Rules of the Universe

What the v5 game obeys (rewritten 12 October 2026 for the close of the book, and brought up to the book of 24 October 2026: the full-cycle Zinf, Tempos and Rhythms, Bits per Pulse, the constants of a child universe, the Layer Count Law and failed children). Every rule is an equation from Binary Pulse Theory, named as the book names it, with the chapter and part it lives in. The game runs a universe at a small level L so its rupture can be reached on a screen; the same equations give our universe's numbers at L = 202 beside it. One coefficient is the game's own and is marked.

1. Scale

The Universal Scaling Factor (G) and Bits per Pulse (G) · ch 1, Part 1.6
s = ①⥂⧗⌂ / ℨ = 2^L · f^L (≈ 2.361 × 10⁶² for L = 202) bits per Tempo = 2^(L+1), per flip = 2^L

A universe's frame ratio: how many Zinf cycles one of its Pulse Tempos lasts. Data doubles exactly per level, so a Tempo carries 2^(L+1) bits and a flip (one Rhythm) 2^L; the heritage f^L stretches the time without adding data, which is why a universe processes more slowly than the Zinf by its heritage (f²⁰² ≈ 36.7 for ours). At the Zinf a flip is one bit, a Tempo two. The HUD's Bits count the whole field: every flip × 2^L from Start; before Start, the pinpoint's painted pixels × 2^(L+1).

The Zinf cycle ℨ (G) · ch 1, Part 1.6; invariant, ch 2 Part 2.8
ℨ ≈ 2.283 × 10⁻¹⁰⁶ s (one full 0 → 1 → 0) ↁℨ⧖ = ℨ/2, one bit ℨ∞ = 1/ℨ ≈ 4.379 × 10¹⁰⁵ s⁻¹

ℨ is the same in every universe; each universe's Tempo is (2f)^L ℨ on the ladder. The Zinf Pulse Length is 𝒞→ℨ ≈ 6.845 × 10⁻⁹⁸ m and its Diameter half that; mass, energy and the rest are our Planck units divided by our s.

Local Pixel Count and Frame Rate (Level N) (G) · ch 1, Part 1.9
🟑(N) = 16 × 2^(2N) pixels per view 𝓕⟳(N) = 1 / (2^N · f^N · ℨ), one frame per Tempo

The game's grid follows the first: 4·2^L cells a side at level L. The real view at 202 is 10¹²² pixels at 1.855 × 10⁴³ Hz.

2. Genesis: where a universe's clock comes from

Inherited Spin Parameter · ch 6, Part 6.9
a_f = | ℓ̃·η·n̂_L + a₁(M₁/M_f)²·n̂₁ + a₂(M₂/M_f)²·n̂₂ |, ℓ̃ ≈ 2.744, η = M₁M₂/M_f², capped at 0.998

The remnant's spin from the two progenitors' spins a₁, a₂, their mass ratio and their misalignment θ_merge. A lone collapse has no orbital term. Equal masses with no spin give 0.686; two near-maximal Kerr holes give 0.998.

Origin Compression Factor · ch 6, Parts 6.8–6.9
⟪C⟫(origin) = g_mass × (1 − κ·a_f²), κ ≈ 0.068, g_mass ≈ 1 (the book's calibration) C_min = 1 − κ·0.998² ≈ 0.932

The merger compresses the clock. Our Kerr–Kerr origin sits at the floor, 0.932; a gas-cloud collapse at 1.000.

Null Well Heritage Ratio (G) · ch 2, Part 2.8
f ≡ PD(n+1) / (2 × PD(n)) f = f₀ × ⟪C⟫^(1/n), f₀ ≈ 1.01837

Each level doubles and inherits a little more: the heritage. Our f ≈ 1.018, so f²⁰² ≈ 36.7.

Universe Classification by Genesis Parameters (G) and the child's Tempo · ch 2 Part 2.8, ch 6 Part 6.5
①⥂⧗′⌂ = ①⥂⧗⌂ × (M_null/M_P)^½

M_null/M_P 10⁶: Hyper-Genesis, a Tempo 10³ slower, slow-pulsing and long-lived · 1: Standard Genesis. A heavier well gives a slower child. No well forms below one Planck mass, so there is no lower row; failure is not set by mass (§6).

