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FTNILO: Explicit Multivariate Function Inversion, Optimization and Counting, Cryptography Weakness and Riemann Hypothesis Solution Equation with Tensor Networks

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Pith's one-line read The paper claims an explicit tensor-network integral formalism, FTNILO, that yields exact solution equations for multivariate inversion and optimization, an integral equation that decides the Riemann hypothesis, and a corollary that every…

desk verdict The paper's central delta identity is false—it drops the Jacobian—so its inversion, counting, crypto, and Riemann claims collapse; what remains correct is the standard Laplace/delta-framework formalism. read the letter →

arxiv 2505.05493 v1 pith:FT2DRWFG submitted 2025-05-03 math.OC math-phmath.MP

Alejandro Mata Ali This is my paper · ORCID
classification math.OCmath-phmath.MP MSC 90C2611M26
keywords fieldtensornetworksfunctioninversionglobaloptimizationDiracdeltaimaginarytimeevolutionRiemannhypothesiszerocountingcryptographicweakness
topics The Riemann Hypothesis
open problems The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces FTNILO, a tensor-network calculus that turns the problem of inverting or optimizing a continuous multivariate function into an explicit integral equation in a single variable. Its central claim is that any function whose computation can be drawn as a logical circuit has its inverse, its optima, and its zero counts given by closed formulas built from Dirac deltas and definite integrals, with no search step. The paper applies this machinery to claim an explicit integral equation that decides the Riemann hypothesis: the equation counts zeros of the zeta function off the critical line, so a zero result confirms the hypothesis and a nonzero result refutes it. It also derives the corollary that every cryptographic protocol based on functions has an exact secret-recovery formula.

What carries the argument

The central object is the Field Tensor Network Integral Logical Operator (FTNILO): a tensor network whose nodes are functions rather than arrays, with sums over discrete indices replaced by integrals over continuous variables. Each logical operator that receives $\vec{x}$ and outputs $\vec{y}=g(\vec{x})$ while multiplying the amplitude by $h(\vec{x})$ is tensorized as $h(\vec{x})\delta(\vec{y}-g(\vec{x}))$; chaining these operators and integrating over internal bond variables enforces the circuit's logic globally. Two extraction devices do the work of turning this network into solution coordinates: connecting One Constant functions to all free indices except the target coordinate, and the imaginary-time weight $e^{-\tau f}$ with the limit $\tau\to\infty$ concentrating the density at the global optimum. For zero counting, the same integrated network is interpreted as the FTNILO number, the number of points satisfying the imposed condition. The Riemann application rests on a circuit for a convergent series representation of $\zeta(s)$ valid in $\mathrm{Re}(s)>0$, with the critical line excluded by a factor $1-\chi_{1/2}(x_n)$; the paper's delta-consistency criterion reduces the Riemann hypothesis to checking whether the product of deltas in this network is consistently nonzero everywhere in the strip.

What would settle it

Evaluate equation (5.13) on a finite truncation of the zeta series over a bounded rectangle inside the critical strip and compare the result with a direct numerical count of the zeros of $\zeta(s)$ in that rectangle; a nonzero value where direct counting gives zero, or a zero where direct counting finds a non-trivial zero off the line, would refute the claimed zero-counting equation. For the inversion claim, apply the FTNILO formula to $f(x)=x^3$ on $\mathbb{R}$ and check whether the integral expression returns the actual root $X=Y^{1/3}$ as a well-defined delta integral.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that a logical circuit for a function $f$ can be tensorized into a field tensor network whose nodes encode the circuit's operations as Dirac deltas, so that integrating out all but one coordinate produces the exact solution coordinate. For inversion of an injective $f:\mathbb{R}^n\to\mathbb{R}^m$, the $j$-th component of the solution is $\Omega_j = \int x_j\,\Phi(\vec{x},\vec{y})\,\delta(\vec{y}-\vec{Y})\,d\vec{y}\,d\vec{x}$, where $\Phi$ is the tensorized circuit; the paper calls this the restricted FTNILO inversion function. For optimization, the same construction is applied to $e^{-\tau f(\vec{x})}$, and the normalized density is taken to converge to a Dirac delta at the minimizer as $\tau\to\infty$, yielding the exact optimum as a limit of expectation values. Counting versions give the number of solutions $N=\int \Phi(\vec{x},\vec{y})\delta(\vec{y}-\vec{Y})\,d\vec{y}\,d\vec{x}$. On this basis the paper asserts that every injective function has an explicit inverse, that non-injective functions can be handled by a region-sampling protocol, that all function-based cryptographic schemes can be broken, and that the Riemann hypothesis is decided by evaluating the corresponding FTNILO zero-counting equation in the strip $0<\mathrm{Re}(s)<1$; the hypothesis is true exactly when that equation returns zero.

