The Formation Condition at Full Scale: densestPeaks()

The BPT law

BPT Part 2.3 (Null Wells and the Birth of New Universes) states the Null Well Formation Condition: ℜ⌂(x, τ, ℨ) ≥ ℜ⨶(ℨ) = k(ℨ) × ℏ(ℨ) × ↁρ(ℨ). The local recursive tension density ℜ⌂ at a place and time must reach the critical recursion threshold ℜ⨶, which the Data Density ↁρ (data bits processed per unit volume per unit time) sets through the quantum of action. In the book's words, "Null Well formation occurs when local computational complexity exceeds substrate processing capacity": the well is "a protective substrate mechanism, creating localized computational voids that prevent recursive overflow." BPT Part 6.3 gives the formal version, ℜ⌂(x, τ) ≥ ℜρ⨶ = k × ρ_P × 𝒞→², k times the Planck energy density. BPT Part 3.13 applies it to the data layer: Proto Wells "form at the Data Nova where the data is densest, as every Null Well forms where the overflow overwhelms the Zinf point." The game reads "critical density" as the densest the data is: wells go where the data is densest and nowhere else, and how many there are is "the data's own answer" (comment at app.js:1557–1561).

The established physics

Halos collapse at the peaks of the early density field (Bardeen, Bond, Kaiser & Szalay 1986, cited in BPT Part 3.14). The peaks predict where structure forms.

The game's math

densestPeaks() app.js:1562 runs when the field is drawn at its true level (uni.Lm = uni.L).

1. The field of rows × cols cells is cut into blocks of B = 8 cells a side. That gives GR = ⌊rows/8⌋ by GC = ⌊cols/8⌋ blocks. Each block counts its pulsing cells: cnt(i, j) = Σ pulserGrid over its 64 cells.
2. Each count is smoothed over its 3 × 3 neighbors on the torus: sm(i, j) = Σ_{a,b ∈ {−1,0,1}} cnt(i+a, j+b).
3. top = max sm. Every block with sm = top exactly is dense. Dense blocks that touch (8-way, on the torus) are joined into one place, as the nova finder joins tied cells.
4. Each place's middle is the mean of its blocks, unwrapped around its first block so a place across the seam is not split. The well's position is then refined to the centroid of the pulsing cells in the 3 × 3 blocks (−B to 2B cells) around that middle block (2026-10-27: "wells lined up on the block grid").
5. The place's mass is Σ cnt over its blocks, in cells, at least 1.

If the field is too small for 2 × 2 blocks, one well is placed at the center holding every pulsing cell.

Output: a list {r, c, mass}. The count is the number of tied maxima. On a smooth field that is usually one place, or a few.

In the game

This runs at the Data Nova of a universe drawn at its own level, which on a desktop means the shallow universes (the field is 4·2^L cells a side, so level 6 or less on a desktop, level 4 or less on a phone; Part 3.3). The banner "Data Nova · the data erupts" comes up at the top of the page with the line "every bit the Pulsecore held bursts onto the lattice at pulse speed · not yet matter: the scaffold of space" (app.js:5240–5242). Where the densest place lies, a Proto Well appears on the torus: a violet core inside a magenta ring with a white-to-magenta glow that breathes about once a second, each at its own phase (setProtoLayer() scene3d.js:1619, pulseProtoWells() scene3d.js:1657). On a phone the mark is drawn 3.4 times its size, on a desktop 2.2 times (PROTO_K, scene3d.js:1514), so a lone Proto Well can be found from across the torus. The HUD label switches from "Wells" to "Proto Wells" (app.js:2140), and the line reads "1 · law 1.00" at level 0, with the law to two decimals. In the drawer's view menu, "2D · the field" shows the same place as the center of the well's rings (Part 3.16).

The scale model

Exact: no scale model. densestPeaks() is used only when the drawn level is the true level, so each place found is one well. When the field is a scale model, densePlaces() is used instead (Part 3.3).

Where the game and the theory differ

• BPT's condition is a threshold: a well forms wherever ℜ⌂ reaches ℜ⨶. The game takes only the global maximum of the smoothed count, so it places wells at the tied densest blocks and none at lesser peaks. The physics it cites, peaks predicting halos, counts every peak above a threshold; the game's version is the single-maximum case of it.

Fixed numbers