HOPE: recall in a library of Hilbert spaces, what it finds and where it stops

1 · What HOPE is

What it does for you. You hand the engine part of something: a passage it may have seen, a sentence with a word missing, a reading taken from a model. HOPE finds the stored thing that part belongs to. It lets the reading settle, the way a ball comes to rest in the lowest hollow of the ground it is dropped on, and the hollow it comes to rest in is the answer. No arm without the settle has been run, so how much of any count below the settle earns, rather than the stored readings it starts from, is not isolated; where a result is read as the settle's, the reading is likely, not measured. The reading and the answer are the same kind of object, a state of the same space, so an answer can be checked by putting it back in and seeing that it stays where it is.

What it keeps. HOPE does not keep one large memory. It keeps a library of separate spaces, one for each open model it has read, and every space has the same small size: ten axes, each of which takes one of two values, so each space has 1,024 possible states. The library holds 99 such spaces in three groups that share no input, 21 text models, 77 image models and 1 audio model, which is 101,376 states as 99 separate spaces, never one space.

How a space is made. A space is made from one model. We run the model on a fixed set of inputs, keep its internal readings, and reduce them to the ten directions that carry most of their variation (Jolliffe 2002). Over those directions a space keeps two things: the readings themselves, as stored patterns, and one pull for each of the 45 pairs of its axes. To recall, a cue is placed on a space's ten directions and handed to a modern Hopfield network that holds the space's stored readings (Krotov and Hopfield 2016; Krotov and Hopfield 2021; Ramsauer and others 2021); the network lowers its energy step by step until nothing moves, and the state it stops in is the recalled pattern. To ask which space a reading came from, the same reading is settled in every space of its group, and the spaces are ranked by how closely the settled state matches the reading.

What went against us comes first. One range we had recorded was falsified when re-run and is corrected (section 6); the separate spaces do not join into one shared space (section 2); a space's ladder does not carry across spaces (section 4); recall of a passage from its own words was refuted when measured, and a ten-axis space holding 40 readings is over capacity (section 3); the missing-word model is calibrated worse than a word count and writes no new text (section 5); and the site's own space loses measured answers, so it is switched off (section 7). Six of the 41 claims we re-ran have moved since they were first written, likely because an input changed, and section 8 lists them.

What this paper contributes, losses first, each with its basis and the sections that carry it:

1. Measured, one run. One recorded range falsified on re-run and corrected, six claims moved, and the separate spaces shown not to join into one shared space (sections 2, 6 and 8).
2. Measured, quoted as recorded. Recall of a passage from its words refuted, a ten-axis space over capacity at 40 readings, the missing-word model's poor calibration and the site space's measured losses, each quoted as recorded and not re-run (sections 3, 5 and 7).
3. Measured, one run. The right text space named first for 372 of 840 held-out readings, 44.29 per cent against 4.76 per cent by chance (section 3).
4. Theorem in the model. A settle never raises its energy, and ten axes with one common pull have one ladder shape whatever the pull (sections 1 and 4).
5. Reading. That 99 small spaces are a library and not one large space, read from the measured agreement (section 2).

2 · The library, and the one large space it is not

We measured whether the spaces make one shared space, and they do not. If the 21 text spaces were views of one large space, one fit that lines them up on the readings it is shown (Schönemann 1966) should keep lining them up on readings it has not seen. On the readings the fit was shown, the text spaces agree 0.324232; on readings held out of the fit they agree 0.03793. The 77 image spaces agree 0.022244 on the readings the fit saw and exactly 0.0 on the readings held out. Held out of the fit, each group's agreement still clears the same procedure run on readings shuffled within each model, over 20 shuffles, by a margin of 20.7861 standard deviations for text and 19.6104 for images: the readings share some structure. That this is far too little to make one shared space is our reading of these numbers, one run with one split.

The groups are not compared. The text and image spaces were read on different inputs, so no reading is common to both, and a fit across the two groups is refused rather than made up: a distance between a text space and an image space is undefined.

Pooling the text spaces keeps nothing. Put together as one space of 210 axes, the 21 text spaces keep 0 of their 945 pulls once the weakest are cut at the floor every space uses, and all 945 are pruned. Every one of the 945 text pulls is below that floor, the largest at 0.047345; in the image spaces 45 of 3,465 clear it, the largest at 0.050144. The pooled space names the space a reading came from 376 times in 840 against the separate spaces' 372, on the same 840 probes in one run; no test of the difference was made.

