Six surrounding spheres
Octah
An octahedral interstice, shaped by six inward-curving spherical surfaces.
Lattice Structure Manifold · A research theory
Two shapes.
Zero parameter.
In LSM, matter begins with geometry. Pack equal spheres together and two shapes emerge between them: the octah and tetrah. Ben Đ Om’s construction builds outward from these shapes to particles, mass and the structure of space. The latest development selects an enclosing proton geometry and sets out an integer framework constrained by observation.
O
Four surrounding spheres
Tetrah
A tetrahedral interstice, shaped by four inward-curving spherical surfaces.
September 29, 2026 · Author-adopted development
A complete cage.
One additional octah.
Ben Đ Om selects 8T + 7O as the leading LSM proton geometry: the complete first shell around a lattice host, plus one exterior octah. Its charge is +e. Its shell contains 88 triangular face incidences.
The exact enclosure minimum, the selected physical interpretation, and the observation-constrained mass calculation are set out separately below. The complete v15 catalog, earlier models and derivations remain part of this publication.
8T + 7O
12T + 9O
| Exact FCC property | Selected 8T + 7O | Retained v15 12T + 9O |
|---|---|---|
| Cell count | 15 | 21 |
| Charge Q in units e/6 | 6 | 6 |
| Raw faces F = 4T + 8O | 88 | 120 |
| Shared faces I | 25 | 32 |
| Exposed triangular faces F − 2I | 38 | 56 |
| Most surrounded host | 8T + 6O · complete cage | 8T + 5O · one O missing |
| Point group | C3v · threefold axis | C4v · fourfold axis |
| Symmetry operations, including reflections | 6 | 8 |
| Single-face connections that disconnect the cluster when removed | 1 | 0 |
The missing first-shell octah is a property of the retained v15 arrangement, not a claim about every possible arrangement with 12T + 9O counts.
The enclosure minimum, from first principles
- A complete FCC first shell around one host contains eight T cells and six O cells: 14 compulsory cells.
- Its integer charge is Q = 2 × 6 − 8 = 4.
- Proton charge is Q = 6. At least one additional cell is therefore required.
- One O supplies exactly 2 charge units and can share a face with the cage. The 15-cell 8T + 7O construction achieves the lower bound.
Result: 8T + 7O is a minimum-cell +e cluster subject to containing this complete FCC first shell. Ben adopts it as the leading proton geometry within LSM. Positive charge alone does not impose enclosure: the retained 2T + 4O positron is a smaller connected +e cluster.
The seventh octah attaches through one triangular face. The v15 proton has more redundant face connections. The selected geometry has less occupied interstitial volume and less full cavity-boundary area, but a dynamical energy and transformation rule must determine resistance to separation and physical stability.
V(v15) − V(selected) = 4VT + 2VO > 0
A(v15) − A(selected) = 4sT + 2sO + 18a > 0
A(v15) − A(selected) = 4sT + 2sO + 18a > 0
Here sT, sO and a are the curved-wall and aperture areas defined in the retained surface proof. A mass rule depending only on T and O cannot distinguish different arrangements with the same counts.
Exact coordinates, flat reference views and reproducibility
Use host positions at integer triples with odd coordinate sum, octah centers at the complementary integer triples, and tetrah centers at half-integer triples. Set the selected host to (1,0,0). Put T at all eight (1 ± ½, ±½, ±½), six O at that host plus or minus each coordinate unit vector, and the seventh O at (2,1,1). Multiply coordinates by √2 for unit host radius. The checker uses doubled integer coordinates, so adjacency does not depend on rendered mesh tolerances.
Two cells share a face when their host-vertex sets share three vertices. The comparison uses the site's original v15 coordinates without rearrangement. The regular flat cells below have vertices at host centers; they extend into the original sphere regions. They are incidence references, not an established compressed-monad tiling with an independent host cell occupying the same volume.
Computed comparison and mass-search results · Standalone Python checker · Exact integer-coordinate inputs. The check verifies contacts, symmetry, connectedness, enclosure, bridge faces and the bounded integer mass search.
The fully packed limit
A true monad has only flat faces
Ben's defining distinction: a true Monad is fully packed and every component boundary is flat. Its four-part inventory remains one host component, two tetrah and one octah. The sphere describes the undeformed host; in the true-monad limit the host is polyhedral.
Mtrue = Sflat + 2Tflat + Oflat
Planar boundaries · disjoint interiors · no unfilled volume · Q = 0
Planar boundaries · disjoint interiors · no unfilled volume · Q = 0
Flat faces do not require the three component types to become identical shapes. Ben proposes deep black-hole interiors as the physical realization of this limit. A deformation law and spacetime metric are needed to establish where that limit occurs. The adopted names Monad, Mobnad and Mocnad and their original T/O inventories remain as defined in the complete composite-unit note.
A constructive route to a flat partition
A weighted Voronoi cell at site pi contains points x satisfying |x − pi|² − wi ≤ |x − pj|² − wj for every j. Pairwise boundaries are planes, and the cells partition space with disjoint interiors. Applying this to host and interstice centers gives a precise family of fully packed constructions to investigate.
