+++ title = “Introduction” section = “root” +++
What is htpy?
htpy is an interactive tool for computing and visualizing spectral sequences in stable homotopy theory. It runs as a single-binary server that embeds both the computation engine and the viewer interface, serving a browser-based frontend.
A persistent instance is available at htpy.app.
What it does
htpy computes $\operatorname{Ext}$ groups over the Steenrod algebra by constructing minimal free resolutions, then provides an interactive environment for working with the resulting spectral sequences. The core capabilities are:
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Computation. Minimal resolutions of modules over the Steenrod algebra $A$ at any prime $p$, with Yoneda products, Massey products, and secondary operations.
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Visualization. Interactive spectral sequence charts with pan/zoom, page navigation, product structlines, differentials, and per-generator detail panels.
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Differential propagation. Enter known differentials and propagate consequences via the Leibniz rule, naturality, the long exact sequence in Ext, and Moss’ theorem. The deduction engine applies 11 distinct mathematical passes in a fixed-point loop to derive all consequences.
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Contradiction detection. Speculatively assume a differential and check whether it leads to a contradiction, narrowing the space of possibilities by elimination.
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Motivic homotopy. Compute the motivic Adams spectral sequence, the $C\tau$ Ext module, and the algebraic Novikov spectral sequence. The $\tau$-LES (iterated long exact sequence from the $C\tau$ cofiber sequence) computes $\tau$-torsion orders and constrains differentials via Chua’s Lemma 12.4.
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Cross-validation. Systematic comparison against reference data from Isaksen, BFCC, LWX, and Bruner–Rognes, with a test suite verifying 7 independent consistency checks across 5 spectral sequences.
Modes
htpy has two main modes:
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The Archive — a curated collection of pre-computed spectral sequence data for the sphere spectrum, organized by spectral sequence type. These are read-only and serve as a reliable reference.
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The Playground — upload your own module definitions, explore the resulting spectral sequences, add labels and differentials, run the deduction engine, and investigate patterns interactively.
Lineage
The computation engine builds on the Ext computation machinery originally developed in ext-rs by Hood Chatham, Dexter Chua, and Joey Beauvais-Feisthauer. The Groebner basis module is ported from SSeqCpp by Weinan Lin. htpy extends these with a web-based viewer, an archival format, a deduction engine for differential propagation, motivic computations, and tools for working with the $C\tau$ bridge.
Source code
htpy is open source. The repository is hosted on sourcehut: