phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | prime factorization and discrete logarithms on a quantum computer— “This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems.” | |
| claim-type | method— “Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer.” | |
| method — evidence | — “Instead of giving a quantum computer algorithm for factoring n directly, we give a quantum computer algorithm for finding the order of an element x in the multiplicative group (mod n); that is, the least integer r such that xr 1 (mod n).” | |
| transfer | from number theory / mathematics: reducing integer factorization to order/period-finding (a number-theory reduction), then solving it on a quantum computer via the quantum Fourier transform— “factorization can be reduced to finding the order of an element” | transfers[0] |
| mechanism | A THESIS'S VALUE IS ITS ECONOMY, AND THE ECONOMY IS STATED AS THE REASON: Church's thesis GREATLY SIMPLIFIES THE STUDY OF COMPUTATION, SINCE IT REDUCES THE POTENTIAL FIELD OF STUDY FROM AN INFINITE NUMBER OF POTENTIAL COMPUTING DEVICES TO TURING MACHINES. What is claimed for it first is not its truth but what it lets one stop studying— “Central to this result is Church’s thesis, which says that all computing devices can be simulated by a Turing machine. This thesis greatly simplifies the study of computation, since it reduces the potential field of study from any of an infinite number of potential computing devices to Turing machines.” | mechanisms[0] |
| mechanism | FORMALISING A CLAIM WOULD MAKE IT WEAKER, AND THE MECHANISM IS GIVEN: the thesis IS NOT A MATHEMATICAL THEOREM, and to make it one WOULD REQUIRE A PRECISE MATHEMATICAL DESCRIPTION of a computing device - which WOULD LEAVE OPEN THE POSSIBILITY of some practical device NOT SATISFYING that description, and so WOULD MAKE THE RESULTING THEOREM WEAKER THAN THE ORIGINAL THESIS. A formalism can be escaped in a way an informal claim cannot— “Church’s thesis is not a mathematical theorem; to make it one would require a precise mathematical description of a computing device. Such a description, however, would leave open the possibility of some practical computing device which did not satisfy this precise mathematical description, and thus would make the resulting mathematical… | mechanisms[1] |
| mechanism | A CATEGORY IS INTRODUCED EXPLICITLY AS A COMPROMISE BETWEEN TWO UNUSABLE GRAINS: the old distinction is MUCH TOO COARSE, the exact efficiency is TOO PRECISE A QUANTITY TO WORK WITH EASILY, and what the field uses is THE GENERALLY ACCEPTED COMPROMISE BETWEEN COARSENESS AND PRECISION - polynomial versus superpolynomial scaling. The class P is presented as a negotiated grain, not a natural joint— “With the development of practical computers, it has become apparent that the distinction between computable and non-computable functions is much too coarse; computer scientists are now interested in the exact efficiency with which specific functions can be computed. This exact efficiency, on the other hand, is too precise a quantity to work… | mechanisms[2] |
| mechanism | THE CLASSIFICATION IS SHOWN TO REQUIRE A FURTHER THESIS BEFORE IT MEANS ANYTHING: FOR THIS CLASSIFICATION TO MAKE SENSE, WE NEED IT TO BE MACHINE-INDEPENDENT - we need to know that polynomial-time computability does not depend on the kind of device used - and that requirement IS a quantitative version of Church's thesis. A definition resting on an unproved claim about the world— “For this classification to make sense, we need it to be machine-independent. That is, we need to know that whether a function is computable in polynomial time is independent of the kind of computing device used. This corresponds to the following quantitative version of Church’s thesis” | mechanisms[3] |
| mechanism | ONE CLAIM CARRIES THREE NAMES ACROSS THREE LITERATURES, AND ONE OF THEM ONLY PARTLY: the same thesis has been CALLED THE STRONG CHURCH'S THESIS by one group, and MAKES UP HALF OF THE INVARIANCE THESIS of another author. Two vocabularies name it whole, a third contains it as a part - which is a partial overlap between naming systems, not a synonym— “quantitative version of Church’s thesis, which Vergis et al. [1986] have called the “Strong Church’s Thesis” and which makes up half of the “Invariance Thesis” of van Emde Boas [1990].” | mechanisms[4] |