Site positions, weights, nonempty-cell conditions and face topology must still be specified. This construction does not automatically retain four physical faces for T and eight for O; the literal-face mass rule must be checked against the resulting cells. The 1S:2T:1O inventory is exact bulk accounting, while a four-whole-piece repeating tile needs an explicit assignment. An HCP primitive translation cell contains two host sites.
Observation-constrained integer foundations
Count in integers. Derive the physical scale.
LSM's proposed microscopic description uses integer inventories and integer updates. Observations constrain which constructions to investigate; their decimal values in conventional units do not require decimal-valued microscopic constituents.
The v14 normalization, checked against the actual formula
In Particle Masses v14, “Particle Masses 2,” cell G2 uses =((D2*(1))+(E2*2))*(51.099895), with D2 = 8 and E2 = 1. Thus 51.099895 keV/c² multiplies T + 2O. The literal face count is four times this weighted count.
W = T + 2O; F = 4T + 8O = 4W
mbare = W × 51.099895 keV/c² = F × 12.77497375 keV/c²
mbare = W × 51.099895 keV/c² = F × 12.77497375 keV/c²
One literal face incidence therefore has the v14 conversion 12.77497375 keV/c². A shared interface contributes two incidences, one for each adjoining cell. This mass inventory is different from the exposed-face count. Updating the calibration to CODATA's electron mass changes the conversion to me/40 = 12.77497376725 keV/c²; calibrating from the electron is not an independent prediction of its mass.
The selected proton beside observation
Holding the existing node contribution of 73,360 face units fixed gives the following comparison. Applying that same contribution to the new cage is a stated assumption to investigate.
| Construction | Integer face-unit mass | Ratio to the 40-unit electron |
|---|---|---|
| Retained v15 proton | 73,360 + 120 = 73,480 | 1,837 |
| Selected 8T + 7O proton | 73,360 + 88 = 73,448 | 9,181/5 = 1,836.2 |
| 2022 CODATA observation | Measured reference | 1,836.152673426(32) |
The selected geometry reduces the discrepancy from about +461 parts per million to +25.775 parts per million. Its remaining excess is about 24.184 keV/c², or 1.89306296 face units with electron calibration. This is larger than the measurement uncertainty.
Why a finer unit alone cannot remove the residual
For any +e T/O inventory, Q = 2O − T = 6, so F = 16O − 24. A neutral T/O core adds a multiple of 16. Relative to the 40-face electron, the bare additive ratios therefore lie on a grid with spacing 2/5. The measured ratio is not on that grid. Subdividing all faces equally leaves the ratios unchanged.
The next physical rule must account for binding, excitation, host coupling, or a different relation between face inventory and rest energy. The microscopic counts can remain integers. The additional contribution must follow from shared local rules rather than a separate adjustment for each particle.
A precise integer unit convention
Let K and H be positive integers: K mass/energy quanta per bare triangular face incidence, and H elementary action quanta in ℏ/2. Define compatible reference units:
q* = e/6; μ* = me/(40K); a* = ℏ/(2H)
E* = μ*c²; τ* = a*/E*; ℓ* = cτ*
In these units: c = 1; e = 6; ℏ = 2H
E* = μ*c²; τ* = a*/E*; ℓ* = cτ*
In these units: c = 1; e = 6; ℏ = 2H
| Inventory | Integer charge Q | Bare mass count |
|---|---|---|
| Tetrah | −1 | 4K |
| Octah | 2 | 8K |
| Electron / positron | −6 / 6 | 40K / 40K |
| Monad / Mobnad / Mocnad interstitial parts | 0 / 0 / 0 | 16K / 32K / 48K |
| Selected / v15 proton shells | 6 / 6 | 88K / 120K |
| Current neutral node | 0 | 73,360K |
M = K Ftotal + Nexcitation − Nbinding
All counts on the right are integers.
All counts on the right are integers.
Position is represented by integer lattice labels and evolution by integer update counts. H permits spatial/time refinement independently of K. A spin projection ℏ/2 has action count H. The unit definitions do not by themselves derive discreteness, the values K and H, or the binding and excitation counts. Host contributions to the Monad classes remain to be specified; the proton's node term is not silently added to every neutral inventory.
The 346 / 655 search result and its exact limits
For the selected cage plus the unchanged node, choose zero electron binding correction as the calibration reference and introduce one integer proton binding count B. Then Me = 40K and Mp = 73,448K − B.
An exhaustive positive-integer search for the smallest K matching the CODATA ratio within one quoted standard uncertainty gives K = 346 and B = 655.
Me = 13,840; Mp = 25,412,353
Mp/Me = 25,412,353 / 13,840
Mp/Me = 25,412,353 / 13,840
This is an observation-constrained arithmetic fit. Neither 346 nor 655 has yet been derived from the geometry or an update rule. It demonstrates compatibility with one measured ratio, not a fit to the full particle spectrum. This quantum is about 36.9 eV/c², too coarse for the observed light-neutrino sector under a universal positive integer rest-mass rule. It is a search candidate, not the final fundamental scale.
