FA-99396 / Embedding vector normalization / Open access

Epsilon normalization adds epsilon to every coordinate divisor · case 01

Epsilon normalization adds epsilon to every coordinate divisor.

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

ROOT CAUSE

The stabilizer is added to each coordinate magnitude instead of to the vector norm.

THE FAILURE

The stabilizer is added to each coordinate magnitude instead of to the vector norm.

Unsuccessful approach: Adding epsilon to the norm changes every ordinary vector even when no stabilization is needed.

Case contract

Divide a vector by max(L2 norm, positive epsilon); zero and empty vectors return all-zero shape.

Why this case matters

A small embedding preparation contract can be checked without a model or retrieval system.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(v, epsilon):
    return [x/(abs(x)+epsilon) for x in v]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('ordinary norm', solve([3,4],0.1), [0.6,0.8])
check('small vector', solve([0.03,0.04],0.1), [0.3,0.39999999999999997])
check('zero shape', solve([0,0],0.1), [0,0])
check('empty', solve([],0.1), [])
check('single above epsilon', solve([N],0.1), [1.0])
check('negative coordinate', solve([-3,4],0.1), [-0.6,0.8])
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
ordinary norm	`[0.9677419354838709, 0.9756097560975611]`	`[0.6, 0.8]`	Failed
small vector	`[0.23076923076923075, 0.2857142857142857]`	`[0.3, 0.39999999999999997]`	Failed
zero shape	`[0.0, 0.0]`	`[0, 0]`	Passed
empty	`[]`	`[]`	Passed
single above epsilon	`[0.9090909090909091]`	`[1.0]`	Failed
negative coordinate	`[-0.9677419354838709, 0.9756097560975611]`	`[-0.6, 0.8]`	Failed

SHA-256 / b62995c073064e8e5d80dd61517d77d02cc225743a4f395edaf7e1303bd94226

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(v, epsilon):
    length=sum(x*x for x in v)**0.5
    return [x/(length+epsilon) for x in v]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('ordinary norm', solve([3,4],0.1), [0.6,0.8])
check('small vector', solve([0.03,0.04],0.1), [0.3,0.39999999999999997])
check('zero shape', solve([0,0],0.1), [0,0])
check('empty', solve([],0.1), [])
check('single above epsilon', solve([N],0.1), [1.0])
check('negative coordinate', solve([-3,4],0.1), [-0.6,0.8])
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
ordinary norm	`[0.5882352941176471, 0.7843137254901962]`	`[0.6, 0.8]`	Failed
small vector	`[0.19999999999999996, 0.2666666666666666]`	`[0.3, 0.39999999999999997]`	Failed
zero shape	`[0.0, 0.0]`	`[0, 0]`	Passed
empty	`[]`	`[]`	Passed
single above epsilon	`[0.9090909090909091]`	`[1.0]`	Failed
negative coordinate	`[-0.5882352941176471, 0.7843137254901962]`	`[-0.6, 0.8]`	Failed

SHA-256 / f9d8ff1835cc761d3ffdb6cb4d6aec5a7f1c416383d7647f828b6078cce493f8

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

Verification & scope

Inputs are finite toy vectors or token rows; this contract makes no claim about model quality or production embedding behavior. 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 2026-09-30T16:48:23.554898+00:00.

Case digest / 19143646de68a8fbc14b01ca0444ae5fdc8bbfb4104a9552f0b37dd8a0edbc47