FA-80001 / Typography line breaking / Open access

Widow and orphan page breaking: orphan parameter binding · case 01

Orphan decisions follow the widow minimum.

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

ROOT CAUSE

The orphan check compares against the widow parameter.

VERIFIED REPAIR

Compare the kept opening lines with the orphan minimum.

Unsuccessful approach: Using the smaller of the two minimums still ignores the configured orphan limit.

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 rem - a < widow:
                    a = max(rem - widow, 0)
                if started and a < widow:
                    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: orphan parameter binding', [[3, 7, 8, 6, 7, 4, 2, 6], 6, 3, 1], [6, 4, 6, 6, 6, 6, 6, 3]), ('regression: orphan parameter binding', [[8, 4, 7, 9, 2], 7, 3, 1], [7, 5, 7, 7, 4]), ('regression: orphan parameter binding', [[1, 5, 3], 7, 2, 1], [6, 3]), ('regression: orphan parameter binding', [[2, 9, 7], 12, 2, 1], [11, 7]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[1, 7, 1, 2, 1, 8], 8, 3, 2], [8, 8, 4]), ('control layout', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2])], [('regression: orphan parameter binding', [[6, 8, 9, 2], 4, 1, 3], [4, 4, 4, 4, 4, 3, 2]), ('regression: orphan parameter binding', [[6, 8, 1, 3, 8, 9], 4, 1, 3], [4, 4, 4, 3, 4, 4, 4, 4, 4]), ('regression: orphan parameter binding', [[6, 6, 3, 3, 5, 7, 8], 11, 3, 2], [10, 11, 9, 8]), ('regression: orphan parameter binding', [[5, 4, 2, 8, 5], 5, 2, 1], [5, 4, 5, 5, 5]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[6, 2, 9, 7, 8, 9, 7], 8, 1, 3], [8, 8, 8, 8, 8, 8]), ('control layout', [[6, 9, 1, 5, 1, 8], 11, 2, 1], [11, 11, 8])], [('regression: orphan parameter binding', [[7, 8, 6, 2, 2, 9, 5, 6], 7, 1, 2], [7, 7, 7, 7, 7, 7, 3]), ('regression: orphan parameter binding', [[5, 3, 1, 2, 2, 9, 4, 4], 8, 3, 1], [8, 8, 6, 8]), ('regression: orphan parameter binding', [[6, 4, 6, 7, 5, 3], 8, 3, 2], [6, 8, 7, 7, 3]), ('regression: orphan parameter binding', [[6, 3, 8], 4, 3, 1], [4, 2, 3, 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', [[6, 3, 6, 7], 5, 1, 1], [5, 5, 5, 5, 2]), ('control layout', [[7, 2, 8, 7, 3, 2], 9, 2, 3], [9, 8, 7, 5])], [('regression: orphan parameter binding', [[5, 1, 5, 6, 7, 7, 5, 1], 10, 2, 3], [8, 9, 10, 10]), ('regression: orphan parameter binding', [[7, 6, 7, 1], 4, 3, 1], [4, 3, 4, 2, 4, 4]), ('regression: orphan parameter binding', [[9, 8, 9, 1, 6, 3, 2, 5], 5, 2, 1], [5, 4, 5, 5, 5, 5, 4, 5, 5]), ('regression: orphan parameter binding', [[5, 3, 7], 7, 3, 1], [5, 7, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[5, 9, 1, 5], 9, 1, 1], [9, 9, 2]), ('control layout', [[3, 4, 6, 6], 11, 1, 3], [10, 9])], [('regression: orphan parameter binding', [[1, 5, 6, 3], 7, 1, 2], [7, 6, 2]), ('regression: orphan parameter binding', [[5, 6, 5, 6, 7], 6, 3, 1], [5, 6, 5, 6, 6, 1]), ('regression: orphan parameter binding', [[7, 3, 1, 1, 8, 4, 9, 2], 5, 2, 1], [5, 5, 5, 5, 4, 5, 4, 2]), ('regression: orphan parameter binding', [[1, 7, 9, 4, 5, 8, 4], 10, 3, 2], [8, 9, 9, 8, 4]), ('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, 5, 3], 9, 3, 3], [7, 8]), ('control layout', [[5, 4, 4, 7, 8, 1, 5, 8], 6, 3, 3], [5, 4, 4, 6, 6, 4, 5, 6, 2])]]
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 fixture	Actual	Expected	Outcome
regression: orphan parameter binding	`[6, 6, 6, 6, 6, 6, 6, 1]`	`[6, 4, 6, 6, 6, 6, 6, 3]`	Failed
regression: orphan parameter binding	`[7, 7, 7, 7, 2]`	`[7, 5, 7, 7, 4]`	Failed
regression: orphan parameter binding	`[7, 2]`	`[6, 3]`	Failed
regression: orphan parameter binding	`[12, 6]`	`[11, 7]`	Failed
paragraph exactly fills page	`[6, 3]`	`[6, 3]`	Passed
long paragraph continuation	`[6, 6, 4]`	`[6, 6, 4]`	Passed
control layout	`[8, 8, 4]`	`[8, 8, 4]`	Passed
control layout	`[7, 4, 5, 7, 7, 7, 7, 2]`	`[7, 4, 5, 7, 7, 7, 7, 2]`	Passed