Further constraints: neutrinos, geometry and couplings
The 2T + 1O interstitial inventory has bare mass 16 face units, about 204.4 keV/c². Its identification with observed light neutrinos needs an additional rest-energy, binding or propagation mechanism: KATRIN's 2025 result bounds the effective electron-neutrino mass below 0.45 eV/c² at 90% confidence. Adding a positive host mass does not resolve that discrepancy.
Integer coordinates can yield exact square-root distances. An octahedron's edge/diagonal ratio is invariant and irrational; changing units cannot make both lengths integer multiples of one common nonzero length. Integer microscopic inventories and update rules do not require every derived ratio or average to be an integer.
The gravitational constant and electromagnetic coupling remain dynamical constraints. This convention does not derive G or the fine-structure constant. The next task is a common local integer rule for binding, excitation and host coupling that reproduces the electron, a proton and light-neutrino behavior with the same parameters.
Particle Masses v14 source workbook · NIST mass reference · KATRIN result · Complete research note
The construction and its tests
Geometry.
Integer counts.
Observation.
LSM is a zero parameter program: the intended final construction determines its physical relationships from shared geometric and dynamical rules, without adjustable parameters. Observations have guided the search: theory was fitted to data to narrow the possible constructions. That discovery process does not replace the remaining task of deriving the selected values from first principles.
The geometric program starts with explicit lattice cells, counts their faces and charge, and tests which connected structures satisfy the chosen conditions. The selected 8T + 7O proton reaches the minimum for a complete FCC first-shell cage at +e.
The mass program keeps microscopic counts integer-valued and uses observations to constrain the remaining rules. The electron calibration and the 346/655 mass-ratio fit are identified as such. Their physical origin, binding law and wider particle agreement remain to be derived.
Inspect the integer construction and its constraintsEarlier premise · September 28 text with the author’s September 29 headline wording
The first-principles objective remains part of LSM. The earlier wording below predates the observation-constrained development. It is preserved as history; it does not describe the new fitted arithmetic example as a parameter-free prediction.
Lattice Structure Manifold · A research theory
Two shapes.
Zero parameter.
In LSM, matter begins with geometry. Pack equal spheres together and two shapes emerge between them: the octah and tetrah. Ben Đ Om’s zero-parameter proposal builds outward from these shapes to particles, mass and the structure of space.
The zero-parameter premise
Geometry sets
the numbers.
Ben Đ Om’s LSM is built on a zero-parameter claim: all model-defining numbers are extracted from first principles. Particle counts, charge assignments and mass relationships follow from the theory’s construction, with no constants fitted to experimental targets.
Start with packing. Follow the structure. Check the numbers. The public models below expose the geometric counts, the particle accounting and the comparison with experiment.
Follow the numerical comparison →The construction, step by step
Space has structure.
Matter follows.
Begin with a close-packed lattice. Build particles in its interstices. Connect their structure to charge, inertia and mass. LSM brings these ideas into one geometric account, extending from the smallest building blocks to cosmic expansion.
Follow the construction from the lattice upward. Exact geometric results, LSM’s physical interpretation and measured reference values are identified at each step.
01 / The substrate
The geometry beneath matter
LSM postulates an effectively unbounded lattice of primordial black holes near the Planck scale. Its nodes form close-packed HCP or FCC arrangements; the spaces between them provide a second level of structure.
The spheres here represent the proposed nodes. HCP repeats two layers, ABAB; FCC repeats three, ABCABC. Their local interstices are the tetrah and octah shown above.
Exact equal-sphere packing fractionπ / (3√2) = 74.048%
This is occupied volume in ideal Euclidean packing. Identifying it with a dark-sector mass or energy fraction is a separate physical hypothesis.
Dark matter, dark energy and the 74% idea
The proposal associates the hidden substrate with dark matter and the expansion mechanism with dark energy. A volume fraction alone does not determine either energy density. Planck 2018’s base ΛCDM fit gives about 31.5% total matter and 68.5% dark energy; total matter includes both ordinary and dark matter. Planck Collaboration reference.
The existence, stability and close packing of Planck-scale black holes are model assumptions here.
HCP and FCC stacking
02 / Growth and expansion
A mechanism for growing space
In the proposed mechanism, densely surrounded nodes cannot accommodate a larger merger product without rearranging their neighbors. Ben proposes that this local constraint, together with a drive to minimize surface area, produces outward pressure throughout the lattice.
Fusion-born neutrinos are proposed as feedstock: as their motion relative to the local lattice tends toward zero, they are incorporated into new nodes. Growth distributed through space would then drive expansion throughout space.
The incorporation process, merger constraint and expansion rate still need a dynamical law. No capture threshold or cosmological rate is assigned in this illustration.
