FA-78541 / Broadcast timecode arithmetic / Open access
2:3 pulldown film frame placement: cadence group size · case 01
Film frames after the first cadence group map to video frames far too early.
ROOT CAUSE
Each four-frame group advances by five units, mixing video frames with fields.
VERIFIED REPAIR
Each group of four film frames spans ten fields.
Unsuccessful approach: Eight fields per group assumes 2:2 pulldown.
Case contract
Film frames follow the A2 B3 C2 D3 field cadence (10 fields per 4 film frames). phase is the cadence position (0=A..3=D) of film frame 0, which starts at video field 0. Negative n returns None. For film frame n return [video frame of its first field, "upper" for an even field index else "lower", number of fields].
Why this case matters
Timecode arithmetic errors misplace edits, commercial breaks and captions against the broadcast clock.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(n, phase):
if n<0:
return None
def sf(a):
return 5*(a//4)+[0,2,5,7][a%4]
a=n+phase
start=sf(a)-sf(phase)
count=[2,3,2,3][a%4]
return [start//2,'upper' if start%2==0 else 'lower',count]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: cadence group size', [4, 0], [5, 'upper', 2]), ('regression variant: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair probe: cadence group size', [8, 0], [10, 'upper', 2]), ('partial repair variant: cadence group size', [26, 2], [32, 'lower', 2]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3]), ('normal control', [-3, 2], None), ('normal control', [-4, 2], None)], [('regression: cadence group size', [2, 3], [2, 'lower', 3]), ('regression variant: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair probe: cadence group size', [9, 1], [11, 'lower', 2]), ('partial repair variant: cadence group size', [36, 1], [45, 'upper', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('normal control', [-2, 3], None), ('normal control', [-6, 1], None), ('normal control', [-3, 0], None)], [('regression: cadence group size', [8, 0], [10, 'upper', 2]), ('regression variant: cadence group size', [10, 1], [12, 'lower', 3]), ('partial repair probe: cadence group size', [35, 1], [44, 'upper', 2]), ('partial repair variant: cadence group size', [38, 3], [47, 'lower', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('normal control', [-1, 1], None), ('normal control', [-3, 3], None), ('normal control', [-5, 3], None)], [('regression: cadence group size', [9, 1], [11, 'lower', 2]), ('regression variant: cadence group size', [22, 2], [27, 'lower', 2]), ('partial repair probe: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair variant: cadence group size', [1, 3], [1, 'lower', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('normal control', [-1, 0], None), ('normal control', [1, 2], [1, 'upper', 3]), ('normal control', [2, 1], [2, 'lower', 3])], [('regression: cadence group size', [35, 1], [44, 'upper', 2]), ('regression variant: cadence group size', [26, 2], [32, 'lower', 2]), ('partial repair probe: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair variant: cadence group size', [3, 2], [3, 'lower', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [-1, 2], None), ('normal control', [0, 3], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3])]]
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: cadence group size | [2, 'lower', 2] | [5, 'upper', 2] | Failed |
| regression variant: cadence group size | [20, 'upper', 2] | [42, 'lower', 2] | Failed |
| partial repair probe: cadence group size | [5, 'upper', 2] | [10, 'upper', 2] | Failed |
| partial repair variant: cadence group size | [15, 'upper', 2] | [32, 'lower', 2] | Failed |
| boundary control | [0, 'upper', 2] | [0, 'upper', 2] | Passed |
| boundary control | [0, 'upper', 3] | [0, 'upper', 3] | Passed |
| normal control | [1, 'upper', 3] | [1, 'upper', 3] | Passed |
| normal control | None | None | Passed |
| normal control | None | None | Passed |
SHA-256 / d2c80f710cf4b181df5b57f391b0c586cf683fd85c9b582aeaddf5f6786e27ca
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(n, phase):
if n<0:
return None
def sf(a):
return 8*(a//4)+[0,2,5,7][a%4]
a=n+phase
start=sf(a)-sf(phase)
count=[2,3,2,3][a%4]
return [start//2,'upper' if start%2==0 else 'lower',count]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: cadence group size', [4, 0], [5, 'upper', 2]), ('regression variant: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair probe: cadence group size', [8, 0], [10, 'upper', 2]), ('partial repair variant: cadence group size', [26, 2], [32, 'lower', 2]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3]), ('normal control', [-3, 2], None), ('normal control', [-4, 2], None)], [('regression: cadence group size', [2, 3], [2, 'lower', 3]), ('regression variant: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair probe: cadence group size', [9, 1], [11, 'lower', 2]), ('partial repair variant: cadence group size', [36, 1], [45, 'upper', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('normal control', [-2, 3], None), ('normal control', [-6, 1], None), ('normal control', [-3, 0], None)], [('regression: cadence group size', [8, 0], [10, 'upper', 2]), ('regression variant: cadence group size', [10, 1], [12, 'lower', 3]), ('partial repair probe: cadence group size', [35, 1], [44, 'upper', 2]), ('partial repair variant: cadence group size', [38, 3], [47, 