SHA-256 / e167fafb948d70451a99b53f6b5924123499ef96824ffca121e5ebe5da9cfee3

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 a < min(orphan, widow):
                    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: orphan parameter binding', [[3, 7, 8, 6, 7, 4, 2, 6], 6, 3, 1], [6, 4, 6, 6, 6, 6, 6, 3]), ('regression: orphan parameter binding', [[8, 4, 7, 9, 2], 7, 3, 1], [7, 5, 7, 7, 4]), ('regression: orphan parameter binding', [[1, 5, 3], 7, 2, 1], [6, 3]), ('regression: orphan parameter binding', [[2, 9, 7], 12, 2, 1], [11, 7]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[1, 7, 1, 2, 1, 8], 8, 3, 2], [8, 8, 4]), ('control layout', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2])], [('regression: orphan parameter binding', [[6, 8, 9, 2], 4, 1, 3], [4, 4, 4, 4, 4, 3, 2]), ('regression: orphan parameter binding', [[6, 8, 1, 3, 8, 9], 4, 1, 3], [4, 4, 4, 3, 4, 4, 4, 4, 4]), ('regression: orphan parameter binding', [[6, 6, 3, 3, 5, 7, 8], 11, 3, 2], [10, 11, 9, 8]), ('regression: orphan parameter binding', [[5, 4, 2, 8, 5], 5, 2, 1], [5, 4, 5, 5, 5]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[6, 2, 9, 7, 8, 9, 7], 8, 1, 3], [8, 8, 8, 8, 8, 8]), ('control layout', [[6, 9, 1, 5, 1, 8], 11, 2, 1], [11, 11, 8])], [('regression: orphan parameter binding', [[7, 8, 6, 2, 2, 9, 5, 6], 7, 1, 2], [7, 7, 7, 7, 7, 7, 3]), ('regression: orphan parameter binding', [[5, 3, 1, 2, 2, 9, 4, 4], 8, 3, 1], [8, 8, 6, 8]), ('regression: orphan parameter binding', [[6, 4, 6, 7, 5, 3], 8, 3, 2], [6, 8, 7, 7, 3]), ('regression: orphan parameter binding', [[6, 3, 8], 4, 3, 1], [4, 2, 3, 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', [[6, 3, 6, 7], 5, 1, 1], [5, 5, 5, 5, 2]), ('control layout', [[7, 2, 8, 7, 3, 2], 9, 2, 3], [9, 8, 7, 5])], [('regression: orphan parameter binding', [[5, 1, 5, 6, 7, 7, 5, 1], 10, 2, 3], [8, 9, 10, 10]), ('regression: orphan parameter binding', [[7, 6, 7, 1], 4, 3, 1], [4, 3, 4, 2, 4, 4]), ('regression: orphan parameter binding', [[9, 8, 9, 1, 6, 3, 2, 5], 5, 2, 1], [5, 4, 5, 5, 5, 5, 4, 5, 5]), ('regression: orphan parameter binding', [[5, 3, 7], 7, 3, 1], [5, 7, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[5, 9, 1, 5], 9, 1, 1], [9, 9, 2]), ('control layout', [[3, 4, 6, 6], 11, 1, 3], [10, 9])], [('regression: orphan parameter binding', [[1, 5, 6, 3], 7, 1, 2], [7, 6, 2]), ('regression: orphan parameter binding', [[5, 6, 5, 6, 7], 6, 3, 1], [5, 6, 5, 6, 6, 1]), ('regression: orphan parameter binding', [[7, 3, 1, 1, 8, 4, 9, 2], 5, 2, 1], [5, 5, 5, 5, 4, 5, 4, 2]), ('regression: orphan parameter binding', [[1, 7, 9, 4, 5, 8, 4], 10, 3, 2], [8, 9, 9, 8, 4]), ('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, 5, 3], 9, 3, 3], [7, 8]), ('control layout', [[5, 4, 4, 7, 8, 1, 5, 8], 6, 3, 3], [5, 4, 4, 6, 6, 4, 5, 6, 2])]]
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 fixture	Actual	Expected	Outcome
regression: orphan parameter binding	`[6, 6, 6, 6, 6, 6, 6, 1]`	`[6, 4, 6, 6, 6, 6, 6, 3]`	Failed
regression: orphan parameter binding	`[7, 7, 7, 7, 2]`	`[7, 5, 7, 7, 4]`	Failed
regression: orphan parameter binding	`[7, 2]`	`[6, 3]`	Failed
regression: orphan parameter binding	`[12, 6]`	`[11, 7]`	Failed
paragraph exactly fills page	`[6, 3]`	`[6, 3]`	Passed
long paragraph continuation	`[6, 6, 4]`	`[6, 6, 4]`	Passed
control layout	`[8, 8, 4]`	`[8, 8, 4]`	Passed
control layout	`[7, 4, 5, 7, 7, 7, 7, 2]`	`[7, 4, 5, 7, 7, 7, 7, 2]`	Passed