Load-bearing premise

The load-bearing assumption is that the delta-consistency test for the Riemann FTNILO equation can actually be carried out, since the paper states that this search is outside its scope and supplies no method, complexity bound, or numerical evidence, and more generally that the integral expressions written with Dirac deltas can be evaluated as explicit equations rather than merely re-encoding the original search problem.

Editorial extensions

If this is right

  • Function inversion would no longer require search: every injective function with a known logical circuit would have its inverse given by a sequence of definite integrals, one per coordinate.
  • Global optimization would reduce to evaluating the limiting expectation values of the imaginary-time density $e^{-\tau f}$, with the minimizer recovered exactly as $\tau\to\infty$ and approximated to arbitrary accuracy for finite $\tau$.
  • Counting versions of inverse problems, such as how many inputs $x$ satisfy $f(x)=Y$, become a single integral, with a sampling protocol for isolating individual solutions in degenerate cases.
  • The Riemann hypothesis would be decidable by evaluating one FTNILO integral, equivalently by checking delta consistency once, since the integral counts zeros in $0<\mathrm{Re}(s)<1$ off the critical line.
  • Every cryptographic protocol whose encryption function can be represented as a circuit would have an exact secret-recovery equation, stated in the paper as Corollary 4.7.1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit equations are formal integrals of products of Dirac deltas; whether they can be evaluated in a way that beats brute force is not addressed, so the practical reading of explicit remains open.
  • The Riemann delta-consistency criterion appears to restate the problem: proving that no consistent assignment of deltas exists across the infinite limit is equivalent to proving an absence of zeros, and the paper's Section 5.3 places that search out of scope.
  • If the formalism succeeds, analogous zero-counting equations could be written for other Dirichlet series or L-functions whose terms admit circuits, potentially connecting the framework to questions beyond the Riemann zeta function.
  • The cryptographic corollary should be read as an existence-of-formula statement; for real encryption functions the corresponding integral may be computationally intractable, so the corollary does not by itself imply practical attacks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces FTNILO, a continuous 'field tensor network' formalism in which inverse problems, optimization problems, zero-counting problems, and the Riemann hypothesis are encoded as integrals over products of Dirac delta functions. The central claims are: Theorem 2.1 gives an explicit equation for the unique solution of f(X)=Y; Theorem 2.2 gives a counting/checker equation; Theorems 2.8–2.10 extend this to multiple solutions; Section 4 concludes that every function-based cryptographic protocol can be broken; and Section 5 derives 'Riemann FTNILO equations' and claims, via a 'delta consistency' criterion, to decide the Riemann hypothesis. The paper also claims to recover the discrete MeLoCoToN tensor-network formalism as a limiting case.

Significance. If the central inversion identity were correct, the framework would provide a uniform representation of inversion, optimization, and counting problems and would constitute a notable contribution. The paper correctly recognizes that δ(y−f(x)) is the natural free inversion density in Eq. (2.19), and the discrete limit in Section 3 is a plausible formal parallel to MeLoCoToN. However, the paper provides no reproducible algorithm, no numerical demonstration, and no rigorous distribution-theoretic development. More importantly, the central step Eq. (2.24) is false for non-unit Jacobians, so the claimed theorems do not follow. Consequently, despite the breadth of claimed implications, the significance of the formal contribution in its current form is low; the mathematical content beyond textbook delta calculus is not established.

