FA-99051 / Learning rate schedules / Open access

Cosine decay remains above zero at the horizon · case 01

Cosine decay remains above zero at the horizon.

Verified by executionVariant 1 · 6 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Dividing normalized progress by two runs only a quarter of the cosine cycle.

THE FAILURE

Dividing normalized progress by two runs only a quarter of the cosine cycle.

Unsuccessful approach: Removing amplitude normalization raises the initial rate to twice the base.

Case contract

For 0<=step<=total return base*(1+cos(pi*step/total))/2; clamp outside.

Why this case matters

A deterministic scalar schedule makes step-to-rate mapping explicit for a small training loop.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(step, base, total):
    import math
    return base*(1+math.cos(math.pi*step/(2*total)))/2 if 0<=step<=total else (base if step<0 else 0)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('quarter',solve(1,8,4),8*(1+2**-0.5)/2); check('start',solve(0,6,3),6); check('middle',solve(2,10,4),5); check('end',solve(4,10,4),0); check('after',solve(5,9,4),0); check('negative',solve(-1,9,4),9)
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
quarter	`7.695518130045147`	`6.82842712474619`	Failed
start	`6.0`	`6`	Passed
middle	`8.535533905932738`	`5`	Failed
end	`5.0`	`0`	Failed
after	`0`	`0`	Passed
negative	`9`	`9`	Passed

SHA-256 / a539504f60fc1352257016703241120f7e00d379a9863a969e740c1ed831d7f4

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(step, base, total):
    import math
    return base*(1+math.cos(math.pi*step/total)) if 0<=step<=total else (base if step<0 else 0)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('quarter',solve(1,8,4),8*(1+2**-0.5)/2); check('start',solve(0,6,3),6); check('middle',solve(2,10,4),5); check('end',solve(4,10,4),0); check('after',solve(5,9,4),0); check('negative',solve(-1,9,4),9)
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
quarter	`13.65685424949238`	`6.82842712474619`	Failed
start	`12.0`	`6`	Failed
middle	`10.0`	`5`	Failed
end	`0.0`	`0`	Passed
after	`0`	`0`	Passed
negative	`9`	`9`	Passed

SHA-256 / 6e28dfd01f622e7c0e808ad70f9bcd7deb88dadbed981a4e89680b419ab1320c

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 6 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

These are toy scalar contracts with integer steps and finite parameters; they do not model framework defaults or optimizer state. 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 2026-09-30T16:48:19.909886+00:00.

Case digest / f497d2eb3428e43dc820780026297f2fbde184f900add5f994b6f7d7178dcab2