FA-99416 / Embedding vector normalization / Open access

Weighted centering subtracts an unweighted mean · case 01

Weighted centering subtracts an unweighted mean.

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

ROOT CAUSE

The center ignores token or sample weights even though the downstream operation is weighted.

THE FAILURE

The center ignores token or sample weights even though the downstream operation is weighted.

Unsuccessful approach: Using only the first row as the center leaves the weighted centroid offset.

Case contract

Return vectors centered by their weighted coordinate mean; zero total weight returns copies unchanged.

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(vectors, weights):
    m=[sum(v[j] for v in vectors)/len(vectors) for j in range(len(vectors[0]))] if vectors else []
    return [[x-m[j] for j,x in enumerate(v)] for v in vectors]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('unequal weights', solve([[0,N],[2*N,3*N]],[3,1]), [[-N/2,-N/2],[3*N/2,3*N/2]])
check('equal weights', solve([[0],[2*N]],[1,1]), [[-N],[N]])
check('zero total', solve([[N],[2*N]],[0,0]), [[N],[2*N]])
check('empty', solve([],[]), [])
check('single vector', solve([[N,N]],[N]), [[0,0]])
check('ignored row weight', solve([[N],[0]],[0,1]), [[N],[0]])
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
unequal weights	`[[-1.0, -1.0], [1.0, 1.0]]`	`[[-0.5, -0.5], [1.5, 1.5]]`	Failed
equal weights	`[[-1.0], [1.0]]`	`[[-1], [1]]`	Passed
zero total	`[[-0.5], [0.5]]`	`[[1], [2]]`	Failed
empty	`[]`	`[]`	Passed
single vector	`[[0.0, 0.0]]`	`[[0, 0]]`	Passed
ignored row weight	`[[0.5], [-0.5]]`	`[[1], [0]]`	Failed

SHA-256 / 9e74cfd86ceb8948888947ba9e08af9a5697596f009692900b7e397553d9b27e

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(vectors, weights):
    m=[vectors[0][j] for j in range(len(vectors[0]))] if vectors else []
    return [[x-m[j] for j,x in enumerate(v)] for v in vectors]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('unequal weights', solve([[0,N],[2*N,3*N]],[3,1]), [[-N/2,-N/2],[3*N/2,3*N/2]])
check('equal weights', solve([[0],[2*N]],[1,1]), [[-N],[N]])
check('zero total', solve([[N],[2*N]],[0,0]), [[N],[2*N]])
check('empty', solve([],[]), [])
check('single vector', solve([[N,N]],[N]), [[0,0]])
check('ignored row weight', solve([[N],[0]],[0,1]), [[N],[0]])
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
unequal weights	`[[0, 0], [2, 2]]`	`[[-0.5, -0.5], [1.5, 1.5]]`	Failed
equal weights	`[[0], [2]]`	`[[-1], [1]]`	Failed
zero total	`[[0], [1]]`	`[[1], [2]]`	Failed
empty	`[]`	`[]`	Passed
single vector	`[[0, 0]]`	`[[0, 0]]`	Passed
ignored row weight	`[[0], [-1]]`	`[[1], [0]]`	Failed

SHA-256 / 43215510a39edc360bc0dede86ee5239dbdf47ee60a0a567bc72b467f287b29b

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 2026-09-30T16:48:24.023786+00:00.

Case digest / d9cd56203412bff332b44370c8c39cb112b9e639dbf12d7a5f5adbab802f6f7d