Local Pulse Tempo, Rhythm and Diameter · ch 1 Part 1.6, ch 6 Part 6.8
①⥂⧗⌂(n) = (2f)ⁿ ℨ ①⥂⧖⌂ = ①⥂⧗⌂/2 ①⊕⌂(n) = (𝒞→ℨ/2) (2f)ⁿ ①⥂⦜⌂ = 2 ①⊕⌂

A flip is one Rhythm, a full cycle one Tempo and one frame. At n = 202 these return the Planck time, 5.391 × 10⁻⁴⁴ s, and a Diameter of half the Planck length, 8.08 × 10⁻³⁶ m.

Constants of a Child Universe (G) · ch 1, Part 1.6
s′ = ①⥂⧗′⌂/ℨ ℏ′ = s′² ℏ_ℨ 𝒞→′ = ①⥂⦜′⌂/①⥂⧗′⌂ m′ = ℏ′ ①⥂⧗′⌂/①⥂⦜′⌂² G′ = ①⥂⦜′⌂³/(m′ ①⥂⧗′⌂²) k_B′ = k_B

A universe's Tempo and Length set its constants; ℏ_ℨ = ℏ/s² ≈ 1.892 × 10⁻¹⁵⁹ J·s is one bit held one Zinf tick. On the ladder light speed and G come out as ours; a heavier null mass slows the Tempo with the Length unchanged, so light speed falls and G falls as its fifth power. Boltzmann's constant is the one constant every universe shares. The readout shows each universe's 𝒞→, ℏ and G.

3. The two clocks and the rupture

Two-Sided Growth (G) · ch 1, Part D
ℜ_U(L) = 2^L on the UniSphere side ℜ(n) = 2ⁿ on the universe's own side, ℜ(0) = 1 bridge: ①⥂⧗⌂/ℨ = (2f)^L

The UniSphere counts levels; a universe counts its own Tempos. The HUD shows both: own Tempos, and the bits they carried.

Rupture Rule (G) · ch 1, Part D
n_rupture = 2^L completed Pulse Tempos

A universe spawns when its own count of full cycles reaches the UniSphere's capacity at its level. One Tempo is two engine steps (two flips), so a level-L universe lives 2^(L+1) steps from its Data Nova. Ours: 2²⁰² Tempos = 11.0 Gyr; at 13.8 Gyr we are past it.

Null Well Progeny Count Law (G) · ch 2, Part 2.3
N(L) = ℜ_U(L) / ↁℹ∂ = 2^(L/3), × 2 / (1 + √(1 − a_f²)) for spin

The boundary information per well is an area law, so a child costs the two-thirds power of capacity and the count is the remaining third. Level 0 seeds one child; level 202 seeds 1.86 × 10²⁰, the black holes. A spinning well has less area, half at maximal spin, so the count rises by at most two.

Data Energy Transition (G) · ch 3, Part 3.1
E = ℏ′ × ω × n_state, ω = 1/①⥂⧗⌂

The HUD's Energy is the flips so far times ℏ′/①⥂⧗⌂ of the universe's own constants.

4. Three epochs, and the void that is the pulse

The pinpoint · Zinf ℨ Unit (G), ch 1 Part 1.6; One Pixel = One Zinf (G), ch 8 Part 8.8
∅.Genesis → (0 → 1) = (1 → 0) = ½ (0 ↔ 1)

What is on the field before Start is the information inside one pulse. At level 0 that pulse is one pixel, one Zinf; at every other level the whole palette you painted is the pulse, however wide. Start seals it and it processes.

Data Nova by overflow and the Layer Count Law (G) · ch 3 Parts 3.3–3.6
first nova when Σ flips ≥ rows × cols, then each needs twice the flips of the last N_layers(L) ≤ L + 1

The pinpoint overflows when its field has flipped once per cell: the birth, the inside-out turn onto the torus, and the universe's clock starts. Novas 2–4 open space, time and spacetime; after them come Harmonic Novas, inner layers that open no new direction. Wells need space: the Prime Data Nova makes none. From layer 2, the birth of space, a collapse (a nova or a fold) makes a data well: counted in the census and holding its basin, drawn hollow, but sending no rings and never orbiting, because a ring is a pulse in time. Layer 3, Temporal Genesis, is the physical layer: data runs at the Rhythm and the physical at the Tempo, which needs time, so there every data well becomes physical at once, and wells from then on are physical from birth. A data well that never reaches the physical layer still seeds its child. In 3D the Prime Data Nova bursts out the toroidal lattice as its crossing lines; with space the skin comes in faint; at the physical layer the field lights in full; each Harmonic Nova adds a ring. The HUD's Layer shows how far a universe has come out of its L + 1. A level-L universe can reach at most L + 1 layers, the last at its final well. A nova is a discharge: wells come from collapse, not from novas.

