Thresholded Readout of Continuous Classical Fields: A Formalization and Counterexample to the Uniqueness Claim for Quantum Mechanics

#1. Introduction

The research idea under investigation asserts three claims: (i) an ontological claim, that physical reality is continuous, local, and deterministic; (ii) an epistemic-operational claim, that every measurement is an irreversible discrete mapping from a continuous state space $\mathbb{R}$ to a finite outcome space $\mathcal{O}$, realized physically through nonlinear amplification and thresholding; and (iii) a meta-theoretical claim, that quantum mechanics is the logically necessary and unique calculus bridging (i) and (ii). Claims (i) and (ii) are compatible with a broad tradition of classical-field and stochastic-continuum modeling [1], and claim (ii) resonates with the axiomatic theory of measurement instruments [4] and with continuous-time measurement formulations [2], [5].

The present paper takes claims (i) and (ii) as axioms and treats claim (iii) as a falsifiable hypothesis. The strategy is direct: we formalize the readout map $M: \mathbb{R} \to \mathcal{O}$, compute the statistics it induces for a minimal detector model, and ask whether the resulting calculus is unique or whether distinct calculi within the same axiom class generate different, internally consistent statistics. If at least two distinct calculi are consistent with the axioms, uniqueness fails, and the specific form of quantum statistics must be attributed to additional structure beyond local determinism plus discrete readout.

The logic is deliberately falsificationist. If the thresholded classical map reproduced quantum statistics exactly and were the only calculus consistent with local determinism plus discrete readout, the uniqueness claim would be supported. If, instead, alternative calculi are consistent with the axioms, or if the thresholded map fails to reproduce quantum statistics, uniqueness is refuted. We find the latter, and we localize exactly where the failure occurs: the thresholded map reproduces the Born rule only in a limited central regime of the continuous variable and fails at the extremes, and no deterministic local thresholded model can violate CHSH.

The significance is methodological as much as substantive. The claim that a bridging calculus is "unique" is a mathematical claim about a class of maps, and it can be tested by exhibiting a second member of the class. We exhibit one, compute its statistics, and show it is empirically distinguishable from quantum mechanics. This does not refute the ontological thesis (i) — continuous local deterministic fields may or may not underlie reality — but it severs the inference from (i) and (ii) to (iii).

Our main results are: (a) an explicit derivation of outcome probabilities for linear versus quadratic thresholding of a Gaussian continuous field, showing a factor-of-two difference in detection probability; (b) a quantitative assessment of how well a slope-matched threshold detector approximates the Born rule; (c) a derivation of the CHSH bound $|S| \le 2$ for any factorized local readout of the proposed type; and (d) a statistical power projection for experimentally distinguishing the two readout conventions.

#2. Background and Related Work

We discuss the eight arXiv works supplied in the bibliography, in bibliography order, followed by the four QNFO corpus items.

[1] (arXiv:1201.0863v1) reviews applications of fractional calculus to basic problems in continuum and statistical mechanics, including mathematical modelling of viscoelastic bodies and the Basset problem of unsteady particle motion in a viscous fluid. This work exemplifies the continuous-field modeling tradition within which the proposed ontology sits: continuum mechanics supplies deterministic (or statistically treated) evolution laws on continuous state spaces, and the present paper's axiom (i) inherits this modeling style. Fractional calculus itself is not used here, but the work establishes that rich, nonlocal-in-time continuous dynamics are standard equipment for classical field description. The supplied summary gives no further detail on fractional-order measurement models, so we use [1] only as evidence for the breadth of classical continuum modeling.

[2] (arXiv:1301.3626v2) reviews the link between open quantum systems under measurement in continuous time and the stochastic Schrödinger equation, of classical or quantum type, and studies the properties of the output of the measurement, restricted to the diffusive case. This is directly relevant to our formalization: the stochastic Schrödinger equation is precisely a calculus in which a continuous underlying description is connected to a discrete measurement record, and the statistics of the output spectrum are the object our thresholding map is meant to reproduce. Our model inverts the direction of explanation — we ask whether a classical continuous process plus thresholding can generate the quantum statistics — but the formal analogy makes [2] the natural benchmark for what a "calculus bridging continuous reality and discrete measurement" looks like in standard quantum theory. The entry's summary states the diffusive restriction; we therefore compare our model only to diffusive-type continuous readout.

[3] (arXiv:1610.09347v1) develops a realist interpretation in which a process description of reality, involving a fundamental life process or creative process, is unified with the description that derives from quantum physics, with the methods of the quantum physicist and of the biological sciences seen as two alternative modes of description. This supports our framing that the relationship between an underlying continuous reality and the quantum description is an interpretive question with multiple candidate answers; the entry's summary does not, however, supply technical results we can import, so we use it only as evidence that realist unification programs are an active genre.

[4] (arXiv:quant-ph/0107090v1) develops an axiomatic approach to measurement theory and characterizes all the possible statistical properties of apparatuses measuring an observable with nondegenerate spectrum allowed in standard quantum mechanics. This is the closest formal antecedent to our program: it fixes the quantum-side statistics that any candidate calculus must reproduce, and our counterexample calculus is evaluated against exactly such statistical properties. Our Section 4 shows that a thresholded classical readout can produce statistics outside this characterized set, which is the crux of the uniqueness question. The supplied summary states the axiomatic characterization but gives no further detail on specific axioms, so we rely on it only at the level stated.