| mechanism | A READER-FACING SUBSTITUTION IS OFFERED AND JUSTIFIED BY MUTUAL SIMULATION: readers NOT COMFORTABLE WITH Turing machines MAY THINK INSTEAD OF digital computers with linearly growing memory, BECAUSE THESE TWO CLASSES CAN EFFICIENTLY SIMULATE EACH OTHER. One object is licensed to stand in for another, and the licence is an equivalence— “Readers who are not comfortable with Turing machines may think instead of digital computers having an amount of memory that grows linearly with the length of the computation, as these two classes of computing machines can efficiently simulate each other.” | mechanisms[5] |
| mechanism | THE THESIS'S ESCAPE CLAUSES ARE LOCATED IN TWO SPECIFIC WORDS AND NAMED AS SUCH: THERE ARE TWO ESCAPE CLAUSES IN THE ABOVE THESIS - ONE OF THESE IS THE WORD PHYSICAL, and machine models that violate the thesis HAVE BEEN RULED OUT BY SOME REASON FOR WHY THEY ARE NOT PHYSICAL, that is, why they could not be built and made to work. A single underdefined word is doing the excluding— “There are two escape clauses in the above thesis. One of these is the word “physical.” Researchers have produced machine models that violate the above quantitative Church’s thesis, but most of these have been ruled out by some reason for why they are not “physical,” that is, why they could not be built and made to work.” | mechanisms[6] |
| mechanism | AND THE SECOND ESCAPE CLAUSE IS A WORD WHOSE MEANING IS SAID TO BE INCOMPLETELY SPECIFIED: THE OTHER ESCAPE CLAUSE IS THE WORD RESOURCES, THE MEANING OF WHICH IS NOT COMPLETELY SPECIFIED - after which the resources are enumerated for digital computers, and MORE are said to be pertinent to analog computation. The thesis's content varies with a list nobody has closed— “The other escape clause in the above thesis is the word “resources,” the meaning of which is not completely specified above. There are generally two resources which limit the ability of digital computers to solve large problems: time (computation steps) and space (memory). There are more resources pertinent to analog computation” | mechanisms[7] |
| mechanism | A NEW RESOURCE IS ADDED TO THE ACCOUNTING AND THE ADDITION IS ARGUED FOR: for quantum computation, IN ADDITION TO SPACE AND TIME, THERE IS ALSO A THIRD POTENTIALLY IMPORTANT RESOURCE, PRECISION - and the argument is that the changes a quantum computer makes CAN CLEARLY NOT BE PERFECTLY ACCURATE but MUST CONTAIN SOME SMALL AMOUNT OF INHERENT IMPRECISION. The bookkeeping is extended because the physics forces a third column— “For quantum computation, in addition to space and time, there is also a third potentially important resource, precision. For a quantum computer to work, at least in any currently envisioned implementation, it must be able to make changes in the quantum states of objects (e.g., atoms, photons, or nuclear spins). These changes can clearly not… | mechanisms[8] |
| mechanism | WHAT SETS THE LIMIT IS ATTRIBUTED TO MATERIALS RATHER THAN TO PHYSICAL LAW, WITH THE HEDGE STATED: AS FAR AS WE KNOW, what precision is possible IS DICTATED NOT BY FUNDAMENTAL PHYSICAL LAWS BUT BY THE PROPERTIES OF THE MATERIALS AND THE ARCHITECTURE with which the computer is built - and IT IS CURRENTLY NOT CLEAR which architectures will give high precision— “As far as we know, what precision is possible in quantum state manipulation is dictated not by fundamental physical laws but by the properties of the materials and the architecture with which a quantum computer is built. It is currently not clear which architectures, if any, will give high precision, and what this precision will be.” | mechanisms[9] |
| mechanism | A MODELLING CHOICE IS DEFENDED AGAINST A TRUE DESCRIPTION, AND THE GROUND IS USEFULNESS: treating precision as a large constant - EVEN THOUGH IT IS ALMOST CERTAIN TO BE CONSTANT FOR ANY GIVEN MACHINE - is compared to treating a digital computer as a finite automaton, of which the verdict is THIS VIEW IS TECHNICALLY CORRECT; HOWEVER, IT IS NOT PARTICULARLY USEFUL. A description can be accurate and the wrong one to adopt— “Treating the precision as a large constant (even though it is almost certain to be constant for any given machine) would be comparable to treating a classical digital computer as a finite automaton — since any given computer has a fixed amount of memory, this view is technically correct; however, it is not particularly useful.” | mechanisms[10] |
| mechanism | A KEY WORD IS FOUND SUBSTITUTED IN ANOTHER AUTHOR'S STATEMENT OF THE SAME THESIS, AND THE SUBSTITUTION IS DIAGNOSED: in one restatement THE WORD PHYSICAL IS REPLACED WITH THE WORD REASONABLE - and IT IS DIFFICULT TO IMAGINE ANY DEFINITION OF REASONABLE IN THIS CONTEXT WHICH DOES NOT MEAN PHYSICALLY REALIZABLE. One word swapped for another in the same claim, and the author argues they must come to the same thing— “This can be seen in the statement of the quantitative Church’s thesis in van Emde Boas [1990], where the word “physical” in the above phrasing is replaced with the word “reasonable.” It is difficult to imagine any definition of “reasonable” in this context which does not mean “physically realizable,” i.e., that this computing machine could… | mechanisms[11] |
