Compute exact complexity
Minimum-layer enumeration supplies exact formula sizes across the complete four-input universe.
Exact synthesis × semantic learning × calibrated bounds
Learning finds slack.
Calibration decides what can be claimed.
A six-paper research program using exact Boolean synthesis, semantic prediction, analytic envelopes, and frozen prospective tests to learn where mathematical bounds are loose—and why aggressive selection can break their calibration.
65,536Every four-input Boolean function, not a sample.
173.9MFive-input functions with exact minima established through formula cost 14.
95–97%Held-out exact-complexity variance explained by the expanded semantic profile.
6 / 120Prospective misses after selecting the largest predicted bound improvements.
The research program
From exact synthesis
to failure geometry.
The project advances through six linked stages. Exact data establishes the phenomenon; semantic structure explains it; analytic bounds turn it into mathematics; prospective selection reveals where the guarantee still fails.
Measure semantic structure
Truth-table invariants record decomposition, certificates, boundary geometry, algebra, and Fourier phase.
Recover exact difficulty
Held-out semantic profiles explain 95–97% of minimum-size variation across three gate bases.
Build analytic envelopes
Khrapchenko obstruction supplies the lower endpoint; explicit trees, restrictions, and covers supply upper endpoints.
Search for removable slack
A calibrated headroom model targets cases where a certified construction appears unnecessarily loose.
Explain the failures
Top-score selection concentrates rare one-headroom cases and exposes an exact calibration boundary.
What works
Intersecting intervals preserves calibrated coverage.
A proved analytic envelope can be intersected with a calibrated prediction interval. The result never widens the theory and inherits the calibration guarantee, while gap normalization certifies a minimum fractional tightening.
I* = [L,H] ∩ [Ŷ − E−, Ŷ + E+]
theory supplies validity; learning supplies location
What broke
Selection exposed a sharp low-headroom boundary.
When a policy subtracts a learned reduction from a constructive upper bound, validity depends on the available headroom. Top-score selection enriched the one-headroom subgroup and pushed six functions across the exact failure threshold.
v(f) = max{0, z(f) − [qg(f) + h(f)]}
failure occurs when predicted gain exceeds calibrated headroom
Established and unresolved
The signal is strong.
The selection problem is real.
Four-input calibration removes most of the uncertainty left by classical bounds. The harder prospective test confirms that semantic ranking finds opportunity, but rejects the claim that the current correction remains safe after top-score selection.
NAND
96.3% COVEREDMean shrinkage82.0%
Guaranteed shrinkage81.4%
NOR
96.6% COVEREDMean shrinkage79.2%
Guaranteed shrinkage79.0%
AND / OR / NOT
96.4% COVEREDMean shrinkage71.5%
Guaranteed shrinkage70.8%
PROSPECTIVE UTILITY
99 / 120 IMPROVEDMean tightening0.8795 gate
Targeted tightening1.1744 gates
FROZEN COVERAGE
CRITERION FAILEDObserved coverage95.0%
Required coverage99.0%
FAILURE GEOMETRY
ONE-HEADROOMMatched boundary group6 / 10 missed
Selection enrichment5.31×
Complete four-input result
The calibrated sandwich
Gray marks the full analytic range; gold shows the gap-normalized 95% envelope; black is exact complexity. Curves summarize equally populated complexity percentiles.
Six-paper sequence
One developing
research program.
The papers move from prediction to structure, from structure to bounds, and from calibrated refinement to a precisely characterized prospective failure. The negative result narrows the next theorem target rather than erasing the earlier signal.
Latest paper · 2026
When Selection Breaks Calibration
The six prospective misses are not unrelated errors. They form an exact one-gate-headroom boundary in which construction has nearly solved the instance, global semantic features still predict difficulty, and top-score selection concentrates the dangerous tail.
Open latest paper · PDF“Calibration must follow the query that will actually be made.”When Selection Breaks Calibration
Open research
Inspect every layer.
Exact targets, folds, predictions, calibration tables, protocol hashes, failure casebooks, figures, and scripts ship with the repository. Downstream analysis runs locally with no model API. Regenerating the full exact cost-14 layer is a substantial cloud-scale computation.