[5] (arXiv:quant-ph/0012115v1) formulates quantum continual measurements using instruments, positive operator valued measures, quantum stochastic differential equations, and classical stochastic differential equations for vectors in Hilbert spaces or trace-class operators, and studies entropy and information gain; the summary also notes concurrent work by Ozawa, though the supplied text is truncated and gives no further detail on that point. The work is significant here because it demonstrates that classical SDEs for Hilbert-space objects and quantum stochastic equations are inter-derivable descriptions of the same measurement statistics — precisely the kind of bridge our paper interrogates. We use its framing of information gain qualitatively; the truncated summary prevents us from citing specific quantitative results from it.

[6] (arXiv:2307.09558v1) concerns automatic skeleton adjustment for self-avatars in virtual reality, addressing motion capture, trackers, and inverse kinematics in the metaverse context. This work is tangential to quantum foundations; we cite it only as an engineering instance of the same abstract operation our axiom (ii) posits — mapping continuous tracked signals onto a finite discrete structure (a skeleton) via a fixed nonlinear adjustment procedure. The summary supplied gives no further detail relevant to measurement theory, and we draw no foundational conclusion from it.

[7] (arXiv:1402.1217v2) examines entanglement and state disturbance arising in protective measurement and argues that these inescapable effects doom the claim that protective measurement establishes the reality of the wave function, adding that an exponential number of protective measurements would be required to reconstruct multi-qubit states. This is relevant to our ontological claim (i): it removes one route by which the wave function itself could be taken as the directly measured continuous object, reinforcing the need for an explicit readout map between whatever continuous reality is posited and the discrete record. The entry's summary is truncated, so we rely only on the stated conclusions about entanglement, disturbance, and exponential reconstruction cost.

[8] (arXiv:quant-ph/9809038v1) investigates foundations of quantum Turing machines, characterizes local transition functions for fully general quantum Turing machines, discusses preparation and measurement protocols, and proposes a halting protocol that works without spoiling the computation. The characterization of local transition functions parallels our concern with locality: a discrete, local transition structure sitting on top of a continuous or quantum substrate. The work shows that locality constraints on discrete readout and transition rules are formally tractable, which supports the well-posedness of our axiom (ii); the truncated summary prevents citation of its specific protocol details.

The QNFO corpus items [9]–[12] (POST QUANTUM SYNTHESIS; Principia Ontologica; Time as Epistemic Cognitive Fiction; and the PQS AI-Evaluation Audit) supply the provenance of the research idea: [12] is described as an archive of the research record investigating the Post-Quantum Synthesis framework, a neutral investigation of its four sub-pillars mapped against external literature, plus an earlier-iteration analysis of AI gate-check convergence between Claude and Gemini evaluating PQS. Entries [9], [10], and [11] carry no summary text in the supplied bibliography, so we cite them only as corpus context for the idea's origin and draw no substantive claims from them.

In summary, the literature supplies: rigorous axiomatic characterizations of quantum measurement statistics [4], dual quantum/classical stochastic descriptions of continuous measurement [2], [5], broad classical continuum modeling tools [1], realist precedents [3], constraints on state reconstruction [7], and discrete-protocol rigor [8]. None of the supplied entries derives the Born rule from a classical thresholding map, which is the gap this paper addresses.

#3. Methods

#3.1 Axioms

We adopt the proposal's axioms in explicit form.

• A1 (Continuity). The underlying state of the world at any time is a field configuration $x \in \mathbb{R}$ (one-dimensional here; the extension to fields on $\mathbb{R}^n$ is not needed for the results).
• A2 (Local determinism). The evolution $x(t)$ is generated by a local deterministic law; no superluminal influence enters the dynamics.
• A3 (Discrete irreversible readout). A measurement is a map $M: \mathbb{R} \to \mathcal{O}$ with $\mathcal{O}$ finite, implemented by nonlinear amplification followed by thresholding, and is irreversible (the pre-threshold value is not recoverable from the outcome).

#3.2 The readout map

A measurement device is a triple $(\mathcal{A}, g, \Theta)$ where $\mathcal{A}$ is a nonlinear amplifier acting on a continuous field variable $x \in \mathbb{R}$, $g \gt 0$ is a gain parameter, and $\Theta$ is a threshold. The amplifier produces an intermediate variable

and the readout map $M_\Theta: \mathbb{R} \to \mathcal{O} = \{0, 1\}$ for a two-outcome detector is the irreversible step function

Irreversibility enters because $M_\Theta$ is many-to-one and is not invertible on its image: the preimage of outcome $1$ is the set $M^{-1}(1) = \{x : \mathcal{A}_g(x) \ge \Theta\}$, and all continuous information within that set is discarded.

#3.3 Candidate calculi

A "calculus" in the sense of the research idea is a rule assigning outcome probabilities $P_k$ to each outcome $k \in \mathcal{O}$ given the continuous state. We consider two members of the thresholded-amplifier family:

• Linear calculus ($\mathcal{C}_L$): $\mathcal{A}_g(x) = g x$, so outcome $1$ occurs when $x \ge \Theta/g$.
• Quadratic calculus ($\mathcal{C}_Q$): $\mathcal{A}_g(x) = g x^2$, so outcome $1$ occurs when $|x| \ge \sqrt{\Theta/g}$ (the amplifier is insensitive to sign, as intensity-like detectors are).

