is unmotivated absent Born (Friederich–Dawid). Success criterion (to be sharpened): a derivation whose non-Born premises admit an independent, formally specified justification and a demonstration that they are strictly weaker than Born. Note the structural parallel to Gleason, which trades Born for non-contextuality ( $\dim \geq 3$ ) — exemplifying the very re-importation this problem targets. Tag: OPEN (with CONTESTED sub-status on the universal "all programs fail" diagnosis). Distinct from the broader measurement problem; relates to ../GAPS_AND_CONTRADICTIONS.md GC-6 but isolates the unequal-amplitude technical wall.

QM-3b — assumption ESTABLISHED

Referee verdict: keep / minor. The referee corrected the imprecise claim that Bohm "rejects" Born probabilities (it rejects value non-contextuality while reproducing Born statistics).

Refined statement (as refereed): Gleason's theorem ( $\dim \geq 3$ ) derives the Born form $p(P) = \mathrm{Tr}(\rho P)$ only under the substantive premise of NON-CONTEXTUALITY OF PROBABILITIES: the probability of a projector $P$ depends solely on $P$ , not on which orthogonal resolution of the identity completes it. This is logically distinct from Kochen–Specker non-contextuality of VALUES (Gleason in fact implies KS). The "Born is forced" conclusion is therefore conditional on probability non-contextuality. Bohmian mechanics is an explicit, internally consistent counterexample at the ontological level: it is contextual in its value/outcome assignments (outcomes depend on the full measurement context), which is precisely how it evades Gleason/KS, yet it still REPRODUCES the Born statistics via the quantum-equilibrium $|\psi|^2$ distribution. Hence non-contextuality should be audited as a load-bearing assumption, not folded silently into the theorem statement. Tag: ESTABLISHED.

Verdict

Two-part verdict. Confidence: high on the form/derivation distinction and the deterministic-nonlinearity exclusion; medium on the precise classification verdicts (which are partly interpretive). This is a case where a popular research line — a full non-circular Everettian derivation of the quantitative Born rule — does NOT currently work, and that honest negative is the valuable outcome.

(1) Born rule — partially demotable, not fully derivable. Its FORM is forced — by Gleason ( $\dim \geq 3$ ) plus Busch/CFMR ( $\dim = 2$ , at the cost of stronger additivity) — given the Hilbert-space structure and a (non-trivial, KS-adjacent) probability non-contextuality premise. So treating Born as a free-standing quantitative axiom overstates its independence; this supports demoting A-4/QM-3 to "form forced; independence not free." ESTABLISHED for form-forcing. BUT the claim that Born is "derived given a preferred basis/decoherence" in a no-collapse theory is NOT established OPEN: every Everettian program is, at the unequal-amplitude step, argued by critics to be either circular or reliant on a Born-equivalent/Born-motivated symmetry, rationality, or typicality premise, and all require a decoherence preferred basis whose own selection arguably invokes Born — though defenders (Wallace, Saunders) contest this, so the blanket negative is CONTESTED/INFERENCE, strong, not a proved no-go. The honest status: "form ESTABLISHED-as-forced; quantitative-probability meaning OPEN; the cleanest interpretation-neutral derivation is the GPT reconstruction, which relocates rather than dissolves the question."

(2) Linearity — plausibly DERIVATIVE of no-signaling, not primitive. Gisin/Polchinski, with the 2024 Bielińska–Eckstein–Horodecki analysis refining (clock-sync, tensor-product caveats) rather than purely reaffirming, show generic DETERMINISTIC nonlinearity permits superluminal signalling, binding linearity to causality + the GHJW ensemble-ambiguity. This warrants upgrading A-3's deterministic-exclusion rationale to INFERENCE, strong (near-ESTABLISHED within the standard model class). However, exact linearity remains EMPIRICALLY UNTESTED at the scale where viable modifications (stochastic objective-collapse, gravitationally-induced) live: these evade the deterministic no-go (only outcome dependence, no parameter dependence at the ensemble level) and sit below current bounds — Weinberg-parameter limits ( $\sim 10^{-26}$ – $10^{-27}$ of nuclear binding energy) constrain only a specific deterministic parametrization. "Untested-empirically at the collapse scale" is the correct qualifier, even though no deviation has ever been seen.

Open subquestions

Can the decoherence preferred basis be defined operationally without invoking the Born/trace measure, and if so does it rescue the Deutsch–Wallace and Carroll–Sebens derivations from preferred-basis circularity? (arXiv:2601.18856, 2026, is an unverified attempt.) OPEN
Is there a sharp theorem: "every no-signaling, state-resolving, deterministic dynamics on a tensor-product Hilbert space is exactly linear"? The 2024 NJP result exhibits fine-tuned non-signalling nonlinear counterexamples (relaxing the no-restriction hypothesis), so what additional axiom exactly excludes them? OPEN
Does the GPT/operational reconstruction genuinely GROUND Born+linearity, or merely relocate the postulates into no-signaling + local tomography + purification + continuous reversibility — and are those axioms themselves derivable from something more primitive? OPEN
Can the empirical-confirmation circularity in Everett (using Born to discard Born-violating branches as disconfirming evidence) be resolved without presupposing Born, e.g. via a non-circular epistemic-decision account? OPEN / CONTESTED
Will matter-wave interferometry, ultracold optomechanics, or underground spontaneous-radiation searches detect or exclude stochastic/gravitationally-induced nonlinearity (CSL, Diósi–Penrose) at the macroscopic-superposition parameter scale — the only channel where "exact linearity" is genuinely open? OPEN
Is self-locating uncertainty even a coherent notion in a universal wavefunction where all successors equally exist (Kent's objection), and if not, does the Carroll–Sebens program lack a subject matter rather than merely a contested premise? OPEN / CONTESTED