--- title: Constrained PSLQ Search url: https://www.emergentmind.com/topics/constrained-pslq-search type: topic --- # Constrained PSLQ Search Searching arXiv for the cited PSLQ-related papers to ground the article. Constrained PSLQ search denotes a family of integer-relation problems in which the classical PSLQ objective—finding a nonzero coefficient vector \(a\) such that \(a_1x_1+\cdots+a_nx_n=0\)—is supplemented by explicit restrictions on the admissible data, coefficient set, search dimension, stopping rule, or candidate basis. Taken together, the literature shows four recurrent forms of constraint: empirical input with certified error bounds and thresholded termination; bounded search over unknown algebraic degree and coefficient height; coefficient-domain restriction to rings of algebraic integers; and theory-driven searches in small logarithmic bases for multiplicative closed forms [1707.05037], [1403.4026], [1809.06063], [2603.27613]. ## 1. Constraint classes and problem formulations The common feature of constrained PSLQ search is that the search is not conducted over unrestricted exact arithmetic on a fixed vector. Instead, one fixes a relation class and then enforces auxiliary conditions that determine what counts as an admissible candidate and what counts as a valid termination. | Constraint class | Representative formulation | Representative source | |---|---|---| | Empirical input | \(\|\bm\alpha-\bar{\bm\alpha}\|_2<\varepsilon_1\) | [1707.05037] | | Coefficient-size bound | Relation norm bounded by \(M\) | [1707.05037], [1403.4026] | | Degree bound | Search over \((1,\alpha,\dots,\alpha^i)\), \(i=2,\dots,d\) | [1403.4026] | | Coefficient-domain restriction | \(a_j\in \mathcal O_{\mathbb K}\) for quadratic \(\mathbb K\) | [1809.06063] | | Basis restriction | PSLQ on \((\ln|C_M|,\ln\Gamma(\cdot),\ln\pi,\ln2,\ln3,1)\) | [2603.27613] | In the empirical-data setting, the search is constrained by finite precision, bounded perturbation of the hyperplane matrix, prescribed stopping thresholds, and output-quality guarantees. In the algebraic-number reconstruction setting, the search is constrained by a degree bound \(d\) and a height or norm bound \(M\). In the algebraic-integer setting, the coefficients are forced to lie in \(\mathcal O_{\mathbb K}\) rather than \(\mathbb Z\) or \(\mathbb Z[i]\). In constant recognition, the search space is deliberately narrowed to a small, theory-motivated logarithmic basis and a prescribed coefficient box [1707.05037], [1403.4026], [1809.06063], [2603.27613]. This suggests that “constrained PSLQ search” is best understood as a design pattern rather than a single algorithmic variant. The core PSLQ mechanism remains recognizable, but the admissible relation set and the correctness criteria are altered by external structural information. ## 2. Empirical-data PSLQ and residual-certified termination For a real vector \(\bm\alpha\in\mathbb R^n\), an integer relation is a nonzero vector \(\bm m\in\mathbb Z^n\setminus\{\bm0\}\) such that \[ \langle \bm\alpha,\bm m\rangle=0. \] The empirical-data problem distinguishes the intrinsic vector \(\bm\alpha\) from an approximation \(\bar{\bm\alpha}\) satisfying \[ \|\bm\alpha-\bar{\bm\alpha}\|_2<\varepsilon_1. \] The practical objective is not to find a relation of \(\bar{\bm\alpha}\), since \(\bar{\bm\alpha}\) typically has no exact integer relation at all, but to recover an exact relation for the unknown \(\bm\alpha\) from inexact input [1707.05037]. The