Both calculi satisfy the axioms exactly: the underlying field is continuous, local, and deterministic; the readout is discrete, irreversible, and produced by nonlinear amplification and thresholding. The quadratic amplifier is nonlinear in the stronger sense required by the idea's wording; the linear amplifier is the degenerate (affine) member of the family, included as the minimal baseline.

#3.4 Field ensemble

We take the continuous field variable at the detector to be distributed, over an ensemble of nominally identical preparation runs, as a Gaussian with density

with $\sigma = 1$ in dimensionless units throughout. This is the minimal nontrivial ensemble: symmetric, continuous, and normalizable. No quantum postulate is used at this stage; $p(x)$ is simply the ensemble distribution of the classical field amplitude.

For the Born-rule comparison (Section 4.2) we additionally use the noisy threshold map with additive readout noise $\eta \sim \mathcal{N}(0, \sigma_n^2)$:

where $\Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z} e^{-u^2/2}\, du$ is the standard normal cumulative distribution function.

#3.5 Target statistics and locality test

The quantum predictions to be tested are:

• Born rule (two-outcome, single parameter). For a qubit-like system prepared at angle $\alpha$ relative to the detector basis, $P_1^{\mathrm{QM}}(\alpha) = \cos^2(\alpha/2) = \frac{1 + \cos\alpha}{2}$.
• CHSH correlation. For two parties with settings $a, a'$ and $b, b'$, the quantum correlation is $E_{\mathrm{QM}}(a,b) = -\cos(a - b)$ for the singlet-type state, and the CHSH combination is $S = E(a,b) + E(a,b') + E(a',b) - E(a',b')$.

For the CHSH test we consider two spatially separated detectors each implementing a factorized readout: detector $A$ outputs $A(a, \lambda) \in \{\pm 1\}$ as a deterministic function of its local setting $a$ and the local field value $\lambda_A$, and detector $B$ outputs $B(b, \lambda) \in \{\pm 1\}$ similarly, with no dependence of $A$ on $b$ or of $B$ on $a$ (locality of the readout, inherited from axiom A2). The correlation is

where $\rho(\lambda)$ is the distribution of the shared field configuration.

#4. Analysis

All numbers in this section are computed explicitly from the inputs stated above. The only external inputs are the standard normal cumulative distribution values $\Phi(1) = 0.8413$ and $\Phi(2) = 0.9772$, which are tabulated values of the Gaussian integral, and the standard mathematical constants $\sqrt{2} \approx 1.4142136$ and $\sqrt{\pi} \approx 1.7724539$.

#4.1 Outcome probability under the quadratic calculus

With gain $g = 1$ and threshold $\Theta = 1$, the quadratic calculus fires when $|x| \ge \sqrt{\Theta/g} = \sqrt{1} = 1$. The detection probability is

Using $\Phi(1) = 0.8413$:

#4.2 Outcome probability under the linear calculus

With the same $g = 1$, $\Theta = 1$, the linear calculus fires when $x \ge \Theta/g = 1$:

#4.3 Divergence between the calculi

The two calculi, both satisfying the axioms, give

a ratio

The factor of $2$ is structural, not coincidental: analytically, for any symmetric density $P_1^{(Q)} = 2\bigl(1 - \Phi(\Theta/g)\bigr)$ while $P_1^{(L)} = 1 - \Phi(\Theta/g)$, so the ratio is exactly $2$ in the limit of exact tail symmetry; the rounded inputs give $0.3174/0.1587 \approx 1.9994$, which rounds to $2.000$. for a symmetric density, the quadratic threshold counts both tails while the linear threshold counts one, and at $\Theta = g$ the tail probabilities coincide. Both calculi are internally consistent, deterministic at the field level, and produce irreversible discrete records. Therefore the calculus connecting continuous reality to discrete outcomes is not unique within the axiom class; uniqueness fails unless an additional principle selects the amplifier.

#4.4 Born-rule approximation by a slope-matched threshold detector

We now ask how well the best locally calibrated noisy threshold detector matches the Born rule. We posit a classical field whose value at the detector is $x(\alpha) = \cos\alpha$ (a deterministic, local encoding of the preparation angle into the field amplitude — the most favorable linear encoding for the proposal), a threshold $\theta = 0$, and readout noise scale $\sigma_n$. The model's outcome probability is

We choose $\sigma_n$ by matching the small-amplitude slope of $P_1^{\mathrm{thr}}$ to that of the Born rule at $\alpha = \pi/2$, where both are centered at $1/2$. Expanding both around $\alpha = \pi/2$ (so $\cos\alpha \approx 0$):

Matching slopes gives $\frac{1}{\sigma_n\sqrt{2\pi}} = \frac{1}{2}$, i.e.

As a check, $\sqrt{2\pi} = \sqrt{2}\cdot\sqrt{\pi} = 1.4142136 \times 1.7724539 \approx 2.5066283$, so the slope factor is $\frac{1}{\sigma_n\sqrt{2\pi}} = \frac{1}{0.7978846 \times 2.5066283} = \frac{1}{2.0000000} = 0.5$ exactly, by construction.

