Publication context: Ben adopted the 8T+7O construction as the leading proton geometry on September 29, 2026, at 00:23 Central. The following research note is retained in its original drafting context. The live site and archive edition include the adopted presentation.

# LSM: proton comparison and integer foundations

Working research note — 29 September 2026. This is an additive proposal for Ben's review, not a replacement of existing theory or a site publication. Preserve all nonredundant information on dclxvi.lc. Retain earlier formulations with their provenance when introducing a revision.

## Sources and scope

- Ben's present definition: Monad = one host component S, two tetrah T, and one octah O; Mobnad and Mocnad name the medium and large neutrino classes. A **true monad** has exclusively flat boundaries and is fully packed.
- [Particle Masses v14](https://docs.google.com/spreadsheets/d/1QSwY5zO6or5F70QLj_V6k2pfMiCCDanhAa_z_D8Oq_g/edit), tab “Particle Masses 2,” formulas G2, G3, G5, G10, G13 and G18:G20; read directly, including formulas.
- Current site source: commit `89c4176263129411720aa45da85cbdd54bf7f797`; exact proton sites in `scripts/public-particle-layouts.json`, counts and node contribution in `scripts/public-particle-catalog.json`, and the 8T+7O construction in `scripts/check-foundations.py`.
- [NIST CODATA 2022 constants](https://physics.nist.gov/cuu/Constants/Table/allascii.txt): electron mass and proton/electron mass ratio. The ratio is 1836.152673426 with standard uncertainty 0.000000032.

The displayed proton layouts are **FCC** layouts. HCP and FCC have the same T:O inventory ratio but different face adjacency. The minimum proof below is specifically for the FCC complete first-shell cage.

## Exact comparison

Use host positions at integer triples whose coordinate sum is odd. Octah centers are the complementary integer triples; tetrah centers are half-integer triples. Multiply coordinates by sqrt(2) to make the host-sphere radius one. Two cells are face neighbors exactly when their host-vertex sets share three vertices.

For the new candidate, place the chosen host at (1,0,0). Take the eight tetrah at (1 ± 1/2, ± 1/2, ± 1/2), the six octah at that host plus/minus each coordinate unit vector, and the extra octah at (2,1,1). The comparator uses the site's actual current 12T+9O coordinates without rearrangement.

| Quantity | 8T+7O candidate | Current 12T+9O layout |
|---|---:|---:|
| Total cells | 15 | 21 |
| Charge in units e/6: 2O − T | 6 | 6 |
| Raw face incidences: 4T + 8O | 88 | 120 |
| Shared triangular faces I | 25 | 32 |
| Exposed triangular apertures: 4T + 8O − 2I | 38 | 56 |
| Cells incident on most surrounded host | 8T+6O, complete | 8T+5O, one O missing |
| Point group | C3v | C4v |
| Symmetry operations, including reflections | 6 | 8 |
| Single face connections whose removal disconnects the graph | 1 | 0 |

These are exact incidence and symmetry checks, independent of rendered mesh resolution. The incomplete cage observation concerns the **current arrangement**, not every possible arrangement with 12T+9O counts.

The complete FCC cage is 8T+6O. It has charge 4 in units e/6. To reach proton charge 6 requires another 2; one O supplies exactly 2. Therefore **8T+7O is a minimum-cell +e cluster subject to containing that complete first shell around one host**: at least 14 cage cells are compulsory, and at least one additional cell is compulsory. This is a constrained geometric minimum. It is not a global minimum over all positively charged clusters; the existing 2T+4O positron inventory also has +e.

The seventh O attaches through only one shared triangular face. The current proton has more redundant face connections. Compactness and resistance to separation consequently need not rank the two models in the same order.

For undeformed curved interstices, the total boundary area is

\[
A=T s_T+O s_O+F_{\rm exposed}a,
\]

where the curved wall areas s_T and s_O and triangular aperture area a are all positive at fixed host radius. Thus

\[
A_{12T+9O}-A_{8T+7O}=4s_T+2s_O+18a>0.
\]

Likewise the occupied interstitial volume difference is 4V_T+2V_O>0. These are occupied volume and boundary-area comparisons, not a proof of greater dynamical stability. A stability law needs contact, strain, binding, confinement and node terms, plus specified permitted transformations. A mass formula depending only on T and O cannot distinguish different arrangements with identical counts.