3 · Recall: naming the space a reading came from

What was refuted first. Recall of a passage from its own words, inside one space, was refuted when measured: over 80 probes in 8 text spaces, the settle landed on the right passage 4 times, against 48 for a plain match on the same words. The likely cause is capacity: a ten-axis space holding 40 readings recalls only 40 per cent of them from an exact cue, and 82 per cent at 12, which is why the library keeps 12. Both are quoted as recorded and not re-run here; no committed runner re-derives them. On the library's own ground, our consultant was measured too: on its set of 26 questions, HOPE switched on was cited on none of them, and the answers were the same with it on as off, 24 of 26 correct either way.

The test. We took the 21 text spaces and each of the 40 readings every one of them stores, 840 readings in all. Each reading was taken out of every space before it was asked about, then settled in all 21 spaces, and the spaces were ranked by how closely each settled state matched it. By chance the right space would come first once in 21 tries, 40 times in 840, and in the top three 120 times.

The result. The space a reading came from was named first 372 times of the 840, 44.29 per cent, against 4.76 per cent by chance, and was in the top three 541 times. That is about nine times chance on one group of spaces, and it is still a weak instrument: more than half the time the first answer is wrong. Ranked by the energy of the settled state instead of by the match, the right space comes first only 92 times and in the top three 194 times, so the energy is the wrong score and is kept only as the control that shows it. A model's name, put in as a reading, does not lead back to its own space: it came first for 1 of the 21 text models, which is chance.

Each group holds its own members. Stored as one memory per group, every member came back when its own pattern was shown again: 21 of 21 text models and 77 of 77 image models, with no change, with a tenth of the pattern changed and with three tenths changed.

Same observation, three vocabularies. Three fields (information theory, systems and graph theory) describe the same situation in different words, and an insight in one usually has a twin in the other two. This series names a twin only where its own model has the object, and says so where a twin is missing or runs the other way. Each twin is our reading, not an identity, unless a result the paper states makes the two one number. Capacity, in this section and in section 6, is one such observation. In information theory it is how many readings a memory can hold and still return from a cue. Read as a graph, in the classical memory of section 6 every pattern is written into the same pulls between the same cells, so each one added moves where the others come to rest. Read as a system, at its last load, which section 6 places at or past its capacity, that memory came to rest as close to another stored pattern as to the cued one. A HOPE space keeps its readings themselves as well as its pulls (section 1), so the graph twin here is the classical memory's, not a HOPE space's.

4 · The ladder, and what does not carry across spaces

A ladder that cannot tell spaces apart. Every space has a ladder, one rung for each axis, which the service page draws for three of them. Read with every pull counted alike, all 99 ladders have the same shape, so 0 of the 4,851 pairs of spaces differ. That is arithmetic, not a finding about the models: ten axes that all pull on each other with one common weight have one ladder shape, whatever the weight. Read with each pull weighted by its own strength, all 4,851 pairs differ. One space, the audio model's, has a flat ladder under that weighting, because its readings never move together.

What does not carry across. We tested whether a space's ladder says anything about another space, and on every test it does not. Spaces of the same kind are no closer on the ladder than spaces of different kinds: the separation reads z = −0.1607 over 3,136 same-kind and 1,715 cross-kind pairs, where −2 would count as a difference. Using the ladder to pick the model that matches a caption to an image, or an image to a caption, does no better than picking at random: over 200 caption and image pairs no choice clears a z of 2 in either direction; the ladder's choice reads −0.1925 and 1.6223, and a random choice −0.4046 and 1.4237. And a ladder cannot be shrunk to one number that keeps the neighbours: the one number that is unique for 99.2866 per cent of a space's points loses the nearest neighbour of 94.6275 per cent of them.

5 · HOPE as a language model

What it does not do. As a model of language, HOPE was measured with a calibration worse than a plain word count, −0.7967 bits per hidden word; it is not autoregressive, and it writes no new text: every sample it produced is a stored window. These are quoted as recorded on 15 September 2026 and not re-run here.

Filling in a missing word. The same settling can fill a gap in a sentence, the missing-word test (Taylor 1953). We stored windows of eight words from the sentences of our own published pages as patterns, hid one word at a time in sentences the memory had not stored, and asked HOPE for it. On the pages as they stood when this was first measured, HOPE named the hidden word first 0.1938 of the time over 418 hidden words, with 1,895 stored patterns, and that number re-derives exactly on the same text. The strongest baseline we know for this test, a count of word sequences that sees both sides of the gap, was recorded at 0.1005 there, and in its top five it was ahead of HOPE, 0.2225 against 0.2201.

On today's pages the numbers moved, and we say why. Our pages have changed since: the same test today hides 1,058 words and stores 4,617 patterns. HOPE names the hidden word first 0.1853 of the time and the sequence count 0.0879; in the top five HOPE reads 0.224 and the sequence count 0.2183. Both first-place numbers fell from what we had written, because the text they read changed, and HOPE stays ahead of the baseline on both texts.

