The universe’s foundational building blocks

• Planck base units
• Four selected irrationals: pi, phi, the square root of 2, and Euler’s number
• 202 base-2 notations
• The infinite: Our Assumptions
• The finite-infinite bridge: Our Assumptions
• Hexarchon (working draft of a Pre-Registration just below)

Preregistration

“Hexarchon” denotes the proposed geometric framework constructed from idealized spheres under the sphere-packing and stacking rules specified below. The study will examine the geometric relationships generated by those rules and evaluate the explicitly-stated hypotheses. The expression “finite–infinite causeway” names the framework’s proposed interpretation; that interpretation is not assumed to be established by the definition or by the geometric results alone. The broader framework hypothesizes that these geometric relationships participate in a physical mechanism; the present preregistration tests only the particular claims identified below.

Generation postulate. The Hexarchon model assumes that one new idealized sphere of specified Planck-scale dimensions is added per Planck-time interval within the defined generation domain. The sphere dimensions, time parameter, generation domain, and placement rule are specified below. This assumption is not presented as an established consequence of Planck units.

Generation postulate. The Hexarchon model assumes that one new idealized sphere of specified Planck-scale dimensions is added per Planck-time interval within the defined generation domain. The sphere dimensions, time parameter, generation domain, and placement rule are specified below. This assumption is not presented as an established consequence of Planck units.”

• Mathematical question: What structures and relationships follow when this rule is applied?
• Physical question: Does a real process follow this rule, and what independently measurable consequences would distinguish it from alternatives?

Proposed preregistration: Sphere-generation rules

1. Status and scope

The Hexarchon model postulates a discrete sequence of sphere-generation events. These rules define a mathematical construction. Whether that construction represents a physical process is a separate hypothesis and is not assumed in the geometric analysis.

In this protocol, “generation” means adding a new sphere to the model’s inventory and assigning it a location. It does not, by definition, establish the creation of matter, energy, or spacetime.

2. Sphere specification

Every generated sphere has the same fixed radius:

$$r=\ell_{\mathrm P},$$

where $\ell_{\mathrm P}$ℓP​ denotes the Planck length.

Each sphere is represented as an idealized closed ball, including its interior and boundary. Once generated, its radius remains unchanged.

The choice $r=\ell_{\mathrm P}$r=ℓP​, rather than a diameter of $\ell_{\mathrm P}$ℓP​, is an explicit model assumption.

3. Initial condition

At model time $t=0$t=0, one seed sphere exists, with its center at the origin:

$$\mathbf{x}_0=(0,0,0).$$

No other spheres are initially present. The seed is an initial condition, not an event counted within the subsequent generation sequence.

4. Generation clock and rate

The model uses a single global time parameter with successive generation times

$$t_n=n\,t_{\mathrm P}, \qquad n=1,2,3,\ldots$$

At each generation time, exactly one new sphere is added to the entire modeled system.

Consequently, immediately after event $n$n, the total number of spheres is

$$N_n=1+n.$$

This is a global generation rule. It is not one new sphere per existing sphere, per contact, or per region.

5. Placement and stacking

Before generation begins, the protocol shall specify a reproducible, ordered inventory of allowed sphere-center locations:

$$\mathbf{x}_0,\mathbf{x}_1,\mathbf{x}_2,\ldots$$

At event $n$, the new sphere is placed at $\mathbf{x}_n$​.

The inventory must satisfy these construction requirements:

• Distinct spheres have non-overlapping interiors:$$\|\mathbf{x}_i-\mathbf{x}_j\|\geq 2r \quad\text{for }i\ne j.$$
• Every new sphere contacts at least one previously generated sphere:$$\|\mathbf{x}_n-\mathbf{x}_j\|=2r \quad\text{for at least one }j<n.$$
• The site-selection procedure, layer sequence, and resolution of ties are fixed before the registered construction is run.
• Sites may not be selected or reordered to improve agreement with a desired result.

The exact packing-and-stacking algorithm must be included as an appendix or executable specification. These constraints alone do not uniquely determine that algorithm.

6. Persistence and permitted changes

After placement, spheres remain at their assigned locations. The baseline construction permits no deletion, splitting, merging, resizing, or movement.

Any later model allowing such changes must specify the additional rules separately. Sphere addition alone is not designated a demonstrated physical mechanism.

7. Physical properties

This baseline construction assigns sphere size, location, and generation time only. It does not automatically assign Planck mass, energy, charge, spin, or other physical properties.

Any such assignments require separately stated assumptions and, where applicable, accounting rules for conserved quantities.

8. Stopping rule and interpretation

Before execution, the preregistration shall specify the final event number $n_{\max}$nmax​, the quantities to be calculated, and the criteria used to evaluate each hypothesis.

Results will be described as consequences of this specified construction. They will not, by themselves, establish that spheres are physically generated or that a finite–infinite causeway exists.

Placement-and-stacking specification
Before the registered construction begins, the protocol will specify how each new sphere’s center is determined from the existing configuration. The specification will identify the initial arrangement, permissible contact relationships, stacking sequence, and a deterministic rule for choosing among otherwise equivalent available positions. Any feature deliberately left variable will be identified as a separate model variant before analysis.

