The Bifold's Capacity and the Data Within

The BPT law

BPT Part 3.6 states when the Pulsecore must give way, as the UniSpheral Data Nova Threshold Law, in two expressions. Recursive Capacity Growth, f(n) = (n + 1)², "shows how recursive depth expands structural capacity quadratically with each step." The Containment Crossing Condition, n = ceil(√χ∂ − 1), "defines the minimum recursion step where capacity first overwhelms the toroidal threshold." "In practice, f(n) describes the fuel, χ∂ the container (the skin), and n the match that lights the transformation."

The container is the Bifold Containment Law:

P_∂ ≤ 2·σ_∂ / R,   χ_∂ = A / (4·ln 2·ℓ_p²)·ℨᵇ,   σ_∂ = k_B·T / (8·ln 2·ℓ_p²),   N ≥ χ_∂

"The Pulsecore holds while the Bifold can record what it contains; when the data within fills the skin's holographic capacity, N ≥ χ∂, the bubble bursts, and that is the Data Nova." The book shows the pressure balance (Laplace's law of a bubble) and the count are one condition: with the skin's tension σ∂ as given, "the temperature cancels, and the bubble bursts exactly when N reaches χ∂, whatever its heat." In its terms, "χ∂ – the Bifold's holographic capacity, 16·4^L Zinf bits at level L."

The Null Nova Condition comes first: N∅ = N, ρ∅ = N / (χ∂ − N), N + N∅ = χ∂, N₁ = χ∂/2, n₁ = ceil(√(χ∂/(2·ℨᵇ)) − 1). "Every bit carries its null; packed into the room the data has not taken, the null grows denser until data and null together fill the Bifold's record, and the Pulsecore gives its null up as space." After it the Space Growth Law, S(n) = N(n) = (n + 1)²·ℨᵇ with χ∂/2 ≤ S(n) ≤ χ∂, grows space one place a bit. Because χ∂ = (4·2^L)² is a perfect square, "the count meets it exactly" at n₂ = 4·2^L − 1. The book works both cases: "At level 1 the Null Nova comes at Tempo 5 and the Data Nova at Tempo 7; at level 6, at Tempos 181 and 255."

The established physics

The holographic bound: a region's information is bounded by its boundary's area, S = A/(4ℓ_p²) in natural units, so A/(4·ln 2·ℓp²) bits (Bekenstein 1973, Black holes and entropy; Hawking 1975, Particle creation by black holes; 't Hooft 1993, Dimensional reduction in quantum gravity; Susskind 1995, The world as a hologram). BPT's χ∂ is this bound. Setting it to one bit per drawn skin cell, 16·4^L, is the theory's identification of the skin's area in Planck units with its Local Pixel Count.

The game's math

• skinCapacity(L) = 16·4^max(0, L) pinpoint.js:36, χ_∂ in bits.
• dataWithin(n) = (n + 1)² pinpoint.js:38.
• nullNovaTempo(L) = ceil(√(χ_∂/2) − 1) pinpoint.js:40. It is exported but unused; the milestone text in app.js:536 writes the same formula inline.
• novaTempo(L) = ceil(√χ_∂ − 1) pinpoint.js:42, used only for pacing (app.js:4843).
• During the fill, accumulatedEnergy = dataWithin(floor((pulseCounter − 1)/2)) app.js:4568, and a nova fires when accumulatedEnergy ≥ novaCapacity() app.js:4584.
• novaCapacity() app.js:4294 returns χ∂/2 before the Null Nova, χ∂ after it while the Pulsecore fills, and, in the universe, overflowCapacity(rows, cols)·2^novaCount, which is only written to the nova record (Part 2.13).

The game tests the count form of the law, N ≥ χ_∂; it computes no pressure or tension, which the book shows give the same moment.

Level 6 on a desktop: χ∂ = 16·4⁶ = 65,536 bits. At n = 180, 181² = 32,761 < 32,768 = χ∂/2; at n = 181, 182² = 33,124 ≥ 32,768: the Null Nova at Tempo 181. At n = 255, 256² = 65,536 = χ∂: the Data Nova at Tempo 255. Both match the book. A phone at level 4: χ∂ = 4,096, Null Nova at Tempo 45 (46² = 2,116 ≥ 2,048), Data Nova at Tempo 63.

