Hidden Zero Uniqueness

• Hidden zero uniqueness is a paradigm where specific vanishing conditions on constrained sets yield rigidity and unique outcomes in fields such as quantum field theory, statistical physics, and analytic structures.
• It utilizes minimal additional information—like locality, pole structure, or translation invariance—to replace traditional criteria, thereby enhancing UV scaling, gauge invariance, and combinatorial analyses.
• By imposing recursive vanishing constraints and combinatorial rules, this approach achieves robust classification results, evidenced in scattering amplitudes, disordered spin systems, number theory, and topological invariants.

Hidden zero uniqueness refers to a family of mathematical and physical phenomena in which the structure, behavior, or classification of objects (such as functions, amplitudes, solutions, or states) is uniquely determined by the vanishing of those objects on special sets—termed "hidden zeros"—together with minimal additional information (such as locality, pole structure, or translation invariance). These hidden zeros are not typically associated with obvious or generic vanishing conditions but instead with constrained, often combinatorial or geometric subsets in kinematic, spatial, or configuration spaces. The consequences are typically uniqueness, rigidity, or strong classification results, often replacing or surpassing previously central criteria (such as unitarity, positivity, or completeness). This principle has emerged in fields as diverse as scattering amplitudes, cosmological wavefunctions, disordered spin systems, Fock spaces, number systems, and partial differential equations.

1. Hidden Zeros in On-Shell Scattering Amplitudes and Cosmological Wavefunctions

In perturbative quantum field theory, "hidden zeros" refer to special kinematic conditions under which on-shell scattering amplitudes or cosmological wavefunction coefficients vanish identically, even though there is no factorization pole or physical singularity. In flat-space $\mathrm{Tr}(\phi^3)$ and related theories, these loci arise as constraints such as $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$ for non-adjacent pairs, leading to vanishing of the amplitude on these codimension-one subspaces. More generally, in cosmological wavefunctions of $\phi^n$ theories, only very mild locality requirements together with the full set of graph-based hidden zeros—now viewed as vanishing on loci $p_{a \in G_L} \cdot p_{b \in G_R} = 0$ for graph splits—suffice to uniquely determine the tree-level wavefunction up to normalization, by enforcing a dual factorization or shuffle property (Li et al., 1 Apr 2026, Rodina, 2024).

Hidden zeros in this context have a direct correspondence with enhanced large-$z$ falloff under Britto-Cachazo-Feng-Witten (BCFW) shifts: imposing vanishing on these special loci is logically equivalent to requiring improved ultraviolet scaling for each subset of amplitude terms. This yields a tri-equivalence (vanishing on zero locus $\Longleftrightarrow$ enhanced BCFW/UV scaling $\Longleftrightarrow$ subset-vanishing structure), and forms the foundation for recursive uniqueness proofs in amplitudes and wavefunctions (Rodina, 2024). In Yang-Mills theory and gravity, gauge invariance and soft limits likewise play the role of hidden zeros, with uniqueness of the amplitude following from the enforcement of these conditions on $n-1$ legs (Rodina, 2016).

2. Hidden Zero Uniqueness in Spin Glasses and Random Surfaces

In disordered statistical physics, the hidden zero uniqueness phenomenon manifests in the classification of ground states and metastates in systems such as the random ferromagnetic Ising model on $\mathbb Z^d$ (Wehr et al., 2014). Here, hidden zeros correspond to the absence of nontrivial interfaces (regions where spin flips occur) in metastable ground states when subject to translation-covariance and measurability. The rigorous result is that for IID positive couplings with finite mean, every translation-covariant zero-temperature metastate is supported only on the two uniform configurations; all attempts to find nonconstant ground states through measurable, covariant selections are obstructed by the energetic hidden zeros associated with interface formation. The same rigidity applies to the random surface interpretation: no infinite minimal surface apart from the empty (all $+1$) or full (all $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$0) can be selected in an equivariant, measurable way. This is intrinsically a hidden zero uniqueness phenomenon: the only invariantly observable zero-energy objects are the trivial ones, and any attempt to detect others through measurable means must fail.

