Formula isomorphism

The question

The lab ingests textbooks: physics, finance, control theory, information theory, ecology, machine learning. Every formula in them becomes a node in the knowledge graph, with the LaTeX it was written in and, where the parser succeeds, a signature of its structure. Two formulas with the same signature have the same shape, whatever they are called. The hope was that a detector over those signatures would surface the correspondences that textbooks keep in separate chapters: a diffusion equation in physics and an option price in finance, an action sum in mechanics and a path integral.

The question this page answers is narrower than the hope. Did the detector find such correspondences, and if not, why not.

What was built

A parser that turns a formula into a syntax tree and hashes it into a signature. A matcher that scores two formulas by weighted overlap of their parts, with rare parts weighted up. A detector that generates candidate pairs across documents and writes an edge into the graph when the score clears a threshold, eleven kinds of edge for eleven kinds of correspondence. Later, a second parser the lab wrote itself; the two off-the-shelf ones had been measured in May at 7 percent and 19 percent of textbook notation, and in July the one in use turned out to return partial trees without saying so. Alongside, a canonical register: one entry per formula that appears in many places, with every occurrence pointing at it.

Underneath all of it, the corpus. It grew from sixteen thousand formula and variable nodes in March to about two hundred thousand after a purge and a full re-ingest in April, and to 274,244 by the end of August.

What was measured

• 15 March: the edges that existed from earlier experiments were checked by hand. 437 of 443 were trivial. All were deleted. The discriminator between a real and a trivial match turned out to be formula length, not score.
• 29 March: with a boost for shared archetypes the detector went from 19 to 84 matches and from 4 to 14 pairs of fields. Over 41,000 formulas that is 0.2 percent with any edge at all.
• 17 May: the first bulk run wrote about two hundred edges per kind, 1,804 in all. By June the count stood at 2,059 and has not changed since.
• 18 May: nine edges scored 0.7 or above. Checked by hand, one was real, a discrete action sum against a Hamiltonian path integral. The other eight were the floor score of a generic bridge, not a match of structure. One kind of edge sat entirely at 0.30, its floor.
• 26 May: three named candidates, chosen because a textbook says the correspondence exists. Black-Scholes against the heat equation fell, because the physics side was not in the corpus. The Einstein field equations fell, because “stress tensor” in the corpus is mostly elasticity, not relativity. A thermodynamic identity closed.
• 25 July: 208,527 formulas, 61,878 of them with a signature. The hypothesis that signing the rest would lift the scores was tested in a dry run over all eleven kinds: average score 0.30 to 0.33, about 17 pairs in 2,200 at 0.7 or above. Signing everything raised coverage from 30 to 81 percent and changed nothing about the scores.
• 25 July, the one real class: seven formulas from six fields with the same signature, and all seven genuinely the same operation. A convolution in deep learning, delta sifting in physics, an inverse Fourier transform, the variational free energy of active inference, a mass integral in finance. The detector had found it, once, among thousands of classes that were not that.
• 26 July: the parser in use returned a partial tree without saying so; 51.5 percent of its successes were fragments, measured on 4,000 cases. The lab’s own parser, built that day, handled 57.9 percent of the 4,000 formulas the old one had failed on, against 1.4 percent for the standard library, and ran about two hundred and fifty times faster. Re-parsing the corpus took 522 seconds. Structure classes went from 18,047 to 64,888. With a minimum formula length of 25 characters, exactly five signatures span three or more fields; four are worthless, the two real ones are common knowledge.
• 26 July: 122,949 annotated variables are 2,465 distinct symbols. Only 16,377 formulas, 6 percent, are linked to the book they came from.
• 28 August: 274,231 nodes, 126,472 distinct LaTeX strings. 53.9 percent of the corpus is duplicates. The cause was a parameter not passed through, at two places. Fixed, and shown fixed by running the same extraction twice on one document: plus 13, then plus 0. The duplicates were kept.
• 2 September: 98,175 entries, 35.8 percent, have a LaTeX field of twelve characters or fewer. A single symbol with two indices is the most common signature across eighteen fields. Of the 2,059 edges, 42 score above 0.5. Six of the eleven kinds have none. The best-scored edge, at 1.0, pairs a field correlation from active inference with a Chernoff bound from information theory. Its variable mapping is empty, and its explanation is the same generic list every other edge carries.
• 3 September, live count as this page was written: 274,244 nodes, 126,472 distinct, 48,559 with a parse error, 2,059 edges, one canonical formula, zero formulas pointing at it.

What fell

• The historical edge set, entirely, in March. One hundred percent noise.
• The idea that signing more formulas would help. Coverage tripled, scores did not move.
• Two of three named candidates, for reasons that were about the corpus, not the method: the other half of the correspondence was not there, or the word meant something else in the books that were.
• The boost from archetypes and the four-archetype substitute for a full primitive vocabulary, rejected as a replacement in March.
• The number on this site. From late July to 2 September the formula page said “isomorphism links: 2,059”. The quality measurements from May and July were on record the whole time. On 28 August the lab asked its own memory whether cross-field structure matching worked and got four July findings back in two seconds. It had not asked before. The page now says “candidate links: 42, of 2,059 generated, sample did not hold”.
• The diagnosis, made and not yet built: candidate pairs are generated by conceptual nearness, and the score checks for exact identity of the structure hash. Two criteria with no relation to each other. That is why the best-scored edge is a bad one.

What stands

One real correspondence class, found and verified. A parser that understands textbook notation, measured against the alternatives. A corpus whose defects are now numbers rather than suspicions: one third fragments, one half duplicates, six percent traceable to a book, parse errors at 17.7 percent. And a decision.

On 26 July the founder stopped the refill and asked for the question first: what is the corpus for, and what would make it usable. The answer, given on 2 September, was to repair the reading side. The canonical register for formulas has held exactly one entry since April, the Shannon entropy, with nothing pointing at it; the variable side of the same register was scaled to the whole corpus on 26 July, at the founder’s word, after its stated purpose had been falsified that same evening: 12,558 canonical variables, 81,708 aliases. Building the formula side is the next step. The ingestion timer is off until then. The agent’s own summary in the handover of that day: we are collecting into a barrel with no tap, and pouring more in changes nothing about that.