Forbidden Zone: Cross-Disciplinary Insights

• Forbidden zone is defined as a region in phase, energy, or parameter space where otherwise allowed processes are suppressed by specific mechanisms such as interference, selection rules, or singularities.
• In collider physics and semiconductor optics, forbidden zones diagnose system dynamics through amplitude cancellations and weak optical transitions, providing actionable experimental insights.
• Across dark matter models, space physics, and discrete dynamics, forbidden zones highlight regimes of viable reactions or singular breakdowns, underscoring their cross-disciplinary significance.

Searching arXiv for the provided topic and cited papers to ground the article in current records. “Forbidden zone” is a domain-dependent technical term denoting a region in phase space, energy, configuration space, or parameter space where an otherwise admissible process is suppressed, undefined, or classically excluded. Across high-energy collider phenomenology, semiconductor optics, dark-matter freeze-out, near-Earth space physics, and discrete dynamical systems, the term consistently marks a boundary imposed by interference, selection rules, thermal kinematics, magnetospheric topology, or singular iteration. In each usage, the forbidden zone is not merely an absence region; it is a structurally informative locus whose boundaries encode the governing dynamics and often furnish experimentally or computationally useful diagnostics (Hagiwara et al., 2012, Woods-Robinson et al., 2023, Yang, 2022, Suvorova et al., 2014, Palladino, 2012).

1. Conceptual structure of forbidden zones

The broadest common definition is a region in which a process is prevented or strongly attenuated by a specific mechanism. In collider physics, photon emission can vanish because tree-level helicity amplitudes cancel through destructive interference under special kinematics (Hagiwara et al., 2012). In semiconductor optics, absorption can be suppressed even when the joint density of states is nonzero because the relevant interband transitions are electric-dipole forbidden or extremely weak (Woods-Robinson et al., 2023). In thermal dark-matter models, annihilation into heavier states is kinematically forbidden at zero temperature yet becomes active at finite temperature through the Boltzmann tail, so the forbidden zone is a viable relic-density region in model parameter space rather than a literal exclusion region (Yang, 2022). In magnetospheric physics, the forbidden zone is a geographic and magnetic region below the inner radiation belt where high-energy electrons are not expected to be stably trapped under quiet conditions (Suvorova et al., 2014). In rational difference equations, the forbidden set—used synonymously with “forbidden zone”—is the subset of initial conditions for which forward iteration hits a singularity in finite time (Palladino, 2012).

These usages differ in ontology but share a common formal role. Each forbidden zone identifies a structured null, suppression, or singular locus produced by constraints intrinsic to the theory. This suggests that the term is best understood not as a single physical concept but as a recurring analytical pattern: the emergence of sharply delineated regions where naive kinematic, topological, or algebraic accessibility fails.

2. Amplitude-cancellation forbidden zones at hadron colliders

In the context of the Large Hadron Collider, the forbidden zone arises from the null radiation zone theorem for processes of the form $a+b\to 1+2+V$, where an Abelian vector current is emitted. The theorem states that all tree-level helicity amplitudes vanish when the charge-weighted eikonal denominators associated with radiation from all external charged legs become equal, together with charge conservation (Hagiwara et al., 2012). In the formulation given for initial legs $i\in\{a,b\}$, final legs $f\in\{1,2\}$, and emitted vector momentum $p_V$,

$$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$

for all $i,f$, with

$\sum_i Q_i \;=\;\sum_f Q_f,$

guarantees destructive interference of all helicity amplitudes (Hagiwara et al., 2012).

For a massless photon, real solutions exist only when all charges have the same sign. In the identical-charge case $Q_a=Q_b=Q_1=Q_2$, the conditions simplify to

$$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$

This is the collider-theory prototype of a forbidden zone: a phase-space region in which emission is “forbidden” by interference, not by conservation-law impossibility (Hagiwara et al., 2012).

The specific manifestation studied at the $\sqrt{s}=8$ TeV LHC is $i\in\{a,b\}$0. At large $i\in\{a,b\}$1, the process is dominated by identical-quark scattering, especially $i\in\{a,b\}$2, and the null radiation zone is characterized by small

$i\in\{a,b\}$3

and near-zero

$i\in\{a,b\}$4

where $i\in\{a,b\}$5 and $i\in\{a,b\}$6 is the rapidity of the di-jet cluster (Hagiwara et al., 2012). Under these conditions, the gauge-invariant $i\in\{a,b\}$7- and $i\in\{a,b\}$8-channel diagram sets cancel. More generally, when $i\in\{a,b\}$9 and $f\in\{1,2\}$0 but $f\in\{1,2\}$1, the amplitude in this region is proportional to $f\in\{1,2\}$2, and the cross section scales like $f\in\{1,2\}$3, so identical-charge scattering exhibits the strongest suppression (Hagiwara et al., 2012).

