“The nail and hammer are fundamental to the manufacture of chairs, tables, stools, cabinets, and other varied items. So are certain ‘Mathematical Formulas or Relationships’ to the solution of diverse technological problems. Yet, hammers can be patented, while ‘Mathematical Formulas or Relationships’ cannot.”

*All footnotes can be viewed here.*

“Mathematical Formulas and Relationships” fall under the “Abstract Idea” exception to the categories of patentable subject matter. Characterizing “Mathematical Formulas and Relationships” as “Abstract Ideas” has led to misrepresentation of mathematical concepts in patent law. A “Mathematical Formula or Relationship” is a means of expression and should be inspected to extract what it expresses. Next, the content that is being expressed may be evaluated to determine whether the “Mathematical Formula or Relationship” is expressing a “Tool” or a “Model,” both of which are used for building machines and devising technological processes and neither of which needs to be categorically excepted from patentability. Additionally, acknowledging “Mathematical Formulas and Relationships” as “Tools” and “Models” modifies a longstanding view of those types of “Algorithms” that consist of “Mathematical Formulas and Relationships.” This modified view leads to the conclusion that one of the inventive features of an “Artificial Intelligence” (AI) machine or process is the collection of algorithms that underlie the AI machine or process.

**Mathematics as a Language for Expressing a Tool or a Model**

“Mathematical Concepts,” as defined by the United States Patent and Trademark Office’s (USPTO’s) “October 2019 Update to Subject Matter Eligibility Guidelines” (“2019 PEG”),[1] include “Mathematical Relationships, “Mathematical Formulas or Equations,” and “Mathematical Calculations.”[2] A bundled version of these terms, “Mathematical Formulas or Relationships,” is used here.

The term “Algorithm,” while broadly including any step-by-step method such as a baking recipe,[3] is often used to refer to a mathematical relationship or formula.[4] Algorithms don’t fare well under the current view of case law[5] either, and need auxiliaries to come to their rescue.[6] Such auxiliaries come to play at the initial step of determining eligibility and are apart from the core requirements of inventiveness such as novelty, nonobviousness, and adequacy of disclosure.[7]

The paper sets forth three arguments regarding the nature and treatment of “Mathematical Formulas and Relationships.”

First, and as a preliminary point of inspection, note that “Mathematical Formulas and Relationships” are a means of expression and a “Language” such that each particular “Mathematical Formula or Relationship” may express something of an entirely different nature.[8] As such, “Mathematical Formulas and Relationships” do not constitute a “category” that can then be “categorically” excluded.

Second, some types of “Mathematical Formulas and Relationships” are “Tools” just as a “Hammer” is a tool.[9] The “Tool” types of “Mathematical Formulas and Relationships” are essential for creating technological processes and systems that are considerably more complex than a chair.[10] Yet, when it comes to patentability, this type of “Mathematical Formulas and Relationships” is not given the same consideration that is accorded a “Hammer.”

Third, there is a category of “Mathematical Formulas and Relationships” that are currently considered unpatentable because they are deemed expressions of “Laws of Nature.”[11] Members of this category represent, at best, approximate “Models” of observed physical phenomena.[12] These “Models” evolve in terms of the domain of their applicability and are replaceable in other domains or other models become more accurate in modeling the same phenomenon and therefore the original model gets replaced.[13]

The so called “Laws of Nature” that may be expressed by a mathematical formula or equation are in turn equated with “Fundamental Truths.”[14] With a certain degree of approximation or within certain domains of applicability, these may represent a workable version of the truth. However, being mere approximate and replaceable “Models” of physical phenomena,[15] these so-called “Laws” could hardly represent an absolute or fundamental truth in general and across time and different domains. Rather, their approximate, domain-dependent applicability that results in their evolving nature points to the inventive mind of man.

Based on the above premises, an analysis must begin with acknowledging a “Mathematical Formula or Relationship” as a language and determining the *content* of what it expresses rather than excluding it categorically just because of the mathematical form of expression.[16]

Once the content expressed by the “Mathematical Formula or Relationship” has been determined, the next step is to see if the content expresses a “Tool” that has applicability across different areas of science or a “Model” of some particular physical phenomena.

