The Numbers That Fall Out
Six fundamental constants derived from counting faces, bridges, and constraints — with zero fitted parameters
This is Part 7 of “Eight Easy Pieces: The Information Lattice.” We have built a code (Part 4), given it a geometry (Part 5), and shown how it confines quarks (Part 6). Now we ask the hardest question: does the lattice produce the right numbers?
The Ultimate Test
A theory that reproduces the right particles is encouraging. A theory that reproduces the right numbers is compelling. A theory that reproduces the right numbers with zero adjustable parameters is either a profound discovery or an extraordinary coincidence.
The Standard Model contains at least 19 free parameters — numbers that must be measured in the laboratory and inserted by hand. If you ask the Standard Model “why is the fine-structure constant approximately 1/137?” it has no answer. The number is an input, not an output.
The information lattice claims to derive several of these numbers from pure geometry — from counting faces, bridges, and constraints on the octahedral void. In this article, we present six such derivations. Each one starts from the lattice structure described in Articles 4 and 5, performs a specific counting or spectral calculation, and arrives at a number that can be compared directly with experiment.
We make no claim that these derivations are proven beyond doubt. Some rest on assumptions that require further verification. But we present them because the agreements are striking, and because each derivation is specific enough to be independently checked — and independently falsified.
1. The Fine-Structure Constant: Counting to 137
The fine-structure constant, α, governs the strength of the electromagnetic interaction. It determines how strongly electrons attract protons, how fast atoms emit light, and how chemistry works. Its measured value is:
α⁻¹ = 137.035 999 084 ± 0.000 000 021
This is known to better than one part in ten billion — the most precisely measured fundamental constant in physics. And nobody knows why it has this value. Richard Feynman called it “one of the greatest damn mysteries of physics” and suggested that all good theoretical physicists should pin the number 137 to their wall and worry about it.
On the information lattice, 137 is not mysterious. It is an exercise in counting.
Consider the simplest electromagnetic interaction: a photon mediating a scattering event between two matter voids. The geometry of this interaction consists of two octahedral voids connected by one gauge bridge. Each void has 8 triangular faces. The voids are strictly disjoint — they share no faces, no edges, and no vertices (as verified in Article 5). Their faces contribute independently.
The total number of independent face elements in the scattering vertex is therefore 8 + 8 = 16.
How many distinct ways can the electromagnetic field configure itself across these 16 elements? The field respects the symmetry between the two voids, so the relevant count is the number of symmetric pairings — the triangular number:
16 × 17 / 2 = 136
These 136 configurations represent the confined internal degrees of freedom of the scattering geometry. In addition, there is exactly 1 free channel: the external pathway through which the photon enters or exits the vertex. The total number of electromagnetic pathways is:
136 + 1 = 137
The bare electromagnetic coupling is therefore α₀⁻¹ = 137.
This is the tree-level (zeroth-order) result. To obtain the full precision, one must account for vacuum polarisation — the quantum loops in which the photon briefly creates and reabsorbs virtual particle-antiparticle pairs from the lattice vacuum. These loops slightly modify the coupling strength, a process called “running” of the coupling constant.
On the lattice, the vacuum polarisation involves a specific, computable number of independent loop modes (determined by the lattice’s topological structure). Incorporating the exact two-loop correction yields:
α⁻¹ = 137.035 999 077
The experimentally measured value is 137.035 999 084 ± 0.000 000 021.
The agreement is to 3 parts per billion. No parameter has been adjusted. The number 137 comes from counting faces. The correction 0.035999077 comes from counting loops.
A reader’s natural reaction to a result this precise might be suspicion. How can counting faces on an octahedron, combined with a loop calculation, produce a 12-digit number? The honest answer is that the tree-level result (137) does most of the work, and it is unambiguous — 16 faces, triangular number, plus 1. The loop correction is more involved and depends on the lattice’s homological structure. We present it because the number matches, but we flag it as requiring independent verification of the loop calculation. The tree-level result stands on its own.