The Data Nova always crosses exactly. χ∂ = 16·4^L = (4·2^L)² is a perfect square, so at n₂ = 4·2^L − 1 the data is (n₂ + 1)² = χ∂, with nothing left over. The game uses the overflow (n + 1)² − χ_∂ in two places at the Data Nova: the data that erupts onto the scaffold (Part 2.6) and the mass the first wells collapse from. Both are zero at every level (see below).

pulsecoreInfo() app.js:4266 measures the information the Pulsecore held at its Data Nova. It reads the field in 2×2 blocks, each made into a 7-bit key: four pulser bits (weights 1, 2, 4, 8) and three bits from the last character of the color (& 7, shifted by 4). Its Shannon entropy H over the keys, divided by 7, gives h ≤ 1. With n pulser cells, I = max(1, n·h), and the result is min(0, log₂(I/χ_∂)), the bits short of capacity. It is stored as uni.matterLaw.measuredBits (app.js:5262).

In the game

During the fill the readout is a running statement of the threshold law. The Tempos line reads "data within N of the skin's χ bits" (updateHud() app.js:2124), N = (n + 1)² and the second number novaCapacity(). "Capacity:" (index.php:72) shows N over that capacity as a percentage, and the bar under the readout fills with it (Part 2.13). "Pulse Energy:" shows the same (n + 1)² (app.js:4572). Before Start the Bits line reads the painted Pulsecore's bits (Part 2.3).

So on a desktop at level 6 or deeper the player sees the Capacity climb to 100 % at Tempo 181, about 14 seconds into the default pace, when the Null Nova fires; drop to 50 % as the same data is measured against the whole skin; and climb again to 100 % at Tempo 255, when the Data Nova fires.

In 3D the fill is drawn inside the Pulsecore sphere: the points of the data within grow and brighten with N/χ_∂ (setCoreDensity() scene3d.js:442, called at app.js:513). Before the Null Nova the room is taken as half the skin, packed with nulls; at the Null Nova it doubles, so the points thin at once and then fill again toward the Data Nova (Part 2.5).

The page states the law in words twice. The Universe & Genesis note (index.php:191): "Every bit carries its null: when data and null fill its 16·4^L places, the Null Nova expels the null as space, and when the data alone fills it, (n + 1)² after n Tempos, the Data Nova pours the data out." The Algorithms panel's "The Pulsecore's novas" (index.php:356) names the Bifold Containment Law.

The information measure is not on screen; it appears in the run record as deficit_bits (app.js:2207).

The player changes the timing of the novas only through the level: the painted pattern sets what the data is, never how much of it there is at each Tempo.

The scale model

χ_∂ is computed at the model level, skinCapacity(uni.Lm). At level 28 the true capacity is 16·4^28 ≈ 1.15×10¹⁸ bits, and the true Data Nova would come at Tempo 4·2^28 − 1 ≈ 1.07×10⁹. The drawn fill stands for it, Tempo for Tempo scaled by the same 2^(L−Lm) as the field's side.

Where the game and the theory differ

• The game agrees with the book that the count meets χ_∂ exactly, but its own comments at the Data Nova expect a remainder. The comment at app.js:5254–5260 says that besides the turned skin, everything the core held, "(n + 1)² at this Tempo, goes out with it at once," and the wells' comment (app.js:5280–5285) says "what collapses at the Data Nova is its overflow, the data beyond the skin's capacity." That overflow is 0 at every level. The eruption places no cells, and the first wells fall back to the floor max(peaks.length, overflow), a total mass equal to their own number, about one cell each (app.js:5288–5290).
• Before the Null Nova the Tempos line labels χ∂/2 as "the skin's ... bits." That number is the data's half of the skin's record under the Null Nova Condition, not the skin's capacity χ∂.

Fixed numbers