**Mathematics and AI**** Inventions**[17]

The arguments regarding the nature of “Mathematical Formulas and Relationships” arrive at an incidental conclusion regarding AI[18] inventions.[19]

Noting that an algorithm is an expression of a process or procedure, the key components of an AI process or machine are the algorithms underlying its operation[20] and algorithms are often expressed with “Mathematical Formulas and Relationships.” If the “Mathematical Formula or Relationship” is recognized as the “Tool” or the “Model” that it may be, it follows that any novel and nonobvious “Tool” or “Model” that creates an AI process or machine constitutes an “inventive” part of the AI process or machine. The results[21] of AI are generated by running data using the inventive “Tool” through the inventive “Model.”[22]

**“Mathematical Formulas or Relationships” Are a Language like English or French**

A 2020 IPWatchdog article by Jose Nunez[23] argued that:

Mathematics provides a

descriptive languagethat can be used to describe virtually everything in a precise manner. Mathematics can be used to describe not only laws of nature, but also many other concepts, such as defining proportions to combine materials, expressing a cost function to be optimized by a machine-learning algorithm … and so forth.[24]

Mr. Nunez is correct.

Consider 2P+3A=5F, where P=Pear, A=Apple, and F= pieces of Fruit, which is a way of saying: 2 pears plus 3 apples is equal to 5 pieces of fruit. The equation 2P+3A=5F is a “Mathematical Relationship” as well as a “Mathematical Formula or Equation” as well as a “Mathematical Calculation,” thus falling under all 3 types of “mathematical concepts” as defined by the USPTO.[25] Yet, this relationship, formula, equation, or calculation is merely a “Language” for expressing a concept about pears and apples; a concept that is hardly an “Abstract Idea.”

When encountering a “Mathematical Formula or Relationship,” first find out what it says in English. It may be expressing something mundane and concrete about the pieces of fruit in a bowl.

“Mathematical Formulas or Relationships” are mere languages for expressing concepts and happen to be exceptionally well-suited for precise and concise expression of our understanding of physical phenomena.

As argued above, “Mathematical Formulas or Relationships” which are types of “mathematical concepts”[26] are capable of expressing widely disparate contents that cannot form a “category.” Nevertheless, the form of expression is “categorically” excluded from the categories of patentable subject matter.[27] “Mathematical formulas or relationships” in a claim are often saved by constructs[28] such as “Practical Applications,”[29] whereas the truly inventive portion of the claim is often the “Mathematical formula or relationship” alone.

**A “Mathematical Formula or Relationship” May be a “Tool” Just Like a Hammer is a Tool**

Most of all “process, machine, manufacture, composition of matter, or any new and useful combination thereof,”[30] that are patented, use or rely on some form of “Mathematical Concept” for their construction or operation. This is akin to considering a chair to be patentable subject matter but excluding the hammer that was used for building it from patentability. The “Mathematical Concepts” that are excluded from the categories of patentable subject matter sometimes operate like a hammer in that they do not become part of the finished product and other times like a nail[31] that becomes part of the finished product. Whether, figuratively, a hammer” or a nail, the “mathematical concept” is not patentable on its own while no one questions the patentability of literal hammers and nails.

Some “Mathematical Formulas or Relationships” can be applied to a wide variety of different tasks in different technological areas. These are often of the “Tool” type. A mathematician must devise them and must recognize their utility for particular analytical tasks. The tasks that utilize these mathematical tools might not have been possible at all without the devised tools or may have been doable, just not as easily or elegantly. A carpenter can drive a nail using a piece of rock but if he has the correct type of hammer, he can do the same job much more precisely, rapidly and easily.

The nail and hammer are fundamental to the manufacture of chairs, tables, stools, cabinets, and other varied items. So are certain “Mathematical Formulas or Relationships” to the solution of diverse technological problems. Yet, hammers can be patented, while “Mathematical Formulas or Relationships” cannot.[32]

The Preface to *Brunton and Kutz*[33], provides an insight to the “Tool” characteristic of a number of mathematical methods and supports the “hammer” analogy for mathematics:

… With modern mathematical methods, enabled by unprecedented availability of data and computational resources,we are now able to tackle previously unattainable challenge problems. A small handful of these new techniques includerobust image constructionfrom sparse and noisy random pixel measurements,turbulence controlwith machine learning,optimal sensor and data actuator placement, discovering interpretable nonlinear dynamical systems purely from data, and reduced order models to accelerate the study and optimization of systems with complex multi-scale physics.[34]

It is likely that almost all of the examples in the “small handful of new techniques” are considered patentable subject matter and are the “Chairs” of our analogy. Yet, the “mathematical methods,” the “hammers” or “nails” of the analogy, that make them possible are excepted from patentability.

