# The gate was measuring the wrong thing

*2026-08-01. Supersedes the correlation threshold used as the project's
go/no-go statistic since the synthetic world results.*

## The correlation does not govern recovery

The standing rule was that the field correlation at the true pairing has to
reach about 0.9, that nothing works below it, and that any new representation
can therefore be priced in minutes without running a search. Natural data sat
at 0.656 and no recovery run was made on it, on the strength of that rule.

**A controlled truncation refutes the rule.** Take the synthetic fields that
recover at 95.3%, project both to rank *r*, and match blind from a spectral
start with exact refinement. The correlation barely moves across the ladder;
recovery moves across its entire range. (What replaces the correlation is
*not* the shared-direction count in the next column — see the correction that
follows this section. The column is reported because it is what motivated the
hypothesis, not because it survived it.)

| Field | ρ at truth | Shared directions | Recovery (256 scenes, chance 0.4%) |
|---|---|---|---|
| synthetic, rank 4 | 0.902 | 11 | 6.2% |
| synthetic, rank 8 | 0.906 | 13 | 12.9% |
| synthetic, rank 16 | 0.928 | 18 | **95.6%** |
| synthetic, rank 32 | 0.928 | 27 | 93.8% |
| synthetic, rank 64 | 0.928 | 27 | 95.8% |
| synthetic, rank 128 | 0.929 | 27 | 96.1% |
| synthetic, full | 0.929 | 27 | 95.8% |

A rank-8 field correlating at 0.906 — comfortably past the supposed threshold
— recovers 13%. The same field at rank 16 recovers 96%. Correlation is held
almost fixed while recovery crosses from failure to success, so **something
other than the strength of the agreement is doing the work.** Identifying that
something is a separate problem, and the next section records two attempts at
it that both failed.

The theory says the same thing in advance and we misread it. Correlated-matching
thresholds are derived for exchangeable full-rank noise, where each of the
N²/2 entries carries an independent constraint on the pairing. A rank-*r*
shared component supplies about *rN*. At 256 scenes with *r* near ten that is
an eighth of what the threshold calculation assumes, so quoting the 0.9 figure
against a low-rank field compares a number to a bound derived for a different
object.

The correction cuts against us as well as for us. With the old gate retired,
the reason for never having run a search on natural data went with it, so the
search was run: five blind trials on the best natural field, spectral start
chosen by energy over five candidates, exact refinement to convergence.
**Recovery is 0.0000 against a chance rate of 0.0039** — the spectral start
lands exactly on chance and refinement moves away from the truth, which is what
descent does when the truth is not the optimum. Natural data does not recover,
and now that is a measurement rather than an inference from a statistic that
does not govern it.

## Correction, same day: width is not the statistic either

The section below proposed the width of the shared spectrum as the replacement
gate. **A control run later the same session refutes it**, and the refutation
is recorded here rather than folded away because it was my own claim and it
lasted four hours.

Adding independent noise to both synthetic fields lowers the correlation
without narrowing the shared signal underneath, which is still full-rank. That
control was run to guard against overcorrecting, and it did more than that:

| Field | ρ | shared width | shared dimension (held out) | recovery |
|---|---|---|---|---|
| synthetic, noise 0.35 | 0.828 | 15.7 | 16.8 | **94.7%** |
| synthetic, caption omits size | 0.830 | 15.0 | 16.6 | **5.6%** |
| Visual Genome, best | 0.731 | 15.0 | 17.2 | **0.0%** |
| synthetic, noise 0.50 | 0.743 | 13.3 | — | 70.7% |

Three fields agree on the correlation to within 0.1, agree on both width
measures to within 1.5, and differ in recovery across the entire range. The
principal-angle count fails because noise rotates eigenvectors and lowers the
measured overlap without narrowing anything. A second instrument built
specifically to fix that — canonical correlations fitted on one third of the
scenes and scored on another, so a direction counts only if it generalises —
fails the same way, and rates Visual Genome *highest* of the three failures.

So **two candidate statistics have now been proposed and refuted in one
session**, and the honest position is narrower than the one I wrote four hours
ago:

- **Established.** The correlation does not govern recovery. Rank-8 truncation
  holds it at 0.906 and drops recovery to 12.9%; independent noise drops it to
  0.743 and keeps recovery at 70.7%. Both directions are controlled, and
  together they retire the 0.9 rule for good.
- **Established.** Genuine rank truncation below about 8 destroys recovery, and
  suppressing a caption factor destroys it while the correlation stays at 0.83.
- **Not established.** That any single width or dimension statistic predicts
  recovery. Neither of the two tried does.