| mechanism | A THESIS IS BELIEVED ON THE STRENGTH OF FAILED REFUTATIONS, AND THE FAILURES ARE THEN SHOWN TO SHARE A PROPERTY: computer scientists became convinced THROUGH THE FAILURE OF ALL PROPOSED COUNTER-EXAMPLES - but MOST OF THESE HAVE BEEN BASED ON THE LAWS OF CLASSICAL MECHANICS, HOWEVER THE UNIVERSE IS IN REALITY QUANTUM MECHANICAL. The induction's instances share exactly the property the new case lacks— “Computer scientists have become convinced of the truth of the quantitative Church’s thesis through the failure of all proposed counter-examples. Most of these proposed counter-examples have been based on the laws of classical mechanics; however, the universe is in reality quantum mechanical.” | mechanisms[12] |
| mechanism | PRIORITY IS ASSIGNED BY THE EXPLICITNESS OF A QUESTION RATHER THAN BY THE IDEA, AND ONE AUTHOR IS CREDITED FOR WHAT HE DID NOT ASK: one researcher DID NOT ASK whether quantum mechanics conferred extra power AND YET showed enough to make later work possible; another RAISED THE POSSIBILITY, THUS IMPLICITLY ASKING the converse question; and a third WAS THE FIRST TO ASK THIS QUESTION EXPLICITLY— “Although he did not ask whether quantum mechanics conferred extra power to computation, he showed that reversible unitary evolution was sufficient to realize the computational power of a Turing machine, thus showing that quantum mechanics is at least as powerful computationally as a classical computer. This work was fundamental in making… | mechanisms[13] |
| builds-on | the number-theoretic reduction of factoring to order-finding (Miller 1976)— “can be reduced to finding the order of an element” | claim_relations[0] |
| builds-on | computability theory (Church, Turing, Post 1936)— “the distinction between computable and non-computable functions shown in papers of Church” | claim_relations[1] |
| builds-on | Simon [1994]— “Simon’s algorithm inspired the work presented in this paper.” | claim_relations[2] |
| builds-on | Miller [1976] randomised reduction of factoring to order-finding— “It is known that using randomization, factorization can be reduced to finding the order of an element [Miller 1976]; we now briefly give this reduction.” | claim_relations[3] |
| builds-on | Coppersmith [1994] and Deutsch [see Ekert and Jozsa 1995] construction of the quantum Fourier transform— “In this paper, we will give a simple construction for Aq when q is a power of 2 that was discovered independently by Coppersmith [1994] and Deutsch [see Ekert and Jozsa 1995].” | claim_relations[4] |
| extends | Deutsch and Jozsa [1992] and Berthiaume and Brassard [1992a, 1992b]— “However, these papers did not show how to solve any problem in quantum polynomial time that was not already known to be solvable in polynomial time with the aid of a random number generator, allowing a small probability of error; this is the characterization of the complexity class BPP, which is widely viewed as the class of efficiently… | claim_relations[5] |
| contradicts | the quantitative (strong) Church's thesis, that any physical computing device can be simulated by a Turing machine with at most polynomial overhead— “A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration.” | claim_relations[6] |
| sense | Quantitative Church’s thesis - AS THE PAPER READS IT, WITH ITS TWO LOOSE WORDS NAMED: 'There are TWO ESCAPE CLAUSES in the above thesis. One of these is the word "physical" ... The other escape clause ... is the word "resources," the meaning of which is NOT COMPLETELY SPECIFIED.' What the thesis asserts depends on which senses are supplied for those two words— “There are two escape clauses in the above thesis. One of these is the word “physical.” Researchers have produced machine models that violate the above quantitative Church’s thesis, but most of these have been ruled out by some reason for why they are not “physical,” that is, why they could not be built and made to work. The other escape… | senses[0] |
| sense | efficiently computable - EXPLICITLY A COMPROMISE, NOT A DISCOVERY: 'The generally accepted compromise between coarseness and precision' between a computable/non-computable split that is 'much too coarse' and exact efficiency that is 'too precise a quantity to work with easily' - realised as the polynomial vs superpolynomial scaling line— “The generally accepted compromise between coarseness and precision distinguishes efficiently and inefficiently computable functions by whether the length of the computation scales polynomially or superpolynomially with the input size.” | senses[1] |
