PrimeMind · structural atlas · build 2.4
Prime bases extend to the right, power levels rise upward from the lower-left origin, and every cell keeps a fixed size. Compare consecutive integer fingerprints, inspect completed gap skeletons, and test the simplest first-unsupported-boundary construction.
The lower-left data cell is A(1,1)=2. Prime bases qᵢ increase to the right; exponent levels a increase upward. Every cell represents the same coordinate at every integer and keeps the same physical size until you deliberately choose another display scale. The dot shows that coordinate's changing recurrence phase.
Every card uses the same lower-left origin, the same prime-base columns, the same upward power levels, and the same fixed cell size. Filled cells are the exact prime-power coordinates contained in that integer. Empty inherited support at a prime is shown as a new-base event rather than as a missing full fingerprint.
Each column is an integer. Each row is a permanent atlas lane. A colored square means that coordinate is active at that integer. Click any square or column to move the current integer and select its atlas coordinate.
Follow one complete prime gap from offset 0 at the lower boundary to the next prime at the top. Every integer keeps its own fixed-height row. The default paired tree places exponent coordinates on the left, distinct prime-factor slots on the right, and the integer's largest-prime-factor seed on the center spine. A synchronized matrix shows the nested prime-power fingerprint. During animation, completed rows remain visible and the sequence can continue automatically into later gaps.
Follow mode scrolls only the two internal sequence panes. With continuous playback enabled, the current integer advances through successive prime gaps; the document viewport remains fixed. Previous/Next gap opens the requested gap at its upper prime so its whole structure is immediately bright.
The lower prime is offset 0 at the bottom. Click any row to inspect it. Future rows are ghosted until the animation reaches them.
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Sequence position rises upward. In the paired encoding, a left coordinate −a records exponent a, a right coordinate +j holds the j-th distinct prime factor, and both branches meet the center spine labeled by the integer's largest-prime-factor seed.
The cumulative curve includes both prime boundary rows; the summary cards label interior-only totals separately.
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| base | integers | occurrences | extra powers |
|---|
| level a | active cells | integers |
|---|
| offset | n | class | factorization | Ω | ω | max exponent | LPF | higher-power cells | cumulative Ω |
|---|
Treat the active prime-power cells in a completed gap as an ordered mass distribution. The cumulative curve is normalized against a uniform left-to-right reference, and the residual bridge shows exactly where the observed mass is front-loaded or back-loaded. The same page separates internal shape balance from absolute excess relative to the permanent recurrence frequencies.
The dashed diagonal is uniform mass per ordered integer. The half-mass crossing and center of mass are measured independently.
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This curve begins and ends at zero. Positive values mean mass accumulated early; negative values mean it accumulated late.
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The same mass is shown before accumulation, preserving every individual integer contribution.
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For recurrence-based metrics, the baseline rate is the sum of the retained coordinate frequencies. The endpoint need not return to zero; it records total excess or deficit.
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This scan keeps the same metric and row scope while moving through neighboring completed gaps. The current gap is highlighted.
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| rank | n | mass | cum. | x | F(x) | bridge | Ω | factorization |
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| base | mass | left | right | center |
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| level | mass | left | right | center |
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Each coordinate qᵃ is compared with its smooth count expectation, interval length divided by qᵃ. A single coordinate differs from that expectation by less than one activation; their signed discrepancies add to the total Ω excess.
| coordinate | period | actual | expected | excess | first hit | start phase |
|---|
Four exact growth calendars are kept separate: global Ω-depth at 2ᴸ, individual prime-power coordinates at qᵃ, independently necessary base coordinates at q², and branch-layer starts at p·2ᴸ⁻¹ for the selected prime p.
| event | integer | coordinate / layer | meaning |
|---|
Begin at a known prime P. Continue a chosen set of previously established base coordinates into the future. Each candidate column is marked wherever an inherited base coordinate divides it. The first completely unsupported column is the predicted boundary. With the complete required base set through √(P+H), this is exact inside the declared horizon; with a reduced set it is an experimental survivor only.
Prime bases rise upward from q=2 at the bottom. Integers advance to the right. A filled cell means that base coordinate divides the candidate. The first empty vertical column is the first unsupported boundary.
The matrix is a fixed coordinate dictionary. Its lower-left origin and cell dimensions never change when the integer changes. Recurrence phase is the moving state. The gap skeleton is a linked representation retained for possible later compression experiments.
The coordinate never moves. The bottom-left data cell is A(1,1)=2; A(2,2) is always 3²=9. Prime index increases rightward and exponent increases upward.
The coordinate is active once every m integers, including inside every composite that contains qᵢᵃ. Larger powers recur less frequently.
The dot inside each cell shows recurrence phase. Advancing n by one advances every coordinate by one step within its own period.
These are related expansion events, but they do not mean the same thing.
A prime is first recognized as an unsupported boundary relative to all earlier prime bases; it then introduces its own new base column.
This preserves the full factor occurrences of the completed gap. It is descriptive unless only past information is used.
CSV exports retain exact integers, exact periods, coordinate labels, phases, and activity states. The JSON package includes all current settings plus the matrix, fingerprint sequence, gap matrix, revealed skeleton, and any locked boundary test.