geometric object underlying PSLQ is the hyperplane matrix \(\bm H_\alpha=(h_{i,j})\in\mathbb R^{n\times(n-1)}\), defined from \[ s_j^2=\sum_{k=j}^n \alpha_k^2,\qquad j=1,\dots,n, \] by \[ h_{i,j}= \begin{cases} 0, & 1\le i2/\sqrt{3}\), BergmanSwap chooses \(r\) maximizing \(\gamma^r|h_{r,r}|\); after swapping rows \(r\) and \(r+1\), the local quantities are \[ \eta=h_{r,r},\qquad \beta=h_{r+1,r},\qquad \lambda=h_{r+1,r+1},\qquad \delta=\sqrt{\beta^2+\lambda^2}. \] The central constraint introduced for empirical data is termination. Exact PSLQ stops when \(h_{n,n-1}=0\), but with empirical input that condition generally never occurs. The thresholded variant \(PSLQ_\varepsilon\) instead terminates when \[ |h_{n,n-1}|<\varepsilon_2. \] The key invariant is that if \[ \bm H(k)=\bm A \bm H_\alpha \bm Q \] and \[ (z_1(k),\dots,z_n(k))=(\alpha_1,\dots,\alpha_n)\bm A^{-1}, \] then \[ |z_{n-1}(k)| \le \sqrt{\alpha_{n-1}^2+\alpha_n^2}\,|h_{n,n-1}(k)|. \] Since the \((n-1)\)-st column of \(\bm B=\bm A^{-1}\) is the candidate relation \(\bm m\), \(z_{n-1}(k)\) is exactly \(\langle\bm\alpha,\bm m\rangle\). Thus thresholded termination yields the certified residual bound \[ |\langle \bm\alpha,\bm m\rangle| \le \sqrt{\alpha_{n-1}^2+\alpha_n^2}\,\varepsilon_2. \] The perturbation analysis is expressed in terms of a perturbed lower-trapezoidal matrix \(\overline{\bm H}_\alpha\) satisfying \[ \|\overline{\bm H}_\alpha-\bm H_\alpha\|_F \le \varepsilon_3. \] For the leading \((n-1)\times(n-1)\) submatrix \(\bm H_{[1..n-1]}\), the exact Frobenius formulas \[ \|\bm H_{[1..n-1]}^{-1}\|_F^2 = (n-2)+\frac{\|\bm\alpha\|^2}{\alpha_n^2}, \qquad \|\bm H_{[1..n-1]}\|_F^2 = (n-2)+\frac{\alpha_n^2}{\|\bm\alpha\|^2} \] govern conditioning. If \[ \varepsilon_3 < \frac{1}{\|\bm H_{[1..n-1]}^{-1}\|_F}, \] then \(\overline{\bm H}_{[1..n-1]}\) is nonsingular and \[ \|\overline{\bm H}_{[1..n-1]}^{-1}\|_F \le \frac{1}{1-\varepsilon_3\|\bm H_{[1..n-1]}^{-1}\|_F} \|\bm H_{[1..n-1]}^{-1}\|_F. \] The main forward error theorem states that if \[ \|\bm H_\alpha-\overline{\bm H}_\alpha\|_F<\varepsilon_3< \frac{\alpha_n}{2\sqrt{(n-2)\alpha_n^2+1}}, \] and \(PSLQ_\varepsilon\) terminates with \(|h_{n,n-1}|<\varepsilon_2\), then for the returned \(\bm m\), \[ |\langle \bm\alpha,\bm m\rangle| < C\bigl(\|\bm m\|\varepsilon_3+\alpha_n\varepsilon_2\bigr), \qquad C=\frac{2\bigl(\sqrt{(n-2)\alpha_n^2+1}+\alpha_n\bigr)}{\alpha_n}. \] The two terms separate input or hyperplane perturbation from threshold-induced residual error. If a separate gap bound \(\delta\) is known such that every nonzero value \(|\langle\bm\alpha,\bm m\rangle|\) is at least \(\delta\), then \[ C\bigl(\|\bm m\|\varepsilon_3+\alpha_n\varepsilon_2\bigr)<\delta \] certifies that \(\bm m\) is an exact relation. The practical error-control theorem specializes to \(\overline{\bm H}_\alpha=\bm H_{\bar\alpha}\). Assuming \(\|\bm\alpha\|=1\), \(|\alpha_n|=\max_i|\alpha_i|\), and a target relation bound \(M\), the sufficient conditions \[ \|\bm\alpha-\bar{\bm\alpha}\| < \varepsilon_1 < \frac{\varepsilon}{16MC\,n^{3/2}}, \qquad \varepsilon_2 < \frac{\varepsilon}{2C\alpha_n}, \qquad \|\bm m\|1\), the current suffix has no integer relation with \(2\)-norm less than \(M\), so the search extends leftward. If it occurs at \(k=1\), the algorithm returns \[ \lambda_1(x)>M, \] meaning there is no integer relation of Euclidean norm at most \(M\). When a relation is found, IPSLQ returns the last column of \(B\). The asymptotic