The proposed growth cycle
- 01Fusion
Rearrangement into denser nuclei
- 02Neutrino emission
Small interstitial structures leave the reaction
- 03Local slowing
Motion decreases relative to the proposed lattice
- 04Incorporation
Neutrinos become lattice feedstock
- 05Node growth
Distributed growth is proposed to drive expansion
What Hawking’s area theorem establishes
For classical black holes under the theorem’s assumptions, total horizon area does not decrease: Afinal ≥ A1 + A2. This allows mergers; it does not prove that a crowded lattice forbids them. Observed mergers have been used to test the area law. Isi et al., 2021.
The diagram compares equal, non-spinning, uncharged holes with a hypothetical mass-additive final Schwarzschild hole, ignoring radiation. The LSM crowding rule requires an additional argument, especially near the Planck scale.
03 / Interstitial matter
Charge, counted in shapes
LSM proposes that smaller black-hole clusters occupy the interstitial regions and assemble into particles. Ideal close packing supplies exactly two tetrahedral sites for every octahedral site in the bulk lattice.
Particle Masses v15 assigns −⅙e to T and +⅓e to O. The candidate counts then reproduce the listed net charges. Selecting stable arrangements and deriving a full particle spectrum require more than this charge bookkeeping.
Explore all nine rotating candidates →The charge ledger
q / e = (−T + 2O) / 6
- Neutral building unit
- 2 T + 1 O 0e
- Electron candidate
- 8 T + 1 O −1e
- Proton candidate
- 12 T + 9 O +1e
04 / Inertia
Inertia begins at the interface
The proposed source of inertia is coupling between a particle’s connected interstices and the surrounding lattice. Their shared planar openings interrupt the local curved geometry; accelerating the cluster would require the lattice response to change.
Ben describes this as “drag” on the lattice. To recover ordinary inertia, the required response is resistance to acceleration, while uniform motion remains possible without continual braking.
Recovering Newton’s laws requires a quantitative response law and momentum conservation. The geometry identifies candidate coupling regions; it does not yet derive those laws.
Where the proposed coupling acts
Target response: Freaction = −meffa
05 / The proton mass increase
A geometric origin for the mass gap
LSM proposes a distinction between a small interstitial cluster and one that encloses a substrate node. Enclosure is hypothesized to tie the particle more strongly to the lattice, producing a large increase in effective inertial mass.
In v15, the proton’s 12T + 9O arrangement contributes 120 chart units. Adding the neutral node’s 73,360 gives 73,480, compared with the electron’s 40.
The rotatable diagram isolates the enclosure idea. Its complete first shell is 8T + 6O, not the full proton recipe. The published proton arrangement has 8T + 5O around its most surrounded host; establishing a complete enclosing geometry remains a separate construction step.
A node inside its surrounding sites
8 T + 6 O · complete first shell
06 / The node hierarchy
4,585: inside the proposed node
The proposal connects the roughly 1,800-fold baryon/electron mass scale to a node with 4,585 smaller constituents, and interprets the factor 2/5 as a clue to a “2, then 5” hierarchy of division or assembly.
An independent FCC count does contain exactly 4,585 lattice sites at r² = 168, with 48 sites on that outer shell. Here the FCC coordinates are integers with even x + y + z; nearest-neighbor distance is √2.
The geometric count is exact. The next part of the public derivation is to show why physical stability selects this shell and why the 2/5 factor follows from the same construction.
The counted FCC sphere
4,585 sites total · r² = 168
Inspect the nearby count plateaus
Here N(166) = 4,537, N(168) = 4,585 and N(170) = 4,729. Every even/odd pair in this window shares a count because squared radii of even-parity FCC points are even integers. A counting plateau alone does not establish physical stability.
Retained v15 numerical baseline. The selected 8T + 7O comparison and integer mass search appear in the September 29 development.
Put the numbers beside nature
Three ratios to keep distinct
The proton-to-electron mass ratio is a concrete place to examine the construction. Here are the node contribution, the complete v15 accounting and the measured reference, side by side: mp/me.
Node term alone
1,8344,585 × 2/5−0.11724% from referenceComplete v15 proton
1,83773,480 / 40 = 1,834 + 3+0.04615% from reference2022 CODATA reference
1,836.152673426Standard uncertainty: 0.000000032NIST reference value ↗These are approximate numerical correspondences, not exact experimental agreement. The proposed hierarchy and merger rule remain hypotheses to derive and test.
Reproduce this retrospective v15 baseline: download the runnable work pack or inspect the exact residual report. The package includes an independent FCC enumeration, source inputs and checksums.
Foundation note · September 28, 2026
The monad.
The closed electron shell.
Ben Đ Om’s question: can one indivisible unit, a completely occupied octah shell, and its positive partner establish LSM from first principles?
This note supplies exact definitions, a conditional electron minimum theorem, and a constructive test of the proton selection claim. It extends the existing theory and retains every published particle candidate. The geometry is established within the stated model; physical particle identity and dynamical stability remain separate requirements.
Author update · September 28, 2026 · 11:37 p.m. Central
Monad = one sphere, two tetrah, one octah
Ben Đ Om’s adopted definition: the Monad is the minimal complete composite unit of the proposed substrate: one HCP host sphere S, two tetrah T and one octah O. It has four pieces of three geometric types, with the tetrah occurring twice. The components remain distinct parts of one unit; they do not need to collapse into a common shape.
