C90 · Koreksi, hak, dan aksesibilitas

{
  "id": "O003-C001",
  "unit": "chapter_01_sets",
  "file": "repo/source/sec_sets_exer.ptx",
  "source_locator": "authority line 111 / derivative lines 108-121",
  "category": "source_markup",
  "source_issue": "literal \\\\item merges an intended fourth identity into Exercise 3(c)",
  "target_action": "split the two identities into separate sibling task elements",
  "status": "verified",
  "evidence": "task count 32 to 33; schema pass; PDF exercise topology inspected"
}

{
  "id": "O003-C002",
  "unit": "chapter_01_sets",
  "file": "repo/source/sec_union_int_comp.ptx",
  "source_locator": "authority lines 109-224",
  "category": "upstream_pdf_typography",
  "source_issue": "the committed upstream PDF drops or weakly renders set-minus glyphs although source retains \\\\setminus",
  "target_action": "preserve every source \\\\setminus and verify the derivative render",
  "status": "verified",
  "evidence": "mathematics sequence preserved; derivative PDF page 10 visibly renders set difference throughout the proof"
}

{
  "id": "O003-C003",
  "unit": "chapter_01_sets",
  "file": "repo/source/sec_basic_top.ptx",
  "source_locator": "derivative line 47",
  "category": "font_glyph",
  "source_issue": "literal Unicode hollow star is absent from the PDF font and rendered as a missing-glyph box",
  "target_action": "represent the same symbol as semantic inline math <m>\\\\star</m>",
  "status": "verified",
  "evidence": "schema pass; derivative PDF page 7 and HTML browser render show the star"
}

{
  "id": "O003-C004",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_fun_set.ptx",
  "source_locator": "authority line 10",
  "category": "mathematical_variable",
  "source_issue": "the definition of f(A) quantifies a in the undefined set C although A was declared as the domain subset",
  "target_action": "replace a in C with a in A in the derivative formula",
  "status": "verified",
  "evidence": "local context declares A subset X and the chapter summary independently gives f(C) with c in C"
}

{
  "id": "O003-C005",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_summ.ptx",
  "source_locator": "authority line 66",
  "category": "mathematical_variable",
  "source_issue": "the summary declares f:A to B but then requires D to be a subset of the unrelated symbol Y",
  "target_action": "replace Y with the declared codomain B in the derivative prose",
  "status": "verified",
  "evidence": "the following preimage formula uses f(a) in D and therefore requires D subset B"
}

{
  "id": "O003-C006",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_fun_set.ptx",
  "source_locator": "authority line 14",
  "category": "mathematical_domain",
  "source_issue": "the definition assumes f:X to Y and B subset Y but defines f inverse of B using a in the unrelated subset A",
  "target_action": "replace a in A with a in the full domain X in the derivative formula",
  "status": "verified",
  "evidence": "by definition a preimage is taken over the full domain of f; the later summary correctly uses a in A only because its function there has domain A"
}

{
  "id": "O003-C007",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_inv_func.ptx",
  "source_locator": "authority line 283",
  "category": "mathematical_direction",
  "source_issue": "for g:B to C the source says g inverse maps B to C instead of C to B",
  "target_action": "replace the stated domain and codomain of g inverse with C and B",
  "status": "verified",
  "evidence": "the surrounding activity assumes bijections f:A to B and g:B to C and derives the inverse of g composed with f"
}

{
  "id": "O003-C008",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_inv_func.ptx",
  "source_locator": "authority line 284",
  "category": "mathematical_typo",
  "source_issue": "the defined inverse composite is printed as f^{_1} composed with g inverse",
  "target_action": "replace f^{_1} with f^{-1}",
  "status": "verified",
  "evidence": "the next line and the theorem goal both state (g composed with f) inverse equals f inverse composed with g inverse"
}

{
  "id": "O003-C009",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 162",
  "category": "mathematical_codomain",
  "source_issue": "choosing D subset B cannot make the already-declared f:A to B surjective unless the codomain is explicitly restricted",
  "target_action": "state that the same rule is regarded as f:A to D",
  "status": "verified",
  "evidence": "the preceding task similarly restricts the domain and the following task explicitly writes a restricted-domain/codomain function"
}

{
  "id": "O003-C010",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 328",
  "category": "mathematical_typo",
  "source_issue": "the displayed identity for the preimage of a composite is missing its final closing parenthesis",
  "target_action": "add the missing parenthesis",
  "status": "verified",
  "evidence": "balanced formula is (h composed with g)^-1(O)=g^-1(h^-1(O))"
}

{
  "id": "O003-C011",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 385",
  "category": "mathematical_subscript",
  "source_issue": "the inverse of f1 times f2 is printed as h1 times h1 although h_i was defined as the inverse of f_i",
  "target_action": "replace the second h1 with h2",
  "status": "verified",
  "evidence": "the product inverse acts coordinatewise and the immediately preceding definition assigns h2 as the inverse of f2"
}

{
  "id": "O003-C012",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 471",
  "category": "mathematical_composition",
  "source_issue": "the surjection question switches from the defined composite g composed with f to the domain-incompatible f composed with g",
  "target_action": "restore g composed with f",
  "status": "verified",
  "evidence": "the exercise introduction declares f:A to B and g:B to C; only g composed with f is defined"
}

{
  "id": "O003-C013",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority lines 625-739",
  "category": "missing_hypothesis",
  "source_issue": "the final true/false exercise uses f:X to Y throughout without introducing f X or Y locally",
  "target_action": "add the hypothesis f:X to Y at the start of the exercise",
  "status": "verified",
  "evidence": "all thirteen tasks then have declared domains for images and preimages"
}

{
  "id": "O003-C014",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 29",
  "category": "source_prose",
  "source_issue": "the source says 'that as exactly two preminages'",
  "target_action": "translate the intended phrase as 'yang memiliki tepat dua prapeta'",
  "status": "verified",
  "evidence": "local grammatical repair only; mathematical request is unchanged"
}

{
  "id": "O003-C015",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 144",
  "category": "source_prose",
  "source_issue": "the source asks 'If f an injection?'",
  "target_action": "translate the intended question 'Apakah f merupakan injeksi?'",
  "status": "verified",
  "evidence": "local grammatical repair only; exercise topology is unchanged"
}

{
  "id": "O003-C016",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 460",
  "category": "source_prose",
  "source_issue": "the source omits 'is' in 'if g composed with f an injection'",
  "target_action": "translate the intended conditional with a complete Indonesian predicate",
  "status": "verified",
  "evidence": "local grammatical repair only; mathematical question is unchanged"
}

{
  "id": "O003-C017",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_exer.ptx",
  "source_locator": "authority line 48",
  "category": "mathematical_terminology",
  "source_issue": "the classification exercise warns about domain and range although its paired examples are designed to differ by codomain",
  "target_action": "replace range with codomain in the derivative instruction",
  "status": "verified",
  "evidence": "surjectivity is determined by comparing the range with the declared codomain; the rational-function pair has the same rule and domain but different codomains"
}

{
  "id": "O003-C018",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_intro.ptx",
  "source_locator": "authority line 189",
  "category": "source_markup",
  "source_issue": "the taxicab figure prompt prints an unmatched opening parenthesis before 'points' and leaves the first coordinate pair without its opening parenthesis",
  "target_action": "place both coordinate parentheses inside the math node and remove the stray prose parenthesis",
  "status": "verified",
  "evidence": "the prompt now names the two points (x_1,x_2) and (y_1,y_2) symmetrically"
}

{
  "id": "O003-C019",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space.ptx",
  "source_locator": "authority line 126",
  "category": "mathematical_domain",
  "source_issue": "the finite example declares X={a,b,c} but gives d the unrelated domain A times A",
  "target_action": "replace A times A with X times X",
  "status": "verified",
  "evidence": "the table and every subsequent quantifier use X"
}

{
  "id": "O003-C020",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space.ptx",
  "source_locator": "authority lines 168-175",
  "category": "missing_case",
  "source_issue": "the finite-metric proof quantifies only x and y although z occurs and then claims only the all-distinct cases remain without treating z=y",
  "target_action": "quantify x y z and add the z=y degenerate triangle case",
  "status": "verified",
  "evidence": "the proof now exhausts x=y x=z z=y and the three all-distinct inequalities"
}

{
  "id": "O003-C021",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_euclid_rn.ptx",
  "source_locator": "authority lines 23 and 277",
  "category": "mathematical_operator",
  "source_issue": "both expanded Euclidean-distance formulas omit the plus sign before the final squared coordinate difference",
  "target_action": "insert the missing plus signs",
  "status": "verified",
  "evidence": "both expanded expressions now match the summation formula beside them"
}

{
  "id": "O003-C022",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_euclid_rn.ptx",
  "source_locator": "authority lines 149-150",
  "category": "missing_hypothesis",
  "source_issue": "the Cauchy-Schwarz proof treats a=sum y_i^2 as a positive quadratic coefficient without handling y=0",
  "target_action": "dispose of y=0 separately and state y nonzero hence a>0 before using the quadratic formula",
  "status": "verified",
  "evidence": "the denominator 2a and discriminant argument now have their required hypothesis"
}

{
  "id": "O003-C023",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_euclid_rn.ptx",
  "source_locator": "authority line 286",
  "category": "mathematical_delimiter",
  "source_issue": "the closing taxicab formula encloses each absolute coordinate difference in literal set braces",
  "target_action": "remove the stray set braces while retaining the absolute-value bars",
  "status": "verified",
  "evidence": "the formula now matches the chapter introduction and summary definitions of d_T"
}

{
  "id": "O003-C024",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_summ.ptx",
  "source_locator": "authority line 53",
  "category": "mathematical_operator",
  "source_issue": "the summary's expanded Euclidean-distance formula omits the plus sign before the final squared coordinate difference",
  "target_action": "insert the missing plus sign",
  "status": "verified",
  "evidence": "the expansion now matches its adjacent summation formula"
}

{
  "id": "O003-C025",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority line 35",
  "category": "mathematical_domain",
  "source_issue": "the modular-metric exercise calls d a metric on R^2 even though d was defined only on X={1,3,5}",
  "target_action": "state the unit-circle sentence as general motivation using a metric space such as R^2 rather than attributing d to R^2",
  "status": "verified",
  "evidence": "the following open-ball definition remains general and d is used only on its declared domain X"
}

{
  "id": "O003-C026",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 99-103",
  "category": "open_ball_boundary",
  "source_issue": "the rational-metric task says distance no more than 1 but immediately identifies the requested set as the open ball of radius 1",
  "target_action": "use distance strictly less than 1 to agree with the displayed open-ball definition",
  "status": "verified",
  "evidence": "B(b,r) was defined with d(y,b)<r in the same exercise set"
}

{
  "id": "O003-C027",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 171-178",
  "category": "mathematical_domain",
  "source_issue": "the max-metric lemma places A and B in R^n although max A and max B require finite subsets of the ordered set R",
  "target_action": "replace R^n with R",
  "status": "verified",
  "evidence": "the scalar inequality is then well-defined and supplies the coordinatewise estimate used for d_M"
}

{
  "id": "O003-C028",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 251-253",
  "category": "mathematical_value",
  "source_issue": "from p=5 m=34 n=7 the source correctly obtains 27=27 times 5^0 but then prints t(43,7)=27",
  "target_action": "replace it with t(34,7)=0",
  "status": "verified",
  "evidence": "the arguments and exponent now follow the immediately preceding factorization"
}

{
  "id": "O003-C029",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority line 313",
  "category": "mathematical_coordinate",
  "source_issue": "the product-metric definition names its first point as (x_1,y_2) instead of (x_1,y_1)",
  "target_action": "replace y_2 with y_1 in the first ordered pair",
  "status": "verified",
  "evidence": "the displayed definition immediately below uses (x_1,y_1) and (x_2,y_2)"
}

{
  "id": "O003-C030",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 436-438",
  "category": "source_prose",
  "source_issue": "the concavity task inserts the fragment 'an interval' after already declaring [0,infinity) and produces an ungrammatical hypothesis",
  "target_action": "state separately that a and b lie in the declared interval",
  "status": "verified",
  "evidence": "the intended domain and inequality are unchanged and all variables are locally bound"
}

{
  "id": "O003-C031",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority line 367",
  "category": "source_prose",
  "source_issue": "the logarithmic-distance exercise asks 'Is d is a metric'",
  "target_action": "translate the intended single-copula question",
  "status": "verified",
  "evidence": "the mathematical function and domain are unchanged"
}

{
  "id": "O003-C032",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 113-142",
  "category": "source_punctuation",
  "source_issue": "two descriptions of Euclidean betweenness open a parenthesis without closing it",
  "target_action": "close the parenthetical explanations in Indonesian",
  "status": "verified",
  "evidence": "both sentences now have balanced punctuation without changing their claims"
}

{
  "id": "O003-C033",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority line 46",
  "category": "source_prose",
  "source_issue": "the open-ball definition says 'is the the set'",
  "target_action": "translate the intended single article",
  "status": "verified",
  "evidence": "the displayed definition is unchanged"
}

{
  "id": "O003-C034",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_summ.ptx",
  "source_locator": "authority line 43",
  "category": "source_prose",
  "source_issue": "the summary says 'These metric are different'",
  "target_action": "translate the intended plural statement",
  "status": "verified",
  "evidence": "all three named metrics remain the subject"
}

{
  "id": "O003-C035",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 472-473",
  "category": "function_type",
  "source_issue": "the proposed product-distance function is declared X times Y to R even though its inputs are ordered pairs of points of X times Y",
  "target_action": "declare its domain as (X times Y) times (X times Y)",
  "status": "verified",
  "evidence": "the displayed rule takes (a,b) and (c,d) as its two arguments"
}

{
  "id": "O003-C036",
  "unit": "chapter_02_functions",
  "file": "repo/source/sec_func_intro.ptx",
  "source_locator": "authority lines 187-230",
  "category": "schema_structure",
  "source_issue": "the restriction definition is placed inside a task introduction that also contains nested tasks; that introduction permits statement blocks but not a definition block",
  "target_action": "move the unchanged restriction definition immediately before the exploration and retain the task introduction and both nested tasks",
  "status": "verified",
  "evidence": "the definition remains before the work that uses it and the cumulative reader validates against the pinned PreTeXt schema"
}

{
  "id": "O003-C037",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_intro.ptx",
  "source_locator": "authority lines 97-182",
  "category": "schema_structure",
  "source_issue": "the absolute-value lemma and proof are placed inside a task statement; a task statement does not permit a lemma block",
  "target_action": "move the unchanged lemma and proof immediately before the exploration and revise the task's directional phrase from 'below' to a position-neutral reference",
  "status": "verified",
  "evidence": "the lemma remains immediately before the activity that uses it and the cumulative reader validates against the pinned PreTeXt schema"
}

{
  "id": "O003-C038",
  "unit": "chapter_03_metric_spaces",
  "file": "repo/source/sec_metric_space_exer.ptx",
  "source_locator": "authority lines 18-59",
  "category": "schema_structure",
  "source_issue": "the open-ball definition is nested in a paragraph inside a list item inside a task-bearing exercise introduction; that introduction permits statement blocks but not a definition block",
  "target_action": "move the unchanged open-ball definition into a new exercises-level introduction before the first exercise, preserve the list-item tail prose, and use a position-neutral connective",
  "status": "verified",
  "evidence": "xml:id def_ms_open_ball and all mathematical content are retained; the corrected file and cumulative reader validate against the pinned PreTeXt 1.7.5 schema"
}

