{"id":"5676d225-71ef-4d91-a4e7-ba467d1868f0","arxiv_id":"1908.06302","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A proof of Lusin's Theorem through near-uniformity of the iterated Turing jump, giving uniform effective versions and a lowness-based account of where discontinuities must occur.","lead":"Russell Miller proves Lusin's Theorem, the classical result that every Borel measurable function is nearly continuous, using the theory of Turing jumps. The proof yields uniform computable versions, so a continuous approximating function can be produced algorithmically from a code for the original function.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 5.2's finite-string oracle computations are not sound for every x in the interval, and Lemma 5.6 relies on that unsupported soundness; the proof of Theorem 6.1 therefore has an internal gap.","rationale":"The reader identified the external Theorem 4.2 as a load-bearing assumption, and it is indeed unproved in the paper. But that theorem is a standard Borel-code/hyperarithmetic fact and is likely repairable by a proper citation or a short proof. The finite-string issue is internal to the construction: Definition 5.2 and the Main Step conflate a halting computation on a finite padded string with a computation valid for all reals in an interval. If a spurious computation is enumerated and applied, it can block genuine bounds, so the equality g=f outside U, which is exactly what the error-set measure bound is supposed to deliver, is not established. This is the most load-bearing point for Theorem 6.1. The Baire-category and Cantor-space results promised in the abstract are indeed absent, and I agree with the reader that the paper is not in final form. I would keep the verdict CONDITIONAL rather than reject, because the gap is local and seems repairable by requiring the use of accepted computations to lie inside the finite string, which would not affect the uniformity argument.","tokens_in":19151,"tokens_out":26121,"duration_ms":301270,"concrete_test":"Formalize Definition 5.2 and the Main Step with the explicit convention that an oracle consisting of a finite string is sigma^0^infinity, and instrument the construction to record the length of the oracle use of each accepted upper or lower computation. Check whether the proof of Lemma 5.6 ever shows that accepted computations for x outside U have use bounded by the length of lambda_{c,c'} xor rho_{c,c'}; if it does not, the step 'once that happens, the computation (c,c',u) will be applied' is unjustified. Independently, rerun the proof of Theorem 5.1 with the added condition that a computation is used only if its oracle use lies within the finite string; if Lemma 5.6 then goes through with the same measure bound, the original text is missing exactly this condition and the proof must be amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 5.2 lets an upper computation be any halting computation of Theta on the oracle S(A) xor lambda_{c,c'} xor rho_{c,c'}, where lambda and rho are finite strings of length l. Under the standard oracle convention, such a finite string is extended by 0s, so the oracle is a finite-stage approximation of some enumeration, not an actual enumeration of the Dedekind cut of any real. In particular, the first rational q_l inside (c,c') is assigned 0 in both lambda and rho in the artificial oracle, while for every real x in (c,c') that rational belongs to L_x or R_x. Therefore Theta may query past l, see 0, and halt with a value that is true for no x in (c,c'). The Main Step applies every listed computation once all three shrinking intervals lie inside (c,c'); Lemma 5.6 then assumes that the computation delivers a valid upper or lower bound for f(x). The lemma proves the forward direction, that genuine bounds for x outside U appear in the enumeration, but it never proves the converse, that every listed computation applied to x outside U is genuine. The parenthetical in the Main Step, that a contradiction would generate an error interval containing x, is not established; x outside U has no error interval around it. Without a use-boundedness condition on accepted computations, the identity g(x)=f(x) outside U is unsupported, and Theorem 6.1 inherits the gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a computability-theoretic proof of Lusin's Theorem: every Borel function f:R->R agrees with some continuous g