3. Uniqueness Criteria in Algebraic and Analytic Structures

Hidden zero uniqueness theorems pervade analytic and algebraic settings via zero and uniqueness sets:

• In Fock spaces $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$1 of entire functions, a sequence $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$2 may be a uniqueness set (only the zero function vanishes on $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$3), a zero set (there exists a nonzero function vanishing on $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$4), or a uniqueness set with zero excess (removal of a single point from $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$5 yields a nontrivial zero). Stable characterization of zero sets requires global summability of $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$6 (Aadi et al., 2022). If this condition fails on a subsequence, hidden zeros emerge that preclude zero set persistence under deletion, exemplifying hidden zero uniqueness constraints in analytic function theory.
• In number systems, unique $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$7-expansion is equivalent to the requirement that a trigonometric mask polynomial $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$8 avoids vanishing on a sufficient set relative to the $c_{ij}=-s_{ij}=-(p_i+p_j)^2=0$9-ary tree. A minimal cut set of zeros for $\phi^n$0 acts as a hidden zero, and its existence determines whether the expansion system is unique, generalizing classical periodicity and density questions in semigroup theory (Konyagin et al., 18 Feb 2025).
• In partial differential equations, hidden zero uniqueness manifests as rigidity results: for nonlinear Schrödinger equations and semilinear elliptic PDEs, if a solution vanishes on sufficiently "thin" discrete, codimension-one sets at two times (not on open sets), it must be identically zero (Kehle et al., 2022). These sets act as hidden zeros and underpin nonlinear analogues of classical uncertainty relations and Fourier uniqueness pairs.

4. Hidden Zeros, Uniqueness, and Combinatorics in Number Expansions

Beyond traditional base expansions, hidden zeros control the breakdown of uniqueness in legal decompositions from linear recurrence sequences with zero coefficients (ZLRRs). When the leading coefficient $\phi^n$1—for instance, in the Lagonacci sequence—the classical local carry rules are absent, leading to exponentially many admissible decompositions of integers despite preservation of some global statistical structure (Gaussian number of summands, geometric gap decay) (Salami, 16 Apr 2026). The forbidden patterns that would enforce uniqueness in positive linear recurrences are now allowed to "hide," generating exponentially many combinatorially distinct solutions. The transfer-matrix method quantifies this nonuniqueness: the number of legal decompositions grows like $\phi^n$2 for sequence length $\phi^n$3, substantially outpacing the growth rate of the recurrence itself.

5. Hidden Zero Modes in Topological Phases of Non-Hermitian Systems

A physical realization of hidden zero uniqueness occurs in the spectral topology of one-dimensional non-Hermitian Hamiltonians (Monkman et al., 2024). In this setting, the "hidden zero modes" are not exact eigenenergy zero modes at any finite chain length, but normalized states $\phi^n$4 such that $\phi^n$5 as $\phi^n$6. These modes are protected by the singular-value spectrum rather than the conventional eigenvalue spectrum, and their number is determined by a topological index (winding number of $\phi^n$7 for Bloch Hamiltonian $\phi^n$8). The exact counting of hidden zero modes is given by the $\phi^n$9-splitting theorem, relating vanishing singular values to the sum of kernel dimensions of $p_{a \in G_L} \cdot p_{b \in G_R} = 0$0 and its reflected counterpart $p_{a \in G_L} \cdot p_{b \in G_R} = 0$1. The hidden zero modes constitute long-lived edge excitations that are insensitive to the fragile eigenvalue crossing criteria, and thus represent a robust topological invariant in non-Hermitian systems.

6. Logical Structure and Methodologies of Hidden Zero Uniqueness

Across domains, the logical flow underpinning hidden zero uniqueness has a common structure:

1. General ansatz construction: Formulating the most general object (amplitude, solution, function, expansion) compatible with minimal structural or locality constraints.
2. Imposition of hidden zeros: Demanding vanishing (or invariance) on precisely determined subspaces, configurations, or sets—hidden zeros—often induced by symmetry, analytic, or combinatorial principles.
3. Recursive constraints: Utilizing the vanishing conditions to infer factorization (e.g., shuffle decompositions in amplitudes), enforce relations among coefficients (e.g., in cubic graph expansions), or eliminate nonphysical solutions.
4. Uniqueness conclusion: Demonstrating that sufficient imposition of hidden zeros reduces the solution space to a unique (or maximally constrained) functional form, often up to an overall scale or trivial redundancy.
5. Equivalences: Establishing that hidden zeros are equivalent to alternative well-known principles, such as enhanced BCFW scaling, stability criteria, gauge invariance, or topological invariants.

The hidden zero uniqueness paradigm allows for derivation or bootstrap of physical and mathematical structures without recourse to traditional completeness, unitarity, or positive-definiteness, emphasizing instead the discriminatory power of vanishing on non-generic, carefully constructed sets.

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Key References:

• Cosmological/amplitude hidden zeros and dual shuffle factorization: (Li et al., 1 Apr 2026, Rodina, 2024, Rodina, 2016)
• Disordered spin systems and metastate triviality: (Wehr et al., 2014)
• Hidden zeros in analytic/number-theoretic structures: (Aadi et al., 2022, Kehle et al., 2022, Konyagin et al., 18 Feb 2025, Salami, 16 Apr 2026)
• Hidden zero modes and topological bulk-boundary correspondence: (Monkman et al., 2024)