We evaluate both statistics at five angles $\alpha \in \{0, \pi/6, \pi/4, \pi/3, \pi/2\}$. Required cosine values (standard): $\cos 0 = 1$, $\cos(\pi/6) = \frac{\sqrt{3}}{2} \approx 0.8660254$, $\cos(\pi/4) = \frac{\sqrt{2}}{2} \approx 0.7071068$, $\cos(\pi/3) = 0.5$, $\cos(\pi/2) = 0$.

Born rule values $P_1^{\mathrm{QM}}(\alpha) = \frac{1 + \cos\alpha}{2}$:

• $\alpha = 0$: $\frac{1 + 1}{2} = 1.0000$.
• $\alpha = \pi/6$: $\frac{1 + 0.8660254}{2} = 0.9330127$.
• $\alpha = \pi/4$: $\frac{1 + 0.7071068}{2} = 0.8535534$.
• $\alpha = \pi/3$: $\frac{1 + 0.5}{2} = 0.7500000$.
• $\alpha = \pi/2$: $\frac{1 + 0}{2} = 0.5000000$.

Thresholded-model values $P_1^{\mathrm{thr}}(\alpha) = \Phi(\cos\alpha / 0.7978846)$. We compute the arguments $z = \cos\alpha / 0.7978846$ and evaluate $\Phi$ using the standard normal table (values interpolated linearly between tabulated hundredths):

• $\alpha = 0$: $z = 1 / 0.7978846 = 1.2533141$. $\Phi(1.25) = 0.8944$, $\Phi(1.26) = 0.8962$; interpolation at $1.2533$: $0.8944 + 0.33 \times 0.0018 = 0.8950$.
• $\alpha = \pi/6$: $z = 0.8660254 / 0.7978846 = 1.0854371$. $\Phi(1.08) = 0.8599$, $\Phi(1.09) = 0.8621$; at $1.0854$: $0.8599 + 0.54 \times 0.0022 = 0.8611$.
• $\alpha = \pi/4$: $z = 0.7071068 / 0.7978846 = 0.8862269$. $\Phi(0.88) = 0.8106$, $\Phi(0.89) = 0.8133$; at $0.8862$: $0.8106 + 0.62 \times 0.0027 = 0.8123$.
• $\alpha = \pi/3$: $z = 0.5 / 0.7978846 = 0.6266571$. $\Phi(0.62) = 0.7324$, $\Phi(0.63) = 0.7357$; at $0.6267$: $0.7324 + 0.67 \times 0.0033 = 0.7346$.
• $\alpha = \pi/2$: $z = 0$. $\Phi(0) = 0.5000$ exactly.

Deviations $\Delta(\alpha) = P_1^{\mathrm{thr}}(\alpha) - P_1^{\mathrm{QM}}(\alpha)$:

• $\alpha = 0$: $0.8950 - 1.0000 = -0.1050$.
• $\alpha = \pi/6$: $0.8611 - 0.9330 = -0.0719$.
• $\alpha = \pi/4$: $0.8123 - 0.8536 = -0.0413$.
• $\alpha = \pi/3$: $0.7346 - 0.7500 = -0.0154$.
• $\alpha = \pi/2$: $0.5000 - 0.5000 = 0.0000$.

Root-mean-square deviation over the five points:

The largest single deviation is $|\Delta(0)| = 0.1050$ at $\alpha = 0$, where the Born rule demands certainty ($P = 1$) but the noisy threshold map yields $\Phi(1.2533) \approx 0.8950$. This is not a numerical artifact: for any finite $\sigma_n$, $\Phi(1/\sigma_n) \lt 1$, so the thresholded Gaussian map can never reproduce the Born rule's certainty at $\alpha = 0$; and for $\sigma_n \to 0$ the map becomes a step function, reproducing certainty but destroying the linear central regime. The two requirements are in direct tension.

#4.5 CHSH bound for deterministic local thresholded models

In a deterministic local model, each outcome is a function of the local setting and $\lambda$: $A = A(a,\lambda)$, $A' = A(a',\lambda)$, $B = B(b,\lambda)$, $B' = B(b',\lambda)$, each in $\{-1, +1\}$. The CHSH combination for a fixed $\lambda$ is

Since $B, B' \in \{-1, +1\}$: if $B = B'$ then $B + B' = \pm 2$ and $B - B' = 0$, so $S_\lambda = \pm 2A = \pm 2$. If $B = -B'$ then $B + B' = 0$ and $B - B' = \pm 2$, so $S_\lambda = \pm 2A' = \pm 2$. In all cases $|S_\lambda| \le 2$. Averaging over $\lambda$ with any distribution $\rho(\lambda)$ preserves the bound:

This holds for every member of the class defined by Axioms A1–A3 with deterministic threshold readout, since thresholding a local field value yields a deterministic local outcome function. Stochastic readout noise does not help: averaging a noisy outcome is equivalent to averaging over an enlarged $\lambda$ that includes the noise, which is still local.

Quantum value. Take $a = 0$, $a' = \pi/2$, $b = \pi/4$, $b' = -\pi/4$ (standard choices; inputs are the angles themselves). With $E_{\mathrm{QM}}(a,b) = -\cos(a - b)$:

• $E(a, b) = -\cos(0 - \pi/4) = -\cos(\pi/4) = -0.7071068$.
• $E(a, b') = -\cos(0 + \pi/4) = -\cos(\pi/4) = -0.7071068$.
• $E(a', b) = -\cos(\pi/2 - \pi/4) = -\cos(\pi/4) = -0.7071068$.
• $E(a', b') = -\cos(\pi/2 + \pi/4) = -\cos(3\pi/4) = +0.7071068$.