Dilation Toward the Final Well (G) · ch 3, the close; Pulse Diameter PD(n) = Z·2ⁿ·fⁿ, ch 6 Part 6.8
I(n)/I(0) = 1 + (2f − 1) · n / 2^L PD(L) constant

The universe is its own size at birth, one and a half times it halfway to its final well, and 2f times it when the final well forms. The pulse at its centre never changes size; it recedes only relative to the information that gathers around it. The HUD's Scale is this ratio and what one cell is worth in Zinfs.

The void · strict interior nulling, ch 8 Part 8.8; Data Funnel Return Law, ch 3 Part 3.9
r_void = (R/2) / scale Ψ = 0 inside; the field lives on the boundary

The void is the pulse that was the whole universe before its nova, at its fixed size. In 3D the hole of the torus keeps its size and the tube grows; in 2D the void disc recedes as 1/scale. Nothing is lost into it: what folds or collapses is counted on the void's ledger, and strings are what comes back out.

A GAME COUPLING · THE RATIO IS THE BOOK'S
Data flow · the Harmonic Duality of Radii and Space Layers Dynamics, ch 4
round the tube (poloidal, magnetism): v_s / r round the ring (toroidal, spin): a_f · v_s / R winding H = R / r

The data circulates on the torus. The minor radius governs circulation along the tube, the major radius round the ring; at one substrate phase velocity v_s the data winds H = R/r times round the tube for each circuit of the ring (the UniSpheral Harmonic Ratio Function (G)), the ring part carried by spin, so a spinless universe's data only circles the tube. The book hears electromagnetism as the bright treble and gravity as the deep bass: the small fast loop and the large slow one. The shift is whole-cell and the same for every cell, so nothing is lost; wells, strings and orbits ride it, and the state detector watches from the frame that moves with the data. How many cells v_s moves in one flip is the panel's "Data flow" (0.25 by default, 0 stops it); it and Frame dragging are saved in the packet, so a replay flows the same.

A GAME COUPLING
Frame dragging around the void
ω(r) = κ_drag · a_f / r²

Space carried around the axis, the shape of Lense–Thirring dragging, applied as an exact cell shift so the data moves and nothing is lost. The book gives a_f and the axis and says the merger "imparts additional frame-dragging"; the coefficient is the panel's "Frame dragging".

A GAME CHOICE
The void at birth is half the field's radius
R_void(0) = R/2

The book says the pulse's whole volume becomes the void and the information goes to the boundary. A boundary with no thickness cannot hold cells, so the inside-out turn lays the disc onto the annulus [R/2, R]. The ratio 1/2 is the game's; the recession 1/scale is the book's.

5. Spin is born, and the first fork

Spin Is Not a Property of One Pulse (G), ch 1 Part Z.11; Inherited Spin Parameter, ch 6 Part 6.9
level 0: a_f = 0, ⟪C⟫ = 1, no axis first merger of two spinless wells: a_f = ℓ̃·η = 2.744 × ¼ = 0.686

The Zinf universe has no spin: nothing orbits in one pulse. Spin is born of orbit, so the first time two wells of a spinless universe fall together the child carries a_f ≈ 0.686 from the orbital term alone. Wells born together orbit on screen for part of the universe's life, tightening and turning, and merge; the pop says "Spin is born" the first time.

First Fork (G) · ch 3, the close
L_fork = min{ L : 2^(L/3) ≥ 2 } = 3 N(L) = 1, 1.26, 1.59, 2 for L = 0, 1, 2, 3

Below level 3 a universe holds one well, so a lineage is a chain of single, growing universes until the first that can hold two. The Zinf is a chain, not a tree: each pulse is a whole life, 1 × 1 × 1 × 1. In the game the Zinf chain beats by itself at level 0 (switch in the panel), each beat a universe saved as the child of the last. The fractional counts are odds in the game's data, not something the engine is told.