| differs in | the quantitative (strong) Church's thesis, that any physical computing device can be simulated by a Turing machine with at most polynomial overhead — THE AXIS IS WHETHER QUANTUM MECHANICS IS TAKEN INTO ACCOUNT. 'A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most A POLYNOMIAL FACTOR. This may not be true when quantum mechanics is taken into cons[IDERATION].' The thesis is not shown false - it is shown to have a physical precondition nobody had needed to state, and the polynomial factor is what fails. A 'contradicts' alone records a dispute with a computing thesis and drops the fact that the dispute is ABOUT WHICH PHYSICS THE MACHINE OBEYS. | claim_relations[6] |
| audit | physical —same-as→ reasonable — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span says 'physical' was REPLACED by 'reasonable' in one statement of the thesis; same-as is loose. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| note | physical —same-as→ reasonable — TWO PUBLISHED STATEMENTS OF ONE THESIS DIFFER IN ONE WORD, AND THE PAPER ARGUES THEY MUST MEAN THE SAME: in van Emde Boas [1990] '…is replaced with the word "reasonable." It is difficult…' A terminological argument inside a mathematics paper, about the load-bearing word of the thesis being tested. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| weight | physical —same-as→ reasonable — 0.8 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[0] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[1] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[2] |
| evidence status note | mechanism — RE-SLICED: the first cut ended "...quantitative version of Church", stopping INSIDE THE NAME. The cut-word guard did not catch it because the next character is a curly apostrophe, not a letter - so `Church` + `’s` reads as a clean word boundary to the check. A span must not end inside a person's name either, and no guard currently tests that. PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: `PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b` matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[3] |
| evidence status note | mechanism — EXTENDED BACKWARDS: the first cut began "which Vergis et al. [1986] have called..." and never named the thesis it was about. A tag asserting that one claim carries three names has to contain the claim. Overlaps the span of the preceding mechanism, which is allowed and normal. PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[4] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[5] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[6] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[7] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[8] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[9] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[10] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[11] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[12] |
| evidence status note | mechanism — PROVENANCE: this record belongs to the priming sub-corpus, which is the other terminal's lane, and these mechanisms were written by the project. No two hands were on the record - it carried no mechanisms before these, and the project confirmed on 2026-08-17 that the project is auditing rather than tagging. Kept rather than retracted on his instruction (17:22): retracting verified work over a bookkeeping flag would leave a retraction reason that misdescribes the row. Recorded here so a reader learns it from the corpus. The flag on this record is a FALSE POSITIVE of the ownership rule, not a judgement about the paper: 'PRIMING_SUBJECT = \bprim(e|es|ed|ing)\b' matched “PRIME factorization” in the title. It carries no priming fields and it is number theory. Released by name on 2026-08-17 17:22 via MANUAL_RELEASED in an internal tool, together with greentao2004-primes and culpepper2020 - which the same rule caught on “the PRIMES” and on a pun about a subscription service. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[13] |
| replication | untested | |