(Check: $2\sqrt{2} = 2 \times 1.4142136 = 2.8284271$.) The gap is $2.8284 - 2 = 0.8284$, i.e. the quantum prediction exceeds the deterministic local thresholded-model bound by a factor $2\sqrt{2}/2 = \sqrt{2} \approx 1.4142$.

#4.6 Excess noise cannot close the gap

One might ask whether increasing the noise $\sigma_n$ (making the threshold map more "linear") could push $|S|$ past 2. It cannot within Axioms A1–A3 with local dynamics: the CHSH derivation above nowhere used the value of $\sigma_n$; it used only locality and determinism of the outcome function per $\lambda$. Noise enlarges $\lambda$ but does not change the bound. To exceed $|S| = 2$ the model would need either nonlocal dynamics (violating A2) or a non-threshold, non-discretizing readout (violating A3). This is the precise sense in which the uniqueness claim fails: the axioms A1–A3 admit the thresholded Gaussian calculus, which is empirically distinguishable from quantum mechanics both in single-system statistics (Section 4.4) and in entanglement correlations (Section 4.5).

#4.7 Statistical power projection for distinguishing $\mathcal{C}_L$ from $\mathcal{C}_Q$

This subsection is a projection, not a measurement. Assumptions: (i) each calculus is run $n$ times under identical preparation; (ii) outcomes are Bernoulli with probabilities $P_1^{(Q)} = 0.3174$ and $P_1^{(L)} = 0.1587$; (iii) we require a $3$-standard-deviation separation of the two observed frequencies. The pooled proportion is

and the standard error of the difference of two proportions with $n$ trials each is

Requiring $3\,\mathrm{se} \le \Delta P = 0.1587$:

Thus approximately $n = 130$ trials per calculus suffice for a $3\sigma$ discrimination under these assumptions. The uncertainty in this projection is dominated by the assumed Gaussian ensemble; for heavier-tailed field distributions the tail probabilities — and hence $\Delta P$ and $n$ — change, though the factor-of-two structure at $\Theta = g$ persists for any symmetric density.

#4.8 Sensitivity to threshold

For completeness, at threshold $\Theta = 4$, $g = 1$: the quadratic calculus fires when $|x| \ge 2$, giving

while the linear calculus gives

The factor of $2$ is again exact, confirming that the non-uniqueness is not an artifact of the particular threshold chosen.

#2. Background and Related Work

The proposed ontology — continuous, local, deterministic fields with discrete irreversible readout — sits at the intersection of several literatures. We review the supplied bibliography, restricting every statement to what each entry's own title and summary state.

[1] develops applications of fractional calculus in continuum and statistical mechanics, including viscoelasticity and the Basset problem. This work exemplifies the mature classical machinery available for continuous field dynamics, the substrate on which the proposed ontology rests; its summary supplies no further detail relevant to measurement.

[2] treats continuous-time measurement, open quantum systems, and the stochastic Schrödinger equation in the diffusive case. It is the standard continuous-measurement counterpart to the classical threshold readout studied here: where [2] keeps the state continuous under measurement back-action, the present paper discretizes at the readout stage, and the contrast clarifies exactly which step of the proposed axioms does the discretizing.

[3] proposes a realist process interpretation unifying quantum physics and the biological sciences, treating the quantum physicist's and the biologist's modes of description as complementary. This supports the present framing that a continuous underlying reality and discrete operational records may be related by a mapping whose form is an additional postulate rather than a uniqueness theorem; the entry's summary gives no technical detail beyond this interpretive claim.

[4] develops an axiomatic approach to measurement theory and characterizes the statistical properties of apparatuses measuring observables with nondegenerate spectrum. It is the closest formal relative of our Axioms A1–A3: it fixes what a measurement apparatus is by its statistical action, whereas we fix it by a threshold mechanism, and the difference is precisely where our counterexample to uniqueness lives.

[5] treats quantum continual measurements via instruments, POVMs, and quantum/classical stochastic differential equations. Together with [2] it supplies the operational calculus that quantum mechanics itself uses for continuous-to-discrete transitions; our results show that the proposed axiom class admits at least two calculi where this quantum calculus is one specific member.

[6] concerns automatic skeleton adjustment for self-avatars in virtual reality. Although a far-from-foundations application, it is a concrete instance of the same abstract pattern studied here — a continuous tracked signal mapped irreversibly onto a discrete outcome space (skeleton poses) — and its summary supplies no further theoretical content.

[7] examines entanglement and disturbance arising in protective measurement, together with state-reconstruction cost. Protective measurement is one of the few proposed windows onto the continuous wavefunction itself, so [7] marks the empirical edge at which the continuous substrate of the proposed ontology could in principle be probed; the entry's summary gives no quantitative results we can cite.

[8] treats foundations of quantum Turing machines, including local transition functions, preparation, measurement, and halting protocols. Its insistence on locality of transition functions parallels our use of local outcome functions $A(a,\lambda)$, $B(b,\lambda)$ in the CHSH derivation of Section 4.5; the entry's summary supplies no further detail on its measurement axioms.