The term “Algorithm” is often used as the quintessential example of mathematical relationships and formulas that are not patentable.[35] However, according to *Brunton and Kutz*, the “machine learning community” are looking for “scalable, fast algorithms” that have high “prediction quality”

[D]ata science has been largely dominated by two distinct cultural outlooks on data. The machine learning community, which is predominately comprised of computer scientists, is typically centered on predication quality and scalable, fast

algorithms. Although not necessarily in contrast, the statistical learning community, often centered in statistics departments, focuses on the inference of interpretable models.[36]

Thus, new or improved “Algorithms” that have the characteristics of being “scalable,” and “fast,” and have better “prediction quality” need to be “devised,” “thought of,” or in fact “invented” to lead to better outcomes for “machine learning.”

Another example of statements indicating that many types of “Mathematical Formulas and Relationships” are “Tools,” which are inventible and improvable and whose nature is determinative in the possibility and desirability of the product that is built, appears in *Brunton and Kutz* as follows:

… Pattern extraction is related to the second theme of finding coordinate transforms that simplify the system. Indeed, the rich history of mathematical physics is centered around coordinate transformations (e.g., spectral decompositions, the Fourier transform, generalized functions, etc.), although these techniques have largely been limited to simple idealized geometries and linear dynamics….[37]

Thus, something as purely mathematical as **Fourier transform** has a direct bearing on whether some physical device can be made. Further, these purely mathematical constructs are created by someone and would not have existed but for their creation by a person, namely before their “invention.”

*Brunton and Kutz* begin with **singular value decomposition (SVD)** and attribute the possibility of existence and the progress in many modern technological fields to the advent of SVD. Under “Historical Perspective,” *Brunton and Kutz* include a paragraph listing a number of references that discuss the history and development of SVD.[38] This is the history of an “invention” and a series of possibly novel and nonobvious improvements on this invention that have not been patentable because of the categorical exclusion of algorithms.

On Fourier Transform, *Brunton and Kutz* provide:

Fourier’s seminal work provided the mathematical foundation for Hilbert spaces, operator theory, approximation theory, and the subsequent revolution in analytical and computational mathematics. Fast forward two hundred years, and the fast Fourier transform has become the cornerstone of computational mathematics, enabling real-time

image and audio compression,global communication networks,modern devices and hardware, numerical physics and engineering at scale, and advanced data analysis. Simply put, the fast Fourier transform has had a more significant and profound role in shaping the modern world than any other algorithm to date.[39]

**Fourier transform and fast Fourier transform** are the “hammers” or “nails” of image and audio compression, communication networks, and other modern processes and hardware.

By their nature, such “Mathematical Formulas and Relationships” are well-defined and fleshed out to the minutest detail[40] which is the antithesis of being “Abstract.”

Their universal utility[41] may have been the enemy of their patentability.[42] It ought not be. For one, we hear only of the useful ones. For every useful mathematical tool that is created by a mathematician there are many that never see the light of day.[43]

**“Mathematical Formulas and Relationships” that Express a “Law of Nature” Can Be Approximate, Domain-Limited, and Evolving Models – Like a Crash Test Dummy Used as a Model of a Person or a Mannequin in a Store Display**

Some “Mathematical Formulas and Relationships” express a “Law of Nature” which is considered a “Fundamental Truth.” Examples include Newton’s Laws that are the foundation of Classical Mechanics, the second of which may be expressed as F=ma, and Einstein’s theory of special relativity expressed as E=mC2 which is the basis of Quantum Mechanics.[44] A “theory” is a scientific hypothesis that is supported by ample empirical evidence.[45] A “law” is the same hypothesis having even more empirical evidence supporting its correctness.[46] Einstein’s theory illuminates and limits the domain of applicability of Newton’s Laws.[47] Therefore, Newton’s “Laws” could not have been “Fundamental Truths” in the broad sense but are workable models within a certain domain of applicability and as certain approximations of a physical phenomenon.

