tab-ipc-math-ladder

15.9 From interaction laws to latent presentations

The IPC ladder asks how much of the missing presentation is supplied in advance. Its early worlds contain two opaque finite-state generators \(g\) and \(h\). A candidate interaction law has the form

\[ h(g(x))=g^e(h(x)), \]

where opaque state names prevent direct arithmetic on labels. Later worlds replace a single exponent by a missing intrinsic property and then by an unknown collection of latent factors. A typed compiler never chooses the mathematical proposal; it checks public support, constructs the declared package, and refuses unsupported extensions.

Rung	Withheld structure	Principal result	Interpretive limit
IPC–1	Exponent in one supplied conjugacy family	Compiled proposals were admitted in \(6/8\) worlds versus \(4/8\) one-shot packages.	The family was given; the model inferred a parameter and completed a declaration.
IPC–2	Choice among commutation, conjugacy, and no-law families	The initial mixed-family proposer admitted \(2/8\). In a fresh paired study, an explicit public residual diagnostic yielded \(8/8\), versus \(5/8\) for language only.	Diagnosis, rather than compilation, was the dominant bottleneck.
IPC–3	One missing intrinsic material property	Diagnostic-assisted construction and controls passed \(8/8\); language only passed \(7/8\).	The property signature was registered.
IPC–4	Number and partition of one to three latent factors	Exact count and partition rose from \(4/8\) to \(8/8\) with pairwise minor residuals.	Exact, noiseless proportionality made the observer strong.
IPC–5	The same variable-size theory under affine noise	Recovery rose from \(0/8\) to \(6/8\); all six supported worlds passed, but both out-of-doctrine worlds were forced into false proposals and rejected by the compiler.	The observer recovered supported structure, but proposal-side abstention failed.

Table 15.3. Generated mathematical construction from a supplied interaction family to a variable-size latent presentation. Protocols and budgets differ by rung, so the entries form a diagnostic ladder rather than one pooled benchmark.

The IPC–2 distinction matters for provenance. The \(2/8\) figure belongs to the original mixed-family proposer–compiler pilot at a 4,000-token ceiling. The later diagnostic study used fresh worlds, paired language-only and diagnostic-assisted conditions, and a matched 2,200-token ceiling; its corresponding figures are \(5/8\) and \(8/8\). The diagnostic result therefore supports a paired treatment comparison, while the earlier pilot remains the failure that motivated the instrument. The two studies must not be read as arms of one experiment.

IPC–4 is the first study in which the extension has variable size. Six observables must be grouped into the smallest collection of proportional directions, and neither the factor count nor the partition is supplied. The result is defined only up to permutation and independent scaling of latent directions. The assisted model receives pairwise minor residuals but must form their transitive closure, respect the doctrine on factor size, and emit the complete projection. Hidden rows test whether the proposed relations persist.

IPC–5 removes exact proportionality. Centered-correlation diagnostics recover all supported one-, two-, and three-factor worlds, yet the proposal engine forces two four-direction controls into the declared limit of three factors. The compiler rejects both, so admission remains safe while discovery is incorrect. This is a central negative result: a closed theory doctrine can create pressure to explain every observation even when the correct conclusion is that the doctrine is inadequate. Constructive creativity therefore requires an explicit representation of unsupported alternatives and abstention, not only a richer generator language.