The study used MadGraph 5 at leading order with CTEQ6L1 PDFs, applying the cuts $f\in\{1,2\}$4, $f\in\{1,2\}$5, photon acceptance $f\in\{1,2\}$6 or $f\in\{1,2\}$7, $f\in\{1,2\}$8, and $f\in\{1,2\}$9 (Hagiwara et al., 2012). A tight null-zone condition,

$p_V$0

enhances the effect. With this selection, the ratio

$p_V$1

with interference, compared with

$p_V$2

when interference between the $p_V$3- and $p_V$4-channel sets is removed (Hagiwara et al., 2012). With $p_V$5, the paper estimates observability at roughly $p_V$6 statistically under the tighter rapidity cut, versus about $p_V$7 for $p_V$8 (Hagiwara et al., 2012).

This forbidden zone is closely related to, but distinct from, the classic radiation amplitude zero in electroweak $p_V$9 production. The latter is tied to gauge cancellations involving non-Abelian structure and a fixed angular locus, whereas the $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$0 null radiation zone is an external-leg bremsstrahlung zero in a $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$1 process requiring rapidity alignment and transverse-momentum balance (Hagiwara et al., 2012).

3. Forbidden optical transitions and transparency windows in semiconductors

In semiconductor optics, the forbidden zone is an energy interval near the fundamental gap where interband absorption is suppressed because the lowest transitions are electric-dipole forbidden or extremely weak (Woods-Robinson et al., 2023). The relevant microscopic quantity is the dipole or velocity matrix element appearing in Fermi’s Golden Rule. In the independent-particle approximation,

$$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$2

and the absorption coefficient is approximately related to $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$3 by

$$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$4

When the matrix element is suppressed, the optical onset can occur substantially above the fundamental direct gap (Woods-Robinson et al., 2023).

The high-throughput study of approximately $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$5 semiconductors showed that over half exhibit forbidden or weak transitions at their band edges, as quantified by $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$6 (Woods-Robinson et al., 2023). About $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$7 have $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$8 and $$\frac{Q_i}{\,2\,p_i\!\cdot p_V - p_V^2\,}\;=\;\frac{Q_f}{\,2\,p_f\!\cdot p_V + p_V^2\,}\;=\;\text{(common value)}$$9 have $i,f$0, indicating a sizable forbidden zone (Woods-Robinson et al., 2023). Two descriptors were introduced:

$i,f$1

the gap between the lowest direct transition and the lowest direct allowed transition, and

$i,f$2

where $i,f$3 is defined by $i,f$4 (Woods-Robinson et al., 2023). The paper further recommends the average visible absorption metric

$i,f$5

arguing that absorption-based figures are more informative for transparent-conductor screening than electronic gap alone (Woods-Robinson et al., 2023).

The physical origin of these forbidden zones lies in selection rules and orbital localization. Electric-dipole transitions require a change of parity and angular momentum $i,f$6. In crystals with inversion symmetry at the band extrema, same-parity transitions are forbidden. Even when symmetry does not strictly forbid the transition, oscillator strength can be strongly reduced by weak spatial overlap and similar orbital character of the initial and final states (Woods-Robinson et al., 2023). The study quantified localization using inverse participation ratios and an orbital-overlap descriptor

$i,f$7

finding that transitions between two localized band-edge states exhibit large average forbidden energy differences, whereas delocalized-edge cases cluster near $i,f$8 (Woods-Robinson et al., 2023).

The practical consequence is a widened transparency window beyond what the nominal band gap would suggest. This is particularly important for transparent conductors because it partially decouples optical and transport requirements. The paper stresses that this mechanism is already known in commercial n-type oxides such as In$i,f$9O$\sum_i Q_i \;=\;\sum_f Q_f,$0 and SnO$\sum_i Q_i \;=\;\sum_f Q_f,$1, and extends the design principle to p-type transparent conductors (Woods-Robinson et al., 2023).