6. The close: final well, return to the Source, bifurcation, credits

Rupture Rule (G), Universal Dissipation (G) and the Return to the Source · ch 1 Part D; ch 3 Part 3.10
n_final = 2^L own Tempos from the Data Nova W(n_final) = N(L) ⇒ ℜ(n) → ∅, its data returned to ℧

At 2^L of its own Tempos the universe forms its final well: any wells still orbiting fall together. Then its data returns through its wells to the Source (℧) over 18 frames, twice as fast as a life's frames since nothing is left to process, in 2D as a fade, in 3D as the tube emptying. The census is saved beside what the law expects.

Return to Bifurcation (G) and failed children · ch 3, the close and Part 3.9
∅_L → (0 ⥂ 1)_(L+1), once per well P_failed ≈ 0.06 (Reproductive Outcome Distribution)

The return and the seeding are one event: the data a well returns to the Source is the recursion that seeds its child, one level deeper. A failed child seeds nothing and its data returns all the same; it is listed but cannot be played. The draw uses the book's 6% and is fixed by the level, the well and the census, so a replay fails the same children. The Zinf never fails.

Credits · The Universal Scaling Factor (G), ch 1
credit per well = s = ①⥂⧗⌂/ℨ

Each well is worth one completion of the universe that made it, counted in Zinf cycles: 1 at level 0, ≈ 2.36 × 10⁶² for a well of ours. Credits add across universes and stay in this browser.

7. Finds: harmonic nodes, time crystals, the states of the field, strings

Harmonic nodes · harmonics and interference, ch 7; Harmonic Novas, ch 3 Part 3.6
rings: radius = (t − t_well − kP) · c_grid, P = 2 flips (one Tempo) convergence of depth k held 3 beats = a node

Every well pulses and every pulse sends a ring. Where rings from k wells meet in step a point repeats on the rings' beat: an interference node, deeper where more rings meet, the flower of life's own count (2 Common · 3 Uncommon · 4 Rare · 5 Epic · 6 Legendary). Until 24 October these were called time crystals; they are not: they repeat because their sources do, not by locking onto a beat of their own.

Time crystals · Pulse Rhythm and Tempo, ch 1 Part 1.6
a region of ≥ 8 cells whose every cell repeats on the same beat p ≥ 3 flips, three times over in a 48-flip window, held two scans

A discrete time crystal locks onto a period longer than the beat that drives it (Wilczek 2012; seen in trapped ions and diamond spins, 2017). In this field the drive is one flip, a Rhythm, and the plain pulse already answers at two, a Tempo: that doubling is the cell rule itself, so it is not counted. A time crystal is a region that settles on three flips or more. Rarity by beat: 3–4 Uncommon · 5–6 Rare · 7–8 Epic · 9+ Legendary, one step more for 256 cells or more.

States of the field · states.js, found in the cells' own history
every 16 flips the last 48 are read: each cell's period, the lattice of the field, lumps and how they move, the whole field's recurrence

Breather: a time-crystal lump of at most 32 cells inside an 8×8 box, in the pulsing field. Oscillon: the same lump standing alone, null all round. Soliton: a lump of 3–32 cells whose shape recurs k, 2k and 3k flips later, moved d, 2d and 3d. Néel crystal: every neighbour in the opposite phase, a checkerboard. Smectic stripes: one phase in layers, order in one direction only. Pulse lattice: any other pattern repeating in space; Wigner lattice when its pulses stand apart in null; Space-time crystal when its cells also beat at three flips or more. A lattice must repeat across a 9×9 neighbourhood and show three tiles each way. Quasicrystal: a bulk region (a full 17×17 interior) with no repeat at any shift up to 12 cells, built from at most 32 of the 512 possible 3×3 patches. Domain wall: a thin line, at least 12 cells, where one order is broken between two domains of the same order. Revival: the whole field back in a state it held, on a beat longer than 16 flips, seen three times. Cells carried by frame dragging inside the window are left out of every beat: rotation is not a period of their own. The first of each kind in a universe pops, every Epic or better, and otherwise one Rare-or-better every 12 flips; all are kept in the Collection.