The “Laws of Nature” are concise capsules that model and express certain observed correlations and have been indispensable in the furtherance of science and technology.[48] They are nevertheless mere “Models” just like a “Dummy” is a model of a human body in certain type of domain. A “Mannequin” is helpful in fitting clothing because it is a model of the outer shape of a human body. Its utility, however, is limited by how closely it models a human being and is not useful, for example, as a dummy for a car crash test. A crash test dummy must reflect the strength of material of the human body. Same is true of the “Laws of Nature” that “Model” our observations of the physical phenomena: some are coarser; some are finer, and each may have a different domain of applicability; each focuses on accuracy in some particular aspect; they are all useful for a limited range of applications; each may reveal some of the truth and none expresses all of the truth.[49]

The accuracy of the “Laws of Nature,” depends on two sets of tools: 1) the physical tools of observation and 2) the mathematical tools of analysis. As our tools of observation and tools of computation have improved, so have our models of natural phenomena. The “Laws of Nature” have been and will continue to change or be altogether replaced as the two sets of tools continue to improve.

**Patentability of AI Intersects the Unpatentability of Mathematics**

Perhaps because “Algorithms” and “Math” have been disqualified from patentability, we have arrived at today’s discussions regarding the patentability of AI inventions[50] that seek to patent the automatic outputs of a machine.[51]

Today’s Artificial Intelligence is focused on “machine learning” and *Brunton and Kutz* say of “machine learning”:

All of machine learning revolves around optimization. This includes regression and model selection frameworks that aim to provide parsimonious and interpretable models for data [266]. … When the model is not prescribed, then optimization methods are used to select the best model.[52]

Someone and not something decides which model or which optimization method to select and someone and not something improves upon the models and the methods. That someone is “the inventor” or at least “an inventor.”

**If it Walks and Talks Like an Invention…**

Whether a tool of analysis or a model of a physical phenomenon, “Mathematical Formulas and Relationships” sound like inventions and walk like inventions, and they are not as divine and infallible as the courts have deemed them to warrant an exalted exception.[53]

Accordingly, at least part of that which is inventive with respect to AI is the “invention” of the “Person” who creates the AI algorithms. The “learning” and “improvement” of AI is the result of its underlying algorithms; it is not organic.[54]

Jointhe Discussion17comments so far. Add my comment.## Lab Jedor

November 20, 2022 12:25 am“Almost all discoveries happen because people are curious and seek new knowledge, not because scientists seek money.”

You did your discovery while being employed and paid by a university. You advanced your career by publishing, hopefully, a much quoted/cited article. You were rewarded for good work. And that is how it should be.

Drs. Rivest, Shamir, Adleman, Diffie and Hellman did not invent their methods because of money. I am sure of it. They did so because they were curious, brilliant, inventive and they were good at their jobs. But they did patent their work. So what?

Anyway, this post is concerned about the use of mathematics in patent claims and the almost automatic presumption of ineligibility only because mathematical expressions are used. This seems unscientific and unreasonable to the authors of this post and most commenters. You have not commented on it. You only want to let us know that you discovered a method for primality test that “could NOT” be patented. That statement raises questions.

You started out by saying that “you could NOT patent” your discovery. I am still curious why you “could NOT patent” your primality test, while others have done so. Now you make it sound like a high and mighty assertion of a principled decision.

So which one is it? Did you try to patent it and failed? Was it rejected as being patent ineligible? Did the Patent Office reject it as being obvious because pseudoprime tests and Lucas sequences are known? Did you discuss/consider the possibility to patent it and decided not to do so? Did you try to patent it, but your boss refused it? Or perhaps in hindsight you should have considered a patent? It would have added to the status of your “discovery” which would have turned out to be “an invention.” (like RSA and DH which are extremely well known.)

## Anon

November 19, 2022 08:10 pmMr. Baillie,

With all due respect, you are a pompous arse, and your attempt to force your beliefs on the larger universe are disgusting.

Applied math is eminently something that is protectable by patent.

You confuse some type of Math

Sphilosophical view with a patent Eligible advance.You (the Royal you of you personally and your team) have every right to not seek patent protection. But get over yourself in asserting something that is clearly outside of what you know — patent law.

## Robert Baillie

November 19, 2022 07:31 pmWe (B, P, S, and W) invented nothing. We built no new device. We merely discovered something that already exists in mathematical reality, as an astronomer discovers a new comet, or (in the old days), as an explorer discovers a “new” island.