Monad = S + 2T + O
q(Monad) = q(S) + 2(−e/6) + e/3 = 0, with q(S) = 0
q(Monad) = q(S) + 2(−e/6) + e/3 = 0, with q(S) = 0
September 29 distinction: the true Monad is the fully packed, entirely flat-faced limit. The sphere below is its undeformed host reference.
The host sphere is taken to be neutral in this definition. The interstitial part, 2T + O, is the existing small-neutrino inventory. Ben identifies the complete unit with the minimal-size LSM neutrino and names it Monad; the medium and large LSM neutrinos are Mobnad and Mocnad.
| Adopted name | Retained size label | Existing T/O inventory | Existing chart mass |
|---|---|---|---|
| Monad | Neutrino (small) | 2T + 1O | 16 |
| Mobnad | Neutrino (medium) | 4T + 2O | 32 |
| Mocnad | Neutrino (large) | 6T + 3O | 48 |
The current gallery renders the interstitial T/O structures. The complete Monad definition additionally includes S. Mobnad and Mocnad have twice and three times the neutral interstitial inventory; their names do not prescribe a new host-sphere arrangement. Their existing geometries and chart mass values are retained. Incorporating a host into the geometric definition does not automatically add the large node mass term to the neutrino mass: background and excitation energy must be accounted for explicitly.
Why the 1 sphere : 2 tetrah : 1 octah ratio is exact
In ideal close packing, each sphere is incident to eight T voids, each shared among four spheres, and six O voids, each shared among six spheres. The per-sphere allocation is therefore 8/4 = 2 tetrah and 6/6 = 1 octah. The nearest-site Voronoi domain consists of the sphere plus its allocated portions of those surrounding voids. This is an exact partition; a single Voronoi cell contains fractional portions of many voids, rather than just three complete holes.
V(Monad allocation) = V(S) + 2V(T) + V(O) = 4√2 R³
Using the curved-void formulas below, the identity follows from 8ΩT + 6ΩO = 4π. Thus the proposed four-piece inventory has exactly the per-sphere space-filling volume. A specific four-whole-piece motif still needs an explicit assignment and boundary if it is to be used as a literal repeating tile. Its total volume alone does not prove that a chosen arrangement tiles space.
For HCP, the primitive translational repeat has a two-sphere basis: 2S + 4T + 2O in bulk accounting, or two Monad inventories. FCC permits a one-sphere primitive repeat. This distinction preserves the exact HCP/FCC geometry while giving the per-sphere unit one name. IUCr close-packing reference.
For a nonempty inventory containing both T and O, charge neutrality requires T = 2O. The least positive integer solution is T = 2, O = 1. This proves the minimum neutral mixed interstitial inventory. Adding the associated sphere supplies the complete unit in Ben’s definition.
Definition history
This author update supersedes the earlier proposed use of “Monad” for an indivisible state-bearing Voronoi unit. The current term denotes the complete S + 2T + O composite. The earlier codification is retained below as history, together with its useful coordinate definitions and compression analysis. The existing 4,585 count is also retained; this naming update does not establish a new assembly relation between that count and the complete Monad.
Earlier Voronoi codification and compression analysis · retained history
The Monad definition in these earlier paragraphs was superseded by the author’s S + 2T + O definition above. Their geometry and original wording remain available.
1 / Exact codification
A finite unit, a cell, and a physical hypothesis
Working codification: a Monad is one indivisible state-bearing unit at a declared terminal level of the LSM construction. Represent unit i by a site xi, its local state si, and its Voronoi domain Vi. “Indivisible” means the model assigns no lower-level constituents to that unit. The domain records its spatial allocation; a rule for si would record its physics. This is a precise proposed convention, not evidence that nature contains such units.
Mi = (i, si); Vi = {x : |x − xi|² ≤ |x − xj|² for every j}
The cells tile space with disjoint interiors. Ties lie on shared boundaries and can be assigned by a fixed index rule. No infinite regress is required: choose one terminal site set. A composite substrate node must then have a specified, finite assembly map from those units. A unit cannot simultaneously be indivisible and contain 4,585 lower-level units. The existing 4,585 count and the node’s 9,170T + 4,585O construction are retained; neither count alone fixes the number of terminal Monads.
Exact HCP and FCC coordinates, with no decimal specification
Let d > 0 be the nearest-neighbor distance, R = d/2 the touching host-sphere radius, and m, n, k integers. For ideal HCP set a1 = d(1,0,0), a2 = d(½,√3/2,0), a3 = d(0,0,√(8/3)), and b = (a1 + a2)/3 + a3/2. The site set is {ma1 + na2 + ka3 + εb : ε ∈ {0,1}}. HCP uses a two-site basis; it is not a one-site Bravais lattice.