except on a set of measure less than epsilon. The proof strategy is to represent a Borel f as an alpha-jump S-computable function via a cited characterization (Theorem 4.2, from Kechris), prove a near-uniform approximation theorem for the iterated jump on Dedekind cuts (Theorem 3.4), and then construct a Turing functional Gamma that computes g from the oracle S(A) plus an enumeration of the cut of x. The main claimed output is Theorem 6.1, a uniform effective version: an index for g is computed uniformly from the index of f and from epsilon, independently of the Borel level of f, once an oracle coding the presentation A is supplied. The paper also states an extended-valued version (Theorem 6.2) and a variant for continuous functions (Theorem 8.1). The abstract additionally promises Baire-category and Cantor-space versions, but the body of the submitted text does not contain those sections.","tokens_in":19405,"tokens_out":13040,"duration_ms":133329,"significance":"If the main proof can be repaired, this is a genuinely striking paper. It connects a classical theorem of real analysis to the near-uniformity of the transfinite Turing jump, and it yields a concrete, falsifiable uniformity statement: the approximating continuous function is obtained uniformly from epsilon and the index of f. The proof is not circular: it does not assume Lusin's Theorem, and the reliance on the external Borel-computability characterization Theorem 4.2 is declared. Theorem 3.4 is the main technical engine and is largely explicit, and the uniformity in epsilon is a real feature. However, the gap detailed below in the soundness of finite-string oracle computations is load-bearing for Lemma 5.6 and therefore for Theorem 5.1 and Theorem 6.1. The missing Baire-category and Cantor-space sections announced in the abstract are also a completeness problem for this version.","major_comments":[{"comment":"The finite-string oracle computations defined in Definition 5.2 are not sound for every x in the interval (c,c'), and Lemma 5.6 never proves the converse direction needed for the Main Step. Definition 5.2 declares (c,c',u) to be an upper computation if Theta^{S(A) xor lambda_{c,c'} xor rho_{c,c'}}(<u,2n+1>) halts and outputs 1. Under the standard convention, the finite strings lambda and rho are padded with zeros beyond their length l. For every x in (c,c'), the rational q_l inside (c,c') is a member of L_x or R_x, so the padded oracle gives the wrong answer at that position to any computation that queries beyond l. Such a computation can therefore halt with an output that is true for no x in (c,c'). The Main Step then applies every listed computation once the three shrinking intervals lie inside (c,c'), without checking that the computation's use is contained in the finite string. Lemma 5.6 proves, in the forward direction, that genuine upper and lower bounds for x outside U appear in the enumeration, but it does not prove that every listed computation applied to x outside U is genuine. The parenthetical in the Main Step that a contradiction would generate an error interval containing x is not established: x outside U has no error interval around it. Without a use-boundedness condition on accepted computations, the identity g(x)=f(x) outside U is unsupported, and Theorem 6.1 inherits the gap.","section":"§5, Definition 5.2 and Lemma 5.6"},{"comment":"The abstract states that 'Easier results, which we prove by the same methods, include versions of Lusin's Theorem with Baire category in place of Lebesgue measure and also with Cantor space 2^N in place of R,' but the submitted body contains no such theorems and no sections devoted to Baire category or Cantor space; the text ends after Section 8. Since the abstract and introduction advertise these variants as part of the paper's contribution, this is not a mere wording issue. Either these results and their proofs must be added, or the abstract and introduction must be revised to state accurately what the paper contains.","section":"Abstract"},{"comment":"Theorem 4.2 is the bridge from Borel measurability to alpha-jump S-computability, and Theorem 5.1 uses it in a strong form: a single oracle S and a single S-decidable presentation A must work for every enumeration A xor B of the cut of every x in R, with the successor and limit relations of the presentation computable from S. This theorem is cited from [2] without a specific theorem