claim is that the IPSLQ-based minimal-polynomial reconstruction algorithm has complexity \[ O(P(d,M)) \] even when the exact degree is unknown. The improvement is therefore not a faster fixed-dimension PSLQ routine, but the elimination of the multiplicative factor caused by restarts across dimensions. The reported experiments use numbers of the form \[ \alpha=3^{1/s}+2^{1/t}, \] with approximations given to \(500\) decimal digits, Maple 15, and `Digits := 500`. The input degree bound is \(d\), the input height bound is \(M+1\), and the exact degree and height are \(d-1\) and \(M\), respectively. Across ten test instances, the observed ratio \(T_{\text{PSLQ}}/T_{\text{IPSLQ}}\) ranges from \(2.00\) to \(11.22\), with the authors noting that the ratio seems to increase with \(d\). Within the constrained-search interpretation, IPSLQ avoids recomputing reductions for a sequence of nested degree-bounded search spaces [1403.4026]. ## 4. Coefficient-domain constraints and algebraic integer relations A different notion of constraint appears when the coefficients themselves are restricted to a ring of algebraic integers. In this setting, classical PSLQ is generalized from coefficient domains \(\mathbb Z\) and \(\mathbb Z[i]\) to the ring of integers \(\mathcal O_{\mathbb K}\) of a quadratic field \(\mathbb K=\mathbb Q(\sqrt D)\) [1809.06063]. For squarefree \(D\in\mathbb Z\), \[ \mathbb K=\mathbb Q(\sqrt D)=\{q_1+q_2\sqrt D\mid q_1,q_2\in\mathbb Q\}, \] and the ring of integers is \[ \mathcal O_{\mathbb K}=\mathbb Z[\omega]=\{\alpha+\beta\omega\mid \alpha,\beta\in\mathbb Z\}, \qquad \omega= \begin{cases} \sqrt D, & D\equiv 2,3\pmod 4,\\[1mm] (1+\sqrt D)/2, & D\equiv 1\pmod 4. \end{cases} \] An algebraic integer relation of \(x\in\mathbb F^n\) is a nonzero vector \[ a\in(\mathcal O_{\mathbb K})^n \] such that \[ a_1x_1+\cdots+a_nx_n=0. \] This is a direct coefficient-domain constraint: the target relation must lie in a prescribed rank-\(2\) \(\mathbb Z\)-module rather than in the ambient coefficient ring of standard PSLQ. The paper studies two approaches. The first is basis-expansion reduction. Writing \[ a_k=a'_{2k-1}+a'_{2k}\omega, \] one replaces \((x_1,\dots,x_n)\) by \[ (x_1,x_1\omega,\dots,x_n,x_n\omega), \] runs ordinary PSLQ, and reconstructs the coefficients from the integer relation returned. This works well in the real case. In the complex case, however, ordinary complex PSLQ returns Gaussian integer coefficients \[ a_j'=\alpha_j+\beta_j i, \] and the reconstructed coefficient \[ a_k=(\alpha_{2k-1}+\alpha_{2k}\omega)+(\beta_{2k-1}+\beta_{2k}\omega)i \] need not lie in \(\mathbb Z[\omega]\). The reduction therefore relaxes the coefficient-domain constraint too far. The second approach, APSLQ, modifies PSLQ directly by replacing nearest-integer reduction with nearest-algebraic-integer reduction in \(\mathcal O_{\mathbb K}\). The standard hyperplane matrix becomes \[ H_x=(h_{i,j}),\qquad h_{i,j}= \begin{cases} 0 & \text{if } ij, \end{cases} \qquad s_i=\sqrt{\sum_{k=i}^n x_k\overline{x_k}}. \] For a lower trapezoidal matrix \(A=(a_{i,j})\), the reducing matrix \(D_A=(d_{i,j})\) is \[ d_{i,j}= \begin{cases} 0 & \text{if } ij. \end{cases} \] The domain dependence enters precisely through the rounding map \(\lceil\cdot\rfloor\). For imaginary quadratic fields, the paper gives explicit nearest-algebraic-integer formulas. If \(z=\alpha+\beta\omega\) and \(D\equiv 2,3\pmod 4\), then \[ \alpha=\Re(z),\qquad \beta=\Im(z)/\sqrt D, \qquad \lceil