{
  "id": "O003-C039",
  "unit": "chapter_04_metric_space_applications",
  "file": "repo/source/sec_hamming.ptx",
  "source_locator": "authority line 18",
  "category": "mathematical_coordinate",
  "source_issue": "the displayed general word repeats x_1 as its second coordinate",
  "target_action": "replace the second x_1 with x_2",
  "status": "verified",
  "evidence": "the corrected tuple now agrees with the indexed n-coordinate notation used in the definition immediately below"
}

{
  "id": "O003-C040",
  "unit": "chapter_04_metric_space_applications",
  "file": "repo/source/sec_levenshtein.ptx",
  "source_locator": "authority line 87",
  "category": "source_typo",
  "source_issue": "the prose says 'a sting x' where the mathematical object is a string",
  "target_action": "translate the intended word string as untai",
  "status": "verified",
  "evidence": "the surrounding paragraph and every operation concern finite strings"
}

{
  "id": "O003-C041",
  "unit": "chapter_04_metric_space_applications",
  "file": "repo/source/sec_levenshtein.ptx",
  "source_locator": "authority lines 56-116",
  "category": "schema_structure",
  "source_issue": "the Levenshtein definition is nested inside a paragraph inside a task statement; statements permit statement blocks but not a definition block",
  "target_action": "retain the first inquiry task in the first activity, move the pre-definition prose and intact semantic definition to section level, and place the two remaining tasks in a second activity",
  "status": "verified",
  "evidence": "the prose, definition, and all three tasks retain their pedagogical order and the corrected cumulative reader validates against the pinned PreTeXt schema"
}

{
  "id": "O003-C042",
  "unit": "chapter_04_metric_space_applications",
  "file": "repo/source/sec_levenshtein.ptx",
  "source_locator": "authority line 120",
  "category": "mathematical_delimiter",
  "source_issue": "the first spell-check candidate opens but does not close its quotation mark inside the displayed mathematics",
  "target_action": "add the missing closing quotation mark to topography",
  "status": "verified",
  "evidence": "all three candidate strings now have balanced quotation marks"
}

{
  "id": "O003-C043",
  "unit": "chapter_04_metric_space_applications",
  "file": "repo/source/sec_levenshtein.ptx",
  "source_locator": "authority lines 87-116",
  "category": "mathematical_domain",
  "source_issue": "the source calls the domain all possible words without fixing an alphabet, so the alleged metric domain is not a single well-defined set",
  "target_action": "fix an alphabet Sigma, state that x and y are strings over it, and formulate the proof task on Sigma-star",
  "status": "verified",
  "evidence": "deletion, insertion, and substitution are then closed operations on the stated set of finite strings"
}

{
  "id": "O003-C044",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_dist_point_set.ptx",
  "source_locator": "authority lines 10-11",
  "category": "mathematical_hypothesis",
  "source_issue": "the completeness-axiom paraphrase says any subset of the reals bounded below has a greatest lower bound but omits nonemptiness",
  "target_action": "require the subset to be nonempty",
  "status": "verified",
  "evidence": "source-fidelity QA and pinned-schema validation pass; the empty set is bounded below but has no real infimum"
}

{
  "id": "O003-C045",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_dist_point_set.ptx",
  "source_locator": "authority lines 60-64",
  "category": "mathematical_hypothesis",
  "source_issue": "the least-upper-bound and supremum assertion omits nonemptiness",
  "target_action": "require the subset to be nonempty",
  "status": "verified",
  "evidence": "source-fidelity QA and pinned-schema validation pass; the empty set has no real supremum"
}

{
  "id": "O003-C046",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_summ.ptx",
  "source_locator": "authority line 8",
  "category": "source_typo",
  "source_issue": "the summary says a lower bound or a subset instead of for a subset",
  "target_action": "translate the intended preposition for",
  "status": "verified",
  "evidence": "source-fidelity QA passes and the adjacent definition supplies the intended relation"
}

{
  "id": "O003-C047",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_summ.ptx",
  "source_locator": "authority line 42",
  "category": "mathematical_variable",
  "source_issue": "the least-upper-bound condition calls k an upper bound for lowercase s instead of the declared set S",
  "target_action": "replace s with S in the derivative formula",
  "status": "verified",
  "evidence": "protected-math allowance is explicit and the surrounding statement declares only the set S"
}

{
  "id": "O003-C048",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_summ.ptx",
  "source_locator": "authority line 50",
  "category": "mathematical_hypothesis",
  "source_issue": "the point-to-set distance summary omits that A must be nonempty",
  "target_action": "require A to be nonempty",
  "status": "verified",
  "evidence": "source-fidelity QA and pinned-schema validation pass; the defining infimum is not a real number for an empty distance set"
}

{
  "id": "O003-C049",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "authority lines 126-131",
  "category": "function_domain",
  "source_issue": "C[a,b] is described as functions from R to R on [a,b], making the displayed supremum rule only a pseudometric when functions differ outside the interval",
  "target_action": "define X as continuous functions f:[a,b] to R for a nonempty compact interval a<b",
  "status": "verified",
  "evidence": "explicit protected-math allowances pass and the displayed supremum separates every distinct pair in X"
}

{
  "id": "O003-C050",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "authority lines 279-303",
  "category": "mathematical_hypothesis",
  "source_issue": "the quoted well-ordering principle omits nonempty and the proof invokes it before establishing the particular integer set S is nonempty",
  "target_action": "state the principle for a nonempty integer subset and explicitly establish S is nonempty",
  "status": "verified",
  "evidence": "source-fidelity QA and pinned-schema validation pass; the Archimedean property supplies an integer greater than nx"
}

{
  "id": "O003-C051",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "authority lines 478-500",
  "category": "mathematical_hypothesis",
  "source_issue": "the final true-false exercise uses infima and suprema after assuming only that S and T are bounded",
  "target_action": "require S and T to be nonempty bounded subsets",
  "status": "verified",
  "evidence": "source-fidelity QA and pinned-schema validation pass; empty bounded subsets have no real infimum or supremum"
}

{
  "id": "O003-C052",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "authority lines 523-524",
  "category": "mathematical_hypothesis",
  "source_issue": "the final point-to-set item uses d(x,A) without requiring A nonempty and leaves the metric on R unspecified",
  "target_action": "require A nonempty and identify the Euclidean metric d_E",
  "status": "verified",
  "evidence": "protected-math allowance and pinned-schema validation pass; the chapter definition requires A nonempty"
}

{
  "id": "O003-C053",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "authority line 340",
  "category": "notation_definition",
  "source_issue": "the exercise uses R positive without defining whether zero is included",
  "target_action": "define R positive as the set of positive real numbers at first use",
  "status": "verified",
  "evidence": "explicit element-insertion allowance and pinned-schema validation pass; the construction requires x greater than zero"
}

{
  "id": "O003-C054",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_intro.ptx",
  "source_locator": "authority lines 24 30 and 40",
  "category": "notation_consistency",
  "source_issue": "the opening example fixes the Euclidean metric d_E but alternates d and d_E for the resulting point-to-set distance",
  "target_action": "use d_E consistently in the three example displays",
  "status": "verified",
  "evidence": "four explicit protected-math allowances pass and the surrounding prose fixes the Euclidean metric"
}

{
  "id": "O003-C055",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/sec_glb_exer.ptx",
  "source_locator": "five authority theorem blocks embedded in exercise-level prose",
  "category": "schema_structure",
  "source_issue": "five theorem blocks reproduce the RelaxNG diagnostics; direct isolation proves the nested ordered and unordered lists are valid",
  "target_action": "retain the five complete theorem subtrees in source order in a permitted exercises-level introduction while preserving IDs mathematics and the exercise/task order",
  "status": "verified",
  "evidence": "cumulative pinned PreTeXt 1.7.5 RelaxNG validation passes; five explicit whole-subtree topology-move allowances pass"
}

{
  "id": "O003-C056",
  "unit": "chapter_05_greatest_lower_bounds",
  "file": "repo/source/chapters_01_05_reader.ptx",
  "source_locator": "Chapter 5 build wrapper",
  "category": "build_configuration",
  "source_issue": "the source uses custom macros glb and lub that are not defined in the current cumulative wrapper",
  "target_action": "define lossless display macros for greatest and least upper bounds before strict PDF preflight",
  "status": "verified",
  "evidence": "cumulative pinned-schema HTML build and strict XeLaTeX PDF build pass with both macros defined"
}

{
  "id": "O003-C057",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_btwn.ptx",
  "source_locator": "authority line 16",
  "category": "mathematical_symbol",
  "source_issue": "the prose defines the Euclidean metric d_E but the displayed rule names d_R",
  "target_action": "replace d_R with d_E in the defining rule",
  "status": "verified",
  "evidence": "the corrected symbol now agrees with the declared function and every subsequent use"
}

{
  "id": "O003-C058",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_btwn.ptx",
  "source_locator": "authority line 37",
  "category": "mathematical_reference",
  "source_issue": "the prose says to replace metric d although the surrounding definition uses d_E",
  "target_action": "replace d with d_E",
  "status": "verified",
  "evidence": "the repaired reference points to the metric actually used in the preceding definition"
}

{
  "id": "O003-C059",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_btwn.ptx",
  "source_locator": "authority line 115",
  "category": "mathematical_choice",
  "source_issue": "C is described as a function of epsilon alone despite being written C(a epsilon) and the proposed delta uses set braces rather than a scalar minimum",
  "target_action": "state that delta depends on a and epsilon and use min of 1 and epsilon divided by 1 plus 2 absolute a",
  "status": "verified",
  "evidence": "the corrected positive scalar choice is exactly the bound used by the proof that follows"
}

{
  "id": "O003-C060",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_exer.ptx",
  "source_locator": "authority lines 54-61",
  "category": "mathematical_domain",
  "source_issue": "the discrete metric is defined on arbitrary X but the prompt says the cited result proves it is a metric on R",
  "target_action": "state that it is a metric on X",
  "status": "verified",
  "evidence": "the translated exercise now uses X and Chapter 6 source QA receipts the sole protected-math change"
}

{
  "id": "O003-C061",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_exer.ptx",
  "source_locator": "authority lines 171-181",
  "category": "mathematical_variable",
  "source_issue": "the exercise fixes continuity at x equals 1 but asks for a bound involving undefined a",
  "target_action": "replace f(a) with f(1)",
  "status": "verified",
  "evidence": "the translated exercise now uses f(1) and Chapter 6 source QA receipts the sole protected-math change"
}

{
  "id": "O003-C062",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_exer.ptx",
  "source_locator": "authority line 181",
  "category": "missing_external_surface",
  "source_issue": "the sentence 'use the applet at to confirm' has neither a link nor an archived object",
  "target_action": "link the prompt to the original dependency-free epsilon-delta lab packaged with the derivative",
  "status": "verified",
  "evidence": "both Chapter 6 reader links resolve to the packaged lab; deterministic HTML preserves exact local bytes and the print prompt remains mathematically complete without it"
}

{
  "id": "O003-C063",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_intro.ptx",
  "source_locator": "authority lines 25 28 and 42",
  "category": "source_prose",
  "source_issue": "the English source has 'all of the value' duplicated 'if if' and 'require us to formally defining'",
  "target_action": "translate the unambiguous intended prose naturally without reproducing the grammatical defects",
  "status": "verified",
  "evidence": "the Indonesian translation states the same mathematical ideas with correct grammar and no protected-math change"
}

{
  "id": "O003-C064",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_intro.ptx",
  "source_locator": "authority lines 138-164",
  "category": "unarchived_interactive_dependency",
  "source_issue": "the first two exploration prompts depend on a remote GeoGebra object absent from the frozen source closure and with no established copy grant",
  "target_action": "replace the dependency with a separately identified original CC BY 4.0 offline-capable epsilon-delta lab while preserving both mathematical prompts",
  "status": "verified",
  "evidence": "standalone live browser QA passed presets controls responsive reflow text and SVG alternatives sample-only disclosure no-network closure and zero lab console errors"
}

{
  "id": "O003-C065",
  "unit": "chapter_06_continuous_functions",
  "file": "repo/source/sec_cont_func_exer.ptx",
  "source_locator": "authority lines 257-266",
  "category": "missing_hypothesis",
  "source_issue": "the integral formula is called a metric on C[a,b] without the necessary ordering assumption on the interval endpoints",
  "target_action": "state a less than b before defining the function space and integral metric",
  "status": "verified",
  "evidence": "with a less than b the nonnegative integral has the orientation required by the metric axioms; independent companion review confirmed the omission"
}

{
  "id": "O003-C066",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_cont_neighborhoods.ptx",
  "source_locator": "authority lines 82-83",
  "category": "source_typo",
  "source_issue": "the theorem says f is a function a metric space X to Y and omits from",
  "target_action": "translate the intended phrase as a function from X to Y",
  "status": "verified",
  "evidence": "the declared domain and codomain plus every adjacent formulation establish the missing preposition"
}

{
  "id": "O003-C067",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_cont_neighborhoods.ptx",
  "source_locator": "authority lines 145-178",
  "category": "schema_structure",
  "source_issue": "the first task of the reverse-implication activity has two sibling statements so the rendered four-task sequence conflicts with the final reference to parts a through d",
  "target_action": "split the second statement into its own consecutive task while preserving both statements and the intended five-part order",
  "status": "verified",
  "evidence": "the resulting parts a through e make the final prompt's reference to a through d coherent and restore one statement per task"
}

{
  "id": "O003-C068",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_open_balls_summ.ptx",
  "source_locator": "authority line 18",
  "category": "source_typo",
  "source_issue": "the summary says is s neighborhood",
  "target_action": "translate the unambiguous intended phrase is a neighborhood",
  "status": "verified",
  "evidence": "the chapter definition and adjacent summary statements establish the intended article"
}

{
  "id": "O003-C069",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_open_balls_exer.ptx",
  "source_locator": "authority lines 141-154",
  "category": "missing_imperative",
  "source_issue": "the quadratic-function task states that the inverse image contains an open ball instead of asking the learner to prove it, unlike the parallel linear task",
  "target_action": "restore the intended imperative Show that before the containment claim",
  "status": "verified",
  "evidence": "the parallel task structure and following instruction to conclude continuity establish the intended exercise prompt"
}

{
  "id": "O003-C070",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_open_balls_exer.ptx",
  "source_locator": "authority line 94",
  "category": "mathematical_variable",
  "source_issue": "the weighted-graph exercise defines a metric on undeclared S although the graph metric space is declared as X immediately afterward",
  "target_action": "replace S with X",
  "status": "verified",
  "evidence": "the exercise introduces only X as the finite metric space and every subsequent reference uses X"
}

