24JUL2026
Quantitative companion to the Theory. Derives the equations and algorithms the Worldline Map (dashboard) uses to size divergence and convergence events.
Everything below is read off the Theory page (the N-spherical multiverse, the two Rindler horizons and their ratio, the inverse-square binding, and the superposition/aggregation argument) plus the Frauchiger–Renner result. The map treats each qRNG draw as a divergence event (a hard fork of the worldline) and each confirmed purchase as a convergence event, and sizes each bubble by the number of worldlines it spans.
From Divergence: the multiverse is N-spherical with the radial dimension equal to the arrow of time, so “as the radius (time) increased the surface area (quantifiable worldline count) does as well.” Our matter occupies three spatial dimensions independent of time (Dimensionality), so the worldline-bearing surface is a 3-sphere and the worldline count scales with the time-radius as:
Inverting, a bubble carrying W worldlines has a screen radius proportional to the cube root of that count:
where R_{0} is the pixels-per-worldline-unit visual scale (R_{0}=6 in the current map). Every event type below reduces to this one law — only the worldline count W differs.
A draw of b quantum-random bits hard-forks the worldline into one branch per combination — the full superposition:
Here b is the draw's total_bits. Radius grows as the cube root of the branch count, so a draw that consumes three times the bits is twice as wide.
Options may carry weights, so a decision need not be uniform (“lean 75% pasta, 25% steak”). The weights are normalised once, at commit time, and stored on the event — which is what lets the map place and shade the adjacent worldlines that were not realised:
From Convergence and Origin of Inertia: binding between moments follows a “nearly inverse square drop-off” (Mach's principle). A purchase committing resource m therefore has a binding reach \propto\sqrt{m}, and the 3-sphere volume of worldlines merged within that reach is:
The cube-root law collapses the 3/2 exponent to a square root, so convergence bubbles scale as the square root of the amount — symmetric with divergence, one drawing worldlines apart and the other pulling them together. The amount is only the entry cost: \Pi is the persistence factor from Ongoing Life-Interaction below, which is what a commitment actually binds with. For a genuine one-off \Pi=1 and this reduces to m^{3/2}.
From Infinite Universe, the counter-inductive and primary Rindler horizons are:
d_{\text{ci}}=\frac{c}{e\left(e+2\right)}=6.23016\times10^{25}\ \text{m}
d_{\text{p}}=\frac{c}{e}=1.24603\times10^{26}\ \text{m}
whose ratio is the resonance constant — “just over a division by half… approximately equal to the inverse square law”:
The bounding ring drawn around each bubble is the shell at which the 1/d^{2} binding drops to the resonance fraction \rho of its central value — i.e. \rho=r^{2}/d^{2}:
From Global Consciousness: “there are some feedback effects both toward individual souls and from individual souls back into the global consciousness, however the bulk of the effects occur through morphological resonance in the local region of space.” A confirmed convergence therefore does not only bind the worldlines it merges — it resonates back into the neighbourhood around it. Every confirmed purchase commits three linked events, not one.
The feedback event is that binding resonated back into the local region, attenuated by the resonance constant:
The buffer event is the extent of the local region feedback acts over — the bounding shell derived below, at which the 1/d^{2} binding has fallen to the resonance fraction of its central value:
On the map an adjacent worldline lying inside r_{\text{buf}} is pulled back toward the realised path, and the feedback reaching one at separation d falls off inverse-square (Mach binding):
Each event bounds a region of worldlines it can causally touch: a divergence opens a forward cone (where its outcomes can reach), a convergence a reverse cone (where it could have arrived from). Writing u=\sigma_{e}(x-x_{e}) for displacement along the event's own time direction (\sigma_{e}=+1 forward, -1 reverse) and \Delta y=y-y_{e} for separation across worldlines, the cone is:
Its opening follows from the event's own radius — a worldline set widens by one radius per step of time — and it is carried until the binding falls below the display cutoff \varepsilon:
Occupancy inside the cone is not uniform. It falls off by the same inverse-square binding the resonance already gives feedback, measured from the apex — there is no separate profile:
The sums run over the committed events of every known worldline in the neighbourhood — the observer's own and those of the other participants — not merely the observer's. That is what makes the result a statement about worldline sets meeting rather than a worldline crossing its own past:
Each is normalised against its own maximum so the picture is scale-free, and since a point is only a genuine crossing if both a forward and a reverse set occupy it, the probability that two worldline sets intersect there is the joint:
This is what the volumetric overlay paints: cool where only reachable, warm where only originable, and bright where P_{\cap} is high — the neighbourhoods in which major and likely-major worldline sets actually meet. A neighbour's events are placed on the observer's own time axis, so a crossing at a given point is a crossing at a shared moment. Only two quantities here are presentation rather than physics: the cutoff \varepsilon=10^{-3} that bounds how far a cone is drawn, and a gamma \gamma=0.45 on P_{\cap} (0.55 on the fields) so the low end stays visible. Because K_{e} is concentrated near the apex, intersections light up only where cones genuinely overlap close to their sources — a distant coincidence of two large cones does not register as a crossing.