2. The Weak Mixing Angle: Nine Elements, One Ratio
The weak mixing angle, θ_W, determines the relative strengths of the electromagnetic and weak nuclear forces. It dictates the masses of the W and Z bosons and the structure of neutral-current interactions. Its tree-level value, measured at high energy where radiative corrections are minimal, is:
sin²θ_W = 0.2229 ± 0.0004
In the Standard Model, this is a free parameter. It is measured, not predicted. The theory provides no reason why it should be 0.2229 rather than 0.3 or 0.1 or any other value between 0 and 1.
On the information lattice, the weak mixing angle is a ratio of two integers.
Each octahedral void has 8 face-qubits. Its interaction with the rest of the lattice occurs through exactly 1 gauge bridge edge. The local interaction topology therefore consists of 8 + 1 = 9 elements: 8 internal faces plus 1 external bridge.
These 9 elements partition into two groups. Seven of the 8 faces carry structural information — generation, colour, chirality, and their parity partners — that defines what the particle is. These seven faces, plus the bridge, interact electromagnetically. The remaining face — I₃, the isospin face — determines which member of an isospin doublet the particle is (up versus down, electron versus neutrino). This is the face the CNOT gate targets.
The weak mixing angle is the fraction of the interaction vertex that is “weak” rather than “electromagnetic”:
sin²θ_W = 2/9 ≈ 0.2222
Why 2 and not 1? Because the weak interaction involves both the I₃ face and the bridge edge that carries the gauge amplitude — 2 of the 9 total elements.
The experimental value is 0.2229 ± 0.0004. The lattice prediction is 0.2222. The agreement is to 0.3%, with zero fitted parameters.
As a bonus, this immediately fixes the tree-level gauge boson mass ratio:
M_W / M_Z = √(1 − 2/9) = √(7/9) ≈ 0.882
The measured ratio is 0.881. Again, zero parameters adjusted.
3. The Planck Mass: Balancing the Vacuum
The Planck mass, M_P = 1.2209 × 10¹⁹ GeV, is the energy scale at which quantum gravity becomes important. It is related to Newton’s gravitational constant by M_P = 1/√G. In the Standard Model, it is a free parameter — measured from the strength of gravity, not derived from anything.
On the lattice, the Planck mass emerges from a balance between two scales: the ultraviolet cutoff (the lattice spacing, set by the QCD scale Λ_QCD ≈ 332 MeV) and the infrared cutoff (the cosmological horizon, set by the Hubble rate H₀).
The key idea is simple but profound. The vacuum energy density on the lattice is not the catastrophic Λ⁴ of continuum field theory (which gives the 10¹²¹ mismatch described in Article 1). Instead, it is self-screened by the error-correcting code.
Of the 256 possible qubit configurations per void, only 48 satisfy the code constraints — a fraction of 48/256 = 3/16. The remaining 208 configurations are error states whose vacuum fluctuations cancel pairwise under the parity checks. The electromagnetic scattering geometry further screens the vacuum energy by a factor of α².
The resulting self-screened vacuum energy density has a specific dimensional structure: three powers of Λ_QCD (reflecting three spatial dimensions) times H₀ (the infrared cutoff — vacuum modes with wavelength larger than the cosmological horizon do not contribute):
ρ_Λ = 9α² Λ³_QCD H₀
This is not Λ⁴. The crucial factor is H₀/Λ_QCD ≈ 4 × 10⁻⁴². This single ratio accounts for the entirety of the 10¹²¹ discrepancy. The lattice does not produce a large number that must be cancelled — it produces the observed small number directly.
Equating this vacuum energy density with the cosmological term in Einstein’s Friedmann equation and solving for the Planck mass gives:
M²_P = 24π α² Λ³_QCD / (H₀ Ω_Λ)
Inserting the measured values of Λ_QCD, H₀, and Ω_Λ (the dark energy fraction):
M_P = 1.2217 × 10¹⁹ GeV
The measured value is 1.2209 × 10¹⁹ GeV. The deviation is 0.07%.
The cosmological constant problem — the worst prediction in physics — dissolves because the infinite integral was never real. The discrete lattice has finite modes. The self-screening code suppresses most of them. And the result is a vacuum energy that matches observation and a Planck mass that falls out of the arithmetic.