{
  "id": "O003-C071",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_cont_neighborhoods.ptx",
  "source_locator": "authority lines 71-75",
  "category": "mathematical_classification",
  "source_issue": "the paragraph after the counterexample activity calls the inverse image f inverse of B an open ball even though the activity has just shown it need not be one",
  "target_action": "refer to the inverse image as a set that contains B(a delta), not as an open ball",
  "status": "verified",
  "evidence": "the preceding activity and the displayed containment in the following theorem establish exactly the intended distinction"
}

{
  "id": "O003-C072",
  "unit": "chapter_07_open_balls",
  "file": "repo/source/sec_neighborhoods.ptx",
  "source_locator": "authority lines 4-6",
  "category": "mathematical_classification",
  "source_issue": "the opening says the section introduces an open neighborhood although openness has not yet been defined and the section defines a neighborhood that need not itself be open",
  "target_action": "refer to the introduced object simply as a neighborhood of a point",
  "status": "verified",
  "evidence": "the displayed definition and the chapter's own continuity theorem use the broader neighborhood concept; open sets are deferred to the next chapter"
}

{
  "id": "O003-C073",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets_intro.ptx",
  "source_locator": "authority line 13",
  "category": "source_prose",
  "source_issue": "the source omits the articles in the phrase to be neighborhood of point a",
  "target_action": "translate the intended relation naturally as a neighborhood of point a",
  "status": "verified",
  "evidence": "the following definition confirms the intended neighborhood relation"
}

{
  "id": "O003-C074",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets.ptx",
  "source_locator": "authority line 97",
  "category": "mathematical_containment",
  "source_issue": "the informal phrase an open ball B in O that contains x can be misread as elementhood of B in O",
  "target_action": "state explicitly that B is contained in O and contains x",
  "status": "verified",
  "evidence": "O is assumed to be a union of open balls so x belongs to a constituent ball contained in O"
}

{
  "id": "O003-C075",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_interior_set.ptx",
  "source_locator": "authority lines 71-78",
  "category": "mathematical_containment",
  "source_issue": "the paragraph calls Int(A) the largest open subset of X that contains A and thereby reverses the required inclusion",
  "target_action": "state that Int(A) is the largest open subset of X contained in A",
  "status": "verified",
  "evidence": "the definition gives Int(A) as a subset of A and the immediately following theorem states the correct largest-open-subset characterization"
}

{
  "id": "O003-C076",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets_exer.ptx",
  "source_locator": "authority lines 108-114",
  "category": "mathematical_variable",
  "source_issue": "the exercise declares the finite point set S but asks about the complement of undefined A",
  "target_action": "replace R squared setminus A with R squared setminus S",
  "status": "verified",
  "evidence": "S is the only finite point set declared in the exercise and its complement is the intended object"
}

{
  "id": "O003-C077",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets_exer.ptx",
  "source_locator": "authority line 195",
  "category": "missing_copula",
  "source_issue": "the phrase if f a continuous function omits is",
  "target_action": "translate the question with the complete predicate f merupakan fungsi kontinu",
  "status": "verified",
  "evidence": "the repaired grammar preserves the stated continuity question without choosing a theorem target"
}

{
  "id": "O003-C078",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets_exer.ptx",
  "source_locator": "authority line 195",
  "category": "dangling_reference",
  "source_issue": "the instruction Use Theorem supplies neither an xref nor a theorem identifier",
  "target_action": "preserve the generic theorem wording and do not attach an xref until the intended target is established authoritatively",
  "status": "unresolved",
  "evidence": "both the open-set continuity theorem and the open-ball corollary are nearby applicable results so the target is not unambiguous"
}

{
  "id": "O003-C079",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_open_sets_exer.ptx",
  "source_locator": "authority lines 158-160",
  "category": "mathematical_classification",
  "source_issue": "the source calls f(x)=ax+b an arbitrary linear function although arbitrary b makes it affine and it is linear only when b=0",
  "target_action": "preserve the formula and quantified scope in translation and flag the classification for authoritative correction",
  "status": "unresolved",
  "evidence": "for b nonzero the map does not send zero to zero and therefore is not linear"
}

{
  "id": "O003-C080",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_cont_open_sets.ptx",
  "source_locator": "authority lines 63-65 and 135-139",
  "category": "source_punctuation",
  "source_issue": "two direct questions end with periods",
  "target_action": "translate the questions with question marks",
  "status": "verified",
  "evidence": "the interrogative syntax and surrounding activity prompt order make the intended punctuation certain"
}

{
  "id": "O003-C081",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_interior_set.ptx",
  "source_locator": "authority lines 22-24 and 84-87",
  "category": "source_prose",
  "source_issue": "the collection form has subject-verb disagreement and Then interior of A omits the article",
  "target_action": "translate both statements with natural Indonesian agreement and grammar",
  "status": "verified",
  "evidence": "the prose repairs do not change mathematical content or XML topology"
}

{
  "id": "O003-C082",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_intro.ptx",
  "source_locator": "authority line 97",
  "category": "mathematical_inequality",
  "source_issue": "the hypothesis uses n greater than or equal to N but the proof writes the stricter n greater than N",
  "target_action": "use n greater than or equal to N consistently",
  "status": "verified",
  "evidence": "the epsilon definition and chosen threshold require the non-strict inequality"
}

{
  "id": "O003-C083",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_intro.ptx",
  "source_locator": "authority line 197",
  "category": "mathematical_variable",
  "source_issue": "the Euclidean-distance equality has d_E(a,y) on the left but absolute value x-y on the right",
  "target_action": "replace the first argument a with x",
  "status": "verified",
  "evidence": "the surrounding comparison uses arbitrary points x and y"
}

{
  "id": "O003-C084",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_intro.ptx",
  "source_locator": "authority line 208",
  "category": "mathematical_variable",
  "source_issue": "the taxicab metric repeats the coordinates y_1,y_1",
  "target_action": "replace the second occurrence with y_1,y_2",
  "status": "verified",
  "evidence": "a point of R squared requires the two coordinates y_1 and y_2"
}

{
  "id": "O003-C085",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_cont_metric.ptx",
  "source_locator": "authority line 4",
  "category": "source_prose",
  "source_issue": "the phrase different ways to characterize is missing its mathematical object",
  "target_action": "state that the section gives different ways to characterize continuity",
  "status": "verified",
  "evidence": "the section title and all following results concern sequential characterizations of continuity"
}

{
  "id": "O003-C086",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_cont_metric.ptx",
  "source_locator": "authority line 96",
  "category": "mathematical_limit",
  "source_issue": "the proof states that the limit of f(a_n) equals a rather than f(a)",
  "target_action": "replace a with f(a)",
  "status": "verified",
  "evidence": "the theorem assumes a_n converges to a and asserts continuity of f"
}

{
  "id": "O003-C087",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_cont_metric.ptx",
  "source_locator": "authority line 201",
  "category": "source_prose",
  "source_issue": "the phrase the f continuity of f contains a stray f",
  "target_action": "remove the stray symbol",
  "status": "verified",
  "evidence": "the repaired phrase is the continuity of f"
}

{
  "id": "O003-C088",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_cont_metric.ptx",
  "source_locator": "authority line 243",
  "category": "source_prose",
  "source_issue": "be finding is a typographical error",
  "target_action": "translate the intended phrase by finding",
  "status": "verified",
  "evidence": "the sentence describes a proof method"
}

{
  "id": "O003-C089",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_summ.ptx",
  "source_locator": "authority line 20",
  "category": "source_prose",
  "source_issue": "to m metric space contains a stray m",
  "target_action": "translate the intended phrase to a metric space",
  "status": "verified",
  "evidence": "the summary statement requires an indefinite article and no variable m"
}

{
  "id": "O003-C090",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority line 80",
  "category": "source_markup",
  "source_issue": "the point-to-set distance formula is missing its closing set brace",
  "target_action": "restore the closing brace",
  "status": "verified",
  "evidence": "the displayed infimum uses set-builder notation opened earlier in the same expression"
}

{
  "id": "O003-C091",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority line 137",
  "category": "mathematical_notation",
  "source_issue": "the ambient real-number symbol is plain R rather than the established real-number macro",
  "target_action": "use the established R macro",
  "status": "verified",
  "evidence": "the exercise concerns a sequence of rational numbers converging in the real numbers"
}

{
  "id": "O003-C092",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority line 306",
  "category": "mathematical_domain",
  "source_issue": "C[0,1] is described as continuous functions from R to R",
  "target_action": "state continuous functions from [0,1] to R",
  "status": "verified",
  "evidence": "the standard space C[0,1] and the following supremum metric both use the compact interval domain"
}

{
  "id": "O003-C093",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority line 312",
  "category": "mathematical_notation",
  "source_issue": "C[01] omits the comma",
  "target_action": "replace C[01] with C[0,1]",
  "status": "verified",
  "evidence": "the same exercise defines and uses C[0,1]"
}

{
  "id": "O003-C094",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority lines 135-145",
  "category": "mathematical_scope",
  "source_issue": "the rational sequence converging to sqrt(2) is called a counterexample to the literal converse although the retained converse assumes the limit a lies in the subspace Y",
  "target_action": "state that the literal converse with all hypotheses retained is true; then present the example only as ambient convergence to a point outside the subspace",
  "status": "verified",
  "evidence": "the subspace metric is the restriction of the ambient metric, so convergence to a point of Y is equivalent in the two spaces; the sqrt(2) example instead demonstrates failure of convergence in Y when the ambient limit lies outside Y"
}

{
  "id": "O003-C095",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_summ.ptx",
  "source_locator": "authority lines 3-23",
  "category": "audit_reversal",
  "source_issue": "an earlier derivative audit incorrectly classified the source list nested in a paragraph as invalid",
  "target_action": "restore the source-valid nested list topology",
  "status": "superseded",
  "evidence": "the pinned PreTeXt 1.7.5 TextParagraph model explicitly permits List; moving the list to section level is invalid and the cumulative Chapters 1-11 wrapper passes only after restoration"
}

{
  "id": "O003-C096",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority line 299",
  "category": "mathematical_notation",
  "source_issue": "the target function f is written as the parenthesized expression (f), which incorrectly resembles a one-term sequence",
  "target_action": "write the convergence target as the function f",
  "status": "verified",
  "evidence": "the sequence is (f_n), while its proposed pointwise limit is the single function f defined immediately above"
}

{
  "id": "O003-C097",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_set_closure.ptx",
  "source_locator": "authority line 194",
  "category": "mathematical_variable",
  "source_issue": "the first case of the boundary-intersection proof concludes that the ambient space X is an element of the two closures rather than the point x",
  "target_action": "replace the membership subject X with x",
  "status": "verified",
  "evidence": "the case assumes x is a boundary point and proves x belongs to both closures; the following sentence states the resulting containment for x"
}

{
  "id": "O003-C098",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_set_closure.ptx",
  "source_locator": "authority line 203",
  "category": "mathematical_variable",
  "source_issue": "the second case of the boundary-intersection proof concludes that the ambient space X is an element of the two closures rather than the point x",
  "target_action": "replace the membership subject X with x",
  "status": "verified",
  "evidence": "the case assumes x is a boundary point and proves x belongs to both closures; the following sentence states the resulting containment for x"
}

{
  "id": "O003-C099",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_limit_bound_iso_seq.ptx",
  "source_locator": "authority lines 102-110",
  "category": "mathematical_variable",
  "source_issue": "the theorem introduces boundary point b but says the two constructed sequences converge to undefined x",
  "target_action": "replace the convergence target x with b",
  "status": "verified",
  "evidence": "the theorem hypothesis names b; its source exercise ex_CS_2 and the chapter summary both state the sequences converge to b"
}

{
  "id": "O003-C100",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_limits_seq.ptx",
  "source_locator": "authority lines 12-15",
  "category": "mathematical_indexing",
  "source_issue": "the sequence a_n=1/n indexed by every positive integer is claimed to lie in A=(0,1), but a_1=1 is not in A",
  "target_action": "use a_n=1/(n+1) for every positive integer n",
  "status": "verified",
  "evidence": "the repaired sequence lies in both A=(0,1) and B=[0,1] and retains the intended limit 0 with standard positive-integer indexing"
}

{
  "id": "O003-C101",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_exer.ptx",
  "source_locator": "authority line 40",
  "category": "mathematical_notation",
  "source_issue": "the set-builder expression for {1/n} omits its bound variable after the vertical bar",
  "target_action": "insert n before the membership relation n in Z+",
  "status": "verified",
  "evidence": "the numerator sequence and positive-integer domain require the missing index n"
}

{
  "id": "O003-C102",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_exer.ptx",
  "source_locator": "authority line 90",
  "category": "mathematical_identity",
  "source_issue": "the asserted identity A union A^i = A union A' is false in general because A^i is already contained in A",
  "target_action": "replace the left side with A' union A^i",
  "status": "verified",
  "evidence": "every point of A is either a limit point or an isolated point in a metric space; hence A' union A^i equals A union A' and the next task states the needed containment"
}

{
  "id": "O003-C103",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_exer.ptx",
  "source_locator": "authority lines 204-207",
  "category": "mathematical_notation",
  "source_issue": "the prose defines the distance from x to A but the displayed left side is d(a,X)",
  "target_action": "replace d(a,X) with d(x,A)",
  "status": "verified",
  "evidence": "the infimum on the right is over a in A of d(x,a), and the following statement uses the same point-to-set notation"
}

{
  "id": "O003-C104",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_sets_metric.ptx",
  "source_locator": "authority line 106",
  "category": "mathematical_attribution",
  "source_issue": "the complement-of-intersection identity is attributed to DeMoivre's Theorem",
  "target_action": "identify it as De Morgan's law",
  "status": "verified",
  "evidence": "the displayed identity is the arbitrary-family form of De Morgan's law, not de Moivre's formula"
}

{
  "id": "O003-C105",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_sets_metric.ptx",
  "source_locator": "authority line 112",
  "category": "source_prose",
  "source_issue": "the clause and so it an open set omits the verb is",
  "target_action": "translate the intended clause as and therefore it is an open set",
  "status": "verified",
  "evidence": "the subject is an arbitrary union of open sets and the conclusion requires the copula"
}

{
  "id": "O003-C106",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_cont_closed_sets.ptx",
  "source_locator": "authority line 15",
  "category": "source_prose",
  "source_issue": "the phrase Let f be a function f repeats the function symbol",
  "target_action": "recast the sentence as f is given and this function f maps X to Y while retaining both source math nodes",
  "status": "verified",
  "evidence": "the sentence introduces one function f from X to Y and the structural translation keeps the source mathematics intact"
}

{
  "id": "O003-C107",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_summ.ptx",
  "source_locator": "authority line 29",
  "category": "mathematical_notation",
  "source_issue": "the ambient metric-space expression is written =(R,d_E) with a spurious leading equals sign",
  "target_action": "remove the leading equals sign",
  "status": "verified",
  "evidence": "the phrase is as a subset of (R,d_E) and no equation occurs at that point"
}

{
  "id": "O003-C108",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_summ.ptx",
  "source_locator": "authority lines 50-51",
  "category": "mathematical_equivalence",
  "source_issue": "the summary states only that closed preimages imply continuity although the chapter theorem proves an equivalence",
  "target_action": "state if and only if",
  "status": "verified",
  "evidence": "thm_closed_sets_continuity_MS in the same chapter gives both directions"
}