number. Please supply the exact statement and location in Kechris, and verify that it indeed gives the uniformity and decidability properties required by Definition 3.2 and Corollary 3.5. If Theorem 4.2 provides only a presentation that is merely S-computable in the weaker atomic-diagram sense, or if the uniformity over enumerations is not explicit, then the subsequent application of Corollary 3.5 and the construction of Gamma do not follow as written.","section":"§5, Theorem 4.2 / proof of Theorem 5.1"}],"minor_comments":[{"comment":"There is a notational clash in Theorem 6.1: A denotes both the presentation of the ordinal alpha and the left Dedekind cut in the join A xor B. Please use different symbols, for example P for the presentation and A xor B for the cut enumeration.","section":"§6, Theorem 6.1"},{"comment":"The paper never states its convention for using a finite string as an oracle. In Lemma 3.3 and Definition 5.2, expressions of the form Phi^{sigma}(e) downarrow and Theta^{S(A) xor lambda xor rho} are used; please state explicitly whether finite strings are padded with zeros, identified with arbitrary extensions, or treated as finite partial functions, and explain how the convergence notion depends on that convention.","section":"§3, Lemma 3.3 / Theorem 3.4"},{"comment":"There are several typos in the final paragraph of Section 5: 'whcih', 'a dn', and 'end poitnts' should be corrected. The same paragraph's final sentence is extremely long and would benefit from being split for readability.","section":"§5"},{"comment":"Lemma 5.4 is stated with proof omitted (the proof is marked only by a square). It is used in the soundness of the main construction, so a short proof or at least a clear justification of the three-string claim should be included.","section":"§5, Lemma 5.4"},{"comment":"The proof of Theorem 8.1 says the cuts L_q and R_q are 'computable uniformly in Q'; this should read 'uniformly in q'. The decision procedure for the set D is plausible but deserves one more sentence explaining why the additional jump A+1 suffices to decide a universal quantification over infinitely many rationals in [a,b].","section":"§8, Theorem 8.1"}],"recommendation":"major_revision","confidential_remarks":"The finite-string oracle issue is the key technical point: if the author can add a use-boundedness condition to Definition 5.2, the proof of Lemma 5.6 may be repairable. Please also verify that the Baire-category and Cantor-space results promised in the abstract actually exist in the submitted version; if they are part of a later version, the abstract must be corrected. The exact reference for Theorem 4.2 should be checked before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the arXiv v1 carefully. The reader's conditional verdict is right, and the stress-test note is not wrong: there is a real gap at Lemma 5.6, and it is load-bearing.\n\nWhat's genuinely new: the uniform effective Lusin Theorem 6.1 (h computable in ε and the index e), the function-independent error set of Prop 7.1, and the continuous-function recovery Theorem 8.1. The method—deriving Lusin from near-uniformity of iterated Turing jumps, with an error set uniform in f—is a new route to a classical theorem. The construction in Section 5 is explicit, and the use of Kechris's Theorem 4.2 as an external bridge is standard and not circular.\n\nThe gap is exactly the one the stress test identifies. Definition 5.2 accepts any halting computation of Θ on S(A)⊕λ⊕ρ, with finite strings extended by zeros in the usual oracle convention. For a good interval (c,c'), the first rational q_l inside lies beyond the strings' length, so the zero-extension puts q_l in neither L nor R—matching only the rational x=q_l, not the continuum-many other x in (c,c'). A computation that queries q_l or anything past the string is therefore not sound for those x. The Main Step applies every listed computation whose interval contains x, and Lemma 5.6 proves only the forward direction (genuine bounds eventually appear), never the converse (every listed computation is genuine). Without a use-boundedness condition—say, requiring the halting computation's use to stay entirely within the finite string, so it never queries the ambiguous rational—g(x) can be driven by spurious bounds. The parenthetical about a later error interval does not rescue x outside U; no error interval contains such x. Theorem 6.1 inherits the gap. This is not obviously fatal: adding an explicit use bound to Definition 5.2 and rechecking the Main Step looks like a plausible repair. But as written, the proof doesn't go through.