If we or the university had patented the algorithm, who in the world besides us or the university, would be better off? I claim: no one. Patenting would merely impose a cost on anyone who wanted to check a large number for primality.

Almost all discoveries happen because people are curious and seek new knowledge, not because scientists seek money.

## Lab Jedor

November 17, 2022 05:30 pmAs best understood, the Baillie-PSW Primality Test was invented (though mathematicians prefer “discovered”) in 1980. The RSA patent application was filed in 1977. So was the Diffie Hellman patent application. Both methods require large prime numbers and presumably large prime number tests. Both applications resulted in issued patents.

A quick search shows a significant number of issued “primality test” related patents over the years.

So when you say “could NOT patent it,” I am wondering what you mean. Were you trying and told it was impossible or perhaps you did file an application, but it was rejected over 35 USC 101 .

But you are absolutely right that many people benefited from your invention. An impressive list of Computer Algebraic Systems that implement your invention is provided on a Wikipedia Page. As to the monetary value of your potential patent, it would have expired around 2000.

This is my belief:

1) you ( or your employer) should have filed a patent application; (perhaps you did and an examiner issued a 101 rejection)

2) certainly in the context of RSA and DH a patent should have and would have been likely if directed to cryptography.

3) if not now then certainly now and in the future, your invention or inventions like it are and should be patent eligible, no ifs and buts. You probably want to look at eligibility criteria and stay away from claiming the pure math.

I cannot derive from your statement if you are against patents for inventions like yours or you regret not obtaining a patent. But it certainly is patent-worthy.

## Anon

November 17, 2022 10:06 amMr. Baillie,

You will have to pardon our disbelief of your obviously biased viewpoint and assertion of an item that is just not falsifiable.

Perhaps you could augment your position with some reasoning as to why, in your instance at least, an innovation related to your developed algorithm not being patented led to all being better off.

Mind you, it is always up to the innovator (or her assigns) to make the choice of seeking a Quid Pro Quo, but to assert as you do that “all are better off” goes well beyond any one innovator’s prerogative and reflects more an unchecked ego that removes what is valuable from others based on that person’s desire to place their own belief system over the rights of others to choose differently.

## Robert Baillie

November 16, 2022 07:44 pmThe algorithm known as the Baillie-PSW Primality Test was not patented. I think we’re all better off because Carl Pomerance, Sam Wagstaff, John Selfridge, and I could NOT patent it.

## Anon

November 16, 2022 03:23 pmLab,

For the fun of it, the date of your reference (1893) prompted me to look up the date of the Michelson-Morley experiments: between April and July 1887.

And all of sudden, the Aether was not a real thing.

(which leads to one of my repeat observations: the map is not the land)

## Lab Jedor

November 16, 2022 12:45 pmA. We agree.

Just for the fun of it. Here is a quote that I believe reflects the opinion of SCOTUS and the Courts on Mathematics.

“True theory does not require the abstruse language of mathematics to make it clear and to render it acceptable. ” Sir William Preece, Chief Engineer of the British Post Office in 1893.

In case people want to agree with the above, Preece continued:

“the principle of the conservation of energy, the undulatory theory of light (i.e. the Maxwell/Heaviside Equations), the thermal equivalence of work, the

development of electrochemical transformation, and all that is solid and substantial in science and usefully applied in practice have been made clear by relegating mathematical symbols to their proper store place—the study.”

And even more astonishing:

“electrical development has been very much retarded by the phantasies of visionary mathematicians who monopolize the columns of our technical literature and fill the mind of the student with false conclusions.” (this about Heaviside who explained the influence of induction on signal transmission and (re)formulated Maxwell’s unworkable 20 equations in the now standard 4 equations).

Paul J. Nahin in his very enjoyable biography of Oliver Heaviside said about Preece : “Preece was a powerful government official, enormously ambitious, and in some remarkable ways, an utter blockhead.”

## Anon

November 16, 2022 07:26 amLab,

I do understand what you are saying and I have no doubt that

youcan indeed “differentiate clearly between these two aspects.”But the point is NOT what you can do, even as your posts DO identify the same problems as I do.

The point is NOT what you or I understand — it is expressly that the courts do NOT understand what we understand.

