FA-99136 / Learning rate schedules / Open access

Cosine decay ends at half the configured floor · case 01

Cosine decay ends at half the configured floor.

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

ROOT CAUSE

Subtracting half the floor after interpolation lowers the terminal rate below its target.

THE FAILURE

Subtracting half the floor after interpolation lowers the terminal rate below its target.

Unsuccessful approach: Removing amplitude normalization doubles the changing portion and can exceed the base rate.

Case contract

Return floor+(base-floor)*(1+cos(pi*progress))/2 for clamped progress.

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, floor, total):
    import math
    progress=min(1,max(0,step/total))
    return floor+(base-floor)*(1+math.cos(math.pi*progress))/2-floor/2
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('start',solve(0,10,2,4),10); check('quarter',solve(1,10,2,4),2+8*(1+2**-0.5)/2); check('mid',solve(2,10,2,4),6); check('end',solve(4,10,2,4),2); check('after',solve(7,10,2,4),2); check('flat',solve(2,5,5,4),5)
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
start	`9.0`	`10`	Failed
quarter	`7.82842712474619`	`8.82842712474619`	Failed
mid	`5.0`	`6`	Failed
end	`1.0`	`2`	Failed
after	`1.0`	`2`	Failed
flat	`2.5`	`5`	Failed

SHA-256 / 301dc265ff2b37834e4b434e2c3b048320a846127b7a595271f431e96d7434b7

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(step, base, floor, total):
    import math
    progress=min(1,max(0,step/total))
    return floor+(base-floor)*(1+math.cos(math.pi*progress))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('start',solve(0,10,2,4),10); check('quarter',solve(1,10,2,4),2+8*(1+2**-0.5)/2); check('mid',solve(2,10,2,4),6); check('end',solve(4,10,2,4),2); check('after',solve(7,10,2,4),2); check('flat',solve(2,5,5,4),5)
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
start	`18.0`	`10`	Failed
quarter	`15.65685424949238`	`8.82842712474619`	Failed
mid	`10.0`	`6`	Failed
end	`2.0`	`2`	Passed
after	`2.0`	`2`	Passed
flat	`5.0`	`5`	Passed

SHA-256 / 626250e73d73ab7126cd3af0b3be9fc269ff2528548a962b742863ca68838dc6

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 2026-09-30T16:48:20.799100+00:00.

Case digest / 5fa18bc475082af9576ed0080254888c280b7ad90cbf1dd4123309f9c4d2fc17