{
  "id": "O003-C109",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_limit_bound_iso_seq.ptx",
  "source_locator": "authority lines 58 and 81",
  "category": "source_prose",
  "source_issue": "plural sequences is paired with singular converges in two sentences",
  "target_action": "translate with plural agreement",
  "status": "verified",
  "evidence": "both clauses have sequences as their grammatical subject"
}

{
  "id": "O003-C110",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_limit_closed.ptx",
  "source_locator": "authority line 74",
  "category": "source_prose",
  "source_issue": "the question What does the conclusion of part (b) tells us has an inflected verb after does",
  "target_action": "translate the intended question What does the conclusion of part (b) tell us",
  "status": "verified",
  "evidence": "English auxiliary does requires the base verb tell"
}

{
  "id": "O003-C111",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_closed_set_summ.ptx",
  "source_locator": "authority lines 3-96",
  "category": "audit_reversal",
  "source_issue": "an earlier derivative audit incorrectly classified the source list nested in a paragraph as invalid",
  "target_action": "restore the source-valid nested list topology",
  "status": "superseded",
  "evidence": "the pinned PreTeXt 1.7.5 TextParagraph model explicitly permits List; moving the list to section level is invalid and the cumulative Chapters 1-11 wrapper passes only after restoration"
}

{
  "id": "O003-C112",
  "unit": "chapter_10_closed_sets_metric",
  "file": "repo/source/sec_limit_bound_iso_seq.ptx",
  "source_locator": "authority lines 58 and 73-82",
  "category": "mathematical_scope",
  "source_issue": "the prose can be read as claiming that the singular constant sequence (a) is the only sequence in A converging to isolated point a, although sequences with arbitrary finite prefixes also converge to a",
  "target_action": "state that every sequence in A converging to a is eventually constant with value a",
  "status": "verified",
  "evidence": "isolation forces all sufficiently late terms to equal a but imposes no restriction on a finite initial segment; the converse follows from the same eventual-constancy condition"
}

{
  "id": "O003-C113",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_intro.ptx",
  "source_locator": "authority lines 28-54 especially 36-49",
  "category": "schema_structure",
  "source_issue": "a semantic definition is nested inside a task statement and violates the pinned PreTeXt content model",
  "target_action": "split the exploration at that point and place the unchanged definition at section level while preserving its exact reading order index and mathematics",
  "status": "verified",
  "evidence": "the resulting cumulative wrapper passes pinned PreTeXt validation and preserves the prompt-definition-prompt reading sequence"
}

{
  "id": "O003-C114",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_open_closed_sub.ptx",
  "source_locator": "authority lines 21-28",
  "category": "mathematical_scope",
  "source_issue": "openness of O_A supplies a ball only for each a in O_A rather than for every a in A",
  "target_action": "replace the quantified set A with O_A",
  "status": "verified",
  "evidence": "the defining union and all subsequent uses already range over O_A"
}

{
  "id": "O003-C115",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_open_closed_sub.ptx",
  "source_locator": "authority lines 63-75",
  "category": "mathematical_containment",
  "source_issue": "the reverse-implication activity asks for a ball in A contained in A which is automatic and does not prove openness",
  "target_action": "ask for the natural ball B_A(a,delta) contained in O_A",
  "status": "verified",
  "evidence": "B_A(a,delta)=B_X(a,delta) intersection A is contained in O_X intersection A=O_A"
}

{
  "id": "O003-C116",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_prod_metric.ptx",
  "source_locator": "authority lines 4 and 12",
  "category": "mathematical_variable",
  "source_issue": "the first factor is introduced twice as (X,d_1) although every product and coordinate uses X_1",
  "target_action": "replace X with X_1 in both factor declarations",
  "status": "verified",
  "evidence": "the paired factor is X_2 and all displayed domains and coordinates use X_1 times X_2"
}

{
  "id": "O003-C117",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_prod_metric.ptx",
  "source_locator": "authority lines 60-75",
  "category": "mathematical_inequality",
  "source_issue": "the triangle proof falsely drops a positive cross term and asserts d(x,z)^2 is at most d(x,y)^2+d(y,z)^2",
  "target_action": "apply Cauchy-Schwarz to bound the cross term and obtain d(x,z)^2 at most (d(x,y)+d(y,z))^2 before taking square roots",
  "status": "verified",
  "evidence": "Cauchy-Schwarz gives d_1(x_1,y_1)d_1(y_1,z_1)+d_2(x_2,y_2)d_2(y_2,z_2) at most d(x,y)d(y,z)"
}

{
  "id": "O003-C118",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_prod_metric.ptx",
  "source_locator": "authority lines 85-90",
  "category": "mathematical_ambient_space",
  "source_issue": "the real intervals [1,2] and [3,4] are called subspaces of R squared without an embedding",
  "target_action": "replace the ambient space with (R,d_E)",
  "status": "verified",
  "evidence": "their Cartesian product is then the intended rectangle in R squared"
}

{
  "id": "O003-C119",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_prod_metric.ptx",
  "source_locator": "authority lines 134-143",
  "category": "mathematical_scope",
  "source_issue": "any number overclaims the displayed unweighted Euclidean construction while the following definition quantifies only i=1 through n",
  "target_action": "state any finite number of metric spaces",
  "status": "verified",
  "evidence": "the displayed sum and product are finite and indexed by a positive integer n"
}

{
  "id": "O003-C120",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_summ.ptx",
  "source_locator": "authority lines 15-17",
  "category": "mathematical_containment",
  "source_issue": "the relative-open summary writes O_A=X intersection O which collapses to O",
  "target_action": "replace X with the subspace A",
  "status": "verified",
  "evidence": "this matches thm_relatively_open_ms in the same chapter"
}

{
  "id": "O003-C121",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_summ.ptx",
  "source_locator": "authority lines 18-21",
  "category": "mathematical_complement",
  "source_issue": "the first closed-set characterization writes C_A=A intersection O_A for open O_A and therefore describes an open set",
  "target_action": "restore the definition C_A=A setminus O_A",
  "status": "verified",
  "evidence": "the subsequent ambient-closed intersection characterization remains correct"
}

{
  "id": "O003-C123",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority line 6",
  "category": "mathematical_classification",
  "source_issue": "the pair (R,d_E) is called a topological space although d_E is a metric rather than a topology",
  "target_action": "replace topological space with metric space",
  "status": "verified",
  "evidence": "the chapter and exercise use the Euclidean metric and metric-subspace definition"
}

{
  "id": "O003-C124",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority lines 65-74",
  "category": "function_domain",
  "source_issue": "the sum-distance rule takes two points of X times Y but its declared function domain contains only one product point",
  "target_action": "declare d from (X times Y) times (X times Y) to R",
  "status": "verified",
  "evidence": "the displayed rule has ordered-pair arguments (x_1,y_1) and (x_2,y_2)"
}

{
  "id": "O003-C125",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority lines 79-89",
  "category": "function_domain",
  "source_issue": "the finite-product metric has only the point space as its function domain and then refers to undefined X",
  "target_action": "define X as the finite product and declare d from X times X to R",
  "status": "verified",
  "evidence": "the displayed rule evaluates d(x,y) for two elements x and y of that finite product"
}

{
  "id": "O003-C126",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority lines 165-170",
  "category": "mathematical_domain",
  "source_issue": "the function f is declared only on E^m but the asserted distance identity quantifies arbitrary elements of H",
  "target_action": "replace H with E^m in that quantifier",
  "status": "verified",
  "evidence": "f((x_n)) and f((y_n)) are defined only when both sequences lie in E^m"
}

{
  "id": "O003-C127",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority lines 226-243",
  "category": "function_domain",
  "source_issue": "both true-false product-distance functions take two product points but are declared only on X times Y",
  "target_action": "declare each function from (X times Y) times (X times Y) to R",
  "status": "verified",
  "evidence": "each displayed rule has two ordered-pair arguments"
}

{
  "id": "O003-C128",
  "unit": "chapter_11_metric_subspaces_products",
  "file": "repo/source/sec_sub_metric_exer.ptx",
  "source_locator": "authority lines 35-37",
  "category": "mathematical_notation",
  "source_issue": "the subspace metric is abbreviated as d restricted to O although its two-variable domain is O times O",
  "target_action": "write the restricted metric as d restricted to O times O",
  "status": "verified",
  "evidence": "the chapter definition declares a subspace metric by restricting d to A times A"
}

{
  "id": "O003-C129",
  "unit": "chapter_08_open_sets",
  "file": "repo/source/sec_cont_open_sets.ptx",
  "source_locator": "authority lines 93-111",
  "category": "schema_structure",
  "source_issue": "the named corollary is nested inside an activity introduction whose content model does not admit theorem-like blocks",
  "target_action": "move the unchanged corollary and its lead sentence immediately before the activity while retaining the proof setup inside the activity introduction",
  "status": "verified",
  "evidence": "the xml:id mathematics statement and reading order are preserved and the cumulative Chapters 1-11 wrapper passes the pinned PreTeXt 1.7.5 RelaxNG schema"
}

{
  "id": "O003-C130",
  "unit": "chapter_09_sequences",
  "file": "repo/source/sec_seq_exer.ptx",
  "source_locator": "authority lines 95-115",
  "category": "schema_structure",
  "source_issue": "a named theorem is nested inside a task statement whose content model does not admit theorem-like blocks",
  "target_action": "move the unchanged theorem to the permitted exercises-level introduction while retaining the source task prompt and all mathematics",
  "status": "verified",
  "evidence": "the theorem subtree is preserved verbatim in the derivative and the cumulative Chapters 1-11 wrapper passes the pinned PreTeXt 1.7.5 RelaxNG schema"
}

{
  "id": "O003-C131",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_neighborhood_top_space.ptx",
  "source_locator": "authority line 4",
  "category": "mathematical_variable",
  "source_issue": "the neighborhood definition says an open ball is centered at X although the point introduced is a",
  "target_action": "replace X with a in the translated reader and preserve the intended neighborhood definition",
  "status": "verified",
  "evidence": "a neighborhood of a point a is defined using a ball centered at a; no other mathematics changed"
}

{
  "id": "O003-C132",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_top_space_exer.ptx",
  "source_locator": "authority line 63",
  "category": "mathematical_notation",
  "source_issue": "the exercise writes x in Z without the intended integer-number-set macro",
  "target_action": "render the set as x in \\\\Z in the translated reader",
  "status": "verified",
  "evidence": "the correction restores the declared ambient set and is the only non-text formula change in the file"
}

{
  "id": "O003-C133",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/companion/chapter_12_source_guides_a.ptx",
  "source_locator": "companion guide 01",
  "category": "mathematical_typing",
  "source_issue": "the source writes tau={a,b}, which is a set of points rather than a collection of subsets; the intended singleton collection is ambiguous",
  "target_action": "state the literal type error and separately solve the intended tau={{a,b}} reading",
  "status": "verified",
  "evidence": "the companion now preserves the source while making both readings explicit"
}

{
  "id": "O003-C134",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/companion/chapter_12_source_guides_a.ptx",
  "source_locator": "companion guide 04",
  "category": "mathematical_closure",
  "source_issue": "adding only {a,b,c} does not close the source collection under unions; {a} union {b,d} forces {a,b,d}, and further missing subsets follow",
  "target_action": "add the missing pair {b,c} and four missing three-element subsets; identify the resulting power-set topology",
  "status": "verified",
  "evidence": "finite closure computation gives all 16 subsets of X and proves the five additions are necessary"
}

{
  "id": "O003-C135",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/companion/chapter_12_source_guides_a.ptx",
  "source_locator": "companion guide 05",
  "category": "coverage_and_naming",
  "source_issue": "the source prompt asks for three finite and three nonfinite examples, and the original title conflated finite-subset collection with the cofinite topology",
  "target_action": "include the requested examples and rename the guide to distinguish the failed finite-subset collection from the cofinite topology",
  "status": "verified",
  "evidence": "the corrected guide answers every source subprompt and preserves the established glossary distinction"
}

{
  "id": "O003-C136",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_base_top.ptx",
  "source_locator": "authority line 38",
  "category": "mathematical_notation",
  "source_issue": "the finite topology collection closes with X) rather than the required set brace X}",
  "target_action": "replace the closing parenthesis with the set delimiter and preserve the listed topology",
  "status": "verified",
  "evidence": "the correction restores the displayed collection as a set and changes no membership or topology claim"
}

{
  "id": "O003-C137",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_base_top.ptx",
  "source_locator": "authority proof paragraph after thm_Basis",
  "category": "logical_scope",
  "source_issue": "the proof begins by calling X a topological space even though the theorem is constructing tau as a topology on a set X",
  "target_action": "translate X as a set in the proof and retain the theorem's stated construction",
  "status": "verified",
  "evidence": "this removes circular wording without changing the basis conditions or proof steps"
}

{
  "id": "O003-C138",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_base_top.ptx",
  "source_locator": "authority line 337",
  "category": "logical_scope",
  "source_issue": "the discrete-topology basis task asks about a symbol tau that is not defined in that task",
  "target_action": "refer directly to the discrete topology on X, whose topology is already defined as the power set",
  "status": "verified",
  "evidence": "the clarification makes the intended topology explicit without changing the candidate basis or requested minimal addition"
}

{
  "id": "O003-C139",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_interior_set_top.ptx",
  "source_locator": "authority line 122",
  "category": "mathematical_variable",
  "source_issue": "the theorem calls (X,d) a topological space although d is not a metric and the chapter uses a topology",
  "target_action": "write (X,\\\\tau) in the translated theorem statement",
  "status": "verified",
  "evidence": "the theorem concerns an arbitrary topological space, so tau is the correct structure symbol"
}

{
  "id": "O003-C140",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_interior_set_top.ptx",
  "source_locator": "authority line 154",
  "category": "mathematical_variable",
  "source_issue": "the follow-up activity repeats (X,d) for an arbitrary topological space",
  "target_action": "write (X,\\\\tau) in the translated activity introduction",
  "status": "verified",
  "evidence": "the activity proves the topological-interior maximality statement and assumes no metric"
}

{
  "id": "O003-C141",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_top_space_exer.ptx",
  "source_locator": "authority line 576",
  "category": "typographical",
  "source_issue": "the Zariski/cofinite equivalence sentence misspells Zariski as Zaariski in its second occurrence",
  "target_action": "retain the corrected Zariski spelling in the Indonesian derivative",
  "status": "verified",
  "evidence": "the translated sentence already uses the standard name consistently; no mathematical content changes"
}

{
  "id": "O003-C142",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_top_space_exer.ptx",
  "source_locator": "authority line 630",
  "category": "mathematical_notation",
  "source_issue": "the finite topology list contains the malformed entry {a,b,c,}",
  "target_action": "close the set as {a,b,c} in the translated exercise",
  "status": "verified",
  "evidence": "the intended subset is unambiguously the three-element set and the surrounding list uses the same notation"
}

{
  "id": "O003-C143",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_top_space_exer.ptx",
  "source_locator": "authority lines 274-278",
  "category": "enumeration_scope",
  "source_issue": "the prose says the display contains 9 topologies but the numbered list has 10 entries; the strict chain instruction excludes item 1 as a target",
  "target_action": "identify item 1 as the indiscrete starting topology and items 2-10 as the nine target topologies",
  "status": "verified",
  "evidence": "the repair reconciles the stated count with all ten displayed entries and preserves the strict inclusion chain requested by the exercise"
}