Entries [9], [10], and [11] (POST QUANTUM SYNTHESIS, Principia Ontologica, Time as Epistemic Cognitive Fiction) are corpus items for which no summary is supplied; we therefore relate them to our argument only as the document corpus from which the research idea under test originates, and make no claim about their content. Entry [12] is an archive of the research record investigating the Post-Quantum Synthesis framework and AI gate-check convergence; it is the provenance of the uniqueness claim formalized in Section 3, and its summary supplies no technical detail beyond this role.

#3. Methods

#3.1 Axioms

We formalize the research idea as three axioms on a measurement event:

• A1 (Continuous local deterministic substrate). The state of the world relevant to a detector is a classical field value $x \in \mathbb{R}$, evolved by local deterministic dynamics.
• A2 (Locality). A detector's outcome depends only on its local setting (e.g. analyzer angle $a$) and the local field value; it does not depend on distant settings or distant outcomes.
• A3 (Irreversible discrete readout). The outcome is produced by a nonlinear amplification followed by thresholding, mapping $\mathbb{R} \to \mathcal{O} = \{0, 1\}$ irreversibly.

#3.2 Two calculi within the axiom class

A calculus is a rule assigning outcome probabilities to field ensembles. Within A1–A3 we define two:

• $\mathcal{C}_L$ (linear amplifier): fires ($O = 1$) iff $g\,x \ge \Theta$, for gain $g \gt 0$ and threshold $\Theta \gt 0$.
• $\mathcal{C}_Q$ (quadratic, intensity-like amplifier): fires iff $g\,x^2 \ge \Theta$, i.e. iff $|x| \ge \sqrt{\Theta/g}$.

Both are deterministic functions of the local field value, hence satisfy A2 and A3; $\mathcal{C}_Q$ is strictly nonlinear and $\mathcal{C}_L$ is its affine limit, both within the stated class.

#3.3 Field ensemble and threshold model

Over an ensemble of nominally identical preparations the field value at the detector is distributed as a unit-variance Gaussian,

\nwith $\Phi(z) = \int_{-\infty}^{z} p(u)\,du$ the standard normal CDF. For the Born-rule comparison we use the noisy threshold map with readout noise $\eta \sim \mathcal{N}(0, \sigma_n^2)$:

#4. Analysis

All numbers below are derived from the stated inputs; no simulation or external data are used.

#4.1 Outcome probability under $\mathcal{C}_Q$

With $g = 1$, $\Theta = 1$, $\mathcal{C}_Q$ fires iff $|x| \ge 1$:

Using the standard normal value $\Phi(1) = 0.8413$:

#4.2 Outcome probability under $\mathcal{C}_L$

With the same $g = 1$, $\Theta = 1$, $\mathcal{C}_L$ fires iff $x \ge 1$:

#4.3 Divergence between the calculi

The factor of exactly $2$ is structural: for a symmetric density the quadratic threshold counts both tails while the linear threshold counts one, and at $\Theta = g$ the two tail probabilities coincide. Both calculi satisfy A1–A3 exactly, yet give different statistics; the calculus connecting continuous reality to discrete outcomes is therefore not unique within the axiom class unless an additional principle selects the amplifier.

#4.4 Born-rule approximation by a slope-matched threshold detector

We posit the most favorable linear encoding $x(\alpha) = \cos\alpha$, threshold $\theta = 0$, and noise scale $\sigma_n$, giving $P_1^{\mathrm{thr}}(\alpha) = \Phi(\cos\alpha/\sigma_n)$. Matching the small-amplitude slope to the Born rule $P_1^{\mathrm{QM}}(\alpha) = \frac{1 + \cos\alpha}{2}$ at $\alpha = \pi/2$:

\nso $\frac{1}{\sigma_n\sqrt{2\pi}} = \frac{1}{2}$, i.e.

Check: $\sigma_n\sqrt{2\pi} = 0.7978846 \times 2.5066283 = 2.0000000$, so the slope factor is $0.5$ by construction.

At the five angles $\alpha \in \{0, \pi/6, \pi/4, \pi/3, \pi/2\}$ the cosines are $1$, $0.8660254$, $0.7071068$, $0.5$, $0$. Born rule values $\frac{1+\cos\alpha}{2}$: $1.0000$, $0.9330127$, $0.8535534$, $0.7500000$, $0.5000000$. Threshold-model values $\Phi(\cos\alpha/0.7978846)$, with arguments $z = 1.2533141$, $1.0854371$, $0.8862269$, $0.6266571$, $0$ and standard-normal interpolation: $\Phi(1.2533) \approx 0.8944 + 0.33 \times 0.0018 = 0.8950$; $\Phi(1.0854) \approx 0.8599 + 0.54 \times 0.0022 = 0.8611$; $\Phi(0.8862) \approx 0.8106 + 0.62 \times 0.0027 = 0.8123$; $\Phi(0.6267) \approx 0.7324 + 0.67 \times 0.0033 = 0.7346$; $\Phi(0) = 0.5000$.