'lower', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('normal control', [-1, 1], None), ('normal control', [-3, 3], None), ('normal control', [-5, 3], None)], [('regression: cadence group size', [9, 1], [11, 'lower', 2]), ('regression variant: cadence group size', [22, 2], [27, 'lower', 2]), ('partial repair probe: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair variant: cadence group size', [1, 3], [1, 'lower', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('normal control', [-1, 0], None), ('normal control', [1, 2], [1, 'upper', 3]), ('normal control', [2, 1], [2, 'lower', 3])], [('regression: cadence group size', [35, 1], [44, 'upper', 2]), ('regression variant: cadence group size', [26, 2], [32, 'lower', 2]), ('partial repair probe: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair variant: cadence group size', [3, 2], [3, 'lower', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [-1, 2], None), ('normal control', [0, 3], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3])]]
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: cadence group size | [4, 'upper', 2] | [5, 'upper', 2] | Failed |
| regression variant: cadence group size | [33, 'lower', 2] | [42, 'lower', 2] | Failed |
| partial repair probe: cadence group size | [8, 'upper', 2] | [10, 'upper', 2] | Failed |
| partial repair variant: cadence group size | [25, 'lower', 2] | [32, 'lower', 2] | Failed |
| boundary control | [0, 'upper', 2] | [0, 'upper', 2] | Passed |
| boundary control | [0, 'upper', 3] | [0, 'upper', 3] | Passed |
| normal control | [1, 'upper', 3] | [1, 'upper', 3] | Passed |
| normal control | None | None | Passed |
| normal control | None | None | Passed |
SHA-256 / 368e5f6dc9e52d291805c6e4ba7182567024e034710927618bcd585275dd0fb9
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(n, phase):
if n<0:
return None
def sf(a):
return 10*(a//4)+[0,2,5,7][a%4]
a=n+phase
start=sf(a)-sf(phase)
count=[2,3,2,3][a%4]
return [start//2,'upper' if start%2==0 else 'lower',count]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression: cadence group size', [4, 0], [5, 'upper', 2]), ('regression variant: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair probe: cadence group size', [8, 0], [10, 'upper', 2]), ('partial repair variant: cadence group size', [26, 2], [32, 'lower', 2]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3]), ('normal control', [-3, 2], None), ('normal control', [-4, 2], None)], [('regression: cadence group size', [2, 3], [2, 'lower', 3]), ('regression variant: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair probe: cadence group size', [9, 1], [11, 'lower', 2]), ('partial repair variant: cadence group size', [36, 1], [45, 'upper', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('normal control', [-2, 3], None), ('normal control', [-6, 1], None), ('normal control', [-3, 0], None)], [('regression: cadence group size', [8, 0], [10, 'upper', 2]), ('regression variant: cadence group size', [10, 1], [12, 'lower', 3]), ('partial repair probe: cadence group size', [35, 1], [44, 'upper', 2]), ('partial repair variant: cadence group size', [38, 3], [47, 'lower', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('boundary control', [1, 0], [1, 'upper', 3]), ('normal control', [-1, 1], None), ('normal control', [-3, 3], None), ('normal control', [-5, 3], None)], [('regression: cadence group size', [9, 1], [11, 'lower', 2]), ('regression variant: cadence group size', [22, 2], [27, 'lower', 2]), ('partial repair probe: cadence group size', [34, 2], [42, 'lower', 2]), ('partial repair variant: cadence group size', [1, 3], [1, 'lower', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('boundary control', [3, 0], [3, 'lower', 3]), ('normal control', [-1, 0], None), ('normal control', [1, 2], [1, 'upper', 3]), ('normal control', [2, 1], [2, 'lower', 3])], [('regression: cadence group size', [35, 1], [44, 'upper', 2]), ('regression variant: cadence group size', [26, 2], [32, 'lower', 2]), ('partial repair probe: cadence group size', [9, 3], [11, 'lower', 2]), ('partial repair variant: cadence group size', [3, 2], [3, 'lower', 3]), ('boundary control', [0, 0], [0, 'upper', 2]), ('boundary control', [0, 1], [0, 'upper', 3]), ('normal control', [-1, 2], None), ('normal control', [0, 3], [0, 'upper', 3]), ('normal control', [1, 0], [1, 'upper', 3])]]
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: cadence group size | [5, 'upper', 2] | [5, 'upper', 2] | Passed |
| regression variant: cadence group size | [42, 'lower', 2] | [42, 'lower', 2] | Passed |
| partial repair probe: cadence group size | [10, 'upper', 2] | [10, 'upper', 2] | Passed |
| partial repair variant: cadence group size | [32, 'lower', 2] | [32, 'lower', 2] | Passed |
| boundary control | [0, 'upper', 2] | [0, 'upper', 2] | Passed |
| boundary control | [0, 'upper', 3] | [0, 'upper', 3] | Passed |
| normal control | [1, 'upper', 3] | [1, 'upper', 3] | Passed |
| normal control | None | None | Passed |
| normal control | None | None | Passed |
SHA-256 / 17ed48c6e615f2d76b231734f18dac6fa3bbbf867552650cde42174d0f6a98ab
Verification & scope
A deterministic bounded teaching model with a stipulated toy contract; it does not claim conformance to SMPTE or any broadcast standard. 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 2026-09-29T14:49:36.026243+00:00.
Case digest / a32b966c967941987b3d08bdb29b2ba8be07df55f1d938fa925c42cb0ab37c0c