{
  "id": "O003-C144",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_base_top.ptx",
  "source_locator": "authority lines 141-177",
  "category": "schema_structure",
  "source_issue": "the lemma element is nested after a statement inside a task, a structure rejected by the pinned PreTeXt schema",
  "target_action": "move the intact lemma subtree outside the containing activity to the following section-level position while preserving its id and content",
  "status": "verified",
  "evidence": "the repair changes only schema-invalid containment; the learner task still asks for the same induction proof and the lemma remains immediately adjacent"
}

{
  "id": "O003-C145",
  "unit": "chapter_12_topological_spaces",
  "file": "repo/source/sec_top_space_summ.ptx and repo/source/sec_top_space_exer.ptx",
  "source_locator": "authority lines 151-176 of sec_top_space_exer.ptx",
  "category": "schema_structure",
  "source_issue": "the coarser/finer definition is nested inside an exercise introduction, which the pinned PreTeXt schema rejects",
  "target_action": "move the intact definition subtree to the immediately preceding chapter summary section and retain the exercise's contextual lead-in",
  "status": "verified",
  "evidence": "the xml:id def_weaker_topologies, mathematical content, terminology, and later cross-chapter xref target are preserved; only the schema-invalid container changes"
}

{
  "id": "O003-C146",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_union_inter_closed.ptx",
  "source_locator": "authority proof of thm_closed_TS part 2",
  "category": "mathematical_notation",
  "source_issue": "the finite-union proof first defines C as an intersection and then switches to an arbitrarily indexed family even though only C_1 through C_n were introduced",
  "target_action": "define C as the finite union and use k=1 through n consistently in De Morgan's identity and the conclusion",
  "status": "verified",
  "evidence": "the repair makes the proof establish the theorem's stated finite-union claim and changes no hypothesis"
}

{
  "id": "O003-C147",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_top_closure.ptx",
  "source_locator": "authority activity introduction",
  "category": "mathematical_variable",
  "source_issue": "the activity calls (X,d) a topological space although no metric d is assumed and the chapter uses a topology",
  "target_action": "write (X,\\tau) in the translated activity introduction",
  "status": "verified",
  "evidence": "the activity concerns arbitrary topological closure and uses no metric hypothesis"
}

{
  "id": "O003-C148",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_top_closure.ptx",
  "source_locator": "authority proof of thm_TS_closure_closed",
  "category": "logical_scope",
  "source_issue": "the proof calls an open set O centered at x, language that is metric-specific and does not state the needed topological containment relation",
  "target_action": "state that the open set O contains x and satisfies O subseteq N",
  "status": "verified",
  "evidence": "this is exactly the neighborhood property used by the proof and adds no separation or metric assumption"
}

{
  "id": "O003-C149",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_separation_ax.ptx",
  "source_locator": "authority lines 28 and 35",
  "category": "mathematical_notation",
  "source_issue": "the metric-space separation activity first names B(x,r) and B(y,s) but then uses undefined B_x and B_y; its next task swaps the named radii in the intersection",
  "target_action": "use B(x,r) intersect B(y,s) in both tasks",
  "status": "verified",
  "evidence": "the repair uses exactly the two balls selected in the preceding sentences and preserves the disjointness argument"
}

{
  "id": "O003-C150",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_separation_ax.ptx",
  "source_locator": "authority line 65",
  "category": "typographical",
  "source_issue": "the Hausdorff index entry is misspelled Hausdoff",
  "target_action": "use Hausdorff consistently in the Indonesian index and prose",
  "status": "verified",
  "evidence": "the proper name and immediately following definition make the intended spelling unambiguous"
}

{
  "id": "O003-C151",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_separation_ax.ptx",
  "source_locator": "authority lines 121-122",
  "category": "mathematical_classification",
  "source_issue": "the K-topology discussion calls (a,b) setminus K an interval although removing K can disconnect it",
  "target_action": "describe (a,b) setminus K as a set of the stated form rather than as an interval",
  "status": "verified",
  "evidence": "the basis element is unchanged and the repair removes a false geometric classification"
}

{
  "id": "O003-C152",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_separation_ax.ptx",
  "source_locator": "authority lines 284-287",
  "category": "definition_scope",
  "source_issue": "the Niemytzki-plane paragraph calls L the boundary without naming the ambient Euclidean space and does not state that ordinary open basis disks stay above L",
  "target_action": "identify L as the Euclidean boundary of X in R^2 and require ordinary basis disks to lie wholly above L",
  "status": "verified",
  "evidence": "these are the conditions needed for the standard Niemytzki-plane basis and do not alter the underlying topology"
}

{
  "id": "O003-C153",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority lines 155-157",
  "category": "mathematical_scope",
  "source_issue": "the density task calls A a subset of a topological space A and quantifies over every open set although the empty open set can never meet A",
  "target_action": "use ambient space X and quantify over every nonempty open set O",
  "status": "verified",
  "evidence": "this is the standard open-set characterization of density and removes both the variable collision and empty-set counterexample"
}

{
  "id": "O003-C154",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority line 288",
  "category": "cross_reference",
  "source_issue": "the exercise prints the raw identifier (ex_aZ_top) instead of linking to the earlier exercise",
  "target_action": "replace the raw identifier with xref ref=ex_aZ_top",
  "status": "verified",
  "evidence": "the target exists in the admitted cumulative source and the link now resolves normally"
}

{
  "id": "O003-C155",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority lines 347-353",
  "category": "definition_scope",
  "source_issue": "the Double Origin neighborhood formulas use radius 1/r without stating the required positivity of r",
  "target_action": "state that r is a positive parameter for both displayed neighborhood families",
  "status": "verified",
  "evidence": "positivity is required for the intended positive disk radius and is shared by both definitions"
}

{
  "id": "O003-C156",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority lines 357 and 360",
  "category": "geometric_description",
  "source_issue": "the formulas y>0 and y<0 omit the x-axis but the prose says the y-axis is omitted",
  "target_action": "identify the omitted axis as the x-axis in both descriptions",
  "status": "verified",
  "evidence": "the correction follows directly from the displayed inequalities and preserves the added origins"
}

{
  "id": "O003-C157",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority lines 430 and 455",
  "category": "mathematical_variable",
  "source_issue": "the discrete-line exercise asks for complements in an undefined ambient X although the entire exercise is in (Z,tau_dl)",
  "target_action": "write Z setminus closure(A) and Z setminus closure(B)",
  "status": "verified",
  "evidence": "the repaired ambient set is fixed by the exercise introduction"
}

{
  "id": "O003-C158",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority line 466",
  "category": "logical_scope",
  "source_issue": "the metric regularity exercise assumes x lies outside C and then states d(x,C)=0 iff x lies in C, a contradictory restriction that erases the cited characterization",
  "target_action": "for nonempty closed C quantify over every x in X before stating d(x,C)=0 iff x is in C",
  "status": "verified",
  "evidence": "this matches the cited point-to-set-distance exercise and supplies the useful closed-set characterization"
}

{
  "id": "O003-C159",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority line 502",
  "category": "mathematical_notation",
  "source_issue": "the K-topology exercise uses the plain letter R rather than the real-number macro in one occurrence",
  "target_action": "write (\\\\R,\\\\tau_K) consistently",
  "status": "verified",
  "evidence": "the preceding and following occurrences identify the intended real line unambiguously"
}

{
  "id": "O003-C160",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority line 489",
  "category": "logical_direction",
  "source_issue": "the introduction says that B generates tau_K because tau_K is finer than the Euclidean topology, reversing the construction and its consequence",
  "target_action": "state that B is a basis for tau_K and then infer fineness from B containing all ordinary open intervals",
  "status": "verified",
  "evidence": "basis generation defines the topology; containment of the Euclidean basis proves the stated comparison in the correct direction"
}

{
  "id": "O003-C161",
  "unit": "chapter_13_closed_sets_in_topological_spaces",
  "file": "repo/source/sec_closed_sets_top_exer.ptx",
  "source_locator": "authority lines 253-261",
  "category": "schema_structure",
  "source_issue": "a definition block is nested inside a divisional-exercise introduction where the pinned PreTeXt schema permits paragraph-level introductory content but not a block definition",
  "target_action": "preserve the complete definition and index term as an explicitly labeled definition paragraph inside the same introduction",
  "status": "verified",
  "evidence": "the repair is reader-equivalent and makes the cumulative source conform to the frozen PreTeXt 1.7.5 RelaxNG model"
}

{
  "id": "O003-C162",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_intro.ptx",
  "source_locator": "authority line 188",
  "category": "mathematical_notation",
  "source_issue": "the inverse-image expression is malformed as f^{-1}O), with a missing opening parenthesis",
  "target_action": "write f^{-1}(O) consistently with the surrounding proof",
  "status": "verified",
  "evidence": "the theorem and every adjacent occurrence identify the same inverse image unambiguously"
}

{
  "id": "O003-C163",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_intro.ptx",
  "source_locator": "authority line 166",
  "category": "mathematical_variable",
  "source_issue": "the basis activity calls both spaces topological but denotes them by metric symbols (X,d_X) and (Y,d_Y), although no metrics are assumed",
  "target_action": "write (X,\\\\tau_X) and (Y,\\\\tau_Y) in the translated activity",
  "status": "verified",
  "evidence": "the task invokes a basis for the topology and the open-set characterization, so topology symbols are the exact structures in scope"
}

{
  "id": "O003-C164",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_intro.ptx",
  "source_locator": "authority line 252",
  "category": "typographical",
  "source_issue": "topological is misspelled as topologicval in the closed-set characterization theorem",
  "target_action": "use ruang topologi in the Indonesian derivative",
  "status": "verified",
  "evidence": "the theorem statement and notation make the intended word certain and no mathematical content changes"
}

{
  "id": "O003-C165",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_intro.ptx",
  "source_locator": "authority lines 111-129",
  "category": "schema_structure",
  "source_issue": "the formal open-function definition is nested inside a task statement, where the pinned PreTeXt schema does not permit a block definition",
  "target_action": "move the intact definition subtree to the section level immediately after the containing exploration",
  "status": "verified",
  "evidence": "all definition text, index entries, mathematical content, and surrounding inquiry tasks are retained; only the invalid container and block position change"
}

{
  "id": "O003-C166",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_metric_equiv.ptx",
  "source_locator": "authority lines 86-100",
  "category": "definition_consistency",
  "source_issue": "the prose calls any distance-preserving function an isometry while the immediately following formal definition requires a distance-preserving bijection",
  "target_action": "state in the prose that the function is bijective, preserving the book's explicit formal convention",
  "status": "verified",
  "evidence": "the repair reconciles adjacent statements without broadening the definition or changing the metric-equivalence criterion"
}

{
  "id": "O003-C167",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_equiv.ptx",
  "source_locator": "authority line 17",
  "category": "mathematical_variable",
  "source_issue": "the definition calls (X,d_X) and (Y,d_Y) topological spaces although no metrics are assumed",
  "target_action": "write (X,\\tau_X) and (Y,\\tau_Y) consistently with the immediately following homeomorphism definition",
  "status": "verified",
  "evidence": "the definition concerns arbitrary topological spaces and therefore requires topology symbols rather than metric symbols"
}

{
  "id": "O003-C168",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_equiv.ptx",
  "source_locator": "authority lines 50-53",
  "category": "typographical",
  "source_issue": "the homeomorphic-spaces sentence says the spaces (X,\\tau_X) to (Y,\\tau_Y) are homeomorphic",
  "target_action": "join the two named spaces with and in the Indonesian sentence",
  "status": "verified",
  "evidence": "the subject consists of the two spaces and the source's second to is an unambiguous grammatical slip"
}

{
  "id": "O003-C169",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_equiv.ptx",
  "source_locator": "authority lines 60-69",
  "category": "mathematical_exposition",
  "source_issue": "the transition claims continuity of a homeomorphism ensures a smooth deformation and that metric distortion must be globally bounded",
  "target_action": "state that a homeomorphism preserves topological structure but need not be differentiably smooth or globally bi-Lipschitz; present two-sided distance bounds as the stronger sufficient condition proved next",
  "status": "verified",
  "evidence": "homeomorphisms need not be smooth or globally Lipschitz, while the immediately following theorem correctly proves that two-sided global Lipschitz bounds imply topological equivalence"
}

{
  "id": "O003-C170",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_equiv.ptx",
  "source_locator": "authority lines 136-139",
  "category": "typographical",
  "source_issue": "the final activity uses awkward interrogative grammar and ends the question with a period",
  "target_action": "render it as a direct Indonesian question with a question mark",
  "status": "verified",
  "evidence": "the following instruction to explain confirms the sentence is a question"
}

{
  "id": "O003-C171",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_relations.ptx",
  "source_locator": "authority lines 49-57",
  "category": "logical_scope",
  "source_issue": "the symmetry and transitivity clauses say 'whenever ... in S' without explicitly quantifying the elements",
  "target_action": "state that the implications range over every relevant pair or triple of elements, while preserving the original math nodes and relation order",
  "status": "verified",
  "evidence": "the definition is intended as a relation on S and the added quantifier scope removes ambiguity without changing its meaning"
}

{
  "id": "O003-C172",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_relations.ptx",
  "source_locator": "authority lines 82-90",
  "category": "logical_scope",
  "source_issue": "the two learner tasks call metric and topological equivalence an equivalence relation without naming the collection on which the relation is considered",
  "target_action": "scope each task to any chosen set of metric spaces or topological spaces",
  "status": "verified",
  "evidence": "an equivalence relation is defined on a set; the scoped wording prevents an unintended proper-class reading and preserves the requested proof task"
}

{
  "id": "O003-C173",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_invar.ptx",
  "source_locator": "authority line 28",
  "category": "grammar",
  "source_issue": "the activity asks whether X and Y are a singular homeomorphic space although it names two spaces",
  "target_action": "render X and Y as homeomorphic spaces in the Indonesian activity",
  "status": "verified",
  "evidence": "the task compares two topological spaces and the plural wording removes an unambiguous number mismatch"
}

{
  "id": "O003-C174",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_top_invar.ptx",
  "source_locator": "authority line 35",
  "category": "terminology_spelling",
  "source_issue": "the Indonesian draft used indisret instead of the established Indonesian topology spelling indiskret",
  "target_action": "normalize the learner task to topologi indiskret",
  "status": "verified",
  "evidence": "indiskret is the established corpus term and the correction changes no mathematical content"
}

{
  "id": "O003-C175",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_summ.ptx",
  "source_locator": "authority line 20",
  "category": "grammar",
  "source_issue": "the metric-equivalence summary says there is a isometry",
  "target_action": "render the article-correct Indonesian sentence as terdapat isometri",
  "status": "verified",
  "evidence": "the source article is grammatically incorrect and the mathematical assertion is unchanged"
}