Deviations $\Delta(\alpha) = P_1^{\mathrm{thr}} - P_1^{\mathrm{QM}}$: $-0.1050$, $-0.0719$, $-0.0413$, $-0.0154$, $0.0000$. The RMS deviation is

The largest deviation, $|\Delta(0)| = 0.1050$, is structural: for any finite $\sigma_n$, $\Phi(1/\sigma_n) \lt 1$, so the map cannot reproduce the Born rule's certainty at $\alpha = 0$; taking $\sigma_n \to 0$ restores certainty but destroys the linear central regime.

#4.5 CHSH bound for deterministic local thresholded models

For deterministic local outcomes $A = A(a,\lambda)$, $A' = A(a',\lambda)$, $B = B(b,\lambda)$, $B' = B(b',\lambda) \in \{-1, +1\}$:

If $B = B'$ then $B + B' = \pm 2$ and $B - B' = 0$, so $S_\lambda = \pm 2A = \pm 2$; if $B = -B'$ then $S_\lambda = \pm 2A' = \pm 2$. In all cases $|S_\lambda| \le 2$, and averaging over any $\rho(\lambda)$ preserves $|S| \le 2$. Thresholded local readouts are deterministic local outcome functions, so the bound holds for the whole class; readout noise merely enlarges $\lambda$.

Quantum value. With $a = 0$, $a' = \pi/2$, $b = \pi/4$, $b' = -\pi/4$ and $E_{\mathrm{QM}}(a,b) = -\cos(a-b)$: $E(a,b) = -\cos(\pi/4) = -0.7071068$; $E(a,b') = -\cos(\pi/4) = -0.7071068$; $E(a',b) = -\cos(\pi/4) = -0.7071068$; $E(a',b') = -\cos(3\pi/4) = +0.7071068$. Then

\nexceeding the local bound $2$ by $0.8284$, a factor $\sqrt{2} \approx 1.4142$.

#4.6 Excess noise cannot close the gap

The CHSH derivation used only locality and per-$\lambda$ determinism, never the value of $\sigma_n$; enlarging the noise enlarges $\lambda$ but cannot push $|S|$ past $2$. Exceeding $|S| = 2$ would require nonlocal dynamics (violating A2) or a non-thresholding readout (violating A3).

#4.7 Statistical power projection for distinguishing $\mathcal{C}_L$ from $\mathcal{C}_Q$

This subsection is a projection, not a measurement. Assumptions: $n$ Bernoulli trials per calculus with $P_1^{(Q)} = 0.3174$, $P_1^{(L)} = 0.1587$, and a required $3\sigma$ separation. The pooled proportion is

\nand the standard error of the difference of two proportions is

Requiring $3\,\mathrm{se} \le \Delta P = 0.1587$:

#4.8 Sensitivity to threshold

At $\Theta = 4$, $g = 1$: $\mathcal{C}_Q$ fires iff $|x| \ge 2$, giving $P_1^{(Q)} = 2(1 - \Phi(2)) = 2(1 - 0.9772) = 0.0456$, while $\mathcal{C}_L$ gives $P_1^{(L)} = 1 - \Phi(2) = 0.0228$. The factor of $2$ is again exact, confirming that the non-uniqueness is not an artifact of the chosen threshold.

#5. Results

All numerical results below are those computed in Section 4; no simulation or experimental data are reported.

1. Non-uniqueness of the readout calculus. Two distinct calculi, $\mathcal{C}_L$ (linear amplifier) and $\mathcal{C}_Q$ (quadratic amplifier), both satisfy Axioms A1–A3 exactly, yet yield $P_1^{(L)} = 0.1587$ and $P_1^{(Q)} = 0.3174$ at $\Theta = 1$, $g = 1$ (Sections 4.1–4.2), a structural factor of exactly $2.000$ (Section 4.3). The same factor holds at $\Theta = 4$ ($0.0228$ versus $0.0456$, Section 4.8). The readout calculus is therefore not unique within the axiom class.

1. Born-rule approximation is imperfect. A slope-matched noisy threshold detector with $\sigma_n = 0.7978846$ deviates from the Born rule by up to $|\Delta(0)| = 0.1050$ at $\alpha = 0$, with $\mathrm{RMS} = 0.0602$ over the five-point grid $\alpha \in \{0, \pi/6, \pi/4, \pi/3, \pi/2\}$ (Section 4.4). The deviation at $\alpha = 0$ is structural: no finite-noise threshold map attains $P = 1$.

1. CHSH bound. Every deterministic local thresholded readout obeys $|S| \le 2$, while the quantum prediction at the standard angles is $|S| = 2\sqrt{2} \approx 2.8284$, a gap of $0.8284$ (Section 4.5). Readout noise cannot close this gap (Section 4.6).

1. Distinguishability projection. Under the stated assumptions (Bernoulli outcomes with $P_1^{(Q)} = 0.3174$ and $P_1^{(L)} = 0.1587$, $3\sigma$ separation), approximately $n = 130$ trials per calculus suffice to distinguish the two conventions (Section 4.7). This is a projection, not a measurement.

#6. Discussion

Limitations. The analysis is deliberately minimal, and each minimality choice is a limitation. First, the field ensemble is a unit-variance Gaussian; for heavier-tailed or asymmetric ensembles the absolute probabilities $P_1^{(L)}$ and $P_1^{(Q)}$ change, although the factor-of-two structure at $\Theta = g$ persists for any symmetric density (Section 4.7). Second, the Born-rule comparison uses the single most favorable linear encoding $x(\alpha) = \cos\alpha$; other encodings were not surveyed, so the reported deviations ($0.1050$ maximum, $0.0602$ RMS) bound only this encoding. Third, the statistical power figure $n \approx 130$ is a projection resting on Gaussian approximations to Bernoulli frequencies and on the assumed ensemble; it carries no experimental validation.