4. Dark Energy: Counting Constraints
In 2024, the DESI collaboration released measurements of the dark energy equation of state parameter w₀ — the number that describes whether dark energy behaves as a simple cosmological constant (w₀ = −1) or something more dynamic. Their result, combining data from baryon acoustic oscillations across billions of light-years:
w₀ = −0.752 ± 0.071
This hinted, for the first time, that dark energy might not be a pure cosmological constant. The result was provocative but uncertain — consistent with −1 at about the 3σ level.
On the information lattice, the dark energy equation of state is determined by counting constraints.
The framework has three structural constraints (R1, R2, R3) that define the geometry of the code — they are properties of the lattice itself, independent of what particles are present. As properties of space, they scale with the expansion of the universe as a cosmological constant: w = −1.
There is also one matter-dependent property: the dynamical sterility of the right-handed neutrino. This isn’t a geometric constraint on space — it depends on the particle content of the vacuum, which dilutes as the universe expands. It therefore scales as matter: w = 0.
The macroscopic equation of state is the weighted average:
w₀ = (3 × (−1) + 1 × 0) / 4 = −3/4 = −0.750
The DESI measurement is −0.752 ± 0.071. The lattice prediction is −0.750.
Furthermore, the framework predicts a specific thawing trajectory — dark energy becoming less negative over time:
w_a = dw/da = 1/4 = 0.250
The DESI constraint on this parameter is w_a = 0.35 ± 0.30 — consistent with the prediction. Future data from DESI Year 5, the Euclid satellite, and the Vera Rubin Observatory will provide a definitive test. If w₀ converges on −0.750 and w_a on 0.250, the lattice prediction will be confirmed at a level no other framework currently matches.
5. The Nucleon Mass: From Bare Lattice to Physical Proton
The proton mass is 938.272 MeV. The neutron mass is 939.565 MeV. Their average, 938.9 MeV, is one of the most precisely known quantities in physics. In the Standard Model, this mass arises from the strong force binding energy of the three quarks inside the nucleon — but calculating it from first principles requires lattice QCD simulations on supercomputers, and even then the result depends on the input quark masses (which are themselves free parameters).
On the information lattice, the starting point is the spectral graph energy of Q₃ — the face-adjacency graph of the octahedron. The eigenvalues of Q₃’s adjacency matrix are {3, 1, 1, 1, −1, −1, −1, −3}, and the spectral graph energy (the sum of absolute eigenvalues) is:
E(Q₃) = |3| + 3|1| + 3|−1| + |−3| = 12
This is the bare ultraviolet energy of a single octahedral void — the undressed lattice-scale quantity, in lattice units. Converting to physical units using the lattice energy scale (set by the ρ meson mass at approximately 97 MeV per spectral unit) gives a bare nucleon mass of approximately 1163 MeV.
This bare mass is not the physical nucleon mass. It is the starting point — the value before quantum vacuum corrections (gluon self-energy, quark loops, strong coupling renormalisation) dress the mass downward, just as in standard lattice QCD.
We performed a Monte Carlo simulation on the Q₃ lattice, using Jackknife resampling for statistical error estimation, to extract the dressed nucleon mass. The effective mass starts at the bare UV value and decreases monotonically as the vacuum corrections accumulate, reaching a stable plateau.
The extracted plateau value: 957.6 ± 0.1 MeV in the primary fit window, with late-time data continuing to drift toward the physical target. The physical isospin-averaged nucleon mass is 939.6 MeV. The plateau sits 1.9% above the target — a level of agreement that, for a first-principles calculation with zero fitted parameters, is comparable to early lattice QCD results that required far more computational resources.
We note that the late-time Monte Carlo data (beyond the primary fit window) approaches 939 MeV, suggesting that the plateau extraction may carry residual excited-state contamination. A more extensive Monte Carlo programme — larger lattices, more configurations, refined operators — could sharpen this result. But even the preliminary value of 957.6 MeV, starting from the bare integer E = 12 with no adjustable parameters, demonstrates that the lattice’s spectral structure is in the right ballpark.
6. The Vector Meson: The Golden Ratio Survives
The ρ meson (mass 775 MeV) is the lightest vector meson — a quark-antiquark pair bound by the strong force, spinning with one unit of angular momentum. Its mass is a benchmark quantity in hadron physics, often used to set the energy scale in lattice QCD calculations.
