ZF Plots: Semantic and Signal Perspectives

• ZF Plots are representations that capture distinct meanings, serving as semantic maps in set theory and error-rate visualizations in communications.
• They enable analysis of algebra-valued models by contrasting Boolean and paraconsistent assignments, impacting validity, equality, and quotient formation.
• In engineering and simulation, ZF Plots diagnose system performance in pilot-aided communication and fast calorimeter simulation, highlighting practical trade-offs and computational gains.

The available literature suggests that “ZF plots” is not a single standardized term. In one technically central usage, it denotes a semantic “map” or plot of ZF validity across algebra-valued models of set theory, where the underlying universe of names is held fixed and the interpretation of $=$ and $\in$ is varied (Jockwich et al., 2024). In a distinct engineering usage, ZF means Zero-Forcing, and the relevant plots are symbol-error-rate visualizations for equalizers built from reconstructed channel matrices under localization mismatch (Han et al., 27 Nov 2025). A further adjacent usage appears in high-energy-physics fast simulation, where “ZDC/ZF-style plots and metrics” are used to assess whether a generative surrogate preserves stochastic detector-response structure and shower morphology while achieving substantial speedup (Majerz et al., 23 Dec 2025). The term therefore requires domain-specific disambiguation.

1. Scope and principal usages

Two meanings dominate the supplied literature. The first belongs to logic and set theory, where “plot” is primarily structural: it is a semantic organization of the class of models in which ZF, or a classically equivalent strengthening, remains valid. The second belongs to signal processing and communications, where “plot” is literal: a figure displaying SER trends for Zero-Forcing receivers under changing system parameters and estimation errors.

Usage	Object organized or displayed	Representative source
ZF as Zermelo–Fraenkel set theory	A semantic “map” or plot of ZF validity across algebraic environments	(Jockwich et al., 2024)
ZF as set theory without choice in topology	Compactifications in ZF; Urysohn-type separation in generalized topological spaces in ZF	(Keremedis et al., 2018, Hejduk et al., 2021)
ZF as Zero-Forcing	SER plots for ZF and MMSE receivers in a PASCAL system	(Han et al., 27 Nov 2025)
Adjacent “ZDC/ZF-style” plotting usage	Sample detector responses, channel summaries, and shower-morphology metrics	(Majerz et al., 23 Dec 2025)

This multiplicity is not merely terminological. In the set-theoretic usage, the “plot” is a classification device for semantic validity. In the communications usage, the plots are empirical or semi-analytic diagnostics of performance. In the detector-simulation usage, the figures and metrics are used to determine whether a fast surrogate preserves the physically meaningful dependence of a calorimeter response on incoming particle features.

2. Semantic plots of ZF in algebra-valued models

The most explicit set-theoretic use of the phrase appears in “Interpretations of ZF,” where the central claim is that what counts as a model of ZF depends not only on the underlying algebra of truth values, but also on how the basic predicates $=$ and $\in$ are interpreted (Jockwich et al., 2024). The universe $V^{(A)}$ is fixed; what varies is the assignment function

$$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$

and validity is taken relative to a designated filter $D \subseteq A$: $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$

The paper compares two recursive assignments. The standard Boolean-style assignment $\cdot_{BA}$ uses the familiar algebra-valued clauses

$$u \in v_{BA}=\bigvee_{x\in \mathrm{dom}(v)}\big(v(x)\wedge x=u_{BA}\big),$$

and

$\in$0

The paraconsistent assignment $\in$1 preserves the same recursive definition of membership but strengthens equality by adding clauses involving the negation-like operation $\in$2. The stated purpose is to encode both assertion and denial in weak or non-classical environments, where classically equivalent formulations can diverge.

This yields the paper’s semantic “map” or plot of ZF validity across different algebraic environments. On Boolean algebras, the two interpretations coincide: $\in$3 so the modified interpretation is conservative over ordinary Boolean-valued semantics. In ultra-designated cobounded algebras, however, the new interpretation validates a strengthened axiom system $\in$4, classically equivalent to ZF, thereby extending the scope of full set-theoretic validity beyond the Boolean-valued case.

A common misconception in this area is that the classical Boolean-valued semantics already exhausts the relevant model-theoretic picture. The paper directly contradicts that view: Boolean-valued models are presented as only one corner of a broader semantic landscape, and the extension is achieved by altering the interpretation function rather than the ontology of names.

3. Equality, quotienting, and the expansion of the ZF model landscape

The semantic plot in the algebra-valued setting is driven by two technical problems: failure of Leibniz’s law and the obstruction this creates for quotienting by equality. The modified paraconsistent assignment is designed to restore Leibniz’s law in ultra-designated cobounded algebras, and the paper states that $\in$5, where

$\in$6

is Leibniz’s law (Jockwich et al., 2024). Because the usual extensionality axiom is insufficient under the stronger equality notion, the paper replaces it by a classically equivalent strengthened form requiring agreement of both membership and negated membership: $\in$7

Once Leibniz transfer is recovered, equality becomes a genuine equivalence relation,

$\in$8

and the quotient $\in$9 can be formed. The induced relations $=$0 and $=$1 are representative-independent, but membership can remain paraconsistent: there may be $=$2 with both $=$3 and $=$4. One of the most striking consequences is therefore a quotient model with familiar set-theoretic structure and non-classical membership behavior.