The chat's optional flat representation consists of regular cells whose vertices are host centers. It preserves incidence and makes the face geometry clear. Those regular cells extend into the host-sphere regions. It is **not** an established compressed-monad geometry in which an independent host cell also occupies its former sphere volume.

## True monad: explicit flat, fully packed condition

Write

\[
\mathcal M_{\rm true}=S_{\rm flat}+2T_{\rm flat}+O_{\rm flat}.
\]

Require:

1. Every boundary patch of every component is planar in the specified spatial geometry.
2. Components have disjoint interiors, with matching boundaries where they meet.
3. Repetition with the appropriate lattice orientations fills the domain: no residual void volume.
4. The inventory per host remains S:T:O = 1:2:1 and the host is neutral, so charge is 0.
5. The metric, component shapes and deformation rule are stated explicitly; “flat faces” does not itself specify these.

The sphere is the undeformed reference for S; S_flat is a polyhedral host component. Planar faces do not require all components to have the same shape. The four-part inventory is not by itself a proof of a four-cell translational fundamental domain; an HCP primitive translation cell contains two hosts.

A constructive family to investigate is a weighted Voronoi partition of the host and interstice centers. A cell i is defined by

\[
|x-p_i|^2-w_i\le |x-p_j|^2-w_j\quad\text{for every }j.
\]

Each pairwise boundary is a plane; the cells fill space without overlaps of interiors. The site positions, weights, nonempty-cell requirement and resulting faces still need to be checked against the desired S/T/O topology. This supplies a precise route to a flat partition, but does not automatically preserve four and eight physical faces for T and O. Retain that distinction when applying the mass rule.

Ben proposes the deep black-hole interior as the realization of the true-monad limit. The flat-cell definition can be adopted as an LSM postulate. Establishing where it occurs requires a pressure/deformation law and a spacetime metric; it does not follow from the inventory equation alone.

## Mass normalization: v14 and literal faces agree after conversion

The actual electron formula in v14 G2 is:

`=((D2*(1))+(E2*2))*(51.099895)`

Here D2=8 and E2=1. The formula assigns 51.099895 keV/c² to one **T+2O weighted counting unit**, meaning a coefficient of one for each T and two for each O. It does not multiply that number by the literal 40 face incidences.

Let W=T+2O and F=4T+8O=4W. The identical mass rule can be written either way:

\[
m_{\rm bare}=W\,(51.099895\ {
m keV}/c^2)
=F\,(12.77497375\ {
m keV}/c^2).
\]

Thus one literal face incidence has the v14 conversion **12.77497375 keV/c²**. The factor of four is normalization, not a change to the electron geometry. Updating only the calibration to NIST's electron mass gives m_e/40 = 12.77497376725 keV/c². Neither calibration is an independent prediction of electron mass.

F counts each cell's faces: a shared physical interface contributes two incidences. It is not the exposed-face count. The distinction must remain explicit in both mass and surface-energy calculations.

## What observation already constrains

With the present v15 node contribution of 73360 face units held fixed:

\[
M_e=40,\qquad M_{p,\rm current}=73360+120=73480,
\qquad M_{p,\rm candidate}=73360+88=73448.
\]

| Conditional result | Current proton | 8T+7O candidate |
|---|---:|---:|
| Mass ratio to electron | 1837 | 1836.2 |
| Relative discrepancy from observed ratio | about +461 ppm | about +25.775 ppm |

The candidate improves this conditional comparison, but transferring the same node contribution is an assumption. It has not been derived from the new cage. Its remaining excess is about 24.184 keV/c², or 1.89306296 face units with electron calibration. This is far larger than the experimental uncertainty.

There is a stronger arithmetic constraint. With bare face mass only, any +e T/O inventory obeys T=2O−6 and consequently F=16O−24. A neutral T/O core adds another multiple of 16. Relative to the 40-face electron, allowed ratios lie on a grid with spacing 2/5. The measured ratio is not on that grid. **Changing the units, or subdividing every face equally without adding new physics, leaves those ratios unchanged.**

The next model must therefore change the physical mass accounting: for example, derive binding/excitation contributions, change the host contribution, or derive a different relation between face inventory and rest energy. This is an observation-led restriction on the search, not a rejection of integer microscopic variables.

The neutrino identification is a further strong restriction. The 2T+1O interstitial inventory has bare mass 16 face units, about 204.4 keV/c². It cannot directly be identified with the measured light electron-neutrino mass scale under this additive rest-mass rule. [KATRIN's 2025 result](https://www.katrin.kit.edu/1271.php) bounds the effective electron-neutrino mass below 0.45 eV/c² at 90% confidence. A distinct binding, propagation or rest-energy mechanism would be needed; adding a positive host mass does not resolve this discrepancy.