{
  "id": "O003-C176",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_summ.ptx",
  "source_locator": "authority line 23",
  "category": "grammar",
  "source_issue": "the topological-equivalence summary says there is a homeomorphism between X to Y",
  "target_action": "render the relation as a homeomorphism antara X dan Y",
  "status": "verified",
  "evidence": "between requires the conjunction and; the correction preserves the intended pair of spaces"
}

{
  "id": "O003-C177",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_summ.ptx",
  "source_locator": "authority line 34",
  "category": "grammar",
  "source_issue": "the invariant summary omits the article in the phrase that topological space X has",
  "target_action": "render the Indonesian clause as sifat yang dimiliki ruang topologi X",
  "status": "verified",
  "evidence": "the repaired clause identifies X as the space possessing the property without changing the definition"
}

{
  "id": "O003-C178",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_exer.ptx",
  "source_locator": "authority line 557",
  "category": "grammar",
  "source_issue": "the discrete-topology true/false statement says X and Y are topological space",
  "target_action": "render the Indonesian statement with the plural ruang topologi",
  "status": "verified",
  "evidence": "the sentence names two spaces and the plural form removes an unambiguous number mismatch"
}

{
  "id": "O003-C179",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_exer.ptx",
  "source_locator": "authority lines 193-218",
  "category": "grammar",
  "source_issue": "the introduction begins with an ungrammatical fragment before defining the two point topologies and their status as topologies",
  "target_action": "render the Indonesian introduction as a complete definition and a factual cross-reference",
  "status": "verified",
  "evidence": "the intended definition and the two cited exercises are explicit; the repair changes no topology or xref"
}

{
  "id": "O003-C180",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_exer.ptx",
  "source_locator": "authority lines 538-540",
  "category": "grammar",
  "source_issue": "the finite-complement true/false statement uses the fragment f(x)=x^2 mapping ... is continuous",
  "target_action": "render it as the map f(x)=x^2 from one named topological space to the other",
  "status": "verified",
  "evidence": "the mathematical map and source spaces are unchanged while the sentence becomes syntactically complete"
}

{
  "id": "O003-C181",
  "unit": "chapter_14_continuity_and_homeomorphisms",
  "file": "repo/source/sec_cont_top_exer.ptx",
  "source_locator": "translated lines 164 and 255-256",
  "category": "terminology_consistency",
  "source_issue": "the first Indonesian exercise draft used Euclides although the established corpus spelling is Euklides",
  "target_action": "normalize all three occurrences to topologi Euklides or topologi metrik Euklides",
  "status": "verified",
  "evidence": "the terminology ledger and prior admitted chapters consistently use Euklides"
}

{
  "id": "O003-C182",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_sub.ptx",
  "source_locator": "authority line 40",
  "category": "logical_scope",
  "source_issue": "the task conclusion refers to item (1) although no stable generated label is guaranteed and then says any subset despite the chapter's explicit nonempty-subspace convention",
  "target_action": "refer to the result above and state every nonempty subset consistently with the adjacent exploration and formal definition",
  "status": "verified",
  "evidence": "the correction removes an unstable display-label dependency and aligns the conclusion with both explicit nonempty hypotheses without changing the constructed topology"
}

{
  "id": "O003-C183",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_base_sub.ptx",
  "source_locator": "authority line 40",
  "category": "mathematical_notation",
  "source_issue": "the basis-intersection task writes B in plain italic although the surrounding section defines and consistently uses the basis family macro \\\\B",
  "target_action": "replace B by the established \\\\B macro in that membership expression",
  "status": "verified",
  "evidence": "the plain symbol is undefined in context; the repair restores the stated basis family without changing the proof task"
}

{
  "id": "O003-C184",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_sub_summ.ptx",
  "source_locator": "authority line 21",
  "category": "grammar",
  "source_issue": "the singular subject a relatively closed set is paired with the plural predicate are the sets",
  "target_action": "render the Indonesian summary with the singular form suatu himpunan tertutup relatif berbentuk A intersection C",
  "status": "verified",
  "evidence": "the number repair preserves the exact intersection characterization"
}

{
  "id": "O003-C185",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_sub.ptx",
  "source_locator": "authority lines 28-35",
  "category": "schema_structure",
  "source_issue": "the raster figure is nested inside an exploration introduction where the pinned PreTeXt schema does not admit that figure model or an accessibility description",
  "target_action": "move the complete figure block to section level immediately before the exploration and add a faithful Indonesian description",
  "status": "verified",
  "evidence": "all figure content and the stable ID are retained; the new location is schema-valid and the xref remains resolved"
}

{
  "id": "O003-C186",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_sub.ptx",
  "source_locator": "authority lines 43-55",
  "category": "schema_structure",
  "source_issue": "a formal definition is nested inside a task statement where the pinned PreTeXt schema does not admit it",
  "target_action": "move the complete definition block to section level immediately after the exploration",
  "status": "verified",
  "evidence": "the stable ID index definition statement and mathematics are unchanged while the source becomes schema-valid"
}

{
  "id": "O003-C187",
  "unit": "chapter_15_subspaces",
  "file": "repo/source/sec_sub_exer.ptx",
  "source_locator": "authority lines 89-97",
  "category": "schema_structure",
  "source_issue": "a formal definition and its statement shell are nested inside an exercise statement where the pinned PreTeXt schema does not admit them",
  "target_action": "remove only the two invalid ancestor shells retain the index entry in the introductory paragraph and retain the defining paragraph as the next paragraph of the same exercise statement",
  "status": "verified",
  "evidence": "all reader-facing definition content terminology mathematics and the exercise prompt remain in order while the chapter validates"
}

{
  "id": "O003-C188",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients.ptx",
  "source_locator": "authority line 18",
  "category": "terminology_typo",
  "source_issue": "the English source says congruence module n although the established arithmetic relation is congruence modulo n",
  "target_action": "render the Indonesian sentence as relasi kongruensi modulo n",
  "status": "verified",
  "evidence": "module is an unambiguous typographical substitution; the correction restores the standard relation without changing any formula"
}

{
  "id": "O003-C189",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotient_top.ptx",
  "source_locator": "authority line 4",
  "category": "mathematical_object",
  "source_issue": "the English source calls the resulting object on Y a function although the construction defines a topology",
  "target_action": "render the object correctly as topologi yang dihasilkan pada Y",
  "status": "verified",
  "evidence": "the surrounding sentences and section title unambiguously define the quotient topology; no function is being introduced at that point"
}

{
  "id": "O003-C190",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotient_space.ptx",
  "source_locator": "authority lines 211-214 and 259-308",
  "category": "equivalence_relation_scope",
  "source_issue": "the listed ordered identifications do not literally include all reflexive symmetric and transitive pairs even though the source calls the result an equivalence relation",
  "target_action": "state that the equivalence relation is generated by the displayed identifications",
  "status": "verified",
  "evidence": "equivalence closure is required for the claimed quotient construction and preserves every displayed generating identification"
}

{
  "id": "O003-C191",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotient_space.ptx",
  "source_locator": "authority line 270",
  "category": "mathematical_notation",
  "source_issue": "part (i) identifies the bottom edge with itself by (x,0) equivalent to (1-x,0) although its first diagram is the Möbius identification of opposite edges",
  "target_action": "replace the second endpoint by (1-x,1)",
  "status": "verified",
  "evidence": "the frozen figure stem Mobius_identification and its opposite arrows unambiguously pair the bottom edge with the top edge in reverse direction; the adjacent parts likewise correspond in order to Torus_identification and Quotient_sphere_1"
}

{
  "id": "O003-C192",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotient_space.ptx",
  "source_locator": "authority lines 275 and 283",
  "category": "mathematical_object",
  "source_issue": "part (ii) identifies both pairs of opposite square edges in matching directions and all four corners but calls the resulting space a Möbius strip",
  "target_action": "restore the Möbius name and index to corrected part (i) and index and name part (ii) as a torus",
  "status": "verified",
  "evidence": "the displayed relations and the paired Mobius_identification and Torus_identification figures unambiguously define the standard square quotient models in that order"
}

{
  "id": "O003-C193",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_find_quotient_space.ptx",
  "source_locator": "authority lines 26-27",
  "category": "mathematical_scope",
  "source_issue": "the source says the equivalence relation identifies the real line with the interval [0,1] although the interval only supplies representatives and its endpoints are identified",
  "target_action": "state that every equivalence class has a representative in [0,1] with the two endpoints identified",
  "status": "verified",
  "evidence": "the quotient is not literally the closed interval; this wording preserves the intended fundamental-domain construction"
}

{
  "id": "O003-C194",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_find_quotient_space.ptx",
  "source_locator": "authority line 30",
  "category": "mathematical_notation",
  "source_issue": "the objective names the unit circle as S_1 although the defining display and every subsequent occurrence use S^1",
  "target_action": "replace S_1 by S^1",
  "status": "verified",
  "evidence": "the immediately preceding display defines S^1 and no object S_1 is defined"
}

{
  "id": "O003-C195",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_find_quotient_space.ptx",
  "source_locator": "authority lines 139-148",
  "category": "ambient_space_error",
  "source_issue": "the proof equates a subset of X with the inverse image under the induced map which is a subset of X/~ and then calls that subset open in X",
  "target_action": "replace the invalid chain by f^{-1}(O)=p^{-1}(overline f^{-1}(O)) and apply the quotient-topology criterion in X/~",
  "status": "verified",
  "evidence": "the corrected identity is type-consistent and directly proves continuity of the induced map"
}

{
  "id": "O003-C196",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_find_quotient_space.ptx",
  "source_locator": "authority lines 190-194",
  "category": "period_error",
  "source_issue": "for f(t)=(cos(2 pi t),sin(2 pi t)) the source translates lifted intervals by 2 pi k and calls A and B angles",
  "target_action": "treat A and B as normalized real parameters and translate the lifted intervals by k",
  "status": "verified",
  "evidence": "the period in the variable t is 1 so the preimage components are (A+k,B+k) for integers k"
}

{
  "id": "O003-C197",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_find_quotient_space.ptx",
  "source_locator": "authority lines 237-242",
  "category": "proof_scope",
  "source_issue": "the source claims every interval contained in O maps to the single endpoint-determined open arc although an interval of length at least 1 can wrap around the circle",
  "target_action": "choose around every point an open subinterval of length less than 1 and take the union of its open-arc images",
  "status": "verified",
  "evidence": "such subintervals cover O and their images cover f(O), proving that f is open without the overbroad interval claim"
}

{
  "id": "O003-C198",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_summ.ptx",
  "source_locator": "authority line 18",
  "category": "mathematical_object",
  "source_issue": "the source calls the function p a quotient topology and the resulting sentence is grammatically malformed",
  "target_action": "state that p in the preceding construction is called a quotient map and Y is called a quotient space",
  "status": "verified",
  "evidence": "p is the surjective function in the preceding bullet while the quotient topology is the displayed family of subsets of Y"
}

{
  "id": "O003-C199",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 59-62",
  "category": "empty_subspace_edge_case",
  "source_issue": "the claimed equivalence classes include A itself even when A is empty although an empty set cannot be an equivalence class and the intuition then says the empty set is collapsed to a point",
  "target_action": "define x equivalent to y exactly when x equals y or both points belong to A and qualify the collapse intuition so the empty case creates no point",
  "status": "verified",
  "evidence": "the corrected formula gives the intended collapse quotient for every subspace and explicitly leaves the empty-subspace case unchanged"
}

{
  "id": "O003-C200",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 134-158",
  "category": "underdetermined_equivalence_relations",
  "source_issue": "reflexivity alone does not fully specify the equivalence closure of the stated identifications and diagonally opposite is ambiguous for disk points",
  "target_action": "declare the equivalence relation generated by exactly the stated identifications and express the antipodal condition as y=-x",
  "status": "verified",
  "evidence": "equivalence closure supplies symmetry and transitivity while the coordinate formula removes geometric ambiguity without adding identifications"
}

{
  "id": "O003-C201",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 167-170",
  "category": "underdetermined_equivalence_relation",
  "source_issue": "the wording separately supplies a positive-height condition and reflexivity but does not explicitly state the complete binary relation",
  "target_action": "state that two points are equivalent exactly when they are equal or have the same positive second coordinate",
  "status": "verified",
  "evidence": "the biconditional is reflexive on the base edge and makes every positive horizontal slice one equivalence class"
}

{
  "id": "O003-C202",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority line 231",
  "category": "mathematical_notation",
  "source_issue": "the excluded-point topology is written with a duplicated nested tau subscript",
  "target_action": "replace tau sub tau sub overline p by tau sub overline p",
  "status": "verified",
  "evidence": "the corrected symbol matches the definition in the same exercise and the established notation throughout the book"
}

{
  "id": "O003-C203",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 241-250",
  "category": "projective_geometry_error",
  "source_issue": "the source posits one point at infinity where all lines intersect although projective completion adds one ideal point for each parallel direction",
  "target_action": "state that distinct projective lines meet once and parallel affine lines of one direction share their corresponding point at infinity",
  "status": "verified",
  "evidence": "this is the incidence structure of the real projective plane defined immediately afterward"
}

{
  "id": "O003-C204",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority line 284",
  "category": "mathematical_notation",
  "source_issue": "one quotient occurrence uses the ordinary relation macro after the book introduced its spaced quotient-relation macro",
  "target_action": "replace S^2 slash sim by S^2 slash ssim",
  "status": "verified",
  "evidence": "the adjacent occurrences at authority lines 274 and 288 use the intended quotient notation"
}

{
  "id": "O003-C205",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority line 347",
  "category": "mathematical_notation",
  "source_issue": "one real-projective-plane occurrence uses bold R rather than blackboard-bold R",
  "target_action": "replace mathbf R P^2 by mathbb R P^2",
  "status": "verified",
  "evidence": "every other occurrence in the exercise uses mathbb R P^2"
}

{
  "id": "O003-C206",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 313-315",
  "category": "classification_scope",
  "source_issue": "the informal statement says every closed surface is made from spheres tori and projective planes without connectedness or the connected-sum classification",
  "target_action": "state the connected closed-surface classification as the sphere a finite connected sum of tori or a finite connected sum of real projective planes",
  "status": "verified",
  "evidence": "the corrected alternatives are the standard classification boundary and avoid treating disconnected surfaces or ordinary unions as connected sums"
}

{
  "id": "O003-C207",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority line 408",
  "category": "lexical_mathematical_error",
  "source_issue": "the source says continually collapse where continuity of the deformation is intended",
  "target_action": "state continuously collapse",
  "status": "verified",
  "evidence": "the surrounding definition requires a continuous map and the corrected adverb expresses that property"
}

{
  "id": "O003-C208",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/sec_quotients_exer.ptx",
  "source_locator": "authority lines 456-489",
  "category": "duplicate_prompts_retained",
  "source_issue": "the first two true-or-false prompts are repeated verbatim as the third and fourth prompts",
  "target_action": "retain both source occurrences to preserve exercise topology and map the later copies as explicit backend aliases of the first pair",
  "status": "verified",
  "evidence": "the repetition is byte-identical and there is no primary evidence for intended replacement questions; transparent aliasing avoids silently inventing or deleting source content"
}