Initial condition and generation postulate
At model time $t=0$t=0, the system contains one idealized sphere centered at the origin, with radius equal to one Planck length, $r=\ell_{\mathrm P}$r=ℓP​.

At each subsequent model time $t_n=n\,t_{\mathrm P}$​, for positive integer $n$, exactly one additional sphere of the same radius is generated within the entire modeled system. The initial seed sphere is not counted as a subsequent generation event.

Immediately after generation event $n$, the system therefore contains $N_n=n+1$ spheres.

The sphere-placement rule will be specified independently of any proposed correspondence with Galois representations. Possible arithmetic or representation-theoretic connections will be investigated separately and labeled exploratory unless explicitly formulated in advance. Computational implementation will reproduce the specified construction and will not introduce unstated placement choices.

Sphere	Model time	Center coordinates
1: seed	$0$0	$(0,0,0)$(0,0,0)
2	$t_{\mathrm P}$tP​	$(2r,0,0)$(2r,0,0)
3	$2t_{\mathrm P}$2tP​	$(r,\sqrt{3}\,r,0)$(r,3​r,0)
4	$3t_{\mathrm P}$3tP​	$\left(r,\frac{r}{\sqrt{3}},\sqrt{\frac{8}{3}}\,r\right)$(r,3​r​,38​​r)

Initial placement rule. The first four spheres are placed at the coordinates listed above, producing a tetrahedral configuration of mutually tangent, equal-radius spheres. These placements are stipulated initial construction rules, not consequences of the Planck-time generation rate. All subsequent placements must follow a separately specified continuation rule.

Decision record — 28 October 2026
The initial configuration consists of four equal, idealized spheres, each with radius $r=\ell_{\mathrm P}$, placed in three-dimensional Euclidean space. Their interiors do not overlap, and their centers remain fixed after placement.

The seed sphere exists at model time $t=0$t=0. The next three spheres are added at $t_{\mathrm P}$tP​, $2t_{\mathrm P}$2tP​, and $3t_{\mathrm P}$3tP​, respectively, using these coordinates:

$$\begin{aligned} \mathbf{x}_0&=(0,0,0),\\ \mathbf{x}_1&=(2r,0,0),\\ \mathbf{x}_2&=(r,\sqrt{3}\,r,0),\\ \mathbf{x}_3&=\left(r,\frac{r}{\sqrt{3}},\sqrt{\frac{8}{3}}\,r\right). \end{aligned}$$

This tetrahedral starting configuration is adopted as a model assumption. No SU(2), SU(3), Galois, or Langlands correspondence is assumed or tested in this initial placement specification.

Packing baseline. FCC is selected as the working packing geometry for the initial construction. This choice reflects the investigators’ longstanding consideration of FCC and HCP and their usual preference for FCC.

Retained alternative. HCP remains available for a separately specified future comparison. No FCC–HCP comparison is undertaken at this stage.

Previously approved assumptions. The construction retains spheres of radius one Planck length, a single initial seed, one additional sphere per Planck-time interval for the entire modeled system, and the approved fixed, non-overlapping tetrahedral first-four-sphere configuration.

Deferred work. Galois and Langlands connections, SU(2), SU(3), physical interpretations beyond the stated postulates, and Python implementation are postponed.

Approved placement specification

1. Prescribed first five spheres

The approved tetrahedral first-four-sphere configuration remains unchanged. Sphere 5 is added at model time $4t_{\mathrm P}$4tP​, at

$$\mathbf{x}_4=(r,-\sqrt{3}\,r,0), \qquad r=\ell_{\mathrm P}.$$

These first five placements are stipulated starting conditions for the subsequent filling procedure.

2. Fixed reference centroid

“Nearest centroid” means nearest to the fixed centroid of the original four-sphere tetrahedron:

$$\mathbf{c} =\frac{\mathbf{x}_0+\mathbf{x}_1+\mathbf{x}_2+\mathbf{x}_3}{4}.$$

This reference point is not recalculated as new spheres are added.

3. Eligible sites

Beginning with sphere 6, an eligible position must:

• Belong to the specified FCC lattice containing the approved starting configuration.
• Be unoccupied.
• Permit placement without overlapping any existing sphere’s interior.
• Place the new sphere in contact with at least one existing sphere.

4. Selection and tie-breaking

At each subsequent generation event:

1. Select the eligible site with the smallest squared Euclidean distance to $\mathbf{c}$c.
2. If multiple sites have the same minimum distance, select by ascending $z$z-coordinate.
3. If still tied, select by ascending $y$y-coordinate.
4. If still tied, select by ascending $x$x-coordinate.
5. Add exactly one sphere and leave all existing centers fixed.

The coordinate tie-break is an ordering convention, not a proposed physical preferred direction.

FCC lattice definition. Allowed sphere centers are all integer combinations of the three vectors from the initial seed center to the other three centers of the approved tetrahedral configuration. The first five placements are prescribed. Beginning with sphere 6, each new sphere occupies the eligible lattice site nearest the fixed centroid of the initial tetrahedron, with ties resolved by ascending 𝑧, then 𝑦, then 𝑥.