For FCC the site set is {(d/√2)(i,j,k) : i,j,k ∈ ℤ and i+j+k is even}. Both definitions and their Voronoi half-spaces are exact. For a site translated to the origin, each domain inequality is 2x · xj ≤ |xj|². Choosing a physical value of d is an additional physical step.
| Object | Exact role | Geometry |
|---|---|---|
| Host sphere | Occupied ball around a site in the packing model | Radius R; occupied fraction π/(3√2) |
| HCP Voronoi cell | Nearest-site domain of the full HCP point set | Trapezo-rhombic dodecahedron; 12 faces, 24 edges, 14 vertices |
| FCC Voronoi cell | Nearest-site domain of the FCC point set | Rhombic dodecahedron; 12 faces, 24 edges, 14 vertices |
| Tetrah / octah | Void within the four-/six-host convex cell, after subtracting host balls | Four / six inward-curving sphere patches; four / eight triangular apertures |
Both ideal Voronoi cells have volume d³/√2 = 4√2R³. A Voronoi domain includes the host’s assigned share of the surrounding voids. A tetrah or octah is a different geometric object. The flat host-center tetrahedron and octahedron are also distinct from the curved interstices displayed on this site. Close-packing reference · HCP Voronoi geometry.
2 / Compression check
Compression alone does not identify the shapes
Under uniform scaling x ↦ λx and R ↦ λR, with λ > 0, all angles, face incidences and volume fractions stay unchanged. The host spheres remain spheres; the Voronoi cells remain polyhedra; the two void types remain different. Their largest inscribed-sphere radii satisfy:
rT/R = √(3/2) − 1; rO/R = √2 − 1
These unequal ratios do not converge under scaling. At λ = 0 all volumes vanish, giving a degenerate point rather than a finite common shape. Compressing center distances while keeping radii fixed instead introduces overlap. Deforming the nodes into space-filling domains can remove voids, but it changes the model and requires a deformation law; it does not prove that the original T, O and node objects become congruent.
Polyhedral black hole: a faceted domain or a polygonal horizon mesh can be specified mathematically, but neither establishes an event horizon. A physical version needs a spacetime metric, matter/stress assumptions and a horizon criterion. A distorted horizon need not be a round sphere. The classical two-sphere topology result, under its stated assumptions, is a statement about topology, not proof for or against a faceted discretization. Its assumptions also cannot simply be imported into a Planck-scale lattice. Galloway & Schoen.
Result: the Voronoi representation makes a finite Monad geometrically well-defined. Identifying it with a physical black hole, or making all three objects a common physical phase, remains a separate LSM hypothesis with an unspecified constitutive law.
3 / Electron construction
The complete octah shell has an exact minimum property
“Fully filled” here means the central octah and all eight face-neighbor tetrah are occupied. It does not mean that the entire surrounding region contains no host spheres or remaining voids. The resulting cluster has full cubic symmetry Oh: 24 proper rotations and 48 operations including reflections/inversion. This is discrete cubic symmetry, not continuous rotational symmetry.
Declare the admissible class first: finite, nonempty unions of complete FCC interstitial cavities on one fixed, undeformed equal-sphere lattice; distinct cells do not overlap; connectedness means a shared triangular aperture of nonzero area. Use the existing v15 rule q/e = (2O − T)/6. FCC face adjacency is bipartite: each T has four O neighbors and each O has eight T neighbors.
- At q = −e, T = 2O + 6. Thus N = T + O = 3O + 6.
- O = 0 would require six T cells. In FCC no two T cells share a face, so that union cannot be face-connected.
- Therefore O ≥ 1 and N ≥ 9. Equality forces O = 1 and T = 8.
- With only one O, every T must directly touch it. The O has exactly eight available T face-neighbors, so all eight must be occupied. This uniquely fixes the shell, up to lattice symmetries.
q = −e and face-connected FCC ⇒ N ≥ 9; N = 9 ⇔ 8T + 1O
The same argument gives minimum total occupied void volume at fixed host radius: V = (2O+6)VT + OVO, with both cell volumes positive. This is a specific compactness result. It does not assert a global optimum for every possible definition of compactness, such as radius of gyration or surface-to-volume ratio across changing volumes. Inspect the existing electron model →
Proof of the connected surface-area minimum
Let I count occupied T–O face contacts and F count exposed triangular apertures. Counting faces gives F = 4T + 8O − 2I. Since every O has at most eight T neighbors, I ≤ 8O. At q = −e:
F = 16O + 24 − 2I ≥ 24
The electron shell achieves I = 8 and F = 24. For touching host spheres of radius R, each aperture has area a = (√3 − π/2)R². Define ΩT = 3 arccos(1/3) − π and ΩO = 4 arcsin(1/3). The curved host-wall areas per cavity are sT = 4ΩTR² and sO = 6ΩOR². All three areas are positive.
A(C) = TsT + OsO + Fa
A(C) − A(e-shell) = (O−1)(2sT+sO) + (F−24)a ≥ 0
A(C) − A(e-shell) = (O−1)(2sT+sO) + (F−24)a ≥ 0
Equality in this admissible class requires the unique nine-cell shell. Thus a proposed energy E = γA with γ > 0 selects that shell as its unique connected −e minimum. A is the full Euclidean boundary of the occupied cavity union, including the curved host interfaces. A counts neither a convex hull nor a relativistic event horizon. If LSM assigns a different energy to these interfaces, includes spectral terms, or permits lattice deformation, that energy requires its own minimization.