{
  "id": "O003-C209",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/chapters_01_16_reader.ptx",
  "source_locator": "authority source/main.ptx line 83",
  "category": "build_configuration",
  "source_issue": "the cumulative reader wrapper omitted the source-defined quotient-relation macro ssim, causing the strict XeLaTeX build to stop at the first quotient-space occurrence",
  "target_action": "restore the exact upstream definition newcommand ssim equals sim in the cumulative wrapper macro block",
  "status": "verified",
  "evidence": "the pinned authority defines the macro globally and Chapter 16 uses it throughout; restoring that lossless definition preserves source notation and closes the strict build dependency"
}

{
  "id": "O003-C210",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/chapters_01_16_reader.ptx",
  "source_locator": "authority source/main.ptx line 13",
  "category": "build_configuration",
  "source_issue": "the cumulative reader wrapper omitted the source-defined complex-number macro C used by the original Chapter 16 self-study companion",
  "target_action": "restore the exact upstream definition newcommand C equals mathbb C in the cumulative wrapper macro block",
  "status": "verified",
  "evidence": "the companion's unit-circle description uses the conventional complex-number set and the pinned authority provides this exact lossless macro definition"
}

{
  "id": "O003-C211",
  "unit": "chapter_16_quotient_spaces",
  "file": "repo/source/chapters_01_16_reader.ptx",
  "source_locator": "authority source/main.ptx lines 171-174 and source/sec_find_quotient_space.ptx lines 189-194 and 239",
  "category": "build_configuration",
  "source_issue": "the cumulative derivative wrapper omitted the upstream unicode-math preamble that supplies overparen, causing strict XeLaTeX failure at the four retained open-arc expressions",
  "target_action": "define overparen locally with one argument as overset frown over that argument in the cumulative wrapper macro block",
  "status": "verified",
  "evidence": "the local definition retains every source math token and gives both MathJax and XeLaTeX conventional arc notation without importing unicode-math and globally changing the established cumulative reader fonts or pagination; all four final uses receive targeted visual QA"
}

{
  "id": "O003-C212",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_intro.ptx",
  "source_locator": "authority line 67",
  "category": "grammar",
  "source_issue": "the source phrase collections of open sets that cover of (0,1) contains an extraneous of",
  "target_action": "translate the intended statement as collections of open sets that cover (0,1)",
  "status": "verified",
  "evidence": "the surrounding construction and both adjacent uses unambiguously require the verb cover with no preposition"
}

{
  "id": "O003-C213",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_intro.ptx",
  "source_locator": "authority lines 120-121",
  "category": "mathematical_reference",
  "source_issue": "O0 O1 and O4 are called a finite subcover of the family {On} indexed by n at least 3 even though O0 and O1 are not members of that family",
  "target_action": "identify them as a finite subcover of the previously described cover extended by O0 and O1",
  "status": "verified",
  "evidence": "the immediately preceding paragraph explicitly constructs the extended cover by adjoining O0 and O1 and the three stated sets then cover [0,1]"
}

{
  "id": "O003-C214",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_app.ptx",
  "source_locator": "authority lines 4 and 16 plus downstream summary",
  "category": "missing_hypothesis",
  "source_issue": "the maximum-minimum claim permits the empty compact topological space even though a function with empty domain has no attained maximum or minimum",
  "target_action": "state the theorem and its proof activity for a nonempty compact topological space and propagate the same scope to the chapter summary",
  "status": "verified",
  "evidence": "the book defines a topological space as any set carrying a topology so the empty space is allowed; nonemptiness makes f(X) nonempty and the least-upper-bound argument valid"
}

{
  "id": "O003-C215",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_cont.ptx",
  "source_locator": "authority lines 135-139",
  "category": "undefined_map",
  "source_issue": "the corollary refers to f(A) after saying only that X and Y are homeomorphic and never introduces f",
  "target_action": "state in prose that f is a homeomorphism from the displayed space X to the displayed space Y",
  "status": "verified",
  "evidence": "the preceding theorem proves preservation under continuous maps and applying it to a named homeomorphism and its inverse gives the asserted equivalence"
}

{
  "id": "O003-C216",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority line 187",
  "category": "index_domain",
  "source_issue": "the nested-cube proof quantifies k over all integers although only cubes with nonnegative indices are constructed",
  "target_action": "restrict k to the nonnegative integers",
  "status": "verified",
  "evidence": "Q_0 is the initial cube and the inductive construction defines Q_k only for subsequent nonnegative indices"
}

{
  "id": "O003-C217",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority line 194",
  "category": "mathematical_notation",
  "source_issue": "the Cartesian-product display omits the multiplication sign before its final interval factor",
  "target_action": "insert the missing Cartesian-product times sign before the final factor",
  "status": "verified",
  "evidence": "an n-dimensional cube is the product of all n displayed closed intervals"
}

{
  "id": "O003-C218",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority lines 217-223",
  "category": "proof_gap",
  "source_issue": "the proof cites only a_i_s less than or equal to b_i_s to conclude that the supremum of all a_i_s is below every b_i_s",
  "target_action": "use nesting explicitly to show that each b_i_s is an upper bound for every a_j_s before applying the least-upper-bound property",
  "status": "verified",
  "evidence": "for j below i monotonicity gives a_j_s no greater than a_i_s no greater than b_i_s and for j above i containment Q_j in Q_i gives a_j_s no greater than b_i_s"
}

{
  "id": "O003-C219",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority lines 247-248",
  "category": "strictness_error",
  "source_issue": "D_K is defined as the maximum distance in Q_K but the source asserts every point has distance strictly below D_K",
  "target_action": "use d_E(q,x) less than or equal to D_K less than epsilon_q",
  "status": "verified",
  "evidence": "equality can hold at an opposite corner and the corrected chain still places Q_K inside the epsilon ball"
}

{
  "id": "O003-C220",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority line 359",
  "category": "dimension_index",
  "source_issue": "the final Heine-Borel containment uses Q_K to the power m although the ambient Euclidean dimension is n",
  "target_action": "replace Q_K^m by Q_K^n",
  "status": "verified",
  "evidence": "m counts the chosen finite subcover while n is the dimension fixed throughout the proof"
}

{
  "id": "O003-C221",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_exer.ptx",
  "source_locator": "authority lines 3-18",
  "category": "duplicate_prompt_retained",
  "source_issue": "the first exercise contains two byte-identical tasks",
  "target_action": "retain both physical occurrences and map the second as an explicit backend alias of the first",
  "status": "verified",
  "evidence": "there is no primary evidence for a missing replacement prompt so transparent aliasing preserves source topology without inventing work"
}

{
  "id": "O003-C222",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_exer.ptx",
  "source_locator": "authority lines 190-223",
  "category": "wrong_ambient_set",
  "source_issue": "the Either-Or exercise defines X as the interval [-1,1] but its last task asks about subsets of the integers",
  "target_action": "ask about subsets of X",
  "status": "verified",
  "evidence": "all preceding tasks and the stated topology concern the fixed underlying set X and no topology on the integers is defined"
}

{
  "id": "O003-C223",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_exer.ptx",
  "source_locator": "authority lines 374-381",
  "category": "missing_hypothesis",
  "source_issue": "the Hausdorff separation task permits x to belong to the compact set A while requiring disjoint open sets containing x and A",
  "target_action": "require x to lie in X minus A",
  "status": "verified",
  "evidence": "the requested disjoint neighborhoods are impossible when x belongs to A and the corrected statement is the standard compact-set separation theorem"
}

{
  "id": "O003-C224",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_exer.ptx",
  "source_locator": "authority lines 433-438",
  "category": "unbound_variable",
  "source_issue": "the local-compactness definition quantifies x but requires the undefined point p to lie in O",
  "target_action": "replace p in O by x in O",
  "status": "verified",
  "evidence": "the neighborhood must contain the point quantified at the beginning of the definition"
}

{
  "id": "O003-C225",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_fractals.ptx",
  "source_locator": "authority lines 140-151",
  "category": "incorrect_affine_map",
  "source_issue": "the stated half-scale affine map adds v_i rather than one half v_i even though the prose says v_i is fixed and every image point is the midpoint with v_i",
  "target_action": "replace the translation vector by one half v_i and describe the scale factor as one half",
  "status": "verified",
  "evidence": "the corrected map sends v_i to itself and sends every x to the midpoint of x and v_i exactly as the construction requires"
}

{
  "id": "O003-C226",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_fractals.ptx",
  "source_locator": "authority line 116",
  "category": "grammar",
  "source_issue": "the source says the collection is denote at H(X)",
  "target_action": "translate the intended phrase as denoted by H(X)",
  "status": "verified",
  "evidence": "denote by is the unambiguous grammatical construction naming the hyperspace"
}

{
  "id": "O003-C227",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_cont.ptx",
  "source_locator": "authority lines 115-132",
  "category": "pretext_structure",
  "source_issue": "the compact-image theorem is nested inside the final activity task statement and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged theorem to section level immediately after the activity while retaining the task prompt in place",
  "status": "verified",
  "evidence": "the repaired section preserves the inquiry order and validates with zero diagnostics in the cumulative Chapters 1-17 context"
}

{
  "id": "O003-C228",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_rn.ptx",
  "source_locator": "authority lines 321-339",
  "category": "pretext_structure",
  "source_issue": "the closed-subset-of-a-compact-space theorem is nested inside the final activity task statement and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged theorem to section level immediately after the activity while retaining the task prompt in place",
  "status": "verified",
  "evidence": "the repaired section preserves the inquiry order and validates with zero diagnostics in the cumulative Chapters 1-17 context"
}

{
  "id": "O003-C229",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_app.ptx",
  "source_locator": "authority lines 4-16",
  "category": "pretext_structure",
  "source_issue": "the maximum-minimum theorem is nested inside an activity introduction and is invalid under the pinned PreTeXt schema",
  "target_action": "move the theorem to section level before the activity and make the introduction explicitly identify the following activity as its proof",
  "status": "verified",
  "evidence": "the theorem remains immediately adjacent to its proof activity and the cumulative Chapters 1-17 source validates with zero diagnostics"
}

{
  "id": "O003-C230",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_fractals.ptx",
  "source_locator": "authority lines 66-164",
  "category": "pretext_structure",
  "source_issue": "the Hausdorff-distance definition and theorem are nested inside task statements in one activity and are invalid under the pinned PreTeXt schema",
  "target_action": "split the activity into three consecutive activities and move the unchanged formal definition and theorem to section level at their original reading positions while retaining all four prompts",
  "status": "verified",
  "evidence": "the repaired section preserves the inquiry sequence and prompt denominator and validates with zero diagnostics in the cumulative Chapters 1-17 context"
}

{
  "id": "O003-C231",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/chap_Compact_topology.ptx",
  "source_locator": "authority XInclude order",
  "category": "pretext_structure",
  "source_issue": "the source places a supplemental fractals section after the chapter divisional-exercises block although the pinned PreTeXt schema requires divisional exercises to be final",
  "target_action": "place the unchanged fractals section immediately before the final exercises include",
  "status": "verified",
  "evidence": "all chapter content remains present in the same topical neighborhood and the cumulative Chapters 1-17 source validates with zero diagnostics"
}

{
  "id": "O003-C232",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/sec_compact_top_exer.ptx",
  "source_locator": "authority lines 433-441",
  "category": "pretext_structure",
  "source_issue": "a formal local-compactness definition is nested inside a divisional-exercise introduction and is invalid under the pinned PreTeXt schema",
  "target_action": "retain the complete definition and index term as an explicitly labeled definition paragraph in the same introduction",
  "status": "verified",
  "evidence": "no mathematical content or prompt is removed and the cumulative Chapters 1-17 source validates with zero diagnostics"
}

{
  "id": "O003-C233",
  "unit": "chapter_17_compact_spaces",
  "file": "repo/source/chapters_01_17_reader.ptx",
  "source_locator": "authority main.ptx macro closure / derivative wrapper macros",
  "category": "build_closure",
  "source_issue": "the cumulative derivative wrapper omitted the upstream definition \\\\newcommand{\\\\E}{\\\\mathbb{E}} although Chapter 17 exercises use \\\\E repeatedly; XeLaTeX therefore emitted an undefined-control-sequence error and only a partial PDF",
  "target_action": "restore the exact upstream \\\\E macro definition in the cumulative reader wrapper",
  "status": "verified",
  "evidence": "the strict PDF gate exposed the hidden partial build; the restored macro preserves every Chapter 17 source and companion expression and is required for a complete deterministic PDF"
}

{
  "id": "O003-C234",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_path_intro.ptx",
  "source_locator": "authority line 10",
  "category": "source_typo",
  "source_issue": "the phrase 'from one point in the spaced to another' uses 'spaced' where the sentence unambiguously requires 'space'",
  "target_action": "render the intended phrase naturally as 'dari satu titik ke titik lain dalam ruang tersebut'",
  "status": "verified",
  "evidence": "the correction changes no mathematical definition or XML topology and preserves the evident meaning of the surrounding path-connectedness paragraph"
}

{
  "id": "O003-C235",
  "unit": "chapter_18_connected_spaces",
  "file": "repo/source/sec_connect_sets.ptx",
  "source_locator": "authority lines 171-187",
  "category": "pretext_structure",
  "source_issue": "the connected-image theorem is nested inside the final task statement of an activity and is invalid under the pinned PreTeXt schema",
  "target_action": "retain the task's explanatory prompt in place and hoist the unchanged theorem to section level immediately after the activity",
  "status": "verified",
  "evidence": "the theorem's complete subtree and xml:id thm_connected_invariant are unchanged; only its invalid parent is repaired, and the cumulative Chapters 1-18 schema gate reports zero diagnostics"
}

{
  "id": "O003-C236",
  "unit": "chapter_18_connected_spaces",
  "file": "repo/source/sec_connect_subset_rn.ptx",
  "source_locator": "authority lines 41-58",
  "category": "pretext_structure",
  "source_issue": "the separation lemma is nested inside the final task statement of an activity and is invalid under the pinned PreTeXt schema",
  "target_action": "retain the task's proof prompt in place and hoist the unchanged lemma to section level immediately after the activity",
  "status": "verified",
  "evidence": "the lemma's complete subtree and xml:id lem_separation_subset are unchanged; only its invalid parent is repaired, and the cumulative Chapters 1-18 schema gate reports zero diagnostics"
}

{
  "id": "O003-C237",
  "unit": "chapter_18_connected_spaces",
  "file": "repo/source/sec_ivt_fpt.ptx",
  "source_locator": "authority lines 77-96",
  "category": "pretext_structure",
  "source_issue": "the fixed-point theorem is nested inside an activity introduction and is invalid under the pinned PreTeXt schema",
  "target_action": "hoist the unchanged theorem to section level immediately before the activity while retaining the introduction's proof lead and hypotheses",
  "status": "verified",
  "evidence": "the theorem remains immediately adjacent to its proof activity, all mathematical content is retained, and the cumulative Chapters 1-18 schema gate reports zero diagnostics"
}

{
  "id": "O003-C238",
  "unit": "chapter_18_connected_spaces",
  "file": "repo/source/sec_connect_top_exer.ptx",
  "source_locator": "authority lines 153-170",
  "category": "pretext_structure",
  "source_issue": "an un-IDed formal definition of local connectedness is nested inside a divisional-exercise introduction and is invalid under the pinned PreTeXt schema",
  "target_action": "retain the complete index term and definition text as a paragraph explicitly labeled Definisi in the same introduction",
  "status": "verified",
  "evidence": "no mathematical content or exercise prompt is removed; only the invalid definition and statement shells are flattened, and the cumulative Chapters 1-18 schema gate reports zero diagnostics"
}