major comments (4)
  1. [§2.3.1, Eq. (2.24)] The identity δ(Y−f(x)) = δ(x−X) for a unique solution X with Y=f(X) is false in general. In one dimension, the composition rule for the Dirac delta gives δ(Y−f(x)) = Σ_i δ(x−x_i)/|f'(x_i)|, and in several variables a Jacobian determinant appears. For the paper's own one-variable setting, f(x)=2x, X=1, Y=2 satisfies Y=f(X), but Eq. (2.32) yields Ω = ∫ x δ(2−2x) dx = 1/2, not 1. This is not a harmless normalization: the missing factor is precisely the quantity that converts a delta integral into a count. Because Eq. (2.24) is used to identify F(x) with δ(x−X) throughout, Theorems 2.1, 2.8, and 2.10, Lemma 4.1, Lemmas 4.6–4.7, and the Riemann counting equations in Section 5 all inherit the error. A repair would require inserting Jacobian factors into every delta composition, which is not done anywhere in the paper.
  2. [§2.4.1 and Theorem 2.2] Theorem 2.2 claims N=∫F(x)dx is positive exactly when a solution exists and zero otherwise. For continuous solution sets this is false as stated. The paper's own ReLU example (Eqs. (2.62)–(2.63)) gives N=δ(0)V(X′), which is not a finite number and is infinite or undefined in the usual sense. The 'renormalized search' in Eqs. (2.64)–(2.67) divides by δ(0) and compares volumes, but no rigorous definition of the quotient of two generalized-function values is supplied. Thus the counting theorem is not established for the degenerate cases that Protocol 2.7 is supposed to handle, and the caveat 'main limitation' in the text does not rescue the universal theorem statements.
  3. [§5.3, Theorem 5.1] The proof of the Riemann delta-consistency theorem is not a proof. Definition 5.1 of delta consistency is not a well-posed property: the product of distributions δ(f_i(x)) is not a pointwise function, and the condition '≠0 for all ρ(x)' is not defined as a rigorous distributional statement. The inference that a consistent product of deltas must have integral at least one is unsupported; standard delta calculus gives no such general implication. Moreover, since Eqs. (5.13) and (5.15) are built on the incorrect identity in Eq. (2.24), they do not count zeros of ζ(s) off the critical line; at best they represent weighted sums with reciprocal Jacobian factors, and they are undefined at multiple zeros where ζ′(s)=0. Therefore the abstract's claim of an explicit equation that 'gives the solution of the Riemann hypothesis' is not supported.
  4. [§2.3.1, Eq. (2.40) and text after Eq. (2.33)] The claimed 'explicitness' of the inversion equation is misleading. Eq. (2.40), after performing the y-integration, reduces to Ω_j = ∫ x_j δ(Y−f(x)) dx, whose evaluation is exactly the enumeration of solutions of f(x)=Y; the solution X does not appear in the integrand only because the delta already encodes the equation to be solved. This is a definitional restatement of the inverse problem rather than a new explicit representation that bypasses the need to solve it. The same observation applies to the optimization extraction in Eq. (2.47), where the τ→∞ limit of the normalized Gibbs expectation is the argmin by definition of the Laplace principle.
minor comments (4)
  1. [§5.1, Eq. (5.7)] The cyclic 'donut' construction with δ(x_{n%N}−x_{n−1}) is not obviously equivalent to the original linear circuit; the text does not justify that identifying the last output with the first input is legitimate for a function whose domain is not periodic.
  2. [§5.2, Eqs. (5.13) and (5.15)] The expressions use R^{2N+1}, R^{2(N+1)}, R^{2N}, and R_+^{2N} inconsistently, and the delta chains impose x_n=x_{n−1} while the integration limits are not written coherently; this makes the equations difficult to parse.
  3. [§2.4.1, Eq. (2.56)] The Heaviside function is defined with H(0)=1, but the treatment of half-open intervals in the partial moment formulas is not explained; the paper should specify the measure-zero conventions used.
  4. [§5.3 and Abstract] The paper states that the consistency search is 'outside the scope of this work' and offers no method, complexity bound, or numerical evidence, yet the abstract claims an 'explicit integral equation that gives the solution of the Riemann hypothesis'; this overstates what is actually shown.

Circularity Check

3 steps flagged · score 9.0 of 10

The FTNILO solution equations read off an X that is inserted by hand into the Dirac delta, and the Riemann 'resolution' restates the hypothesis as a zero-search that the paper leaves undone.

  1. self definitional [Section 2.2.1, Eq. (2.24); Section 2.3.1, Eqs. (2.31)-(2.32); Theorem 2.1, Eq. (2.40)]
    "If ⃗Y = f(⃗X), the FTN represents the function F(⃗ x) = (1 if ⃗ x=⃗X, 0 if ⃗ x≠⃗X) ... or with the deltas F(⃗ x)=δ(⃗ x−⃗X). ... Ωj = ∫ Xj xjFj(xj)dxj = ∫ Xj xjδ(xj−Xj)dxj = Xj."

    The solution X is introduced as the center of the Dirac delta in Eq. (2.24) by the assertion 'if Y=f(X)' rather than being derived from the field tensor network. The later 'extraction' equations (2.31)-(2.32) then recover X by the sifting property of the delta, so the output is exactly the X that was placed into the delta by hand. Theorem 2.1's formula X=(Ω0,...,Ωn−1) with Ωj=∫∫ xjΦ(x,y)δ(y−Y)dydx is therefore a definitional tautology: it returns the unknown solution because that unknown solution was already assumed as the center of δ(x−X). No independent computation from f is performed.