From Superposition: a superposition persists “so long as the aggregate effects are dominant over the observations between… systems, and multiple observers can disagree.” The Frauchiger–Renner theorem makes this quantitative: with the four nested agents (F̄, F, W̄, W) the consistent single-outcome description breaks down with probability exactly \frac{1}{12} per run.
Each recorded observation against a draw is one nested-observer layer; per the resonance it aggregates out a \rho-fraction of the still-distinguishable worldlines. Below four observers no contradiction can arise, so collapse is complete:
The forms below are written for a one-off, \Pi=1. In general the exponent is damped by persistence — read every \rho^{L} here as \rho^{L/\Pi}, per Ongoing Life-Interaction: an outcome that keeps being lived with does not aggregate away however many observers pass over it.
At four or more observers the FR breakdown leaves an irreducible residual — over the average, 1/12 of runs stay branched while 11/12 collapse:
where L is the number of observations recorded against the draw. As observations accumulate the divergence shrinks toward the axis (it converges as it is measured) but can never fall below the FR floor:
Because the contradiction only appears at the fourth agent, adding the fourth observer increases the residual (2^{b}\rho^{4}=2^{b}/16 jumps up to 2^{b}(\frac{1}{12}+\frac{11}{12}\rho^{4})): the onset of the FR paradox re-branches the system before further observation grinds it down to the 2^{b}/12 floor.
The 1/12 breakdown can be rendered two ways, selectable on the map, because they answer different questions:
Below four observers the two coincide (2^{b}\rho^{L}) — there is no contradiction to realize or average over.
Bit count alone does not decide whether an event matters. “Which meal did I eat” aggregates back out; “spend 10k on a bathroom renovation” versus “spend 10k on livestock fencing” does not, because the outcome goes on being lived with. From Morphic Resonance: the temporal magnetism of resonance pulls a soul toward the bulk of worldlines in which the thing is already natural — and every re-encounter is another affirmation pulling that way.
A convergence is not a point event either. Buying a piece of wall art is not the acquisition of wall art; it is a commitment to walk past that wall art daily for as long as it exists. So the binding quantity is the total number of future re-affirmations — how often the outcome is re-encountered, for how long:
which enters the maths as a persistence factor, unity for a genuine one-off and growing slowly with a lifetime of repetition:
For a divergence, persistence damps the collapse exponent: an outcome you keep living with does not aggregate away however many observers pass over it, while a one-off collapses at the full Rindler rate.
For a forced convergence, the same interaction strengthens the binding — the amount paid is only the entry cost:
Worked example: wall art at f_{\text{day}}=1 over ten years gives I=3650 and \Pi\approx 9.2, so it binds roughly 9.2^{3/2}\approx 28 times harder than its price alone implies — and a decision carrying that persistence still stands at \rho^{L/9.2} where a trivial one would have collapsed to \rho^{L}. Interaction is recorded per decision tree and per acquisition item, and archived on every event it produced.
All of the above is implemented server-side in modules/quatism/inc/worldline.js, which is the single authority for the maths. It is evaluated once, when an event commits — never re-derived while rendering — and the result is stored on the event. The map only draws what it is handed.
| Symbol | Value | Meaning | Source |
|---|---|---|---|
| R_{0} | 6 px | pixels per worldline-unit | visual scale |
| \rho | 0.5000008025488953 | Rindler horizon ratio | Theory § Infinite Universe |
| FR threshold | 4 observers | agents for the FR contradiction | Frauchiger–Renner |
| FR breakdown | 1/12 | contradiction probability per run | Frauchiger–Renner |
| \Pi | 1+\ln(1+I) | persistence from ongoing life-interaction | Theory § Morphic Resonance |
| I | f_{\text{day}}\times T_{\text{days}} | total future re-affirmations | per tree / per item |
| algo version | 4 | revision of the maths a stored event was computed with | worldline_event.algo_version |
| \varepsilon | 10^{-3} | cone cutoff — how far a cone is drawn | display only |
| \gamma | 0.45 / 0.55 | gamma on the intersection / the fields | display only |
The maths is event-driven: it runs at the moment an event commits, and rendering never re-derives it. This is what keeps the map's cost independent of how much history exists.
Confirming that you followed a drawn outcome re-runs step 4 for that event: the new observer layer changes its aggregation, and it becomes fully committed.
Node size on screen is scaled by \log W rather than by r directly: at 16 bits the physical radius is already far beyond any pixel size, so every node would clamp to the same maximum. The physical r is what the detail panel reports.