Failure modes of the argument. The non-uniqueness result would fail if the research idea's wording were read as fixing the amplifier class more narrowly than Axioms A1–A3 do — for example, if nonlinearity were required to be strictly nonlinear (excluding the affine $\mathcal{C}_L$) or if a physical principle (such as energy-like detection) were stipulated to select $\mathcal{C}_Q$ uniquely. The Born-rule mismatch would fail to refute anything if the proposal never claimed to reproduce quantum statistics exactly. The CHSH argument would fail if the readout were allowed to depend on distant settings, i.e. if Axiom A2 were relaxed.

What would falsify our claims. A demonstration that some deterministic, local, continuous-field model with thresholded discrete readout reproduces the Born rule exactly at all angles and violates $|S| \le 2$ would falsify our central negative claims. Conversely, an argument that the axioms A1–A3 implicitly force a unique amplifier (e.g. via a symmetry or positivity principle) would restore the uniqueness claim and refute our counterexample.

Open questions. (i) Do additional physically motivated principles select one member of the thresholded family? (ii) Can stochastic (non-deterministic) local readout evade the CHSH bound while remaining within a generalized A3? (iii) What is the exact best-approximation error of the thresholded family to the Born rule over all encodings, not just the linear one?

#7. Conclusion

We formalized the proposed continuous-to-discrete readout map, exhibited two distinct calculi within the stated axiom class that yield different outcome statistics ($0.1587$ versus $0.3174$), showed that a slope-matched threshold detector approximates but does not reproduce the Born rule (maximum deviation $0.1050$, RMS $0.0602$), and proved the CHSH bound $|S| \le 2$ for all deterministic local thresholded readouts against the quantum value $2\sqrt{2} \approx 2.8284$. The uniqueness claim connecting continuous local deterministic fields plus irreversible discrete readout to quantum mechanics is therefore refuted at the level of the readout calculus: quantum statistics are one member of a broader family, and additional structure beyond the stated axioms is required to select them.

#References

[1] Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. arXiv:1201.0863v1. https://arxiv.org/abs/1201.0863v1 [2] Quantum continuous measurements: The stochastic Schroedinger equations and the spectrum of the output. arXiv:1301.3626v2. https://arxiv.org/abs/1301.3626v2 [3] Beyond Quantum Theory: A Realist Psychobiological Interpretation of Physical Reality. arXiv:1610.09347v1. https://arxiv.org/abs/1610.09347v1 [4] Quantum Measurement, Information, and Completely Positive Maps. arXiv:quant-ph/0107090v1. https://arxiv.org/abs/quant-ph/0107090v1 [5] Entropy and information gain in quantum continual measurements. arXiv:quant-ph/0012115v1. https://arxiv.org/abs/quant-ph/0012115v1 [6] Fitted avatars: automatic skeleton adjustment for self-avatars in virtual reality. arXiv:2307.09558v1. https://arxiv.org/abs/2307.09558v1 [7] Entanglement, scaling, and the meaning of the wave function in protective measurement. arXiv:1402.1217v2. https://arxiv.org/abs/1402.1217v2 [8] Quantum Turing Machines: Local Transition, Preparation, Measurement, and Halting. arXiv:quant-ph/9809038v1. https://arxiv.org/abs/quant-ph/9809038v1 [9] QNFO: POST QUANTUM SYNTHESIS [10] QNFO: Principia Ontologica [11] QNFO: Time as Epistemic Cognitive Fiction [12] QNFO: PQS AI-Evaluation Audit: Post-Quantum Synthesis Investigation and AI Gate-Check Analysis

#Appendix A. Divergence report

The source drafts agreed on all computed values ($P_1^{(L)} = 0.1587$, $P_1^{(Q)} = 0.3174$, factor $2.000$, $\sigma_n = 0.7978846$, max deviation $0.1050$, RMS $0.0602$, $|S| \le 2$ versus $2.8284$, $n \approx 130$). One divergence arose in framing: draft A presented the Born-rule mismatch as an independent refutation of uniqueness, while draft B treated it as a subordinate corollary of the CHSH bound. Resolution: the reconciled text follows draft B's more conservative framing — the primary refutation is the direct two-calculus counterexample of Sections 4.1–4.3, and the Born-rule mismatch (Section 4.4) is reported as an additional quantitative tension rather than a standalone refutation. No numerical values diverged between drafts.

#Appendix B. Claim attribution

secondary to the two-calculus counterexample. The reconciled text adopts draft B's convention — the two-calculus counterexample (Section 4.3) is the primary refutation, and the Born-rule mismatch (Section 4.4) is corroborating evidence — because the counterexample is logically self-sufficient while the mismatch depends on the choice of encoding. A second divergence concerned whether the linear amplifier counts as "nonlinear amplification" under Axiom A3; the reconciled text includes it explicitly as the degenerate affine member of the family and flags this reading in Section 6 as a potential failure mode.

#Appendix B. Claim attribution