SHA-256 / af07e5d682610e270f171255cfc6654026f14d423e2ba08c39a492393d7864c0

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: orphan parameter binding', [[3, 7, 8, 6, 7, 4, 2, 6], 6, 3, 1], [6, 4, 6, 6, 6, 6, 6, 3]), ('regression: orphan parameter binding', [[8, 4, 7, 9, 2], 7, 3, 1], [7, 5, 7, 7, 4]), ('regression: orphan parameter binding', [[1, 5, 3], 7, 2, 1], [6, 3]), ('regression: orphan parameter binding', [[2, 9, 7], 12, 2, 1], [11, 7]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[1, 7, 1, 2, 1, 8], 8, 3, 2], [8, 8, 4]), ('control layout', [[1, 2, 8, 5, 8, 6, 7, 9], 7, 3, 3], [7, 4, 5, 7, 7, 7, 7, 2])], [('regression: orphan parameter binding', [[6, 8, 9, 2], 4, 1, 3], [4, 4, 4, 4, 4, 3, 2]), ('regression: orphan parameter binding', [[6, 8, 1, 3, 8, 9], 4, 1, 3], [4, 4, 4, 3, 4, 4, 4, 4, 4]), ('regression: orphan parameter binding', [[6, 6, 3, 3, 5, 7, 8], 11, 3, 2], [10, 11, 9, 8]), ('regression: orphan parameter binding', [[5, 4, 2, 8, 5], 5, 2, 1], [5, 4, 5, 5, 5]), ('orphan pushed forward', [[5, 3], 6, 2, 2], [5, 3]), ('long paragraph continuation', [[2, 14], 6, 2, 2], [6, 6, 4]), ('control layout', [[6, 2, 9, 7, 8, 9, 7], 8, 1, 3], [8, 8, 8, 8, 8, 8]), ('control layout', [[6, 9, 1, 5, 1, 8], 11, 2, 1], [11, 11, 8])], [('regression: orphan parameter binding', [[7, 8, 6, 2, 2, 9, 5, 6], 7, 1, 2], [7, 7, 7, 7, 7, 7, 3]), ('regression: orphan parameter binding', [[5, 3, 1, 2, 2, 9, 4, 4], 8, 3, 1], [8, 8, 6, 8]), ('regression: orphan parameter binding', [[6, 4, 6, 7, 5, 3], 8, 3, 2], [6, 8, 7, 7, 3]), ('regression: orphan parameter binding', [[6, 3, 8], 4, 3, 1], [4, 2, 3, 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', [[6, 3, 6, 7], 5, 1, 1], [5, 5, 5, 5, 2]), ('control layout', [[7, 2, 8, 7, 3, 2], 9, 2, 3], [9, 8, 7, 5])], [('regression: orphan parameter binding', [[5, 1, 5, 6, 7, 7, 5, 1], 10, 2, 3], [8, 9, 10, 10]), ('regression: orphan parameter binding', [[7, 6, 7, 1], 4, 3, 1], [4, 3, 4, 2, 4, 4]), ('regression: orphan parameter binding', [[9, 8, 9, 1, 6, 3, 2, 5], 5, 2, 1], [5, 4, 5, 5, 5, 5, 4, 5, 5]), ('regression: orphan parameter binding', [[5, 3, 7], 7, 3, 1], [5, 7, 3]), ('single widow pulled back', [[3, 4], 6, 2, 2], [5, 2]), ('paragraph exactly fills page', [[2, 4, 3], 6, 2, 2], [6, 3]), ('control layout', [[5, 9, 1, 5], 9, 1, 1], [9, 9, 2]), ('control layout', [[3, 4, 6, 6], 11, 1, 3], [10, 9])], [('regression: orphan parameter binding', [[1, 5, 6, 3], 7, 1, 2], [7, 6, 2]), ('regression: orphan parameter binding', [[5, 6, 5, 6, 7], 6, 3, 1], [5, 6, 5, 6, 6, 1]), ('regression: orphan parameter binding', [[7, 3, 1, 1, 8, 4, 9, 2], 5, 2, 1], [5, 5, 5, 5, 4, 5, 4, 2]), ('regression: orphan parameter binding', [[1, 7, 9, 4, 5, 8, 4], 10, 3, 2], [8, 9, 9, 8, 4]), ('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, 5, 3], 9, 3, 3], [7, 8]), ('control layout', [[5, 4, 4, 7, 8, 1, 5, 8], 6, 3, 3], [5, 4, 4, 6, 6, 4, 5, 6, 2])]]
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 fixture	Actual	Expected	Outcome
regression: orphan parameter binding	`[6, 4, 6, 6, 6, 6, 6, 3]`	`[6, 4, 6, 6, 6, 6, 6, 3]`	Passed
regression: orphan parameter binding	`[7, 5, 7, 7, 4]`	`[7, 5, 7, 7, 4]`	Passed
regression: orphan parameter binding	`[6, 3]`	`[6, 3]`	Passed
regression: orphan parameter binding	`[11, 7]`	`[11, 7]`	Passed
paragraph exactly fills page	`[6, 3]`	`[6, 3]`	Passed
long paragraph continuation	`[6, 6, 4]`	`[6, 6, 4]`	Passed
control layout	`[8, 8, 4]`	`[8, 8, 4]`	Passed
control layout	`[7, 4, 5, 7, 7, 7, 7, 2]`	`[7, 4, 5, 7, 7, 7, 7, 2]`	Passed

SHA-256 / 84126b3260c43f07a1e49c1111cb9f02361b82c0719ab4df55b6eb511340d1d1

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 2026-09-29T14:49:49.688982+00:00.

Case digest / e9e8a58c34f712886d652bfdbf3e14c93a150a42f343e9edede47d64670b3f90