  2. self definitional [Section 2.3.2, Eqs. (2.42)-(2.47); Theorem 2.4]
    "lim_{τ→∞} F(x,τ)/∫X F(x,τ)dx = lim_{τ→∞} e^{-τf(x)}χ_R(x)/∫X e^{-τf(x)}χ_R(x)dx = δ(x−X) ... Ωj(τ)=∫X∫X xjΦ(x,y,τ)dydx / ∫X∫X Φ(x,y,τ)dydx."

    Eq. (2.42) states that the normalized Gibbs density concentrates on the argmin X, which is the standard Laplace principle. Eq. (2.47) then defines each variable as the τ→∞ expectation of x under that same density. The 'general optimization equation' X = limτ→∞(Ω0(τ),...) thus presupposes X as the concentration point of the limit distribution; it is the definition of the argmin rewritten as a weighted average, not an independent construction of the minimizer. The result is built into the exponential weight e^{-τf}, so the equation cannot fail to return the argmin in the limit, but it also cannot be said to derive it.

1 more flagged steps
  1. renaming known result [Section 5.3, Definition 5.1 and Theorem 5.1]
    "The Riemann hypothesis is false if and only if all its deltas in the Riemann FTNILO equation are consistent. This means that if some subset of the deltas is not consistent, the Riemann hypothesis is true. ... This search is outside the scope of this work, but we can give some possible approaches."

    Definition 5.1 declares deltas consistent when their product is nonzero for some assignment of the non-shared variables, which is exactly the existence of a simultaneous solution of the encoded equations. For the Riemann FTNILO equation, that existence is precisely the existence of a zero of the zeta function outside the critical line. Theorem 5.1 therefore renames the Riemann hypothesis as a 'delta consistency' statement: the equation is nonzero exactly when such a zero exists, and zero exactly when none exists. The abstract's promised 'explicit equation that gives the solution of the Riemann hypothesis' is just an indicator of whether the original zero-search succeeds, and the paper concedes that the search itself is outside the scope of the work.

full rationale

The derivation chain is circular in its central extraction step. Eq. (2.22) correctly obtains F(x)=δ(Y−f(x)), but Eq. (2.24) then replaces this with δ(x−X) by asserting the existence of a unique solution X with Y=f(X). That replacement inserts the unknown solution into the definition of the object that is supposed to find it; the subsequent integrals (2.31)-(2.32) and Theorem 2.1 merely use the sifting property of the delta to output the same X. The optimization version is no different: Eq. (2.42) states the normalized Gibbs density concentrates on the argmin X, and Eq. (2.47) computes the τ→∞ expectation, so the formula is the standard Laplace representation of the argmin with X presupposed as the concentration point. For the Riemann claims, Theorem 5.1 defines 'delta consistency' as nonvanishing of the product of deltas, which is equivalent to existence of a solution of the encoded equations, i.e., existence of a zero outside the critical line. Thus the abstract's 'explicit integral equation that gives the solution of the Riemann hypothesis' reduces to a delta-integral that is zero iff no such zero exists, and the consistency search is admitted to be outside the scope of the work. Separately, Eq. (2.24) omits the Jacobian factors required by the identity δ(f(x)−Y)=Σδ(x−x_i)/|det J|, so the counting equations (2.34), (2.68), (5.13), and (5.15) do not even compute the stated counts; that is a mathematical correctness failure independent of, and in addition to, the circularity. Self-citations to MeLoCoToN [30] are not load-bearing for the circularity finding; the circularity is internal to the delta construction and to the Riemann consistency restatement.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The formalism depends on the assumption that arbitrary functions can be tensorized into delta-products, that the Laplace limit selects the global minimizer, and that the zero-counting integrals are well-defined for generalized functions. The RH section adds the ad hoc delta consistency criterion with no evidence of checkability. There are no fitted constants; the only adjustable parameter is the inverse temperature τ, taken to infinity.