What separates the noise case from the omission case is a live question with a
specific candidate. Suppressing a factor makes scenes that differed only in
that factor *exactly* interchangeable in the text field, so the energy acquires
an exact symmetry and the truth stops being a unique minimum — it becomes one
member of a degenerate orbit. Independent noise creates no such ties. That
predicts a three-way split which the gate can read directly: the truth deeper
than everything found (recoverable), tied with what is found (symmetry), or
shallower than what is found (information deficit). The gate is running on both
fields; the sections below should be read against this correction.

## What the shared spectrum measures

Each field is eigendecomposed and the principal angles between the two leading
32-dimensional eigenspaces are computed; directions with cosine above 0.7 count
as shared. A scene-shuffled null puts the chance count at 1.0 in every
condition below, so the counts are not an artefact of subspace dimension.

| Field | ρ | vision eff. rank | text eff. rank | shared (net of null) |
|---|---|---|---|---|
| synthetic watershed (recovers 95.3%) | 0.928 | 14.5 | 11.8 | **26** |
| synthetic v0 (fails, 5.1%) | 0.508 | 31.3 | 27.9 | 18 |
| Visual Genome, previous best | 0.656 | 18.7 | 13.9 | 10 |
| Visual Genome, best today | 0.716 | 39.7 | 48.0 | 15 |

The failing synthetic field settles the sufficiency question in the other
direction: 18 shared directions with a correlation of 0.508 also fails.
**Recovery needs both a wide shared spectrum and a strong correlation, and
neither alone predicts it.** *(Superseded by the correction above: the pair
does not predict it either. The counts below remain as measured; what they do
not support is the inference that they govern recovery.)*

## Natural data is narrow in the intersection, not in either modality

The diagnosis is sharper than "the features are not good enough". Each side is
individually rich — vision effective rank 40, text 48 — while the two agree on
15 directions. **Each modality is rich about something, and they are rich about
different things.**

That intersection is close to invariant under everything tunable on the vision
side, all measured at matched width:

| Vision configuration | ρ | shared |
|---|---|---|
| 6 spectral segments | 0.692 | 15 |
| 16 spectral segments | 0.679 | 14 |
| annotated region boxes (oracle) | 0.703 | 15 |
| DINOv2-base, 12 layers, 768-dim | 0.731 | 15 |
| DINOv2-large, 24 layers, 1024-dim | 0.709 | 15 |

The backbone row was a pre-registered prediction rather than a hope: the width
had failed to respond to every other vision-side change, so a larger encoder
was predicted to leave it alone. **Doubling the depth and widening the
representation leaves the shared count at exactly 15** and moves the
correlation by less than the segmentation noise floor, in the wrong direction.
The prediction survives, and the row closes on a measurement.

Handing the pipeline ground-truth boxes buys one direction over unsupervised
segmentation and nothing over using fewer segments. This confirms from a new
angle what the earlier oracle-box comparison found: **segmentation is not the
constraint on photographs.**

The correlation column in that table should be read against a noise floor.
Moving the segmentation eigendecomposition from CPU to GPU changes nothing in
the recipe, yet re-deriving segments through it takes the baseline correlation
from 0.6559 to 0.6767 — the eigenvectors differ in sign and, where eigenvalues
are near-degenerate, in rotation, so the clustering that follows lands
differently. **Segmentation reseeding is worth about 0.02 in correlation**, so
the 6-versus-16-segment and oracle-box differences in that table are inside
the noise and only the shared counts distinguish them. The session's headline
movements, 0.656 to 0.716 to 0.725, are three times the floor. The pipeline
itself is unchanged: run against the original CPU-derived segments it
reproduces 0.6559 exactly.

The text side does move it, and saturates:

| Text vector dimension | text eff. rank | ρ | shared |
|---|---|---|---|
| 24 | 19.0 | 0.661 | 10 |
| 48 | 31.8 | 0.692 | 15 |
| 128 | 48.0 | **0.716** | 16 |
| 256 | 53.0 | 0.700 | 17 |

Tripling the text representation's own rank from 19 to 53 buys seven shared
directions and then stops. Enlarging the scene population does not help either
— at matched width the shared count goes 23, 20, 19 for 256, 512 and 1024
scenes, so the ceiling is a property of what the two corpora are about rather
than of how many scenes are sampled.

## Corpus overlap sets the width, by intervention

The invariance results say the width does not come from the encoders. A
controlled intervention says where it does come from. In the synthetic world
the caption is encoded into separate factor blocks, so a block can be
suppressed — the caption then never states that property while vision
continues to see it. Nothing else changes: same images, same segmentation,
same kernel.

| Caption content | ρ | shared | recovery |
|---|---|---|---|
| states everything | 0.928 | 20 | **96.4%** |
| never states size | 0.830 | 15 | **5.6%** |

Deleting one word class costs five shared directions and the entire result.
The correlation is still 0.830 — higher than anything we have achieved on
photographs — and recovery is at chance. **The synthetic world degraded to
Visual Genome's shared width fails exactly the way Visual Genome fails**, at a
correlation Visual Genome never reaches.

