FA-99111 / Learning rate schedules / Open access

Piecewise interpolation is anchored at the upper knot · case 01

Piecewise interpolation is anchored at the upper knot.

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

ROOT CAUSE

Interpolation starts from the upper knot and then moves farther in the segment direction.

THE FAILURE

Interpolation starts from the upper knot and then moves farther in the segment direction.

Unsuccessful approach: Returning a segment endpoint without interpolation creates a staircase.

Case contract

Given sorted [step,rate] knots, linearly interpolate within each segment and clamp to endpoint rates.

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, knots):
    if step<=knots[0][0]: return knots[0][1]
    for (x0,y0),(x1,y1) in zip(knots,knots[1:]):
     if step<=x1: return y1+(y1-y0)*(step-x0)/(x1-x0)
    return knots[-1][1]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
k=[[0,0],[N,10],[N+2,2]]; check('first segment',solve(1,k),10/N); check('first knot',solve(N,k),10); check('between',solve(N+1,k),6); check('last knot',solve(N+2,k),2); check('before',solve(-1,k),0); check('after',solve(N+5,k),2)
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
first segment	`20.0`	`10.0`	Failed
first knot	`20.0`	`10`	Failed
between	`-2.0`	`6`	Failed
last knot	`-6.0`	`2`	Failed
before	`0`	`0`	Passed
after	`2`	`2`	Passed

SHA-256 / f586483b631bec0b5c113f17b785c6373497de688f96e9d5a8c8bc910762b0e3

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(step, knots):
    if step<=knots[0][0]: return knots[0][1]
    for (x0,y0),(x1,y1) in zip(knots,knots[1:]):
     if step<=x1: return y1
    return knots[-1][1]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
k=[[0,0],[N,10],[N+2,2]]; check('first segment',solve(1,k),10/N); check('first knot',solve(N,k),10); check('between',solve(N+1,k),6); check('last knot',solve(N+2,k),2); check('before',solve(-1,k),0); check('after',solve(N+5,k),2)
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
first segment	`10`	`10.0`	Passed
first knot	`10`	`10`	Passed
between	`2`	`6`	Failed
last knot	`2`	`2`	Passed
before	`0`	`0`	Passed
after	`2`	`2`	Passed

SHA-256 / 8b063e0bf8508714c8f666e33ee77e2c977acec34357662ee18c31aad3aa166e

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 2026-09-30T16:48:20.352541+00:00.

Case digest / f9ac3a6e918c10aaa6321bcc6d704e2d6fc73316b4f158eae66b02ff7a672f6d