\n\nSmaller issue: the abstract promises Baire-category and Cantor-space versions; those are absent from this v1. The abstract should be trimmed or the versions supplied.\n\nFor whom: computability theorists and descriptive set theorists interested in effective measure and uniform versions of classical theorems. This deserves peer review: the ideas are substantial, the results are new, and the gap is concrete and likely repairable. I'd send it out with a request for major revision focused on the use-boundedness issue.","headline":"New and uniform computable versions of Lusin's Theorem with a genuinely new proof strategy, but the central Lemma 5.6 has a real gap: finite-string oracle computations are not sound for all reals in the interval, so the uniform theorem as written is not established.","tokens_in":19947,"tokens_out":8801,"would_cite":false,"duration_ms":86735,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03D30","03D78","03E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lusin's theorem can be effectivized uniformly: for every Borel measurable function and every ε>0, a continuous approximant is computed from a code for the function and ε, using an oracle for its Borel level.","keywords":["Lusin's theorem","computable analysis","Turing jump","near-uniformity","Borel functions","Dedekind cuts","generalized lowness","effective measure"],"falsifier":"Theorem 3.4 asserts that for every $S$, every $S$-decidable presentation $A$, and every rational $\\epsilon>0$, the functional $\\Psi_\\epsilon$ agrees with the true iterated jump $(S\\oplus L_x\\oplus R_x)^{(A)}$ for every real $x$ outside the enumerated open set $U_{\\epsilon,S,A}$. Exhibit a single real $x\\notin U_{\\epsilon,S,A}$ and a coded input $\\langle k,n\\rangle$ for which $\\Psi_\\epsilon^{S(A)\\oplus L_x\\oplus R_x}(\\langle k,n\\rangle)\\neq (S\\oplus L_x\\oplus R_x)^{(A)}(\\langle k,n\\rangle)$, and Theorem 3.4—and with it the uniform version of Lusin's Theorem—would be refuted; a computable search over rational intervals would suffice to look for such a disagreement.","tokens_in":18900,"feed_emoji":"📐","tokens_out":10349,"duration_ms":89246,"temperature":0.7,"pith_summary":"This paper proves Lusin's Theorem—every Borel-measurable function $f:\\mathbb{R}\\to\\mathbb{R}$ agrees with some continuous function off a set of measure less than $\\epsilon$—by a route through computability theory, and the route yields more than the classical statement. The approximation is uniform: there is a computable map $h$ such that from a Borel-code index $e$ of $f$ and a rational tolerance $\\epsilon$ one obtains an index $h(\\epsilon,e)$ of a Turing functional that, with an oracle coding the ordinal level of $f$, computes a continuous $g$ agreeing with $f$ off a set of measure $<\\epsilon$. The key is that the iterated Turing jump is nearly uniform: a single functional $\\Psi_\\epsilon$ approximates $(S\\oplus L_x\\oplus R_x)^{(A)}$ from $S(A)\\oplus L_x\\oplus R_x$ except on an open error set of measure $<\\epsilon$ that can be enumerated uniformly from an $S(A)$-oracle and patched by piecewise-linear functions. The proof recasts a classical 'near continuity' theorem as a statement about the jump operator, exposing a single error set that accounts for the discontinuity of every function at a given Borel level.","feed_headline":"Every Borel function has a uniformly computable continuous twin","feed_subtitle":"From a code for a Borel function and a tolerance ε, one index computes a continuous g agreeing with f off a set of measure less than ε.","key_machinery":"The load-bearing mechanism is the near-uniformity of the iterated Turing jump, together with a careful 'error set' repair. For an $S$-decidable presentation $A$ of a countable ordinal $\\alpha$, the $A$-jump $C(A)$ codes all lower jumps as columns. Theorem 3.4 supplies, uniformly in the rational tolerance $\\epsilon$, a Turing functional $\\Psi_\\epsilon$ such that for every real $x$, $\\Psi_\\epsilon^{S(A)\\oplus L_x\\oplus R_x}$ is total and agrees with $(S\\oplus L_x\\oplus R_x)^{(A)}$ for all $x$ outside an $S(A)$-effectively open set $U_{\\epsilon,S,A}$ of measure $<\\epsilon$, containing all rationals; moreover the open intervals