For reproduction, VT = [4/(3√2) − 4ΩT/3]R³ and VO = [8√2/3 − 2ΩO]R³. These are the curved void volumes, not chart mass units.
Why the HCP/FCC distinction matters
In ideal HCP, an O has six T and two O face-neighbors; a T has three O and one T face-neighbor. The FCC proof above therefore does not automatically apply to HCP. A fully regular octah with eight regular face-sharing tetrah is an FCC-compatible local cluster. Embedding that full arrangement in an otherwise HCP substrate would require a local stacking change or another explicitly constructed defect. The current gallery already uses FCC, and is preserved.
What the theorem does not yet establish
Connectedness is a restriction, not a derived confinement law. Six mutually separated T cavities have total charge −e and area 6sT + 24a, less than the electron shell by 2sT + sO. They lie outside the theorem. If charge alone is conserved and such fractional fragments are allowed, the surface functional alone does not ensure electron stability. If T and O inventories are separately conserved, that particular comparison is inadmissible, and the allowed moves must say so. A physical completion must supply confinement/conservation rules and test all permitted fragmentation channels.
The bulk neutral ratio gives 2qT + qO = 0. Declaring the eight-face shell to carry −e then gives qT = −e/6 and qO = +e/3. This explains the current normalization conditional on that identification; it is not an independent derivation of charge from geometry alone. Spin ½, fermionic statistics, inertial mass, propagation and measured electromagnetic behavior also require a dynamical construction.
September 28 proton-selection analysis · retained before adoption of 8T + 7O
Ben adopted the smaller enclosing construction on September 29. The preceding selection question and its complete reasoning are preserved here; the current development is above.
4 / Positive partner
Charge and enclosure do not yet select the proton
Equal and opposite charge establishes neutrality of a pair. It does not determine the positive partner’s structure or mass. The electron’s charge-conjugate partner in measured physics is the positron. The proton is a distinct positive particle. The v15 catalog likewise already includes a 2T + 4O positron candidate as well as the 12T + 9O proton candidate.
At q = +e, T = 2O − 6. The first nonnegative count is 0T + 3O, which is disconnected in FCC. The next is 2T + 4O, and the published positron supplies a connected six-cell example. Positive charge alone therefore cannot make the 21-cell proton the smallest cluster.
| Construction | T, O | q/e | What it establishes |
|---|---|---|---|
| Published electron | 8, 1 | −1 | Unique smallest connected negative-charge candidate under the stated FCC assumptions |
| Published positron | 2, 4 | +1 | Six-cell connected positive candidate; retained |
| Complete node first shell | 8, 6 | +2/3 | All 14 incident cells around a host node |
| First shell + exterior octah | 8, 7 | +1 | 15-cell connected counterexample to charge + enclosure uniquely selecting the proton |
| Published proton | 12, 9 | +1 | 21-cell v15 candidate; original geometry and mass accounting retained |
Explicit smaller enclosing competitor
Use local dimensionless coordinates with a host at (0,0,0) and host nearest-neighbor spacing √2. Put T centers at all eight (±½,±½,±½), O centers at the six signed coordinate unit vectors, then add O at (1,1,1). The first 14 cells cover all incident directions around the central host; the added O shares one face with T at (½,½,½). All are valid FCC cells.
This cluster has 25 internal contacts and 38 exposed apertures. The published proton has 32 internal contacts and 56 exposed apertures. Under the same full-cavity-area functional, the proton exceeds this smaller competitor by 4sT + 2sO + 18a > 0. This does not identify the competitor as a real particle. It demonstrates that an additional physical selection condition is necessary; even requiring a complete first-shell enclosure is insufficient.
The existing site also records that the published proton has 8T + 5O at its most surrounded host, rather than the complete 8T + 6O first shell. That observation is preserved, not silently repaired by relabeling a different shape as the proton. A different topological definition of enclosure must be specified and checked explicitly.
A viable proton derivation must define an independently conserved baryon-like or topological sector, or another physical rule that distinguishes it from a positron, then demonstrate why 12T + 9O wins over admissible competitors. Merely adding a common node mass to every enclosing candidate does not change their ordering.
The retained v15 ratio is 73,480/40 = 1,837; the 2022 CODATA reference is 1,836.152673426(32). The existing approximate correspondence is not converted into an exact match here. NIST mass reference · PDG electron/positron properties.
5 / What is complete, and what remains
A defined geometric foundation and a specific dynamical gap
The finite Voronoi codification, compression result, FCC shell theorem and positive-charge counterexamples are complete mathematical statements within the stated assumptions. They do not yet establish the whole physical theory from shape alone. The next theoretical requirement is an explicit state/update law or energy operator, its allowed moves and conserved quantities, followed by stability and fragmentation calculations in separately defined electron and proton sectors.
