The Universal Scaling Factor and the Zinf Units

The BPT law

BPT Chapter 1, Part Z (The Universal Scaling Factor), states s = 2^L · f^L = 2²⁰² × f²⁰² ≈ 2.361 × 10⁶². It "quantifies the total binary dilation separating substrate Level 0 from observational Level 202". Read against time, s is the frame ratio ①⥂⧗⌂/ℨ: one Local Pulse Tempo lasts s Zinf cycles. The Zinf cycle itself is ℨ = ①⥂⧗⌂ / (2^L · f^L) (Substrate Full-Pulse Duration, Part D, and Part 1.6), and its frequency is ℨ∞ = 1/ℨ (Part 1.7). Base Units at Substrate ℨinf Scale (Part Z.1) divide each Planck unit by s: ℨ⧗ = t_p/s, ℨ⦜ = ℓ_p/s, ℨ⚛ = m_p/s. "Time, length, mass and energy shrink by s; force, power, charge, spin and the constants stay the same; acceleration and current grow by s", and density grows by s². BPT Part 2.8 adds that ℨ is "the clock of the Zinf universe itself, the same for every universe".

The established physics

Dimensional analysis of the Planck units. Acceleration is l/t², so dividing l and t by s multiplies it by s. Density is m/l³, so it scales by s². ℏ/2 is the spin of a spin-½ particle (standard quantum mechanics). The rest is BPT.

The game's math

• scalingFactor(L, f) universe.js:59: s = (2f)^L.
• zinfTime(L, f) universe.js:64: ℨ = t_p / s. zinfFrequency() universe.js:69: ℨ∞ = s / t_p.
• zinfUnits(L, f) universe.js:81 returns time, length, mass, energy and temperature, each the Planck unit ÷ s. It also returns acceleration = (l_p/t_p²)·s, density = (m_p/l_p³)·s², spin = ℏ/2 (the same at every layer) and angular = ℏ/s².

describeUniverse() always calls zinfUnits(OURS.L, OURS.f) (universe.js:294), so ℨ is ours for every universe:

s_ours = (2 × 1.018)^202 = 2.3611 × 10⁶²
ℨ      = 5.391247e-44 / 2.3611e62 = 2.2834 × 10⁻¹⁰⁶ s
𝒞→ℨ    = 1.616255e-35 / 2.3611e62 = 6.8454 × 10⁻⁹⁸ m   (Zinf Pulse Length)

A universe's own frame ratio uni.s is consts.s = Tempo / ℨ (universe.js:301). With a mass ratio of 1 this is (2f)^L for that universe's own f. At level 28 on the panel's Kerr–Kerr default, f = 1.015810 and s = (2.03162)^28 = 4.164 × 10⁸.

uni.s also sets the game's credits. Each well at the final well is worth s credits (rupture() app.js:2564: credits = lineage.total × uni.s). BPT does not state this; it is a game method that rewards a deeper universe by its dilation. The HUD prints s as "per well" (app.js:2139).

In the game

The readout card shows "Frame ratio s" with the tooltip "The Universal Scaling Factor: s = ①⥂⧗⌂/ℨ = (2f)^L on the ladder" (app.js:2074), and "Zinf cycle ℨ 2.28 × 10⁻¹⁰⁶ s" with the tooltip "invariant: the same in every universe" (app.js:2075). The ℨ row reads the same at every level, which is the theory's point made visible: move the slider, and every row changes except this one.

The HUD shows s as money. The line "Depth:" (index.php:80) reads "4.16 × 10⁸ per well" at level 28, with the tooltip "The frame ratio s = (2f)^L: the credit each well of this universe earns". Before the rupture, the Lineage section says "Each is worth 4.16 × 10⁸ credits" (app.js:3315). At the final well the HUD's "Credits:" line (app.js:2150) and the end card's "Credits earned" (app.js:2779) show wells × s, with the running total kept in the browser. The About panel's "Two clocks" card (index.php:305) tells the player that each Tempo "lasts s Zinf cycles and carries 2^(L+1) bits".

The scale model

Exact: no scale model. s and ℨ are computed at the true level.

Where the game and the theory differ

The HUD labels s "Depth". BPT uses "depth" for L itself ("temporal dilation depth") and calls s the scaling factor or dilation index.

Fixed numbers