and 'buffer sets' witnessing the openness are enumerated uniformly. The proof of Theorem 5.1 then runs $\\Theta = \\Phi_e^{\\Psi_\\epsilon}$, collects upper and lower computations on rational intervals, and whenever the three approximations to the input cut fall inside an error interval, invokes a secondary step that defines $g$ piecewise-linearly, so that $g$ is total, single-valued, independent of the enumeration of the cut of $x$, and continuous by the S-computability criterion for continuity.","core_discovery":"The central claim is Theorem 6.1: there exists a computable total function $h:\\mathbb{Q}\\times\\mathbb{N}\\to\\mathbb{N}$ such that whenever $f$ is $\\alpha$-jump $S$-computable, given by $\\Phi_e^{(S\\oplus A\\oplus B)^{(A)}}$ for all enumerations $A\\oplus B$ of cuts of $x$, a continuous $g$ satisfying Lusin's Theorem for $f$ with tolerance $\\epsilon$ is computed by $\\Phi_{h(\\epsilon,e)}^{E(A)\\oplus S(A)\\oplus A\\oplus B}$, where $E(A)$ is the elementary diagram of the presentation $A$ of the ordinal $\\alpha$. The proof builds $g$ by running a near-uniform approximation $\\Psi_\\epsilon$ to the iterated jump $(S\\oplus L_x\\oplus R_x)^{(A)}$ from an $S(A)\\oplus L_x\\oplus R_x$-oracle, applying the original functional $\\Phi_e$ to the output, and repairing the rare failures inside the enumerated open error set $U_{\\epsilon,S,A}$ by declaring $g$ piecewise linear there. Since every Borel function is $\\alpha$-jump $S$-computable for some countable $\\alpha$ and some $S$-decidable $A$, the uniform procedure applies to every Borel-measurable $f$. A further corollary is that the same error set $U_{\\epsilon,S,A}$ works for all such $f$ at once, and that no uniform procedure can achieve agreement off a measure-zero set.","pith_inferences":["Because a single error set $U_{\\epsilon,S,A}$ serves every function at a fixed Borel level, one can read the paper as attaching to each countable ordinal $\\alpha$ a canonical 'defect set' in $\\mathbb{R}$; comparing these sets as $\\alpha$ grows would give a measure-theoretic profile of the Borel hierarchy itself.","The paper notes that any dense computable set could replace $\\mathbb{Q}$ in the construction; this suggests a family of variants in which the continuous approximant is forced to take prescribed values on a chosen countable dense set, which might sharpen where the error set sits.","If the conjecture of Section 8 is correct, there is an exact trade-off: no uniform Lusin operator can also leave continuous functions unchanged, and the single additional jump in Theorem 8.1 is the minimal price of correctness.","A concrete testable extension would be to instantiate the construction for $\\alpha=1$ and simple functions such as characteristic functions of intervals, and verify that the computed $g$ agrees with $f$ off the predicted $U_{\\epsilon,S,A}$; this would give a finite computational check of Theorem 6.1 at the lowest nontrivial Borel level."],"forward_implications":["There is a single computable function $h$ such that, whenever $f$ is given by an $\\alpha$-jump $S$-computation with index $e$, the functional with index $h(\\epsilon,e)$ and oracle $E(A)\\oplus S(A)\\oplus A\\oplus B$ computes a continuous $g$ with $\\mu(\\{x: g(x)\\neq f(x)\\})<\\epsilon$, uniformly in $\\epsilon$ and independently of the Borel level $\\alpha$ (Theorem 6.1).","The same open set $U_{\\epsilon,S,A}$ of measure $<\\epsilon$ contains the disagreement set of every $\\alpha$-jump $S$-computable function at once, so each such $f$ is continuous on the complement of $U_{\\epsilon,S,A}$; in this sense the failures of continuity at a fixed Borel level are confined to a single universal 'bad' set (Proposition 7.1).","Lusin's theorem cannot be pushed to equality off a measure-zero set: the characteristic function of $(0,+\\infty)$ shows that no continuous $g$ can agree with it except on a set of measure 0.","The Lusin procedure need not preserve an already continuous $f$, and this is conjectured to be unavoidable for any uniform procedure; however, with one additional jump, an $S(A+1)$-oracle computes $f$ itself uniformly from its $\\alpha$-jump computation (Theorem 8.1).","The same methods are claimed to yield simpler analogues with Baire category in place of Lebesgue measure and with Cantor space $2^{\\mathbb{N}}$ in place of $\\mathbb{R}$."],"supporting_citations":[{"why":"Supplies the theorem that