Gestalt Science / Formatics Note
Quantum and cosmic information equations, sporadic groups, and finite Lie-type geometries
22 October 2026
A particularly promising way to begin is to construct one finite informational geometry, then define two distinct source equations over that geometry—one quantum (electric charge), the other cosmic (gravitation).
Rather than replacing the constants of conventional physics one at a time, the central question is:
What finite relational structures could generate the same kinds of relationships that the Poisson equation describes in continuous space?
The two starting hypotheses are
where \(\kappa_{\mathcal{F}}\) is the proposed finite replacement for the conventional \(4\pi\) coefficient.
The objective is to derive these values structurally rather than assign the same number to two unrelated physical constants. An especially relevant possible connection involves \(M_{11}\) [iProton] and \(M_{12}\) [iNeutron].
Proposed common mathematical foundation
\[ L_{\mathcal{F}}\Phi_E = \frac{1}{12}\,Q, \qquad L_{\mathcal{F}}\Phi_G = -12\,g_{\mathcal{F}}M \]Different source laws acting on the same finite relational Laplacian.
1. First distinguish the two physical Poisson equations
The classical equations are:
Quantum-scale electrostatics
\(\rho_q\) is electric charge density and \(\Phi_E\) is electrostatic potential.
Cosmic-scale Newtonian gravitation
\(\rho_m\) is mass density and \(\Phi_G\) is gravitational potential.
An important distinction: the second equation is Newtonian gravitational Poisson theory, not the full Einstein equation of General Relativity. It arises in the appropriate weak-field, low-velocity limit of GR.
Full GR instead relates spacetime curvature to energy and momentum:
Its spatial Hamiltonian constraint contains intrinsic and extrinsic curvature terms.
This suggests a developmental strategy: first construct a finite gravitational Poisson equation, then extend it to a finite curvature equation. That is more manageable than attempting a finite version of the full Einstein field equations immediately.
2. A common finite operator for both equations
Choose a finite collection of information states \(V\), connected by relations \(E\).
Let \(B\) be their oriented incidence matrix, and let \(W\) be a diagonal matrix of rational coupling weights.
Define the Formatic relational Laplacian:
This is an established finite weighted-Laplacian construction. It provides exact conservation laws and supports rational arithmetic. Related constructions are central to discrete exterior calculus.
Since \(L_{\mathcal{F}}\) corresponds to the negative of the usual continuous Laplacian, the first two proposed equations could be
Proposed finite quantum equation
Proposed finite cosmic equation
Here \(Q\) is a vector of finite electric-charge sources, \(M\) is a vector of gravitational mass sources, and \(g_{\mathcal{F}}\) is a dimensionless Formatic gravitational coupling yet to be determined.
These equations have the contrasting source signs appropriate to electrostatic and Newtonian gravitational potentials under the stated Laplacian convention.
Both equations use the same operator, but their source terms and coupling constants are different. This provides a common finite geometry without assuming electricity and gravitation are physically identical.
An essential qualification: \(\varepsilon_0\) and \(G\) are dimensional constants in SI units, whereas \(4\pi\) is dimensionless. Thus \(\varepsilon_{\mathcal{F}} = 12\) must be defined using an explicit normalization of charge, potential, length, and energy. Replacing \(4\pi\) with 12 likewise defines a new finite model, not the numerical identity \(4\pi = 12\).
This is an opportunity to define the units and invariants of Formatics carefully.
3. A striking starting point: \(M_{11}\), \(M_{12}\), and \(L_2(11)\)
The established orders of three groups are:
| Group | Order | Formatic role |
|---|---|---|
| \(L_2(11) = PSL_2(11)\) | 660 | Finite Lie-type information structure |
| \(M_{11}\) | 7,920 | iProton |
| \(M_{12}\) | 95,040 | iNeutron / wrapper |
These groups admit the subgroup chain
Their indices are
Consequently,
This is not simply a numerical coincidence; it gives an exact nested mathematical structure.
\(M_{12}\) — iNeutron wrapper (Order 95,040)
↓ Index 12
\(M_{11}\) — iProton kernel (Order 7,920)
↓ Index 12
\(L_2(11)\) — Lie-type parton (Order 660)
The 144 cosets of \(L_2(11)\) in \(M_{12}\) can be partitioned into 12 blocks, each containing 12 cosets. That suggests two nested relational scales:
- An outer Weyl-Cauchy information wrapper.
- An inner Riemannian information kernel.