What you label as “smart-alecks” (and what they do) are defeated with my linguistic handling (which you would rather not employ). I would prefer to “meet the enemy” and defeat them with the language they have chosen to use — and my call for inte11ectual honesty (read that as consistency IN THAT LANGUAGE) would do just that. It is only when we (the royal we of attorneys and our MULTIPLE ethical responsibilities)

allowshenanigans (or worse, perpetuate them) that innovation suffers.As noted, the problem is exasperated by “stakeholders” who have vested interests in denying patent protection to certain types of innovation.

## Lab Jedor

November 14, 2022 01:44 pmA. I think I understand what you mean with the three types of math. However, I have worked enough with math and machine computations to differentiate clearly between these two aspects. The Courts decided that math is abstract (and thus patent ineligible). As an engineer I know that a computer cannot do abstract, it can only process physical signals by physical devices. Hence, a computer doesn’t do math. (That is why these smart-alecks came up with “directed-to”).

## Anon

November 14, 2022 09:38 amLab,

Your statement of, “

To be clear: a machine doesn’t and cannot do math.” is not clear on its face.“Doing” math is exactly what the machine

does.Here, the linguistics would be better understood with my past differentiation of the three “maths” kept in mind (see also my post below).

1) math (on its face — in a

per sesense)2) applied math (that is, math put to use, and here would include most ALL of the computing arts, as well as most all of engineering)

3) Math

S(this is more the philosophy behind both 1) and 2) – which tends to ensnare casual discussions of patents, as most non-patent professionals [think those at Tech-Dirt and the like] tend to confuse themselves in this third zone.Overall though, I think that we are more in agreement (vis a vis eligibility) than not.

## Lab Jedor

November 13, 2022 05:04 pmVery good and timely article. But what is the conclusion as related to patent claims?

“By their nature, such “Mathematical Formulas and Relationships” are well-defined and fleshed out to the minutest detail[40] which is the antithesis of being “Abstract.”” I do not agree.

Mathematics, like spoken and written language, is abstract. There is nothing we can do about that.

Mathematics as provided in symbols and formulas are ideas. They only have meaning when provided with an explanation. But, a machine that performs an operation in accordance with a mathematical expression is real, concrete, non-abstract. They exist and operate outside our presence, no matter how they are described. To be clear: a machine doesn’t and cannot do math.

I would say that a machine or method performing a computation that is described by one or more mathematical expressions is not patent ineligible for that reason. Also, a machine or method performing a task described by a mathematical expression is not patent ineligible for that reason.

Benson and Alice as criteria for patent eligibility have to go as being wrongly decided based on a misconception of what a computer is and does.

Just to make sure we don’t bring in a Trojan Horse on eligibility: A computer implemented invention is determined by the data flow caused by instructions. To say that a computer is a device that is merely executing and storing (what? data? instructions?) as suggested in the Tillis proposal is just too childish and uninformed. It is like saying that a motorized machine is a device that merely applies rotary movement.

## B

November 11, 2022 12:52 pm@ Dwadasi Sarma “How does one address preemption and how can one enforce such a patent ?”

Judge Reyna the Clueless, divorced Alice-Mayo from preemption in Ariosa. Preemption of the basic tools of human endeavor was the policy behind Bilski, Mayo, and Alice Corp. In one brain-dead swoop the CAFC eliminated the only policy restraining Alice-Mayo.

Anyway, no one had to write this (wonderful) article to convince me that the black-robed technical neophytes were wrong in Benson.

## Anon

November 11, 2022 10:11 amDwadasi Sarma,

Your question is not clear, as ALL claims “preempt.”

That is the purpose of claims.

## Anon

November 11, 2022 10:10 amI would add — that in the context of patent law — there are three very distinct ‘takes’ on math:

1) math

2) applied math

3) Math

S1) being the traditional aspects of descriptors of ‘what is there.”

2) being indicative of what most all consider patent eligible, and includes the notions of most all engineering.

3) being the philosophies of BOTH 1) and 2), and (quite sadly) is an item often misunderstood, conflated, and confused in discussions of innovation protection.

## Dwadasi Sarma

November 11, 2022 09:46 amHow does one address preemption and how can one enforce such a patent ?

## Anon

November 11, 2022 09:45 am“This is not a pipe.”

(the map is not the land)

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