{
  "id": "O003-C239",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_path_connect.ptx",
  "source_locator": "authority theorem thm_path_connected",
  "category": "pretext_structure",
  "source_issue": "the path-connected-implies-connected theorem is nested inside an activity introduction and is invalid under the pinned PreTeXt schema",
  "target_action": "hoist the unchanged theorem to section level immediately before the activity while retaining the activity as its proof inquiry",
  "status": "verified",
  "evidence": "the complete theorem subtree and xml:id thm_path_connected are preserved; only its invalid parent is repaired, and the isolated Chapter 19 schema gate reports zero diagnostics"
}

{
  "id": "O003-C240",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_path_connect_equiv.ptx",
  "source_locator": "authority path-product definition",
  "category": "pretext_structure",
  "source_issue": "the formal definition of the product of two paths is nested directly inside an activity and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged definition to section level immediately after the activity and retain the activity's direct prompt paragraph referring to the following definition",
  "status": "verified",
  "evidence": "the original prompt-before-definition reading order and physical prompt carrier are preserved, and the isolated Chapter 19 schema gate reports zero diagnostics"
}

{
  "id": "O003-C241",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_finite.ptx",
  "source_locator": "authority definition def_min_nghb",
  "category": "pretext_structure",
  "source_issue": "the formal minimal-neighborhood definition is nested inside a task statement and is invalid under the pinned PreTeXt schema",
  "target_action": "retain the complete index term, xml:id def_min_nghb, and definition text as an explicitly labeled paragraph inside the same task statement",
  "status": "verified",
  "evidence": "no mathematical content or prompt is removed; only the invalid definition shell is flattened, and the isolated Chapter 19 schema gate reports zero diagnostics"
}

{
  "id": "O003-C242",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_finite.ptx",
  "source_locator": "authority lemma proof after selecting p in P",
  "category": "variable_reference",
  "source_issue": "the proof refers to the neighborhood O of x even though it has just selected the point p and the next sentence uses p in O",
  "target_action": "render O as a neighborhood of p",
  "status": "verified",
  "evidence": "the corrected reference matches the quantified point and the following membership statement without changing the proof's argument"
}

{
  "id": "O003-C243",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_finite.ptx",
  "source_locator": "authority minimal-neighborhood proof",
  "category": "grammar",
  "source_issue": "the source says 'any neighborhood of N of x', with an extraneous 'of' before N",
  "target_action": "render the intended phrase as 'any neighborhood N of x'",
  "status": "verified",
  "evidence": "the correction is grammatical and preserves the quantified neighborhood and point"
}

{
  "id": "O003-C244",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_finite.ptx",
  "source_locator": "authority final theorem reverse implication",
  "category": "mathematical_logic",
  "source_issue": "the reverse implication assumes that X is path connected and repeats the forward direction instead of proving that connectedness plus equality of components and path components implies path connectedness",
  "target_action": "assume X is connected, use its single component C=X and the equality of components and path components, then conclude that X is path connected",
  "status": "verified",
  "evidence": "the repaired argument proves the stated converse and uses exactly the theorem's hypotheses"
}

{
  "id": "O003-C245",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_infinite.ptx",
  "source_locator": "authority distance display in connectedness proof",
  "category": "formula_parenthesis",
  "source_issue": "the displayed Euclidean-distance expression lacks the closing parenthesis that pairs with the opening delimiter around its two point arguments",
  "target_action": "restore the missing closing parenthesis after the second point",
  "status": "verified",
  "evidence": "the repaired expression is syntactically balanced and leaves every mathematical symbol and argument unchanged"
}

{
  "id": "O003-C246",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_infinite.ptx",
  "source_locator": "authority connectedness proof constructed point",
  "category": "formula_reference",
  "source_issue": "the proof concludes that (z,arcsin(z)) lies in the graph B although B is the graph of sin(1/x) and the constructed point has second coordinate sin(1/z)",
  "target_action": "render the constructed point as (z,sin(1/z))",
  "status": "verified",
  "evidence": "the corrected point lies in B by its definition and is the same point used in the preceding distance estimate"
}

{
  "id": "O003-C247",
  "unit": "chapter_19_path_connected_spaces",
  "file": "repo/source/sec_connect_infinite.ptx",
  "source_locator": "authority final proof sentence",
  "category": "punctuation",
  "source_issue": "the source opens a parenthetical with '(In fact,' at the end of the proof but never closes it",
  "target_action": "close the parenthetical at the end of the sentence",
  "status": "verified",
  "evidence": "the repair restores balanced punctuation without altering the mathematical claim"
}

{
  "id": "O003-C300",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top.ptx",
  "source_locator": "authority line 53",
  "category": "grammar",
  "source_issue": "the question says 'what is the smallest collection of sets would we need'",
  "target_action": "render the intended question without the extraneous auxiliary construction",
  "status": "verified",
  "evidence": "the Indonesian question is grammatical and asks for the same smallest added collection"
}

{
  "id": "O003-C301",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_top_prod_space.ptx",
  "source_locator": "authority line 16",
  "category": "notation_consistency",
  "source_issue": "the first factor is written (X tau) although the section consistently names its topology tau_X",
  "target_action": "render the first factor as (X tau_X)",
  "status": "verified",
  "evidence": "the corrected subscript matches every subsequent use in the basis construction"
}

{
  "id": "O003-C302",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_exam.ptx",
  "source_locator": "authority line 10",
  "category": "ambient_space",
  "source_issue": "the real intervals [1,2] and [3,4] are described as subspaces of R^2",
  "target_action": "render them as subspaces of R",
  "status": "verified",
  "evidence": "both intervals are subsets of the real line and their product is then identified with the rectangle in R^2"
}

{
  "id": "O003-C303",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_exam.ptx",
  "source_locator": "authority line 68",
  "category": "grammar",
  "source_issue": "the source says V is 'the intersections' of S^1 with an open ball",
  "target_action": "render V as the singular intersection",
  "status": "verified",
  "evidence": "the repaired number agreement preserves the described relatively open arc"
}

{
  "id": "O003-C304",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_exam.ptx",
  "source_locator": "authority lines 75-116",
  "category": "formula_reference",
  "source_issue": "the radius-two-circle activity repeatedly substitutes S^1 and unit-circle sample points for its declared first factor 2S^1",
  "target_action": "use p in 2S^1; S^1_p as a subset of 2S^1 times S^1; graph the first factor 2S^1; use sample points (2,0) (sqrt(2),sqrt(2)) and (0,2); and take U from 2S^1",
  "status": "verified",
  "evidence": "every repaired reference now lies in the declared radius-two first factor while the second factor remains S^1"
}

{
  "id": "O003-C305",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_proj_cont_prod.ptx",
  "source_locator": "authority line 87",
  "category": "grammar",
  "source_issue": "the source says 'a product of topological space'",
  "target_action": "render the plural 'ruang-ruang topologi'",
  "status": "verified",
  "evidence": "the quantified finite product has multiple topological-space factors"
}

{
  "id": "O003-C306",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_proj_cont_prod.ptx",
  "source_locator": "authority lines 130-152",
  "category": "pretext_structure",
  "source_issue": "the theorem thm_subbasis_continuous is nested inside an activity statement and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged theorem to section level immediately after the activity while retaining the direct proof prompt and hint",
  "status": "verified",
  "evidence": "the theorem ID and forward xref are preserved and isolated Chapter 20 schema validation reports zero diagnostics"
}

{
  "id": "O003-C307",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_proj_cont_prod.ptx",
  "source_locator": "authority line 166",
  "category": "formula_parenthesis",
  "source_issue": "the inverse-image identity lacks the final closing parenthesis in g^{-1}(h^{-1}(O))",
  "target_action": "restore the missing closing parenthesis",
  "status": "verified",
  "evidence": "the repaired expression has balanced function arguments and states the standard inverse-image identity"
}

{
  "id": "O003-C308",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority lines 38-45",
  "category": "pretext_structure",
  "source_issue": "the Hausdorff product theorem is nested inside a task statement and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged theorem to section level immediately after the activity and retain the direct completion prompt",
  "status": "verified",
  "evidence": "the theorem remains adjacent to the generating activity and isolated Chapter 20 schema validation reports zero diagnostics"
}

{
  "id": "O003-C309",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority lines 83 and 88",
  "category": "xref_reference",
  "source_issue": "the proof of lem_Connected_union twice cites the lemma itself for the two-set union result it is using",
  "target_action": "cite the earlier foundational activity act_connected_compenent",
  "status": "verified",
  "evidence": "the replacement xref supplies the needed prior result and removes circular self-citation"
}

{
  "id": "O003-C310",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority line 122",
  "category": "variable_reference",
  "source_issue": "the proof defines X_a but then says the undefined X-prime is connected",
  "target_action": "render X_a as connected",
  "status": "verified",
  "evidence": "the corrected symbol is the subspace introduced in the preceding sentence"
}

{
  "id": "O003-C311",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority lines 120-126",
  "category": "map_type",
  "source_issue": "the factor-identifying homeomorphisms are called inclusion maps although their displayed domains and codomains make them parameterizations into product slices",
  "target_action": "render them neutrally as mappings",
  "status": "verified",
  "evidence": "the formulas and homeomorphism claims are preserved without the incorrect map-type label"
}

{
  "id": "O003-C312",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority line 212",
  "category": "variable_reference",
  "source_issue": "the compactness proof concludes y is in undefined Y_k after defining the finite family V_k",
  "target_action": "render y in V_k",
  "status": "verified",
  "evidence": "the corrected membership refers to the selected member of the finite cover of Y"
}

{
  "id": "O003-C313",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prop_prod_top.ptx",
  "source_locator": "authority lines 221-229",
  "category": "grammar_and_logic",
  "source_issue": "the final compactness paragraph contains 'the is'; duplicates 'is an open cover of X' after a period; and uses singular agreement for multiple covering sets",
  "target_action": "render a coherent finite-subcover argument with plural agreement",
  "status": "verified",
  "evidence": "the corrected prose still selects finitely many W_i covering X and then their product strips covering X times Y"
}

{
  "id": "O003-C314",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_summ.ptx",
  "source_locator": "authority line 37",
  "category": "definition_reference",
  "source_issue": "the subbasis summary calls the collection of all finite intersections 'the set CS' even though CS itself is the subbasis",
  "target_action": "describe the finite-intersection collection as the collection built from CS",
  "status": "verified",
  "evidence": "the repaired wording matches the preceding chapter theorem and preserves the definition"
}

{
  "id": "O003-C315",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_exer.ptx",
  "source_locator": "authority line 87",
  "category": "grammar",
  "source_issue": "the source says X_1 and X_2 are 'topological space'",
  "target_action": "render the plural 'ruang topologi'",
  "status": "verified",
  "evidence": "both quantified factors are topological spaces"
}

{
  "id": "O003-C316",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_exer.ptx",
  "source_locator": "authority lines 184-186",
  "category": "punctuation",
  "source_issue": "the compact-factor question ends with a period rather than a question mark",
  "target_action": "render it as a direct question",
  "status": "verified",
  "evidence": "the interrogative meaning is explicit without altering the mathematical claim under assessment"
}

{
  "id": "O003-C317",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority lines 46-50",
  "category": "index_chain",
  "source_issue": "the k-path sequence begins at p_0 but its adjacency chain begins with p_1 adjacent to p_2 and omits p_0 adjacent to p_1",
  "target_action": "start the adjacency chain with p_0 adjacent to p_1",
  "status": "verified",
  "evidence": "the repaired consecutive-pair condition covers the full displayed sequence through p_m"
}

{
  "id": "O003-C318",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority lines 73 and 75",
  "category": "grammar",
  "source_issue": "the source says 'we us' and repeats 'if if'",
  "target_action": "render the intended verbs and single conditional",
  "status": "verified",
  "evidence": "both repairs are deterministic typographical normalization"
}

{
  "id": "O003-C319",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority line 134",
  "category": "macro_consistency",
  "source_issue": "the digital-line activity writes plain Z although every surrounding integer-space reference uses the protected macro backslash-Z",
  "target_action": "render the integer space with backslash-Z",
  "status": "verified",
  "evidence": "the corrected notation matches the basis definition and all later occurrences"
}

{
  "id": "O003-C320",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority line 172",
  "category": "definition_consistency",
  "source_issue": "the source recalls a path as a homeomorphism from [0,1] although the book's path definition and the immediately following digital-path definition use the range of a continuous function",
  "target_action": "render a path as the range of a continuous function from [0,1]",
  "status": "verified",
  "evidence": "the repaired recall matches the established definition while preserving the subsequent path-versus-arc distinction"
}

{
  "id": "O003-C321",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority lines 169-239",
  "category": "pretext_structure",
  "source_issue": "two formal definitions are nested inside task statements and one additionally places a ul inside a p; both constructions are invalid under the pinned PreTeXt schema",
  "target_action": "move the prerequisite interval prose and both indexed definitions to section level immediately before the same activity; split the invalid list-bearing paragraph into valid definition paragraphs; retain the direct tasks in the activity",
  "status": "verified",
  "evidence": "all definition text and index terms remain contiguous with the activity and isolated Chapter 20 schema validation reports zero diagnostics"
}

{
  "id": "O003-C322",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority lines 247-275",
  "category": "pretext_structure",
  "source_issue": "the theorem thm_digital_Jordan_curve is nested inside an activity task statement and is invalid under the pinned PreTeXt schema",
  "target_action": "move the unchanged theorem to section level immediately after the activity while retaining the task's source citation consequence and direct assessment prompt",
  "status": "verified",
  "evidence": "the theorem ID and forward xref are preserved and isolated Chapter 20 schema validation reports zero diagnostics"
}

{
  "id": "O003-C323",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority lines 293-294",
  "category": "formula_glyph",
  "source_issue": "the theorem contains question-mark glyphs in the subscripts P_{j?1} and P_{i?1} where adjacent indices require subtraction",
  "target_action": "render P_{j-1} and P_{i-1}",
  "status": "verified",
  "evidence": "the corrected indices form the stated triples of consecutive points"
}

{
  "id": "O003-C324",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority line 348",
  "category": "symbol_case",
  "source_issue": "the application task switches from the theorem's point labels P_j to undefined lowercase x_j",
  "target_action": "retain the theorem's uppercase P labels throughout",
  "status": "verified",
  "evidence": "the corrected symbols refer to the enumerated points of the digital Jordan curve"
}

{
  "id": "O003-C325",
  "unit": "chapter_20_product_topology",
  "file": "repo/source/sec_prod_top_app.ptx",
  "source_locator": "authority line 316",
  "category": "grammar",
  "source_issue": "the source says 'let N(x) to be'",
  "target_action": "render the intended definition 'misalkan N(x) adalah'",
  "status": "verified",
  "evidence": "the correction is grammatical and preserves the neighborhood-intersection definition"
}