The full ladder is monotone in how much the caption states:

| Caption content | ρ | shared | recovery |
|---|---|---|---|
| states everything | 0.928 | 20 | 96.4% |
| never states size | 0.830 | 15 | 5.6% |
| never states count | 0.678 | 13 | 1.3% |
| states neither size nor count | 0.544 | 10 | 1.3% |
| never states colour | 0.736 | 9 | 0.0% |

What this establishes is that **corpus overlap controls recovery**, which is
the claim that matters for what to do next. What it does not establish is the
intermediate step — that it does so *through* the width, since the noise
control reaches the same width with recovery intact. The mechanism is under
test; the corpus-level conclusion does not depend on which way that test goes,
and it reframes the natural-data deficit as a statement about Visual Genome
rather than about our pipeline: a region description simply does not say as
much about a photograph as the synthetic captions say about their scenes.

(Widths in this table are computed on standardised fields and are not
comparable entry-for-entry with the tables above, which use raw fields; within
the table the convention is identical, which is what the comparison needs.)

## Two smaller results, one refuted hypothesis

**The projection improved.** Scaling each discriminant direction by its own
eigenvalue to the power 0.5 lets a wide basis be kept without the weak
directions drowning the strong ones. It dominates the unweighted projection at
every setting tested and, unlike the unweighted version, improves with width —
which is the direction that raises rank. With text vectors at 128 dimensions
this takes the field correlation from 0.656 to **0.716** and the shared count
from 10 to 16, at no cost.

**Hubness was the wrong suspect.** The hypothesis was that the correlation is
inflated by a shared "this scene resembles everything" component carrying no
matching information. It is not: the additive row-and-column model accounts for
0.9% of the visual field's variance and 4.9% of the text field's, and removing
it *raises* the correlation slightly, from 0.656 to 0.671. The headline
statistic was honest; it was simply not the statistic that governs recovery.

## Shrinking the problem is dead, and now by measurement

Small blocks were the last standing non-correlation route, on the reasoning
that the hard phase is joint in correlation and population size. The reasoning
had the sign backwards. The information-theoretic threshold is
ρ_IT ≈ √(4 log N / N), which **rises** as N falls: 0.29 at 256 scenes, 0.51 at
64, 0.83 at 16. Shrinking the population raises the bar.

The project's own gate confirms it directly. At every size tested, the best
state a strong searcher reaches — spectral start plus 200 random restarts, each
run to a local optimum under exact steepest descent — is **deeper than the
truth**, in three replicates out of three:

| N | ρ | E(truth) | E(best found) | accuracy of best | truth deepest? |
|---|---|---|---|---|---|
| 16 | 0.705 | 2.95 | 2.37 | 0.271 | 0/3 |
| 24 | 0.703 | 3.24 | 2.29 | 0.194 | 0/3 |
| 32 | 0.679 | 3.69 | 2.38 | 0.073 | 0/3 |
| 48 | 0.706 | 3.48 | 2.65 | 0.125 | 0/3 |
| 64 | 0.700 | 3.24 | 2.15 | 0.151 | 0/3 |
| 96 | 0.698 | 3.65 | 2.57 | 0.115 | 0/3 |

When the truth is not the optimum, no searcher of any cost finds it, and the
question of algorithm class does not arise. Accepting exponential cost buys
nothing here — this is the same conclusion tempering reached by equilibrating
below the truth, now established at sizes small enough that search cannot be
blamed.

Combined with the earlier rank-ladder result — at natural field quality every
truncation from rank 4 to full returns chance — three of the five routes in the
candidate register are closed: reshaping the landscape, searching harder, and
shrinking the problem. Raising the shared spectrum and adding orthogonal signal
are what remain.

## What this changes

The target moves from a scalar to a pair. Raising the correlation from 0.656 to
0.716 was worth having, but the measurement that matters is that it came with
the shared count going from 10 to 16, and 16 is inside the interval where the
synthetic ladder crosses from 13% to 96%.

The corpus, not the encoder, is where the width comes from. The synthetic world
has 26 shared directions because its captions state exactly the world state;
Visual Genome has 15 because a region description and a patch descriptor
overlap on roughly that many aspects of a photograph, and no amount of
segmentation quality, backbone capacity, or scene count moves it. That reframes
the next step as a question about which corpora have naturally wide overlap —
dense descriptions, product listings with photographs and full specifications,
screenshots paired with their accessibility trees — rather than a question
about better features for this one.