free parameters (1)
  • tau (inverse temperature) = Infinity (limit)
    Appears in the optimization density (2.28) and extraction (2.44-2.47). The exact equations require the τ→∞ limit; any finite implementation tunes τ by hand, but no data are fitted.
assumptions (5)
  • domain assumption Every target function f can be represented by a logical circuit and tensorized into a field tensor network of Dirac deltas.
    Invoked in Section 2.2 and Theorem 2.1; without a circuit the inversion equation cannot be written.
  • domain assumption The normalized Boltzmann weight e^{-τ f} χ_R / ∫ e^{-τ f} χ_R converges to a Dirac delta at the unique minimizer as τ → ∞.
    Used in Eq. (2.42) and Theorem 2.4; no regularity conditions are stated.
  • domain assumption The zero-counting integrals over products of Dirac deltas are well-defined and finite for isolated solutions.
    Theorems 2.2 and 2.8 assume N = ∫ F(x) dx counts solutions; continuous solution sets make the integral divergent (Section 2.4.1).
  • standard math The series representation ζ(s) = (1/(s-1)) Σ_n [n/(n+1)^s - n^{-s}/n^s] is valid for 0 < Re(s) < 1 and the factor 1/(s-1) can be neglected for zero counting.
    Eq. (5.9)-(5.10), citing [34]; standard, but the paper uses it to justify the FTNILO construction.
  • ad hoc to paper The delta consistency property of a product of deltas is a meaningful and checkable criterion for the Riemann hypothesis.
    Definition 5.1 and Theorem 5.1; the paper says the consistency search is 'outside the scope of this work', so no evidence of checkability is provided.
invented entities (1)
  • Delta consistency (Riemann criterion)
    purpose: Postulated criterion claimed to decide the Riemann hypothesis by checking whether a product of Dirac deltas in the Riemann FTNILO equation is nonzero somewhere.
    Introduced in Definition 5.1 and Theorem 5.1. It restates the existence of a zero in the forbidden strip and has no falsifiable handle outside the paper; no computation or test is provided.

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Cite this review

Pith. "Pith review of FTNILO: Explicit Multivariate Function Inversion, Optimization and Counting, Cryptography Weakness and Riemann Hypothesis Solution Equation with Tensor Networks." pith.science (2026). https://pith.science/paper/FT2DRWFG

@misc{pith2026250505493,
  author       = {Pith},
  title        = {Pith review of: FTNILO: Explicit Multivariate Function Inversion, Optimization and Counting, Cryptography Weakness and Riemann Hypothesis Solution Equation with Tensor Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FT2DRWFG}},
  note         = {Machine review of arXiv:2505.05493}
}
abstract

In this paper, we present a new formalism, the Field Tensor Network Integral Logical Operator (FTNILO), to obtain the explicit equation that returns the minimum, maximum, and zeros of a multivariable injective function, and an algorithm for non-injective ones. This method extends the MeLoCoToN algorithm for inversion and optimization problems with continuous variables, by using Field Tensor Networks. The fundamentals of the method are the conversion of the problem of minimization of $N$ continuous variables into a problem of maximization of a dependent functional of a single variable. It can also be adapted to determine other properties, such as the zeros of any function. For this purpose, we use an extension of the imaginary time evolution, the new method of continuous signals, and partial or total integration, depending on the case. In addition, we show a direct way to recover both the tensor networks and the MeLoCoToN from this formalism. We show some examples of application, such as the Riemann hypothesis resolution. We provide an explicit integral equation that gives the solution of the Riemann hypothesis, being that if it results in a zero value, it is correct; otherwise, it is wrong. This algorithm requires no deep mathematical knowledge and is based on simple mathematical properties.

Figures

Figures reproduced from arXiv: 2505.05493 by the authors.

Figure 1
Figure 1. Tensor Network with four tensors. As we know, a tensor network represents a summation of products of tensor elements. For example, the tensor network shown in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Logical circuits for a) Computing the zeros from the sum of a sequence, b) Optimizing a quadratic function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Logical Circuit for Optimizing a quadratic function with linear chain interaction to one neighbor with the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Tensorized circuits for a) Computing the zeros from the sum of a sequence, b) Optimizing a quadratic function [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Iteration process to determine first, second and third variable values in the inversion problem for (2.3). [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Iteration process to determine first and second variable values in the optimization problem. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Example of iteration process to determine first and second variable values in an inversion problem with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Iteration process to determine first and second variable values in the optimization problem with multiple [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Iteration process to determine first and second variable values in the inversion of the function (2.3). [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Example of iteration process to determine first and second variable values in the inversion of a vector-valued [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: Example of iteration process to determine first and second output values of a vector-valued injective function [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Logical Circuit for the Riemann Zeta Function. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FTNILO of the Riemann Zeta Function zeros. This FTN returns the amount of zeros of the Riemann Zeta [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: FTNILO of the Riemann Zeta Function zeros in donut shape. This FTN returns the amount of zeros of the [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: Donut Slice for the Riemann Zeta Function. [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.