## Integer LSM units: a precise proposed convention

Keep the fundamental state description discrete. Let K and H be positive integers, to be constrained and ultimately derived:

- K = mass/energy quanta assigned to one bare triangular face incidence.
- H = elementary action quanta in ħ/2.

Choose charge unit q_* = e/6, mass unit μ_* = m_e/(40K), and action unit a_* = ħ/(2H). Define compatible energy, time and length units by

\[
E_*=\mu_*c^2,\qquad \tau_*=a_*/E_*,\qquad \ell_*=c\tau_*.
\]

In this convention the reference constants are **c=1, e=6, ħ=2H**. Every elementary unit has value 1. A spin projection ħ/2 has action count H. H allows the spatial/time unit to be refined independently of K; the choice H=1 gives a natural conversion length, not a proven minimum lattice spacing.

| Inventory | Integer charge Q | Bare integer mass count |
|---|---:|---:|
| T | −1 | 4K |
| O | 2 | 8K |
| Electron 8T+1O | −6 | 40K |
| Positron 2T+4O | 6 | 40K |
| Monad interstitial part 2T+1O | 0 | 16K |
| Mobnad interstitial part 4T+2O | 0 | 32K |
| Mocnad interstitial part 6T+3O | 0 | 48K |
| Candidate proton shell 8T+7O | 6 | 88K |
| Current proton shell 12T+9O | 6 | 120K |
| Current modeled neutral node | 0 | 73360K |

Host mass in the monad classes remains a separate question. The source does not establish that the v15 enclosed-proton node contribution must be added to every neutral host-based inventory.

A strictly integer candidate energy law has the form

\[
M=K F_{\rm total}+N_{\rm excitation}-N_{\rm binding},
\]

where all quantities on the right are integers and F_total includes a core only when the model specifies one. Position can be represented by integer lattice labels, evolution by integer update counts, and charge by integer Q. This is a proposed microscopic rule, not a demonstrated consequence of the existing geometry. The binding/excitation integers must be calculated from local rules, not chosen separately for each particle.

Unit definitions alone do not prove discreteness. Euclidean distances can involve exact square roots even when vertex coordinates are integers, and measured ratios and averages need not be integers. For example, the face edge and diagonal of a regular octahedron have an invariant irrational ratio. The integer requirement is coherent for the microscopic inventory and update rules; requiring every geometric distance to be an integer multiple of one length would exclude that exact regular geometry.

Gravity and the electromagnetic coupling remain additional dynamical constraints. This convention does not derive G or the fine-structure constant, and does not assert that all dimensionless constants become integers by changing units.

## A bounded arithmetic example, not a derived fundamental scale

For the candidate plus the unchanged node, choose electron binding correction zero as the calibration reference and allow one integer proton binding count B:

\[
M_e=40K,\qquad M_p=73448K-B.
\]

An exhaustive positive-integer search for the smallest K that matches the CODATA proton/electron ratio within one quoted standard uncertainty gives **K=346 and B=655**. Then

\[
M_e=13840,\qquad M_p=25412353,
\qquad \frac{M_p}{M_e}=\frac{25412353}{13840}.
\]

All counts are integers. The ratio agrees with the observed interval. This example shows that an integer microscopic mass description is arithmetically compatible with that one measurement. It does **not** explain why a face should contain 346 subunits or why the cage should bind 655 of them. It also does not fit the full particle spectrum: this mass quantum is about 36.9 eV/c², too coarse for the observed light-neutrino sector under a universal positive integer rest-mass rule. Do not adopt it as the final fundamental scale.

The productive next derivation is a local integer rule for binding, excitation and host coupling which produces the electron, a proton candidate, and the light-neutrino behavior with the same parameters. Use measured masses and radii as constraints, retain exact integer candidate counts, and reject candidates whose predictions lie outside the uncertainty intervals.

## Archive timer

An hourly retry task was created, starting 29 September 2026 at 00:55 America/Chicago (05:55 UTC). It permits at most one new archive submission in a rolling 60-minute period across the two public URLs, with publication.html first. It verifies actual updated content and archived figures, rejects the older 04:33 UTC capture, skips live error pages, and stops after verified captures or a blocker requiring user intervention. No new archive success is claimed in this note.