6 · How much a space holds

One range we had recorded was falsified when re-run, and is corrected. We had written that, after the reweighting of section 4, the three spaces we sampled keep between 6 and 29 of their 45 pulls above the floor, and 0 before it. Re-run on the same data, they keep 0 before and 15, 29 and 29 after, so the range is 15 to 29, not 6 to 29. The data are byte for byte the ones the 6 was measured on; the code under the number has changed since, in two files over three commits, and which change, or the original run's own sample, made the 6 is not yet traced. The direction holds: none kept before the reweighting, and some kept after it in all three spaces.

A memory has a size, and we measured where this one ends. Capacity is easiest to see in a classical Hopfield memory of 2,500 two-valued cells, the kind the textbook capacity is stated for; it is a classical memory, not a HOPE space, one run, one cue in one memory, on 7 October 2026. Holding 125 patterns, a load of 0.05 patterns per cell, it recalled the cued pattern with all 2,500 cells right; holding 250, a load of 0.1, it recalled it with 2,499 right. Holding 345, a load of 0.138, it did not: it came to rest as close to another stored pattern as to the cued one, an overlap of 0.1808 with the cue against 0.1792 with the other, a spin glass rather than a memory. So this memory's capacity at this cue lies above 0.1 and at or below 0.138, which is consistent with the textbook value of 0.138 for a large random memory.

7 · The site's own space

A space built from our own pages. HOPE can also hold a body of text. On 6 October 2026 we built one space from our own published pages as they stood that day: 1,727 passages, read into a space of 32 directions. Each passage was held out with a quarter of the passages, the space was rebuilt from the other three quarters, and the passage was used as a cue. The space is rebuilt whenever the pages change, so the numbers below are that day's.

What went against it first. Its basin floor did not discriminate: it refused 0 of 90 in-house questions and counted 11 of 13 off-corpus probes as inside a basin. Asked back in its own words, a passage held in memory landed on its own page for only 784 of 1,166 passages. And answers dropped with the space switched on: in our consultant's set of 26, correct answers fell from 18 to 13 with the space's top 12 as candidates, and in the sales set of 31, first answers fell from 24 to 22. All were measured in one run on 6 October 2026 and are quoted as recorded; the space does not re-derive at today's pages.

What it found. Given a hidden passage's own reading, the space found that passage's page first 900 times of the 1,727, 52 per cent, and within its top five 67 per cent of the time. Given only the passage's words, the way a question arrives, it found the page first 16 per cent of the time, below the 24 per cent a plain match on the same words reached.

Why the switch is off. Our assistant can settle a question in this space before it answers, and that switch is off by default and stays off. With the space switched on, the space's passage is quoted in 12 of 18 answers on our anchored set of 20 questions, but the right first answer falls from 19 of the 20 to 17. A switch that costs measured answers stays off until it costs none. The house report with the switch on, and the consultant on the site space, were not run.

8 · Limits, and what this paper does not say

A "Hilbert space" in this paper is a name for the set of states of a small space of two-valued axes. Every number here is a classical computation on an ordinary processor; nothing was run on quantum hardware, and no advantage over any other method is claimed. Every figure is one machine's processor with two threads, and no graphics-card figure was read; it would close with the same fixtures read on a graphics card and on a second machine. Every number is as of its run's date in its record.

What moved, and why. Six numbers moved between the day they were written and the re-run, each because the text it reads changed since: the consultant's set, from 23 to 24 of 26 answered correctly with HOPE off and with it on; HOPE's missing-word number, from 0.1938 to 0.1853, as the hidden words went from 418 to 1,058; the sequence count's, from 0.1005 to 0.0879; and our anchored set with the reweighting of section 4 switched on, from 17 to 19 of 20.

The open limits come first, each with what would close it.

• The image group's full recall test has not been run. It would close with the full sweep of 4,928 readings run under a rule committed before the run.
• The routing test had no power: every arm sits at chance on its two text and two image models. It would close with a routing test on more models, with its power stated before the run.
• The code change behind one count moving from 6 to 15 is not attributed (section 6). It would close with the change found by commit and the count re-derived on either side of it.
• The library's groups share no inputs, so no comparison across groups is made (section 2). It would close with a set of inputs read by every group, and the comparison run on it.
• A ten-axis space holds about a dozen readings (section 3). It would close with capacity measured at each size under a rule committed before the run.
• The site space's basin floor does not separate on-corpus from off-corpus probes (section 7). It would close with a floor re-set and re-tested on held-out off-corpus probes.
• The site space has sets that were not tested, the house report and the consultant among them (section 7). It would close with those sets run under the same rule.