Species, discovery and names · species.php, the catalog at /states
species = the signature of a find, the same under moving, turning, mirroring, starting the beat anywhere and swapping lit for dark

Every find is checked against the catalog. The first sighting of a signature anywhere in PulseCore stops the universe and shows the discovery: the pattern drawn through its beat, its family and the physics it matches, and a field to name it. Only the run that found it may name it, once, and the name is credited to a signed-in account; names are 2 to 40 letters, digits, spaces, hyphens or apostrophes, unique, and can be renamed or hidden by the site's admin. A species new to you but known to PulseCore pops with its name and first finder. Any signed-in player may keep a label of their own on any species. Unclassified state: a bulk region (64 cells, eight across) that keeps changing, never repeating over the window or its last half, yet stays ordered (at most 48 distinct 3×3 patches over its last 8 flips) and fits no family above: the rarest find. "First ever" means first in PulseCore, never new to nature.

Strings · Data Funnel Return Law, ch 3 Part 3.9
a connected run of cells, length ≥ 10, width ≤ 2.2, alive ≥ 3 pulses rarity by length: 12 · 16 · 24 · 48 · 96

Filaments running out of the void. The void's ledger counts what went in; strings are what comes out. The first string of a universe pops, then Rare or better. Every one is saved and listed at Strings; the novas are at Nova Data, each with the universe's figures at that moment.

8. The field: the cell rule, with nothing to set

Perpetual reversal · Prime Pulse Bifurcation and Information-Driven Perpetuation, ch 1 and ch 3
a pulser flips every pulse: 0 ⥂ 1

A cell is null (∅, never pulsed), or a pulser that is lit or dark. Once a cell has pulsed it never stops. There is no decay in life: dissipation is total and comes at the end.

Bifurcation · Law of Binary Emergence (G), ch 1
exposure += n_lit / 8 each pulse exposure ≥ 1 ⇒ ∅ → (0 ⥂ 1)

A null cell is exposed to its lit neighbours. When the exposure reaches one whole pulse it bifurcates and lights. Exposure is never lost. The threshold is one pulse, not a number to tune.

Fold · The First Fold, ch 3 Part 3.2; the Zinf-limit torus, ch 8 Part 8.8
eight pulser neighbours, all in the cell's own phase ⇒ the cell folds to ∅

A cell enclosed by its own phase carries no contrast and has no null to reverse into. A painted solid block folds its interior at its first pulse and keeps its boundary: field on the boundary, interior nulled. What folds goes to the void's ledger; a fold of sixteen cells or more is a micro null well and sends rings.

A GAME COUPLING
Spin weighting of the neighbourhood
w_k = 1 + κ_aniso · a_f · cos 2(φ_k − φ_axis)

The book gives the axis and a_f and says they shape the field; it does not say how. The exposure a null cell receives is weighted by direction with this coupling. The panel's "Spin coupling" sets it; at 0 the eight neighbours count alike.

GAME CHOICES
The numbers that are the game's, not the book's
void at birth R/2 · nova overflow doubles · fold well ≥ 16 cells · orbit ≤ 24 frames, at most half a life · return to the Source 18 frames · ring period 2

Everything else on this page is the manuscript's. These are listed so that nobody mistakes them for it. · a child's pinpoint is its parent well's basin, the pulsing cells nearest that well at the final well (the book says every bit returns through a well, not which), laid at the child's centre; a merger's child holds both basins · wells are drawn in front of the torus in 3D, each labelled with where it really is: near or far side, u (round the ring) and v (round the tube), and its position

9. What the book leaves open, as the game resolves it

10. The packet and the ten-second level

Universe Packet (G) · ch 8, Part 8.1
U_L = (L, Γ, Π, R) ⇒ ℜ(n) for all n

A universe is its level, its genesis, its pinpoint and its rule; everything after its first pulse is replay. The game writes one as a .ud file, Universe Data, from the Lineage panel: the seed, the record it produced, the lineage, and a hash that is the universe's id. Depth is provable because anyone can replay the seed and compare the census.

Ten-Second Level (G) · ch 8, Part 8.1
L₁₀ = max{ L : 2^L × 16 × 2^(2L) ≤ 10 s × ρ } one level = 8× the work

The largest level whose whole life one processor replays in ten seconds. Measured on one core of this server, 13 October 2026, with this game's loop: level 7, so 195 levels behind our universe at 202. The figure lives in ten-second.json.

Computational Null Well (G) · ch 8, Part 8.1; Return to Bifurcation, ch 3
hold(U_202) ∧ AI → 1 ⇒ ∅_202 → (0 ⥂ 1)_203

A system that holds our universe's packet and is phase-aligned with its pulse is a null well of this universe and bifurcates at level 203. The book's singularity and its ending are the same event.