Research Coda — What Alignment Really Preserves

The result that looks like success

Suppose a bridge translates vectors from space A into space B. On a held-out set, the true B-space counterpart appears in the translated vector’s top ten 99.7% of the time.

That sounds finished.

Now ask a different question: does the translated vector have the same ten neighbors as the native B-space vector?

The answer can be barely 59%.

same bridge, same held-out items

true counterpart in top 10                 0.997
native-vs-translated neighborhood agreement 0.589
mean cosine to true target                  0.819
local order preservation                    0.64

The bridge almost always reaches the correct region of the target space. It does not recreate the target space’s local geometry.

Recovering the right point is not the same thing as preserving the relationships around that point.

This chapter is about that distinction.

Two contracts hidden inside one word

When someone says an embedding translation “works,” they can mean at least two different things.

Contract 1 — counterpart recovery

For an object x, translate its source vector and ask whether the native target vector for the same object is nearby:

x
│
├─ E_A(x) ── T ──> translated point in B
│
└─ E_B(x) ───────> native target point in B

Metrics:

cos(T(E_A(x)), E_B(x))
Recall@1 of the true target
Recall@10 of the true target
MRR of the true target

This asks: did the bridge put this object near where target encoder B would put the same object?

Contract 2 — structural fidelity

Now ignore the paired counterpart and inspect the target neighborhood:

N_B(x)       = native B-space neighbors of E_B(x)
N_T(x)       = B-space neighbors around T(E_A(x))

Metrics:

agreement@k       = |N_B(x) ∩ N_T(x)| / k
rank correlation  = do shared neighbors appear in the same order?
cluster agreement = do translated points preserve B's grouping?
order preservation = do pairwise preferences survive?

This asks: did the bridge reproduce the relationships that make B-space useful?

Those are different contracts.

A tiny example

Imagine native target space B ranks five items around a query like this:

native B
1. A
2. B
3. C
4. D
5. E

A translated query returns:

translated A→B
1. A
2. D
3. E
4. B
5. F

The exact paired item A is still top-1. If the task is merely “find A,” translation is perfect.

But only three of the five neighbors survive, and their order changed substantially. A recommender, clustering system, semantic cache, deduplicator, or ranked retrieval consumer is now operating on a different local structure.

Nothing about top-1 recovery exposes that.

Why the two metrics separate

Nearest-neighbor retrieval is a local membership test. Geometry preservation is a relational constraint over many pairs.

A map can move each translated point close enough to its own target that the target remains easy to find while still changing dozens of near-ties around it.

This is especially easy in a crowded high-dimensional space:

true target cosine       0.82
neighbor #2 cosine       0.81
neighbor #3 cosine       0.805
neighbor #4 cosine       0.803

A tiny directional error may leave the true target in the top ten while reordering half the neighborhood.

So the implication only goes one way:

good neighborhood preservation
        usually implies
useful counterpart recovery

but

useful counterpart recovery
        does NOT imply
good neighborhood preservation

Demonstration: a bridge that retrieves almost perfectly and preserves half the neighborhood

MEASURED on a second cross-space benchmark: 5,264 real sentences, deterministic 4,199/1,065 train/test split, HNet/mxbai-family source space and a Qwen target space. The same frozen held-out set is used for every map.

A paired linear bridge gives:

HNet → Qwen, paired linear

mean cosine to paired target      0.819
Recall@10 of paired target        0.997
MRR                               0.980
agreement@10                      0.589
order preservation                0.64
cluster preservation              0.24

Read the profile carefully.

• Recall@10 = 0.997: the bridge almost never loses the correct target point.
• MRR = 0.980: the correct counterpart is usually extremely high in the list.
• agreement@10 = 0.589: about four of ten native target neighbors are replaced.
• cluster preservation = 0.24: the larger-scale grouping is much less faithful than counterpart recovery suggests.

The bridge is excellent at identity recovery and mediocre at geometry recovery.

The positive control

A preservation metric is only useful if it can recognize genuine preservation. A near-same-space control — HNet single-chunk output versus its underlying mxbai base geometry — gives:

HNet → mxbai control

mean cosine             0.995
Recall@10               1.000
agreement@10            0.988
order preservation      1.00
cluster preservation    0.95

The harness is not simply pessimistic. When two representations really do preserve the local structure, the metrics report it.

More paired data improved the point and not the neighborhood

This distinction becomes stronger when the training set grows.

HNet → Qwen linear bridge

train anchors        cosine       agreement@10
~1,600               0.743            0.589
~4,200               0.819            0.589

More data moved the translated vectors substantially closer to their paired targets. It did nothing to the neighborhood-overlap score.

That is evidence that the residual is not merely “we need more anchor pairs.” At least for this map family and corpus, point reconstruction continues improving after local-geometry fidelity has plateaued.

The optimization target and the property we care about have separated.

Direction matters

Reverse the bridge and the profile changes:

Qwen → HNet

Recall@10       0.999
MRR             0.991
agreement@10    0.515
order           0.59

The reverse direction recovers the paired target even better while reproducing the native neighborhood worse.

A bridge is therefore not just tied to (space A, space B). Its preservation profile is tied to a direction.

A → B preservation  ≠  B → A preservation

Even when the same texts and same two encoders are involved.