A broader consistency-theoretic expansion of the ZF landscape appears in Krivine’s classical realizability construction, which builds models of $=$5 from realizability algebras consisting of terms, stacks, and a subset $=$6 of bad processes (Krivine, 2010). The method is explicitly described as analogous to forcing, but computational rather than order-theoretic. The paper realizes ZF, realizes dependent choice via non-extensional choice, and produces models in which $=$7 is not well-orderable, $=$8 is nontrivial and atomless, and there exist sequences of infinite subsets of $=$9 with strictly increasing and strictly decreasing cardinal behavior, as well as incomparable cardinalities.

Taken together, these results show that a “ZF plot” in the set-theoretic sense is not confined to a single semantics or a single method of model construction. One branch varies the interpretation of the language while fixing $\in$0; another uses realizability algebras to obtain new models of $\in$1 with highly non-classical behavior of the continuum.

4. ZF as ambient theory in topology and generalized topology

The topological papers do not use “plot” as a formal term, but they organize another structural picture of ZF: a map of which classical theorems survive, fail, or bifurcate in the absence of the axiom of choice. In “Hausdorff compactifications in ZF,” the algebra

$\in$2

is the basic invariant (Keremedis et al., 2018). In ZFC, equality of $\in$3 and $\in$4 classifies Hausdorff compactifications. In ZF, that can fail. The paper states that it is relatively consistent with ZF that there exist a Tychonoff space $\in$5 and Hausdorff compactifications $\in$6 and $\in$7 such that

$\in$8

It also states that Glicksberg’s theorem may fail in some models of ZF by using amorphous sets, and that the existence of certain $\in$9-embedded remainders implies van Douwen’s choice principle.

The same paper gives exact ZF criteria for when a family $V^{(A)}$0 generates a compactification through the evaluation embedding

$V^{(A)}$1

and defines a functional Čech–Stone compactification by the requirement

$V^{(A)}$2

The existence of classical $V^{(A)}$3 for all Tychonoff spaces is shown to be choice-theoretic: the paper states that UFT is equivalent to every Tychonoff space having a Čech–Stone compactification.

A related separation of classical implications occurs in generalized topology. “On Urysohn’s Lemma for generalized topological spaces in ZF” works with strong generalized topological spaces $V^{(A)}$4, where $V^{(A)}$5 is stable under arbitrary unions and contains $V^{(A)}$6 and $V^{(A)}$7 (Hejduk et al., 2021). For the generalized codomain topology

$V^{(A)}$8

the paper gives an exact ZF criterion for $V^{(A)}$9: for every pair of non-empty disjoint closed sets, there must exist a rational-indexed family $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$0 satisfying closure nesting, inclusion of one closed set, and avoidance of the other. It proves that every effectively normal GT space satisfies $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$1 in ZF, and that in $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$2, every U-normal GT space satisfies the classical $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$3.

The counterexample $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$4 is especially instructive. It is normal and satisfies both $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$5 and $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$6, yet $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$7 fails. This sharply separates generalized Urysohn phenomena in ZF from their classical topological analogues. A plausible implication is that, in topology as in algebra-valued semantics, the “ZF plot” is best understood as a validity landscape: standard ZFC equivalences split apart, and the surviving structure depends sensitively on the ambient choice principles and on the exact form of the semantics.

5. Zero-Forcing performance plots in pilot-aided simultaneous communication and localization

In communications, ZF means Zero-Forcing. “On the SER Performance of ZF and MMSE Receivers in Pilot-Aided Simultaneous Communication and Localization” studies a multiuser uplink MU-SIMO PASCAL system with $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$8 single-antenna drones and a base station with $$\cdot^{A} : \{\text{sentences of } \mathcal{L}_A\} \to A$$9 antennas (Han et al., 27 Nov 2025). The base station uses pilot symbols to estimate each drone’s DOA or angle $D \subseteq A$0, range $D \subseteq A$1, and Doppler $D \subseteq A$2, reconstructs the channel matrix from these estimated parameters, and then uses that reconstructed channel in ZF or MMSE equalization. The ZF equalizer is defined as the Moore–Penrose pseudoinverse of the estimated channel.

The paper’s central plotting problem is that the exact SER is analytically intractable because the estimated channel matrix depends on multiple random localization errors and ZF requires a matrix inverse. The adopted solution is a hybrid approximation method incorporating Neumann approximation and Taylor approximation. The resulting figures are explicitly “ZF plots” in the standard engineering sense: plots of SER against localization accuracy, SNR, or number of users.

