Analysis Companion
The Analysis page is easiest to use when you begin with one question and let that question choose the tab. This page gives the short, practical route before the fuller guide takes over.
This page is the shorter companion to the full Analysis Guide. It is meant for visitors who want a fast, plain-language answer to what each Analysis tab is for, which one to open first, and how to move from one tab to the next without treating every metric as equally important at once.
Last reviewed: April 2026
The Analysis page becomes much easier once you decide what you are asking. If you want structural rules, start with Modular. If you want spacing behavior, start with Gaps. If you want center arithmetic, start with Factors. If you want local clustering, start with Density. If you only want a rough benchmark, open Expected last rather than first.
Use Modular when you want the quickest explanation of why the same residue-class patterns keep appearing. It is usually the cleanest first stop because it turns the range into a small number of structural buckets rather than a long list of counts. If the site keeps mentioning Mod 6 or residue classes, this is the fastest tab to make that language feel concrete.
Use Gaps when your question is about how twin-prime pairs are spread across the range. This tab helps separate close clustering from wider separation and works especially well after the prime-gaps pages. It is the right entry point when the question in your head sounds like, 'Are the pairs bunching up or spreading out here?'
Factors is most useful when you want to compare twin-prime centers with other nearby even numbers. It is less about generic factor tables and more about whether centers look structurally unusual. If twin centers are still a fuzzy idea, this tab makes more sense after Why Twin Centers Matter or after a quick pass through Explorer.
Density and Expected are comparison tabs rather than entry tabs. Density asks whether twin-prime neighborhoods look locally richer than the surrounding range. Expected asks how the observed count compares with a rough benchmark. Both are more useful once you already know what the range looks like structurally, because otherwise the summaries can float free of the actual pattern.
A reliable sequence is: start with Modular for structure, move to Gaps for spacing, then use Factors if twin centers are part of the question. After that, open Density if you want neighborhood comparison and Expected if you want a rough benchmark. This order keeps the more interpretive tabs from arriving before the underlying pattern is clear.
Use this page as the quick companion when you land on Analysis and want orientation fast. Use the full Analysis Guide when you want the longer walkthrough. Move back to the live Analysis page when you are ready to test a real range, and use the Lab or Explorer if the numbers still need a more visual or row-by-row reading first.
These links support the theorem, history, and heuristic claims summarized on this page.
Use these links to keep reading or jump back into the live number views.
Use the actual Analysis surface once you know which question you want the tabs to answer.
The longer guide explains each tab in more depth and works better when you want the fuller walkthrough.
The prime-gaps page helps when your Analysis question is really about spacing language first.
The bounded-gaps explainer helps when the Gaps and Expected tabs raise theory questions about near misses versus exact twin primes.