FAILURE MAP
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FA-79991 / Typography line breaking / Open access

Widow and orphan page breaking: widow before orphan ordering · case 01

Curing a widow can create an orphan that is left on the page.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The orphan check runs before the widow adjustment, so a reduced opening is never rechecked.

VERIFIED REPAIR

Resolve widows first and then re-apply the orphan rule to the reduced amount.

Unsuccessful approach: Rechecking the orphan rule against the page room instead of the kept lines misses the new orphan.

Case contract

Input [paragraph line counts, page height, min orphan lines, min widow lines]. Lines flow onto pages. When a paragraph must break on a page that already holds lines, at least min widow lines must carry to the next page (move lines forward) and, if the paragraph starts on this page, at least min orphan lines must stay (else move it all). A paragraph that exactly fits stays. Constraints are waived on an empty page. Return lines per page.

Why this case matters

Line breaking decides where paragraphs wrap on screen and in print; a wrong decision point shifts every following line.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    paras, H, orphan, widow = x
    pages = []
    used = 0
    for p in paras:
        rem = p
        started = True
        while rem > 0:
            room = H - used
            if rem <= room:
                used += rem
                rem = 0
                continue
            a = room
            if used > 0:
                if started and a < orphan:
                    a = 0
                if rem - a < widow:
                    a = max(rem - widow, 0)
            used += a
            rem -= a
            if a > 0:
                started = False
            pages.append(used)
            used = 0
    if used > 0:
        pages.append(used)
    return pages
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[5, 2, 7, 5, 9], 6, 3, 3], [5, 6, 3, 5, 6, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[7, 1, 3, 8, 6, 3], 12, 3, 2], [11, 12, 5]), ('control layout', [[4, 1, 5], 9, 3, 1], [9, 1])], [('regression: widow before orphan ordering', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2]), ('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[6, 3, 5, 3, 1, 6, 5, 7], 10, 2, 3], [9, 9, 8, 10]), ('control layout', [[2, 5, 2], 11, 3, 1], [9])], [('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[2, 8, 4, 5, 9, 5, 8, 2], 9, 3, 3], [7, 7, 9, 5, 9, 6]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[7, 4, 1, 1, 2, 6], 6, 3, 2], [6, 6, 6, 3]), ('control layout', [[7, 8, 5, 1], 10, 1, 3], [10, 10, 1])], [('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[7, 6, 7, 4], 8, 2, 3], [7, 8, 5, 4]), ('regression: widow before orphan ordering', [[8, 4, 3, 3, 3, 1, 4], 5, 2, 2], [5, 5, 5, 3, 4, 4]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('control layout', [[3, 3, 8, 5, 7, 7], 5, 1, 3], [3, 5, 5, 3, 5, 5, 5, 2]), ('control layout', [[6, 9, 7, 1, 7, 9, 8], 4, 3, 2], [4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1])], [('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[5, 5, 9, 6, 5], 5, 3, 3], [5, 5, 5, 4, 5, 1, 5]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[2, 9, 5, 6, 7, 3, 5, 6], 9, 1, 2], [9, 9, 9, 8, 8]), ('control layout', [[2, 6, 6, 3, 7, 5], 11, 2, 1], [11, 11, 7])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: widow before orphan ordering[4, 3, 3, 4, 4][4, 2, 4, 4, 4]Failed
regression: widow before orphan ordering[3, 4, 3, 4, 4, 3, 4, 3, 4][3, 4, 1, 4, 2, 4, 3, 4, 3, 4]Failed
regression: widow before orphan ordering[9, 8, 6][9, 7, 7]Failed
regression: widow before orphan ordering[5, 6, 5, 6, 6][5, 6, 3, 5, 6, 3]Failed
single widow pulled back[5, 2][5, 2]Passed
long paragraph continuation[6, 6, 4][6, 6, 4]Passed
control layout[11, 12, 5][11, 12, 5]Passed
control layout[9, 1][9, 1]Passed

SHA-256 / b3734dc6595a52e8517ba3407722b19294b25eea8e01091dc548466531800212

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    paras, H, orphan, widow = x
    pages = []
    used = 0
    for p in paras:
        rem = p
        started = True
        while rem > 0:
            room = H - used
            if rem <= room:
                used += rem
                rem = 0
                continue
            a = room
            if used > 0:
                if rem - a < widow:
                    a = max(rem - widow, 0)
                if started and room < orphan:
                    a = 0
            used += a
            rem -= a
            if a > 0:
                started = False
            pages.append(used)
            used = 0
    if used > 0:
        pages.append(used)
    return pages
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[5, 2, 7, 5, 9], 6, 3, 3], [5, 6, 3, 5, 6, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[7, 1, 3, 8, 6, 3], 12, 3, 2], [11, 12, 5]), ('control layout', [[4, 1, 5], 9, 3, 1], [9, 1])], [('regression: widow before orphan ordering', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2]), ('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[6, 3, 5, 3, 1, 6, 5, 7], 10, 2, 3], [9, 9, 8, 10]), ('control layout', [[2, 5, 2], 11, 3, 1], [9])], [('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[2, 8, 4, 5, 9, 5, 8, 2], 9, 3, 3], [7, 7, 9, 5, 9, 6]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[7, 4, 1, 1, 2, 6], 6, 3, 2], [6, 6, 6, 3]), ('control layout', [[7, 8, 5, 1], 10, 1, 3], [10, 10, 1])], [('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[7, 6, 7, 4], 8, 2, 3], [7, 8, 5, 4]), ('regression: widow before orphan ordering', [[8, 4, 3, 3, 3, 1, 4], 5, 2, 2], [5, 5, 5, 3, 4, 4]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('control layout', [[3, 3, 8, 5, 7, 7], 5, 1, 3], [3, 5, 5, 3, 5, 5, 5, 2]), ('control layout', [[6, 9, 7, 1, 7, 9, 8], 4, 3, 2], [4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1])], [('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[5, 5, 9, 6, 5], 5, 3, 3], [5, 5, 5, 4, 5, 1, 5]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[2, 9, 5, 6, 7, 3, 5, 6], 9, 1, 2], [9, 9, 9, 8, 8]), ('control layout', [[2, 6, 6, 3, 7, 5], 11, 2, 1], [11, 11, 7])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: widow before orphan ordering[4, 3, 3, 4, 4][4, 2, 4, 4, 4]Failed
regression: widow before orphan ordering[3, 4, 3, 4, 4, 3, 4, 3, 4][3, 4, 1, 4, 2, 4, 3, 4, 3, 4]Failed
regression: widow before orphan ordering[9, 8, 6][9, 7, 7]Failed
regression: widow before orphan ordering[5, 6, 5, 6, 6][5, 6, 3, 5, 6, 3]Failed
single widow pulled back[5, 2][5, 2]Passed
long paragraph continuation[6, 6, 4][6, 6, 4]Passed
control layout[11, 12, 5][11, 12, 5]Passed
control layout[9, 1][9, 1]Passed

SHA-256 / 1bd4a59cc737760a1a40e1a2a68c2585a5e24c0dc8980bdd2804f19146da840f

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    paras, H, orphan, widow = x
    pages = []
    used = 0
    for p in paras:
        rem = p
        started = True
        while rem > 0:
            room = H - used
            if rem <= room:
                used += rem
                rem = 0
                continue
            a = room
            if used > 0:
                if rem - a < widow:
                    a = max(rem - widow, 0)
                if started and a < orphan:
                    a = 0
            used += a
            rem -= a
            if a > 0:
                started = False
            pages.append(used)
            used = 0
    if used > 0:
        pages.append(used)
    return pages
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[5, 2, 7, 5, 9], 6, 3, 3], [5, 6, 3, 5, 6, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[7, 1, 3, 8, 6, 3], 12, 3, 2], [11, 12, 5]), ('control layout', [[4, 1, 5], 9, 3, 1], [9, 1])], [('regression: widow before orphan ordering', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2]), ('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[6, 3, 5, 3, 1, 6, 5, 7], 10, 2, 3], [9, 9, 8, 10]), ('control layout', [[2, 5, 2], 11, 3, 1], [9])], [('regression: widow before orphan ordering', [[7, 9, 3, 9, 7, 5], 5, 3, 3], [5, 5, 5, 4, 5, 4, 5, 2, 5]), ('regression: widow before orphan ordering', [[3, 4, 1], 6, 3, 3], [3, 5]), ('regression: widow before orphan ordering', [[3, 5, 5, 1, 7, 7, 3, 1], 4, 3, 3], [3, 4, 1, 4, 2, 4, 3, 4, 3, 4]), ('regression: widow before orphan ordering', [[2, 8, 4, 5, 9, 5, 8, 2], 9, 3, 3], [7, 7, 9, 5, 9, 6]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('control layout', [[7, 4, 1, 1, 2, 6], 6, 3, 2], [6, 6, 6, 3]), ('control layout', [[7, 8, 5, 1], 10, 1, 3], [10, 10, 1])], [('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[7, 6, 7, 4], 8, 2, 3], [7, 8, 5, 4]), ('regression: widow before orphan ordering', [[8, 4, 3, 3, 3, 1, 4], 5, 2, 2], [5, 5, 5, 3, 4, 4]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('control layout', [[3, 3, 8, 5, 7, 7], 5, 1, 3], [3, 5, 5, 3, 5, 5, 5, 2]), ('control layout', [[6, 9, 7, 1, 7, 9, 8], 4, 3, 2], [4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1])], [('regression: widow before orphan ordering', [[9, 7, 4, 1, 2], 10, 2, 3], [9, 7, 7]), ('regression: widow before orphan ordering', [[2, 2, 5, 5, 7, 6, 4, 2], 8, 2, 3], [6, 8, 7, 6, 6]), ('regression: widow before orphan ordering', [[2, 2, 1, 1, 4, 8], 4, 2, 3], [4, 2, 4, 4, 4]), ('regression: widow before orphan ordering', [[5, 5, 9, 6, 5], 5, 3, 3], [5, 5, 5, 4, 5, 1, 5]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[2, 9, 5, 6, 7, 3, 5, 6], 9, 1, 2], [9, 9, 9, 8, 8]), ('control layout', [[2, 6, 6, 3, 7, 5], 11, 2, 1], [11, 11, 7])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression: widow before orphan ordering[4, 2, 4, 4, 4][4, 2, 4, 4, 4]Passed
regression: widow before orphan ordering[3, 4, 1, 4, 2, 4, 3, 4, 3, 4][3, 4, 1, 4, 2, 4, 3, 4, 3, 4]Passed
regression: widow before orphan ordering[9, 7, 7][9, 7, 7]Passed
regression: widow before orphan ordering[5, 6, 3, 5, 6, 3][5, 6, 3, 5, 6, 3]Passed
single widow pulled back[5, 2][5, 2]Passed
long paragraph continuation[6, 6, 4][6, 6, 4]Passed
control layout[11, 12, 5][11, 12, 5]Passed
control layout[9, 1][9, 1]Passed

SHA-256 / c82f20e7ba00a89fa597364b8822270fa95d23699d5cffa977bd7170a9da6369

Verification & scope

A deterministic toy typesetting model with integer widths and a stipulated rule set; it does not claim conformance to any engine. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:49:49.649519+00:00.

Case digest / 441ec6448bbfef4ebe1213666ad1313dccad2883547df491985cb924a116590d