Analysis Guide
A fuller explanation of what each analysis tab is measuring, what question it answers, and how to interpret the results together.
This guide explains how to read the site's structured analysis outputs and how to interpret the mathematical signals they highlight. It is meant for visitors who want more than a tab label but less than a full theory article.
The Analysis page takes one selected range and reuses the same twin-prime dataset across multiple views. That means the tabs are not separate calculations with separate inputs. They are coordinated interpretations of the same analyzed range.
Residue means the remainder a number leaves after division by a chosen modulus. Prime gap means the difference between one prime and the next. Twin center means the number between two twin primes. Expected on this site means a rough benchmark, not a theorem and not a promise that the selected range must behave a certain way.
Start with the question you are asking. If you want structural rules, begin with Modular. If you want spacing behavior, begin with Gaps. If you want to compare centers against other numbers, begin with Factors. If you want local clustering, begin with Density. If you want a rough heuristic benchmark, begin with Expected.
Open Modular first when your question is, what structural pattern do twin-prime pairs seem to follow? The Modular tab shows residue patterns for twin-prime pairs and their centers, and it is the fastest way to inspect whether the usual 6k minus 1 and 6k plus 1 structure is appearing clearly in the selected range.
Open Gaps first when your question is, how are twin-prime pairs spaced? The Gaps tab measures spacing between consecutive twin-prime pairs and between their centers. Use it when you want to see whether pairs appear tightly clustered, widely separated, or distributed in repeating gap sizes.
Open Factors first when your question is, do twin-prime centers look arithmetically unusual? The Factors tab compares twin-prime centers against other even numbers. It helps answer whether centers show unusually simple or distinctive factorization behavior relative to a nearby baseline.
Open Density first when your question is, do twin primes live in locally richer prime neighborhoods? The Density tab measures how many primes and twin-prime pairs appear in local windows around each pair. This view is best when you want to compare local clustering against the global average for the same range.
Open Expected when your question is, how does the observed count compare with a rough benchmark? The Expected tab compares observed twin-prime counts to N divided by log squared N. Treat it as a supporting benchmark, not as proof or as a full explanation of the range.
The rough idea behind that expression is that primes thin out as numbers grow, and the chance of seeing two prime-like events together is much smaller than the chance of seeing one. That is why log terms keep appearing in prime-density discussions. On this site, Expected is intentionally framed as a quick comparison layer, not as a full Hardy-Littlewood model and not as a proof-backed prediction for every interval.
If you are exploring a new range for the first time, start with Modular, then Gaps, then Factors. Move to Density when you want to test whether the local environment around pairs looks special, and use Expected last when you want a coarse benchmark instead of a structural explanation.
If your question is why does Mod 6 keep appearing?, start with Modular and then compare the same range in the Lab. If your question is are twin-prime pairs clumping or spreading out?, start with Gaps and then use Density to see whether local neighborhoods also look unusual. If your question is does this range look high or low for twin primes?, start with Expected, then move back to Gaps and Modular so the benchmark does not float free of the actual structure.
A useful pattern is to move from structural questions to comparative questions. Start with Modular and Gaps to see visible patterns, then move to Factors, Density, and Expected to test whether those patterns also show up in the broader summaries.
Use the live Analysis page when you want to compare a real range, then move into the Lab or Explorer if you want to see the same idea from a visual or row-by-row angle.