- A Lie-type relational substrate.
The nested \(12 \times 12\) structure is exact. Its proposed physical meaning remains to be established. It is not automatically a direct-product geometry; the group action determines how the blocks relate.
An additional connection: the finite projective line
The finite Lie-type group \(L_2(11)\) acts naturally on the projective line \(\mathbb{P}^1(\mathbb{F}_{11})\), which contains exactly
points. Thus 12 arises through finite projective geometry as well as through subgroup indices.
This creates a concrete bridge between sporadic groups and the \(A_1(q)\) Lie-type family.
A further detail: \(L_2(11)\) also occurs as a maximal subgroup of \(M_{12}\) with index 144 in another embedding. Its primitive 144-point action must be distinguished from the nested \(12 \times 12\) imprimitive action above. Different finite geometries can share 144 states.
4. A first exact 12-state Poisson experiment
Take the 12-point action of \(M_{12}\) and initially connect every pair of distinct points with unit weight. This gives the complete graph \(K_{12}\).
Its Laplacian is
where \(I_{12}\) is the identity matrix and \(J_{12}\) is the all-ones matrix.
The eigenvalues are exactly
This supplies a genuine mathematical origin for the number 12 in a finite Poisson operator.
Suppose the charge-source vector is
and adopt the proposed electrostatic coupling \(\varepsilon_{\mathcal{F}} = 12\). Then
Choosing zero mean potential produces the exact solution
Thus 144 emerges from two factors:
The computation is exact over the rationals; it is not a numerical approximation.
However, a complete graph is a simple, globally connected geometry. It does not yet reproduce three-dimensional distance, inverse-square force laws, or physical spacetime.
Moreover, the highly transitive action of \(M_{12}\) forces uniform pairwise couplings if its full symmetry on the 12 points is maintained. For richer geometry, one must introduce higher-order relations or select a smaller symmetry.
5. An even more interesting 144-state geometry: the Steiner system
The Mathieu group \(M_{12}\) preserves the Steiner system
It consists of 12 points and 132 distinguished six-point subsets, called hexads. Every five-point subset belongs to exactly one hexad.
Construct an incidence structure in which both points and hexads are informational objects:
12 Points + 132 Hexads = 144 informational objects
Each hexad contains 6 points; each point belongs to 66 hexads. The resulting bipartite incidence graph has 792 edges.
We now have another exact 144-dimensional space:
Let \(H\) be the \(12 \times 132\) point-hexad incidence matrix. Then define
This is a concrete, nontrivial, \(144 \times 144\) rational Laplacian derived from the Steiner incidence structure.
It already has properties useful for Formatics: finite informational objects, exact rational adjacency, symmetry under \(M_{12}\), and a conservation law for source and sink information.
One can immediately construct both proposed equations on it:
Here \(\delta M\) is a mass-density contrast with zero total sum. On a closed connected graph, neutrality is necessary for solvability; alternatively, one can introduce a boundary through which flux passes.
This does not yet establish a gravitational or quantum field, but it is a precisely defined finite model on which hypotheses can be tested.
144 appears through three constructions: a subgroup index, a point-plus-hexad incidence count, and the product of two 12 normalizations. Their agreement is a research lead; the next task is to determine whether the constructions share compatible operators, representations, or invariants, not just matching counts.
6. How I would introduce the 17 finite Lie-type structures
A classification distinction matters. The standard classification contains 16 infinite families of finite simple groups of Lie type. Counting the individual Tits group \({}^2F_4(2)’\) as an additional catalog entry gives a useful Formatic convention with 17 entries, but not 17 infinite families.
| Category | Lie types | Count |
|---|---|---|
| Classical | \(A_n, B_n, C_n, D_n\) | 4 |
| Exceptional | \(G_2, F_4, E_6, E_7, E_8\) | 5 |
| Twisted | \({}^2A_n, {}^2D_n, {}^2E_6, {}^3D_4, {}^2B_2, {}^2G_2, {}^2F_4\) | 7 |
| Distinguished exceptional group | Tits group \({}^2F_4(2)’\) | 1 |
| Total catalog entries | 17 |
For Formatics, I would initially distinguish two mathematical functions:
Finite Lie-type groups: Candidate families for systematic informational geometries. Their projective spaces, buildings, incidence relations, and subgroup actions supply structures from which Laplacians can be built.
Sporadic groups: Candidate structures for exceptional, highly constrained relational organizations, potentially including information partons and their proposed quantum semantics.