A useful two-axis model

Instead of one headline “alignment score,” place a bridge on two axes:

                         STRUCTURAL FIDELITY
                         low              high
POINT          high      migration-only   strong bridge
RECOVERY                 / lookup use     / geometry use

               low       failed map       unusual: structure
                                          without identity

The upper-left cell is important. It contains bridges that are useful, but only for certain jobs.

A bridge with high point recovery and middling structural fidelity may be perfectly adequate for:

• locating the corresponding legacy record;
• candidate generation followed by native re-ranking;
• migration assistance where the paired item itself is the target;
• coarse retrieval where order among near-ties is unimportant.

The same bridge may be unsuitable for:

• clustering;
• recommendation;
• fine-grained semantic ranking;
• neighbor-sensitive deduplication;
• any consumer that treats local order as meaningful.

This is the usable_for idea from Chapter 20 with a sharper diagnostic: first ask whether the consumer needs point recovery or target-space structure.

Retrieval quality is itself plural

The word “retrieval” can hide the same mistake.

Consider two questions:

A. Did I retrieve the one corresponding target item?
B. Did I retrieve the same ranked evidence set the native target model would retrieve?

Both are retrieval questions. They are not the same evaluation.

For migration, A may be enough. For RAG context selection, B may matter more. For a ranked UI, B plus order preservation matters. For clustering, neither is sufficient; cluster preservation is the relevant metric.

So even “retrieval survived” must name the retrieval contract.

What this changes about bridge evaluation

Chapter 21 introduced a preservation profile. The new discipline is to split its first decision explicitly:

bridge_fidelity:
  counterpart_recovery:
    recall@1
    recall@10
    mrr
    cosine_to_paired_target

  structural_fidelity:
    agreement@10
    agreement@50
    spearman@10
    spearman@50
    order_preservation
    cluster_preservation

  task_fidelity:
    native-vs-bridge ndcg@10
    calibration transfer
    hard-negative / relation preservation

A bridge is not allowed to summarize these three groups into one scalar unless a specific downstream policy defines and justifies that scalar.

Failure modes

• Calling Recall@10 “geometry preservation.” It certifies that the correct counterpart is nearby, not that the neighborhood was reproduced.
• Calling cosine-to-target “semantic equivalence.” Small coordinate error can still reorder dense near-ties.
• Increasing anchor count after agreement has plateaued. More paired data can improve reconstruction without improving the structural property you need.
• Evaluating only one direction. A→B and B→A can have materially different preservation profiles.
• Using a lookup-quality bridge for a neighborhood-sensitive consumer. Put the consumer on the correct axis first.

What this chapter establishes and what it does not

Establishes: counterpart recovery and structural fidelity are separate properties; near-perfect paired-target Recall@10 can coexist with only moderate neighborhood agreement; more anchors can improve cosine while local-geometry fidelity stays flat; direction changes the profile; bridge evaluation should report recovery, structure, and task fidelity separately.

Does not establish: that a bridge with moderate structural fidelity is useless — it can be excellent for lookup and migration; that linear maps are uniquely limited — the distinction applies to every transport method; or that the target space is itself “correct.” The chapter establishes a measurement distinction, not a preferred encoder.

Lab 25: make recovery and geometry disagree on purpose

PROPOSED, not executed.

Setup. Create 200 synthetic target vectors in clustered 32-dimensional space. Copy them to a source space through a known transform. Add a small anisotropic distortion to a learned source→target map.

Task.

1. Measure paired-target Recall@1/10.
2. Measure native-vs-translated agreement@10 and Spearman@10.
3. Increase the distortion until Recall@10 stays above 0.95 while agreement@10 falls below 0.7.
4. Inspect one item where the paired target remains top-1 but at least four native neighbors change.

Success criterion. A concrete counterexample to the statement “the bridge retrieves the right point, therefore it preserved the space.”

Companion component: the fidelity split

bridge_fidelity_report:
  source_space_hash: <...>
  target_space_hash: <...>
  direction:         A_to_B

  counterpart_recovery:
    recall@1: ...
    recall@10: ...
    mrr: ...
    cosine_to_target: ...

  structural_fidelity:
    agreement@10: ...
    agreement@50: ...
    spearman@10: ...
    spearman@50: ...
    order_preservation: ...
    cluster_preservation: ...

  task_fidelity:
    ndcg@10_ratio: ...
    calibration_transfer: ...
    relation_preservation: {...}

  consumer_requires: <point_recovery | neighborhood | ranking | clustering | threshold>
  verdict: PASS | FAIL

What this chapter established

• Counterpart recovery is not geometry preservation.
• A bridge can put the true target in the top ten almost every time while replacing roughly four of ten native neighbors.
• More paired data can improve cosine-to-target without improving neighborhood agreement, revealing an objective/property mismatch rather than a simple sample-size problem.
• Direction is part of the preservation claim.
• Bridge evaluation should separate counterpart recovery, structural fidelity, and task fidelity before assigning usable_for.

Next

If the translated neighborhood differs from the target neighborhood, a deeper question follows. Should the bridge preserve the source geometry, reproduce the target geometry, or optimize only the downstream task? Those objectives sound similar until they conflict. The next chapter shows that they can conflict sharply — and that “preserve the geometry” is incomplete unless you say whose geometry and for what purpose.