What no test has faced. Naming the space a reading came from was measured on the 21 text spaces with every one of their readings; for the 77 image spaces it was read on a sample of 231 readings, and the full test, 4,928 readings each settled in all 77 spaces, has not been run. Apart from 200 COCO caption and image pairs, used verbatim under CC BY 4.0 (cocodataset.org), and 20 one-second windows of LibriSpeech dev-clean, used under CC BY 4.0 and trimmed to 1.0 s, every input is our own: the text calibration lines, the image probe and our published pages; each of the 99 models is credited by its role in the library's catalogue. The routing test rests on two text and two image models.

What Day did. Day re-ran every experiment this paper reports and drew the figure from their record.

9 · What we will measure next

None of these is registered yet. Each will be written down, with the result that would count against it, before it is run, and each says what it would decide.

• Naming the space a reading came from on all 77 image spaces with every one of their readings, the full test of section 8: it would decide whether the image group's recall beats 1 in 77 on every reading.
• Which change of code moved the low end of the corrected range in section 6 from 6 to 15: it would decide whether the first number was a defect of the old code or of its sample.
• Whether the site's own space can lead an answer without losing a measured one, the condition for switching it on in section 7: it would decide the switch.
• A set of inputs read by both text and image models, without which no fit across the two groups can be made or refused on evidence (section 2): it would decide whether the two groups share any frame at all.

10 · Companion papers

Seven papers published beside this one take the same approach: each traces its figures to a recorded measurement and says what it did not measure.

• The Operational Unit: the model behind the engine, and how it was built.
• The periodic table as a ground-state ladder: each element read as the lowest-energy filling of its electrons, with every answer NIST lists checked against it.
• Closing the gaps: four sustainability problems an engine can schedule, and what it cannot do about them.
• Sense, remember, act: what the engine senses, remembers and learns.
• The halo of a decision: a two-way choice, the partner it is tied to and the event that tied them, in textbook terms, with what an experiment could measure.
• The Earth as a two-state recorder: the Earth's dipole read as a two-way choice from public records, with a prediction written before the data, and the planet's clocks.
• The unit of work: a receipt for one settled problem, booked on a hash-chained ledger, and what it is not, with the economy's measured losses first.

Acknowledgements

• LibriSpeech (Panayotov et al., ICASSP 2015), CC BY 4.0. The probe the audio space was read over: twenty utterances of the clean development set, each trimmed to one second.
• COCO (cocodataset.org), captions CC BY 4.0. The routing test between text and image models: paired captions and images; the captions are copied verbatim and the images are not redistributed.

The library's spaces were read from open models. The library's own page, The Hilbert library, lists the licence of every source model, and its LICENSE.txt carries one attribution block for each.

Replicate this paper, or tell us what is wrong

We invite replication and correction. The measured results come from records committed in our repository. The records are not yet published; until they are, write to us through the contact page, naming the paper and the number you want to rebuild or the error you found. We correct an error we can reproduce on the paper's web page, with its date.

This paper registers no test yet. What we will measure next is listed, with the date each prediction was written, on the research page.

The records behind this paper's numbers are deploy/hilbert_experiments.json, deploy/hope_figure.json and docs/HOPE_GROUND_1_2026_10_06.md.

Our own open dataset for this paper is the Hilbert library, published under CC BY 4.0 with its own citation file.

How to cite this paper

Please cite this paper as:

Agile Integrations LLC. 2026. "HOPE: recall in a library of Hilbert spaces, what it finds and where it stops." Version 1, 8 October 2026. https://agileintegrations.net/blog/hope-recall-in-a-library-of-hilbert-spaces.html

Citation file: CITATION.cff.

11 · References

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• COCO Consortium. n.d. COCO val2017 caption annotations. cocodataset.org/#termsofuse, accessed 17 September 2026
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• Panayotov, V., Chen, G., Povey, D. and Khudanpur, S. 2015. "LibriSpeech: an ASR corpus based on public domain audio books." In ICASSP 2015. www.openslr.org/12, accessed 16 September 2026
• Ramsauer, H., Schäfl, B., Lehner, J. and others. 2021. "Hopfield Networks is All You Need." In International Conference on Learning Representations 2021. The modern Hopfield network and its one-step retrieval. arXiv:2008.02217
• Schönemann, P. H. 1966. "A generalized solution of the orthogonal Procrustes problem." Psychometrika 31, 1. The fit that lines one set of readings up on another.
• Taylor, W. L. 1953. "Cloze procedure: a new tool for measuring readability." Journalism Quarterly 30, 415. The missing-word test.