z\rfloor=\lceil \alpha\rfloor+\lceil \beta\rfloor\,\omega. \] If \(D\equiv 1\pmod 4\), then \[ \beta=\frac{2\Im(z)}{\sqrt D},\qquad \alpha=\Re(z)-\beta/2, \] and one chooses the closer of two candidate lattice points. The uniform rounding error is bounded by \[ |z-\lceil z\rfloor|\le \epsilon, \] with \[ \epsilon= \begin{cases} \frac12\sqrt{|D|+1}, & D\equiv 2,3\pmod 4,\\[2mm] \frac14\sqrt{\dfrac{|D|+1}{|D|}}, & D\equiv 1\pmod 4, \end{cases} \] leading to \[ \rho= \begin{cases} \frac{2}{\sqrt{|D|+1}}, & D\equiv 2,3\pmod 4,\\[2mm] \frac{4\sqrt{|D|}}{|D|+1}, & D\equiv 1\pmod 4. \end{cases} \] The classical PSLQ parameter constraints are \[ \frac{1}{\rho}\ge |x-\lceil x\rfloor|,\qquad 1<\tau\le \rho,\qquad \frac{1}{\tau^2}=\frac{1}{\gamma^2}+\frac{1}{\rho^2}. \] As \(|D|\) increases, \(\rho\) decreases; for many fields \(\rho<1\), so \(1<\tau\le\rho\) becomes impossible. This is why the standard proof framework extends only to \[ D=-2,\,-3,\,-7,\,-11. \] The practical APSLQ loop normalizes \(y\leftarrow x/\|x\|\), performs initial Hermite reduction, \[ H'\gets D_{H_y}H_y,\qquad B\gets D_{H_y}^{-1},\qquad y\gets yD_{H_y}^{-1}, \] repeatedly selects \[ r\gets \arg\max_{1\le r\le n-1}(\gamma^r|H'_{r,r}|), \] swaps rows and columns, restores lower trapezoidal form, updates \[ H'\gets D_{H'}H',\qquad B\gets BD_{H'}^{-1},\qquad y\gets yD_{H'}^{-1}, \] and declares a candidate relation when \[ \frac{|y_k|}{\|\operatorname{col}_k(B)\|}<\epsilon, \qquad k\gets \arg\min_{1\le k\le n}|y_k|. \] This thresholded form is practical, but the returned relation does not necessarily preserve the standard bound \(\|a\|\le \gamma^{n-2}M\). The experiments use \(1000\) problems per test set, \(k\in\{2,\dots,10\}\), coefficient ranges \([-9,9]\) and \([-999999,999999]\), and precisions \(75\) and \(175\) decimal digits. APSLQ matches ordinary PSLQ perfectly for \(\mathbb Q\) and \(\mathbb Q(i)\). For real quadratic fields \(D\in\{2,3,5,6,7,10,11\}\), only the reduction method is used and all reported cases are \(1000\) good. For \(D\in\{-2,-3,-7,-11\}\), APSLQ is essentially perfect, with sensitivity at \(D=-11\) when \(\gamma<\gamma_1=\sqrt{22}/2\). For \(D\in\{-5,-6,-10\}\), APSLQ often performs well on real-valued constants but can fail badly on genuinely complex instances, while reduction remains strong empirically. In constrained-search terms, the paper shows both the promise and the geometric limitations of coefficient-domain restriction [1809.06063]. ## 5. Theory-driven basis constraints in constant recognition A fourth form of constrained PSLQ search appears in computational constant recognition. The target constants are the Stokes multipliers \(C_M\) for the one-dimensional anharmonic oscillators \[ H=\frac{p^2}{2}+\frac{x^2}{2}+g x^{2M}, \] with perturbative expansion \[ E(g)=\sum_{k=0}^\infty a_k g^k. \] The large-order growth is modeled as \[ a_k \sim C_M\,(-1)^{k+1}\,A_M^k\, \Gamma\!\big((M-1)k+b+1\big) \left[1+\frac{c_1}{k}+\frac{c_2}{k^2}+\cdots\right], \] where \[ S_0(M)=2^{-1/(M-1)}\frac{1}{M-1}B\!