4. Screening methodology and material realizations in transparent conductors

The transparent-conductor study implemented a three-stage workflow combining optical screening, HSE-refined band-edge assessment, and defect and transport evaluation (Woods-Robinson et al., 2023). Screen 1 used GGA IPA spectra and selected materials by either HSE-shifted $\sum_i Q_i \;=\;\sum_f Q_f,$2 or $\sum_i Q_i \;=\;\sum_f Q_f,$3 in the nominal visible range $\sum_i Q_i \;=\;\sum_f Q_f,$4–$\sum_i Q_i \;=\;\sum_f Q_f,$5 at the GGA level, or by $\sum_i Q_i \;=\;\sum_f Q_f,$6 below ITO’s value $\sum_i Q_i \;=\;\sum_f Q_f,$7. Effective-mass filters were $\sum_i Q_i \;=\;\sum_f Q_f,$8 for n-type and $\sum_i Q_i \;=\;\sum_f Q_f,$9 for p-type materials (Woods-Robinson et al., 2023). Screen 2 refined $Q_a=Q_b=Q_1=Q_2$0, $Q_a=Q_b=Q_1=Q_2$1, and IPA spectra with HSE06 scissor shifts and imposed $Q_a=Q_b=Q_1=Q_2$2, $Q_a=Q_b=Q_1=Q_2$3, or $Q_a=Q_b=Q_1=Q_2$4 below ITO, together with branch-point-energy constraints $Q_a=Q_b=Q_1=Q_2$5 for p-type and $Q_a=Q_b=Q_1=Q_2$6 for n-type (Woods-Robinson et al., 2023). Screen 3 evaluated defect formation and transport using pycdt, HSE band-edge corrections, selected hybrid defect calculations in CP2K/HSE06, and amset transport with polar optical phonon, ionized impurity, and acoustic deformation potential scattering (Woods-Robinson et al., 2023).

Representative candidates identified through forbidden or weak band-edge transitions span ambipolar, p-type, and n-type classes (Woods-Robinson et al., 2023).

Class	Material	Selected descriptors
Ambipolar	BeSiP$Q_a=Q_b=Q_1=Q_2$7	$Q_a=Q_b=Q_1=Q_2$8, $Q_a=Q_b=Q_1=Q_2$9, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$0, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$1
Ambipolar	Zr$$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$2SN$$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$3	$$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$4, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$5, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$6, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$7
Ambipolar	KSe	$$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$8, $$y(p_1{+}p_2)-y(p_V)=0,\quad p_{T1}=p_{T2},\quad p_V^2=0.$$9, $\sqrt{s}=8$0, $\sqrt{s}=8$1
p-type	BAs	$\sqrt{s}=8$2, $\sqrt{s}=8$3, $\sqrt{s}=8$4
p-type	Au$\sqrt{s}=8$5S	$\sqrt{s}=8$6, $\sqrt{s}=8$7, $\sqrt{s}=8$8
p-type	AuCl	$\sqrt{s}=8$9, $i\in\{a,b\}$00, $i\in\{a,b\}$01
n-type	Ba$i\in\{a,b\}$02InGaO$i\in\{a,b\}$03	$i\in\{a,b\}$04, $i\in\{a,b\}$05, $i\in\{a,b\}$06
n-type	GaSbO$i\in\{a,b\}$07	$i\in\{a,b\}$08, $i\in\{a,b\}$09, $i\in\{a,b\}$10
n-type	KSbO$i\in\{a,b\}$11	$i\in\{a,b\}$12, $i\in\{a,b\}$13, $i\in\{a,b\}$14

The study also reported that the canonical TCOs SnO$i\in\{a,b\}$15 and In$i\in\{a,b\}$16O$i\in\{a,b\}$17 are correctly retained by the absorption-based metrics despite modest GGA gaps. For SnO$i\in\{a,b\}$18, $i\in\{a,b\}$19 and $i\in\{a,b\}$20; for In$i\in\{a,b\}$21O$i\in\{a,b\}$22, $i\in\{a,b\}$23 and $i\in\{a,b\}$24, which the paper notes compares favorably with the literature-quoted $i\in\{a,b\}$25 of suppressed absorption near the valence-band edge in ITO (Woods-Robinson et al., 2023).

Several methodological limits are explicitly noted. The optical spectra are IPA-level and neglect excitonic effects, phonon-assisted indirect absorption, spin-orbit coupling, free-carrier Drude absorption, and defect-related subgap absorption (Woods-Robinson et al., 2023). Even so, the work argues that $i\in\{a,b\}$26, $i\in\{a,b\}$27, and $i\in\{a,b\}$28 are cheap, stable, and predictive descriptors at high throughput (Woods-Robinson et al., 2023).