every Borel function is alpha-jump S-computable for some S and S-decidable presentation A; this is the initial bridge from analysis into computability.","marker":"[2]"},{"why":"Establishes the almost-all reduction for the jump operator (A' <=T 0' + A on a measure-1 set), which Section 3 adapts to Dedekind cuts and iterated jumps.","marker":"[6]"},{"why":"Provides the theorem that a real function is continuous iff it is S-computable, used to conclude that the constructed g is continuous.","marker":"[7]"},{"why":"Underlies the framework of iterating the jump through countable ordinals and computable presentations, used in Definition 3.2 and Theorem 3.4.","marker":"[1]"},{"why":"Supplies the standard facts about the Turing jump and the Padding Lemma used to make the uniform function h injective.","marker":"[5]"}],"fun_headline_variants":["Uniform effective Lusin: one recipe for every epsilon","One computable index builds continuous twins for all Borel functions","Effective Lusin: every Borel function gets a computable continuous twin","Computable proof: Borel functions have continuous twins uniformly","Lusin's theorem made effective: one index for all tolerances"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the cited characterization that every Borel function $f:\\mathbb{R}\\to\\mathbb{R}$ is $\\alpha$-jump $S$-computable for some countable ordinal $\\alpha$ with an $S$-decidable presentation $A$; if that bridge from analysis to computability carried hidden uniformity conditions, or failed to supply an $S$-decidable $A$, the near-uniform approximation and the uniform Lusin procedure would not get off the ground.","fun_headline_variants_meta":{"raw":{"variants":["Uniform effective Lusin: one recipe for every epsilon","One computable index builds continuous twins for all Borel functions","Effective Lusin: every Borel function gets a computable continuous twin","Computable proof: Borel functions have continuous twins uniformly","Lusin's theorem made effective: one index for all tolerances"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00157,"raw_usage":{"total_tokens":6309,"prompt_tokens":1024,"completion_tokens":5285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":5198}},"tokens_in":640,"tokens_out":5285,"duration_ms":31964,"temperature":1.0,"reasoning_tokens":5198,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:50:52.229540+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Theorem 3.4 asserts that for every $S$, every $S$-decidable presentation $A$, and every rational $\\epsilon>0$, the functional $\\Psi_\\epsilon$ agrees with the true iterated jump $(S\\oplus L_x\\oplus R_x)^{(A)}$ for every real $x$ outside the enumerated open set $U_{\\epsilon,S,A}$. Exhibit a single real $x\\notin U_{\\epsilon,S,A}$ and a coded input $\\langle k,n\\rangle$ for which $\\Psi_\\epsilon^{S(A)\\oplus L_x\\oplus R_x}(\\langle k,n\\rangle)\\neq (S\\oplus L_x\\oplus R_x)^{(A)}(\\langle k,n\\rangle)$, and Theorem 3.4—and with it the uniform version of Lusin's Theorem—would be refuted; a computable search over rational intervals would suffice to look for such a disagreement.","supporting_citations":[{"cited_title":"Kechris; Classical Descriptive Set Theory (New York: Springer-Verlag, 1995)","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that every Borel function is alpha-jump S-computable for some S and S-decidable presentation A; this is the initial bridge from analysis into computability."},{"cited_title":"almost all","cited_arxiv_id":null,"evidence_quote":"Establishes the almost-all reduction for the jump operator (A' <=T 0' + A on a measure-1 set), which Section 3 adapts to Dedekind cuts and iterated jumps."},{"cited_title":"Weihrauch; Computable Analysis: An Introduction (Berlin: Springer, 2000)","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that a real function is continuous iff it is S-computable, used to conclude that the constructed g is continuous."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underlies the framework of iterating the jump through countable ordinals and computable presentations, used in Definition 3.2 and Theorem 3.4."},{"cited_title":"Soare; Recursively Enumerable Sets and Degrees (New York: Springer-Verlag, 1987)","cited_arxiv_id":null,"evidence_quote":"Supplies the standard facts about the Turing jump and the Padding Lemma used to make the uniform function h injective."}],"review_version":1}