\left(\frac{1}{M-1},\frac32\right), \qquad A_M=S_0(M)^{\,M-1}, \qquad b=-\frac12. \] Here PSLQ is only the final stage of a three-stage constrained workflow: Rayleigh–Schrödinger recursion in the harmonic oscillator basis, Richardson extrapolation of order \(40\)–\(100\), and then PSLQ on a small candidate basis [2603.27613]. The recursion uses \[ a_k=E^{(k)}=\langle 0|V|\psi^{(k-1)}\rangle, \] and \[ |\psi^{(k)}\rangle = \sum_{n\ge 1}\frac{1}{E_0-E_n} \left( \langle n|V|\psi^{(k-1)}\rangle -\sum_{j=1}^{k-1} a_j \langle n|\psi^{(k-j)}\rangle \right)|n\rangle. \] At order \(k\), the support extends up to oscillator level \(n=2Mk\). The implementation uses Python 3.11 with `mpmath`, working precision set to `dps + 30` guard digits, up to \(1200\) perturbation coefficients, and working precision up to \(300\) decimal digits. The ratio sequence \[ T_k=\frac{a_k}{a_{k-1}}, \qquad \widetilde A_k=-\,T_k\prod_{j=2}^{M-1}\big((M-1)k-j\big) \] extracts \(A_M\), and \[ \widetilde C_k = a_k\,(-A_M)^{-k}\,\Gamma\!\left((M-1)k+\frac12\right)^{-1} \] extracts \(C_M\). Richardson extrapolation is applied to sequences of the form \[ s_k=s+\frac{c_1}{k}+\frac{c_2}{k^2}+\cdots, \] using \[ R_N=\sum_{j=0}^N (-1)^{N-j}\binom{N}{j}\frac{(k_0+j)^N}{N!}\,s_{k_0+j}. \] The constrained PSLQ step is formulated additively by taking logarithms. The generic search vector is \[ \big(\ln |C_M|,\ \ln \Gamma(1/(M-1)),\ \ln \pi,\ \ln 2,\ \ln 3,\ 1\big), \] or an expanded version with more Gamma values. A relation \[ n_0 \ln |C_M| + n_1 \ln \Gamma(\alpha_1)+\cdots+n_r \ln \Gamma(\alpha_r) +n_\pi \ln \pi+n_2\ln 2+n_3\ln 3+n_{r+4}=0 \] implies \[ |C_M|^{n_0}\prod_j \Gamma(\alpha_j)^{n_j}\pi^{n_\pi}2^{n_2}3^{n_3}=e^{-n_{r+4}}. \] In all successful cases the constant term is \(0\), so the output is a pure monomial identity. The basis is strongly constrained by the instanton action, Gamma identities at denominator \(M-1\), and the restricted algebraic prefactor subgroup generated by \(2\) and \(3\). This search strategy produces the identities \[ C_2^2 \pi^3 = 6, \qquad C_3^2 \pi^4 = 32, \qquad C_5^4 \Gamma(1/4)^4 \pi^5 = 2^{12}3^2, \qquad C_7^6 \Gamma(1/3)^9 \pi^6 = 2^{20}3^3. \] For \(M=5\), the basis \[ \big(\ln |C_5|,\ \ln \Gamma(1/4),\ \ln \pi,\ \ln 2,\ \ln 3,\ 1\big) \] yields the coefficient vector \[ [-4,\,-4,\,-5,\,12,\,2,\,0]. \] For \(M=7\), the initial basis \[ \big(\ln |C_7|,\ \ln \Gamma(1/6),\ \ln \pi,\ \ln 2,\ \ln 3,\ 1\big) \] yields \[ [24,\ 18,\ 33,\ -74,\ -21,\ 0], \] and the resulting identity is simplified using \[ \Gamma(1/6)=\frac{\Gamma(1/3)^2 \sqrt3}{2^{1/3}\pi}. \] The searches are decisive only because the input values are stabilized before PSLQ. The paper reports, for example, \(500\) coefficients and \(200\)-digit arithmetic for \(M=2\) and \(M=3\), \(600\) coefficients and \(200\)-digit arithmetic for \(M=5\), \(400\) coefficients and \(180\)-digit arithmetic for \(M=7\), and \(1200\) coefficients with \(300\)-digit arithmetic for \(M=4\). For \(M=4\), \[ C_4=-0.740051498259358506401511491622\ldots \] is extracted with about \(30\) reliable digits, and two Richardson windows agree to \(24\) digits: start \(=600\), \(N=100\), and start \(=700\), \(N=100\). The most important negative result is the exhaustive search for \[ |C_4|^{n_1}\Gamma(1/3)^{n_2}\pi^{n_3}2^{n_4}3^{n_5}=1 \] using the logarithmic basis \[ (\ln|C_4|,\ln\Gamma(1/3),\ln\pi,\ln2,\ln3), \] with precision levels \(30,35,40\) digits and coefficient bounds \(200,500,1000,2000\). No relation is found. The paper interprets this not as a theorem of nonexistence, but as strong evidence that there is no low-complexity relation of the searched type. The search uses the heuristic detection limit \[ \text{maxcoeff} \lesssim 10^{D/N}, \] so for \(30\) digits and a \(5\)-element basis, \[ 10^{30/5}=10^6, \] well above the tested coefficient bounds. In this framework, null PSLQ output becomes bounded exclusion rather than a