5. Kinematically forbidden annihilation in hidden-sector dark matter

In hidden-sector dark-matter phenomenology, the forbidden zone is the region of parameter space in which dark matter annihilates primarily into heavier final states during freeze-out. The process is kinematically forbidden in the zero-temperature limit but proceeds at finite temperature due to the high-energy tail of the thermal distribution (Yang, 2022). The model studied consists of a hidden $i\in\{a,b\}$29 gauge boson $i\in\{a,b\}$30 as dark matter and a hidden scalar $i\in\{a,b\}$31 that breaks the gauge symmetry and mixes with the Standard Model Higgs via a scalar portal (Yang, 2022). The vector mass is

$i\in\{a,b\}$32

and the dominant depletion channel is

$i\in\{a,b\}$33

with

$i\in\{a,b\}$34

The thermal average of the forbidden process is obtained from detailed balance. For $i\in\{a,b\}$35,

$i\in\{a,b\}$36

The defining suppression factor is therefore $i\in\{a,b\}$37 with $i\in\{a,b\}$38 (Yang, 2022). The standard Boltzmann equation,

$i\in\{a,b\}$39

then yields a freeze-out condition with $i\in\{a,b\}$40 numerically in the range $i\in\{a,b\}$41–$i\in\{a,b\}$42 for $i\in\{a,b\}$43 (Yang, 2022). The relic abundance condition selects the required reverse-channel cross section $i\in\{a,b\}$44, which spans roughly $i\in\{a,b\}$45–$i\in\{a,b\}$46 depending on $i\in\{a,b\}$47 and $i\in\{a,b\}$48 (Yang, 2022).

The principal boundary of the forbidden zone is set by perturbative unitarity,

$i\in\{a,b\}$49

because the coupling cannot be increased indefinitely to compensate the Boltzmann penalty (Yang, 2022). The paper’s main quantitative conclusion is that larger mass splittings are possible in the sub-GeV regime, with viable $i\in\{a,b\}$50 up to $i\in\{a,b\}$51–$i\in\{a,b\}$52 for $i\in\{a,b\}$53, whereas in the TeV regime the splitting must be within the percent level (Yang, 2022). For $i\in\{a,b\}$54 and $i\in\{a,b\}$55, the required $i\in\{a,b\}$56 is about $i\in\{a,b\}$57 and is weakly dependent on the scalar-Higgs mixing angle $i\in\{a,b\}$58; for larger splittings such as $i\in\{a,b\}$59, thermal contact with the Standard Model can require larger $i\in\{a,b\}$60 or else cannibal dynamics become relevant before freeze-out (Yang, 2022).

Direct detection proceeds through the scalar portal with spin-independent cross section

$i\in\{a,b\}$61

Because the relic-density determination fixes $i\in\{a,b\}$62 over much of the viable region, current experimental limits can be mapped into upper bounds on $i\in\{a,b\}$63 (Yang, 2022). The paper states that PandaX-4T excludes sizable $i\in\{a,b\}$64 for $i\in\{a,b\}$65, that the bound approaches the Standard Model neutrino floor at $i\in\{a,b\}$66–$i\in\{a,b\}$67, and that LZ projections will exclude additional slices of the forbidden zone (Yang, 2022). Mediator searches from LEP/L3, LHCb, kaon decays, CHARM, SHiP projections, and BBN lifetime constraints further carve the parameter space (Yang, 2022).

This usage of “forbidden zone” is therefore unusual among the cases considered here: it names a viable, not excluded, region in which forbidden-at-zero-temperature annihilation controls cosmological freeze-out.

6. Forbidden zones in near-Earth radiation environments and discrete dynamics

In low-Earth-orbit space physics, the forbidden zone denotes the equator-to-low-latitude region below the inner radiation belt where drift shells for energetic electrons do not stably exist under quiet conditions because they intersect the atmosphere in the South Atlantic Anomaly (Suvorova et al., 2014). Using NOAA/POES measurements from 2003–2005, the study documented major storm-related enhancements of $i\in\{a,b\}$68–$i\in\{a,b\}$69 electrons in this region (Suvorova et al., 2014). Quiet-time fluxes at $i\in\{a,b\}$70 and altitudes $i\in\{a,b\}$71–$i\in\{a,b\}$72 are about $i\in\{a,b\}$73–$i\in\{a,b\}$74, but during major geomagnetic storms and extreme solar-wind compressions, $i\in\{a,b\}$75 fluxes increased by $i\in\{a,b\}$76–$i\in\{a,b\}$77 orders of magnitude, reaching intensities as high as $i\in\{a,b\}$78 (Suvorova et al., 2014). Enhancements occurred at any local time and could persist from about $i\in\{a,b\}$79 hour up to about $i\in\{a,b\}$80 hours for the $i\in\{a,b\}$81 channel, whereas $i\in\{a,b\}$82 bursts were typically shorter, on the order of about $i\in\{a,b\}$83 hours (Suvorova et al., 2014).