Earlier surface/spectral and lattice-response ideas remain part of the research record. They must not be treated as a completed operator merely because a compact formula exists: its domain, coefficients, boundary conditions and relation to the v15 states must be specified before it can decide between these candidates.
The requested simple attempt to falsify the proton-fusion animation is the next work stage. No physical fusion test or new experimental outcome is claimed by this foundation note. Existing fusion work remains behind staff access.
Inspect the reproducible geometry checks · Read the Python checker · Download its v15 geometry inputs. The checker verifies FCC incidences, all 48 electron symmetries, the smaller enclosing competitor, current catalog contacts, and an independent numerical HCP/Voronoi cross-check. The unrestricted minimum theorem is supplied by the inequalities above, not inferred from a finite search.
Particle Masses v15
One geometry.
A family of particles.
Turn the structures. Count their constituents. Follow their charge and mass accounting across nine LSM candidates: three neutrino sizes, electron, positron, the neutral node, proton, anti-proton and neutron.
T: tetrah, −⅙e · O: octah, +⅓e · q = (−T + 2O) / 6
Adopted names: Monad (small), Mobnad (medium), Mocnad (large). These views show the T/O structures; the complete Monad includes one host sphere. Read the complete unit definition.
This is the complete retained v15 catalog. The September 29 selected proton is shown above alongside its v15 reference; these nine original rows and their mass accounting remain intact.
LSM candidate · Particle Masses v15
Monad
Neutrino (small)
Threefold axis and inversion · D₃d
Connected FCC arrangement
2 T + 1 O0e
Mass · chart units16
2 × 4 + 1 × 8
LSM candidate · Particle Masses v15
Mobnad
Neutrino (medium)
Twofold symmetry · C₂v
Connected FCC arrangement
4 T + 2 O0e
Mass · chart units32
4 × 4 + 2 × 8
LSM candidate · Particle Masses v15
Electron
Cubic symmetry · Oₕ
Eight tetrah share all eight faces of one octah
8 T + 1 O−1e
Mass · chart units40
8 × 4 + 1 × 8
LSM candidate · Particle Masses v15
Positron
Threefold symmetry · C₃v
Connected FCC arrangement
2 T + 4 O+1e
Mass · chart units40
2 × 4 + 4 × 8
LSM candidate · Particle Masses v15
Mocnad
Neutrino (large)
Twofold symmetry and inversion · C₂h
Three octah in a chain · six face-sharing tetrah
6 T + 3 O0e
Mass · chart units48
6 × 4 + 3 × 8
LSM candidate · Particle Masses v15
HCP/FCC node
Threefold axis and inversion · D₃d
13,755 curved cells · finite FCC realization
9,170 T + 4,585 O0e
Mass · chart units73,360
4,585 × (2 tetrah + 1 octah)
Retained proton reference · v15
Proton
Fourfold symmetry · C₄v
Connected arrangement · node term in mass accounting
12 T + 9 O+1e
Mass · chart units73,480
73,360 node + 120 arrangement
LSM candidate · Particle Masses v15
Anti-Proton
Threefold axis and inversion · D₃d
Connected arrangement · node term in mass accounting
18 T + 6 O−1e
Mass · chart units73,480
73,360 node + 120 arrangement
LSM candidate · Particle Masses v15
Neutron
Fourfold symmetry · C₄v
Connected arrangement · node term in mass accounting
20 T + 10 O0e
Mass · chart units73,520
73,360 node + 160 arrangement
Read the counts, charge and mass
The chart’s charge column gives −T + 2O: −6 for the electron and anti-proton, +6 for the positron and proton, and 0 for the neutral entries. Dividing by six gives the charge shown above.
The mass numbers are retained in chart units. The rows imply 4 units per tetrah and 8 per octah. The proton, anti-proton and neutron each include an additional 73,360 units, matching the neutral node. Their views show the listed T/O arrangement. The theory proposes node enclosure; a complete enclosure for each baryon remains to be constructed.
All particle views use FCC interstices. The node is one finite FCC realization of the HCP/FCC row. These are geometric candidates, with each view scaled independently; the chart does not assign a physical length or a unique spatial arrangement.
About these geometries
Each model shows the volume inside the convex hull of four or six touching, equal-radius spheres after subtracting the spheres. The inward-curving surfaces are spherical; flat triangular openings mark the boundaries shared with neighboring interstices. The same unit host-sphere radius defines the local geometry in every view. The electron places eight tetrah against all eight triangular faces of a central octah, giving full cubic symmetry. The large neutrino forms an elongated chain of three octah, joined and capped by six tetrah. The other small configurations are connected selections of complete FCC symmetry orbits, chosen within a bounded search for symmetry and even spatial spread. The node contains 4,585 octah and 9,170 tetrah, with a threefold, inversion-symmetric finite boundary. The node uses a coarser curved mesh to keep all 13,755 cells interactive. Symmetry refers to the complete oriented cells, including their host shells. The views are candidate arrangements, without assigned physical dimensions or relative particle sizes.