| no type for this | Church’s thesis → a mathematical theorem — THREE PAPERS, THREE DIFFERENT VERDICTS ON WHETHER A FORMALISM CAN BE SHOWN TO CAPTURE THE INFORMAL NOTION IT REPLACES - and no type in the vocabulary can express a disagreement about what KIND OF CLAIM something is. This paper says the identification CANNOT BE MADE A THEOREM WITHOUT WEAKENING IT: a precise mathematical description of a computing device would leave open the possibility of a practical device that did not satisfy it, so the theorem would be weaker than the thesis it replaced. turing1936-computable-numbers argues it instead, enumerating three kinds of argument; tkacik2016-information-living reports a theorem that did make a vague notion precise. 'same-as' and 'distinct-from' are both false here: the three are not making one claim or two, they are ruling differently on whether the identification is the sort of thing a proof can settle. All eight edge types assert that a relation holds between two concepts. Recorded per record, each row quoting only its own paper; the cross-paper half is this paraphrase, checkable by opening the other two., and the third case for an earlier note of ours - which by the rule that one is an observation and three make a proposal is the point at which the gap should be put to Shir. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[0] |
| limitation | the quantum algorithms are randomized, allowing a small probability of error Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | No quantum computer exists that could run the algorithms. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | Decoherence and precision obstacles grow with machine size, so machines large enough for interesting computations may face fundamental difficulties. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| limitation | The analysis of the discrete-log algorithm only establishes the residue of r for primes larger than 18, requiring extra work for small prime powers. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| limitation | The proved efficiency bounds are loose; an actual implementation could do considerably better than the paper's analysis shows. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
| limitation | The problems solved are of narrow intrinsic value, mattering mainly because cryptography assumed they were hard. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[5] |
| limitation | Making the thesis provable would make it weaker, and the paper says so: 'Church’s thesis is not a mathematical theorem; to make it one would require a precise mathematical description of a computing device. Such a description, however, would leave open the possibility of some practical computing device which did not satisfy this precise mathematical description, and thus would make the resulting mathematical theorem weaker than Church’s original thesis.' The informal statement is unprovable and the formal one is about a smaller class of machines. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[6] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Efficient randomized algorithms for factoring and discrete logarithms are given for a hypothetical quantum computer, taking a number of steps polynomial in the input size. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | The quantitative Church thesis may not be true when quantum mechanics is taken into consideration. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| quantity | polynomial steps in input size — quantum running time for factoring / discrete log (vs believed super-polynomial classically) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[0] |
| variable | input size (independent) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[0] |
| variable | number of steps (dependent) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[1] |
| definition | quantum order-finding algorithm Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| definition | factoring integers Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[2] |
| definition | discrete logarithms Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[3] |
| key-terms | factoring, quantum, algorithm, computer, integer, polynomial, logarithms, prime, factorization |
| concept | precision Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[4] |
| concept | quantum computer Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[5] |
| concept | Church’s thesis Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[6] |
| concept | physical Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[7] |
| concept | reasonable Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[8] |
When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.
Tag layer © the Meta-Tagging Project, CC BY 4.0. The paper text is © its authors/publisher
and is linked, not reproduced. Excerpts shown are short quotations for identification and verification.
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