The electrons are interpreted as quasi-trapped within the drift loss cone rather than stably trapped. A likely source is rapid inward transport from the radiation belts during storms or compressions, plausibly driven by strong electric fields of order $i\in\{a,b\}$84 at $i\in\{a,b\}$85, although the study emphasizes that direct measurements in this region are sparse (Suvorova et al., 2014). For X-ray and gamma-ray astronomy, this forbidden zone becomes operationally important because the resulting low-latitude electron populations can rival auroral and SAA intensities and contaminate detectors even on orbits chosen specifically to avoid classical high-radiation regions (Suvorova et al., 2014).

In discrete mathematics, the corresponding notion is the forbidden set for rational difference equations (Palladino, 2012). For an order-$i\in\{a,b\}$86 rational difference equation,

$i\in\{a,b\}$87

the forbidden set $i\in\{a,b\}$88 is the set of initial conditions for which forward iteration becomes undefined at some finite step because a denominator vanishes (Palladino, 2012). The paper treats six second-order cases and introduces six algebraic invariants that reduce each equation to a first-order Riccati or linear recurrence, allowing both explicit closed forms and explicit forbidden sets (Palladino, 2012). For example, in Case 1,

$i\in\{a,b\}$89

with invariant

$i\in\{a,b\}$90

the dynamics reduce to

$i\in\{a,b\}$91

and the forbidden set is the union of initial conditions whose reduced Riccati iterate reaches the pole $i\in\{a,b\}$92 (Palladino, 2012). Across all six examples, the forbidden set is a countable union of algebraic curves in $i\in\{a,b\}$93, obtained as preimages of poles of Möbius or linear maps under iteration (Palladino, 2012).

These two cases—magnetospheric transport and rational recurrence singularity—illustrate opposite meanings of “forbidden.” In the former, the region is classically devoid of persistent trapping yet episodically populated under disturbance. In the latter, the region is a singular locus in initial-condition space beyond which the dynamics cease to be defined.

7. Comparative interpretation and recurring themes

Despite their disciplinary diversity, the forbidden zones described in these works share a small set of recurring structural motifs. First, they are defined relative to a baseline accessibility criterion: allowed radiation, allowed optical absorption, allowed annihilation, stable trapping, or well-defined forward iteration (Hagiwara et al., 2012, Woods-Robinson et al., 2023, Yang, 2022, Suvorova et al., 2014, Palladino, 2012). Second, the mechanism creating the forbidden zone is highly specific: destructive interference in $i\in\{a,b\}$94, dipole selection rules and suppressed oscillator strengths in semiconductors, thermal activation of kinematically forbidden channels in the early universe, disturbed geomagnetic transport below the inner radiation belt, or denominator singularities in rational maps (Hagiwara et al., 2012, Woods-Robinson et al., 2023, Yang, 2022, Suvorova et al., 2014, Palladino, 2012). Third, each forbidden zone is diagnostically useful. In collider phenomenology it provides a test of gauge-invariant interference patterns (Hagiwara et al., 2012). In materials discovery it furnishes absorption-based descriptors that recover known TCOs and expose overlooked p-type and ambipolar candidates (Woods-Robinson et al., 2023). In dark-matter phenomenology it identifies a relic-density mechanism testable through direct detection and mediator searches (Yang, 2022). In space instrumentation it flags a nontrivial source of charged-particle background at low latitudes (Suvorova et al., 2014). In difference equations it enables a geometric classification of singular initial data and a reduction-of-order program (Palladino, 2012).

A common misconception is that a forbidden zone is always an absolute exclusion region. The cited works show otherwise. The collider and optical cases involve suppression rather than impossibility (Hagiwara et al., 2012, Woods-Robinson et al., 2023). The dark-matter case is explicitly a viable mechanism activated by finite temperature (Yang, 2022). The space-physics case concerns a region forbidden to steady trapping but not to transient occupancy during disturbed conditions (Suvorova et al., 2014). The dynamical-systems case is the strictest, since the forbidden set marks truly undefined forward evolution (Palladino, 2012).

Taken together, these usages establish “forbidden zone” as a cross-disciplinary analytical term for structured absence. Whether it denotes a null-amplitude locus, a transparency window, a thermally activated relic-density regime, a disturbed low-latitude radiation environment, or a singular set of initial conditions, its scientific value lies in making the governing constraint visible.