FA-99411 / Embedding vector normalization / Open access

Feature standardization divides by variance · case 01

Feature standardization divides by variance.

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

ROOT CAUSE

The feature residual is divided by variance rather than standard deviation.

THE FAILURE

The feature residual is divided by variance rather than standard deviation.

Unsuccessful approach: Using sample variance changes the scale contract for small batches.

Case contract

Standardize each column with population mean and standard deviation; zero-variance columns become zeros.

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(rows):
    return [[(x-sum(r[j] for r in rows)/len(rows))/max(sum((r[j]-sum(q[j] for q in rows)/len(rows))**2 for r in rows)/len(rows),1) for j,x in enumerate(row)] for row in rows] if rows else []
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('population scale', solve([[0],[2]]), [[-1.0],[1.0]])
check('offset feature', solve([[N,10],[N+2,14]]), [[-1.0,-1.0],[1.0,1.0]])
check('constant column', solve([[1,5],[3,5]]), [[-1.0,0],[1.0,0]])
check('singleton', solve([[N,N]]), [[0,0]])
check('empty', solve([]), [])
check('three points', solve([[0],[3],[6]]), [[-3/(6**0.5)],[0],[3/(6**0.5)]])
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
population scale	`[[-1.0], [1.0]]`	`[[-1.0], [1.0]]`	Passed
offset feature	`[[-1.0, -0.5], [1.0, 0.5]]`	`[[-1.0, -1.0], [1.0, 1.0]]`	Failed
constant column	`[[-1.0, 0.0], [1.0, 0.0]]`	`[[-1.0, 0], [1.0, 0]]`	Passed
singleton	`[[0.0, 0.0]]`	`[[0, 0]]`	Passed
empty	`[]`	`[]`	Passed
three points	`[[-0.5], [0.0], [0.5]]`	`[[-1.2247448713915892], [0], [1.2247448713915892]]`	Failed

SHA-256 / b682a43e8415ddbdbdc396267924a643481be7f182f3d553779cf4ece06c76d9

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(rows):
    return [[(x-sum(r[j] for r in rows)/len(rows))/(max((sum((r[j]-sum(q[j] for q in rows)/len(rows))**2 for r in rows)/max(len(rows)-1,1))**0.5,1)) for j,x in enumerate(row)] for row in rows] if rows else []
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('population scale', solve([[0],[2]]), [[-1.0],[1.0]])
check('offset feature', solve([[N,10],[N+2,14]]), [[-1.0,-1.0],[1.0,1.0]])
check('constant column', solve([[1,5],[3,5]]), [[-1.0,0],[1.0,0]])
check('singleton', solve([[N,N]]), [[0,0]])
check('empty', solve([]), [])
check('three points', solve([[0],[3],[6]]), [[-3/(6**0.5)],[0],[3/(6**0.5)]])
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
population scale	`[[-0.7071067811865475], [0.7071067811865475]]`	`[[-1.0], [1.0]]`	Failed
offset feature	`[[-0.7071067811865475, -0.7071067811865475], [0.7071067811865475, 0.7071067811865475]]`	`[[-1.0, -1.0], [1.0, 1.0]]`	Failed
constant column	`[[-0.7071067811865475, 0.0], [0.7071067811865475, 0.0]]`	`[[-1.0, 0], [1.0, 0]]`	Failed
singleton	`[[0.0, 0.0]]`	`[[0, 0]]`	Passed
empty	`[]`	`[]`	Passed
three points	`[[-1.0], [0.0], [1.0]]`	`[[-1.2247448713915892], [0], [1.2247448713915892]]`	Failed

SHA-256 / 012537b43f3e2f95ba9b97d190a45de68bf0489177319ed17887e01bd4fa9bbe

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

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 2026-09-30T16:48:24.023910+00:00.

Case digest / c061b6b12edd61b311774e7daf25e17fe8c4b48c0a19204d9317ccb37ae70a14