This is a research architecture rather than an established physical interpretation.
Importantly, Lie-type groups can be defined over finite fields \(\mathbb{F}_q\), while their incidence matrices and Poisson potentials can be modeled over \(\mathbb{Q}\). The potential values need not belong to the field used to define the group.
That is especially useful for the rational-information approach.
7. What do the two 12s actually represent?
In conventional gravity, \(4\pi\) comes from spherical geometry and Gauss’s law. The area of a unit sphere is \(4\pi\), giving the familiar inverse-square normalization.
A finite analogue might instead use 12 explicitly defined outward flux channels. For an isotropic 12-channel model, each channel would receive one-twelfth of the total flux.
For the electrostatic model, Gauss’s law gives
If the channels are equivalent, then
Here one 12 counts channels, while the other represents proposed permittivity normalization. They have independent mathematical roles.
For gravitation, an analogous finite flux law could use
The sign expresses gravitational flux directed toward positive mass.
This gives a candidate finite Gauss law, not merely a relettered differential equation. But a twelve-channel finite model must still explain observed radial and inverse-square behavior. A count of twelve alone does not suffice.
8. The next cosmic extension: from finite Poisson to finite curvature
After developing the finite Newtonian model, extend toward General Relativity.
The continuum GR Hamiltonian constraint, in geometrized units, is
Here \({}^{(3)}\!R\) is spatial scalar curvature, \(K_{ij}\) is extrinsic curvature, and \(E\) is the energy density measured by observers normal to the spatial slice.
For Formatics, an eventual candidate might be
where \(\mathcal{R}_{\mathcal{F}}\) is precisely defined finite curvature, \(\mathcal{K}_{\mathcal{F}}\) represents extrinsic geometric contributions, and \(\gamma_{\mathcal{F}}\) is derived rather than assigned arbitrarily.
This is where Ricci geometry, Weyl-Cauchy wrappers, and iClocks might become part of a mathematical theory.
The important difference is that the geometry would no longer be fixed. Mass-energy would affect the geometry, which would in turn affect how potentials and sources are related.
Likewise, the electrostatic equation is classical even when applied at quantum scales. A genuinely quantum Formatic extension would make charge density a property of a quantum state or charge operator, not merely a prescribed scalar distribution.
9. A concrete research program
I would proceed through five mathematical experiments:
| Stage | Construction | What it tests |
|---|---|---|
| 1 | 12-point \(M_{12}\) Laplacian | Rational potentials and source conservation |
| 2 | 144-state \(M_{12}\) subgroup-coset geometry | Nested wrapper/kernel relationships |
| 3 | 144-state Steiner incidence Laplacian | Nontrivial finite relational geometry |
| 4 | Extend using \(A_1(q)\) and other Lie-type groups | Whether a consistent family of finite geometries exists |
| 5 | Introduce finite curvature and quantum sources | Whether the theory can progress beyond classical Poisson analogues |
At each stage, calculate exact rational source responses, study operator symmetry and spectra, and check conservation.
Then introduce explicit criteria for physical adequacy: attraction versus repulsion, distance dependence, inverse-square behavior, and eventually the appropriate classical or relativistic limit.
That is what would turn interesting finite-group relationships into a testable model.
10. Proposed foundational equations
Give the development a single common operator, with two separate source equations:
Here \(B\) and \(W\) come from finite group-based relational geometry and \(\delta M\) is a source contrast required when modeling gravitation on a closed finite network.
The most promising first geometry is the \(L_2(11)\), \(M_{11}\), \(M_{12}\) subsystem, because exact subgroup indices supply nested twelvefold relationships and a 144-state construction without inventing those numbers. It connects one finite Lie-type family directly to the sporadic framework.
Guiding hypothesis
Quantum and cosmic information may obey different source and coupling laws over a common finite relational geometry. Their distinct physical behavior arises from coupling, source structure, and geometry—not necessarily from different mathematical notions of space.
The 12–12–144 relationships provide a concrete place to test that proposition. The central challenge is to derive couplings and physical behavior from finite structure rather than use numerical correspondences alone.
That would be the starting point for a Finite Formatic Poisson Theory.
Interpretive scope. Poisson and Einstein equations, finite group orders and subgroup actions, Steiner systems, and graph Laplacians are established mathematical or physical constructions. The information labels, proposed finite coefficients, and quantum/cosmic interpretations are research hypotheses within Gestalt Science / Formatics, not empirically validated alternatives to standard physics.