mere computational failure [2603.27613]. ## 6. Certification, negative evidence, and limitations Across these variants, constrained PSLQ search is characterized as much by its acceptance criteria as by its search mechanics. Positive output is not accepted merely because a numerical relation is small; it must satisfy a problem-specific certification condition. In empirical PSLQ, the key quantity is the residual \[ |\langle \bm\alpha,\bm m\rangle|, \] controlled by \[ |\langle \bm\alpha,\bm m\rangle| < C\bigl(\|\bm m\|\varepsilon_3+\alpha_n\varepsilon_2\bigr), \] and exactness requires an external gap bound. In IPSLQ, the central certificate is bounded nonexistence, \[ \lambda_1(x)>M, \] for the current degree-constrained search space. In algebraic-integer PSLQ, returned coefficients must be checked for membership in the target ring \(\mathcal O_{\mathbb K}\). In logarithmic constant recognition, “no relation found” means only that no relation was found in the chosen basis, within the chosen coefficient bounds, at the chosen precision [1707.05037], [1403.4026], [1809.06063], [2603.27613]. Several common misconceptions are addressed directly by this literature. A small residual with respect to empirical data is not enough to certify an exact relation of the underlying exact vector. A null PSLQ result is not, by itself, a proof of transcendence or algebraic independence. Enlarging the coefficient domain through a basis-expansion trick can solve the wrong search problem by allowing coefficients outside the intended ring. High input precision alone is not sufficient if the stopping policy or the search basis is poorly chosen. The higher-degree algebraic example in empirical PSLQ, where Bailey’s estimate still fails but the theorem-driven \(\varepsilon_1\) and \(\varepsilon_2\) succeed, and the \(M=4\) Stokes-multiplier study, where exhaustive bounded searches still return no relation, both underscore this point [1707.05037], [2603.27613]. The limitations are equally explicit. The empirical-data analysis assumes exact arithmetic on the empirical hyperplane matrix after the input is fixed and does not fully analyze floating-point roundoff during the PSLQ iterations themselves. It does not address explicit coefficient-bound pruning or custom branch-and-bound mechanisms, and it does not provide a complete exactness theory without external gap bounds. IPSLQ requires known degree and height bounds and delegates detailed precision control to earlier work. Algebraic-integer PSLQ lacks a satisfactory direct treatment of real quadratic fields and loses the standard proof framework once \(\rho<1\). The constant-recognition pipeline is intentionally narrow: it uses small, theory-motivated logarithmic bases, explicit coefficient boxes, and post-verification against all available digits, but it does not convert bounded nonrecognition into formal impossibility [1707.05037], [1403.4026], [1809.06063], [2603.27613]. A plausible synthesis is that constrained PSLQ search becomes effective when external structure is strong enough to specify a narrow admissible relation class and a rigorous or at least carefully bounded acceptance criterion. Under those conditions, PSLQ functions less as an unconstrained discovery engine than as a certification-oriented search method for discrete relations compatible with precision bounds, degree bounds, coefficient-ring restrictions, or basis-level ansätze. Source: https://www.emergentmind.com/topics/constrained-pslq-search