One figure studies “The effect of estimation errors of a) DOA $D \subseteq A$3, b) Doppler frequency $D \subseteq A$4, c) Range $D \subseteq A$5 on SER.” The setup is two drones, a BS with $D \subseteq A$6 antennas, SNR $D \subseteq A$7 dB, QPSK, and pilots varying from 22 to 40. Both simulated and analytical curves are plotted. The observed trends are that SER increases as localization RMSE increases, MMSE outperforms ZF across the whole range, MRC shows an error floor, and the analytical and simulated results match closely using 3rd-order Neumann approximation and 8th-order Taylor approximation. The paper’s theoretical conclusions are that the average SER of ZF is unaffected by range estimation errors, that angle estimation errors are the dominant source of degradation, and that the SER of drone $D \subseteq A$8 depends on localization errors from all drones, not only drone $D \subseteq A$9.

A second figure isolates the effect of other users’ localization errors. With $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$0, 2 drones, 30 pilots, and drone 1 observed in the presence of drone 2 as interferer, the paper reports for BPSK at $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$1 dB an ideal ZF SER of $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$2 and a ZF SER of $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$3 when drone 2’s localization error is present. MRC remains unchanged at $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$4. This is a direct graphical demonstration that ZF depends on interferers’ localization errors because mismatch enters through the inverse of the estimated channel.

A third figure compares actual results, perfect DOA estimation, and perfect localization. At $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$5 dB and $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$6, the reported ZF SER values are $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$7 with full localization errors, $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$8 with perfect DOA, and $$V^{A,\cdot^A} \models_D \varphi \quad \text{iff} \quad \varphi^A \in D.$$9 with perfect localization. The interpretation given is that once DOA errors are removed, performance becomes close to the ideal case, so $\cdot_{BA}$0 is the dominant source of degradation. Additional figures with 3 and 4 drones show that ZF can perform worse than MRC at low SNR and that the gap between ZF and MMSE widens as the number of drones increases. One common misconception is therefore rejected explicitly: ZF is not always the best linear equalizer in PASCAL, and under significant estimation errors it may even be worse than MRC.

6. “ZDC/ZF-style” plots and metrics in fast calorimeter simulation

A different plotting practice appears in “Inverse Autoregressive Flows for Zero Degree Calorimeter fast simulation,” which develops a fast surrogate for the ALICE Zero Degree Calorimeter, specifically the neutron sub-detector ZN, using an Inverse Autoregressive Flow student trained from a Masked Autoregressive Flow teacher in a teacher-student normalizing-flow framework (Majerz et al., 23 Dec 2025). The ZDC is located at a distance of 112.5 m from the ALICE interaction point, and the full GEANT4-based transport simulation is physically accurate but computationally expensive. The ZN response is represented as a 1-channel 44×44 image whose pixel values correspond to the number of detected photons in each fibre, and the stochasticity of the detector response makes the problem a natural target for generative modeling.

The explicit figure in the paper is Figure 1, captioned: “Sample results generated with IAF students. We present two generated samples for each input vector to show that the responses differ between runs.” The point of the figure is not merely visual plausibility. It shows that the same input can produce different valid outputs and therefore that the student remains stochastic rather than collapsing to a deterministic map. This is treated as physically important because transport can involve decays and other physical processes.

The paper supplements the figure with geometry-aware diagnostics. The main global-distribution metric is the mean 1-Wasserstein distance between five physically defined channel sums: a checkerboard-like sum over every second fibre and four quadrant sums over a $\cdot_{BA}$1 partition. Shower morphology is summarized by the shower centre and the radius containing 90% of the total photons, where centre is the energy-weighted centre of mass and radius is the 90% containment radius. The metrics $\cdot_{BA}$2 and $\cdot_{BA}$3 test whether the learned conditional dependence between each unique input vector $\cdot_{BA}$4 and the averaged channel outputs is physically correct.

The methodological novelty lies in two physics-based additions. The first is the channel loss

$\cdot_{BA}$5

introduced to encourage the student to learn the global shower position and shape rather than only pixel-level similarity. The second is an output-variability-based scaling mechanism that downweights rare artefacts while preserving emphasis on common, physically important regions of the dataset. The paper explicitly warns that a lower Wasserstein distance does not necessarily mean better physics, because a model can match coarse output distributions while breaking the dependence between particle inputs and detector responses.

The quantitative results connect the plots to practical deployment. The headline result is a 421× speedup, with generating time per sample of 0.38 ms versus a previously reported ZDC NF time of 160.0 ms. For the full dataset, the bs+ch+div setup achieves $\cdot_{BA}$6, $\cdot_{BA}$7, and Wasserstein $\cdot_{BA}$8. The same setup is reported as best in all compared position and shape metrics for the full dataset, including centre error MAE 1.63 ± 0.01 and radius error MAE 2.26 ± 0.01. In the paper’s own summary, these “ZDC/ZF-style plots and metrics” are interpreted as evidence that the model captures both the global response distribution and the local shower physics while delivering a major computational gain.

A final methodological caution parallels the communications literature. In both settings, a single scalar score is explicitly treated as insufficient. In the PASCAL system, approximate SER must be read together with its dependence on angle, Doppler, and cross-user mismatch. In the ZDC surrogate, Wasserstein distance must be read together with conditional channel fidelity and shower morphology. The shared lesson is that “ZF plots,” whether semantic or graphical, are most informative when they preserve the structure that the underlying domain regards as physically or logically essential.