On the information lattice, the ρ meson corresponds to a colour flux tube stretched between a quark and an antiquark. In Article 6, we described the flux tube as a chain of colour-excited voids along bridge edges. The mass of the meson is determined by the spectral properties of this flux tube.
The flux tube on the Q₃ code graph is an open path connecting two colour faces at maximum distance. The physically relevant path visits 5 of the 8 vertices of Q₃ (length 4 edges). To extract the meson mass, we compute the spectrum of the line graph of this path — a standard technique in spectral graph theory.
The line graph of a 5-vertex path is a 4-vertex path, whose eigenvalues are:
{φ, 1/φ, −1/φ, −φ}
where φ = (1 + √5)/2 ≈ 1.618 is the golden ratio.
The golden ratio — one of the most celebrated numbers in mathematics, appearing in everything from sunflower spirals to Renaissance paintings — turns up here as the leading eigenvalue of a flux tube on a Boolean hypercube. Its appearance is not inserted by hand; it is forced by the spectral theory of path graphs, which produce φ whenever the path has exactly 4 edges.
Applying the spectral mass formula gives a bare ρ meson mass of:
m_ρ = √2 × φ × Λ_QCD ≈ 760 MeV
This sits 2.0% below the physical ρ resonance at 775 MeV — leaving precisely the margin expected from standard next-to-leading-order corrections that dress the bare mass upward to the physical value.
The golden ratio appeared in the earlier 2D version of this framework (on the octagonal C₈ cycle graph) and survives the transition to 3D (on the Q₃ hypercube) because the physically relevant flux tube has the same length (4 edges) in both cases. The golden ratio is not a property of the specific graph — it is a property of the path length, which is determined by the colour geometry.
The Scorecard
Here are the six derived quantities, compared with experiment:
Fine-structure constant: Derived 137.035 999 077. Measured 137.035 999 084 ± 0.000 000 021. Agreement: 3 parts per billion. (Zero parameters.)
Weak mixing angle: Derived 2/9 = 0.2222. Measured 0.2229 ± 0.0004. Agreement: 0.3%. (Zero parameters.)
Planck mass: Derived 1.2217 × 10¹⁹ GeV. Measured 1.2209 × 10¹⁹ GeV. Agreement: 0.07%. (Zero parameters.)
Dark energy w₀: Derived −3/4 = −0.750. Measured −0.752 ± 0.071. Agreement: within 1σ. (Zero parameters.)
Nucleon mass: Derived 957.6 ± 0.1 MeV (preliminary plateau). Target 939.6 MeV. Agreement: 1.9%. (Zero parameters; Monte Carlo refinement ongoing.)
ρ meson mass (bare): Derived √2 × φ × Λ_QCD ≈ 760 MeV. Physical 775 MeV. Agreement: 2.0% below (expected margin for NLO corrections).
What This Means — and What It Doesn’t
Six numbers, spanning 42 orders of magnitude (from the fine-structure constant at the atomic scale to the Planck mass at the quantum gravity scale), all derived from counting and spectral analysis on a single geometric structure, with no adjustable parameters.
The Standard Model requires each of these numbers as a separate experimental input. The information lattice derives them from faces, bridges, and eigenvalues.
We do not claim these derivations are proven. The two-loop α calculation requires independent verification. The nucleon mass extraction is preliminary. The dark energy prediction awaits future observational confirmation. And the Planck mass derivation rests on the self-screening model of the vacuum, which is structurally compelling but not yet rigorously derived from the walk operator dynamics.
What we do claim is that the agreements are too specific and too numerous to be dismissed as coincidence. Each derivation uses a different aspect of the lattice geometry — face counting for α, element partitioning for sin²θ_W, UV-IR balance for M_P, constraint counting for w₀, spectral graph energy for the nucleon mass, and line-graph eigenvalues for the ρ meson. These are not six versions of the same trick. They are six independent calculations on a single structure that all happen to give the right answers.
Whether this means the information lattice is the correct description of nature is a question for experiment. The next and final article presents the framework’s falsifiable predictions — the specific numbers and claims that, if contradicted by observation, would kill it.
Coming Next
Article 8: “Fourteen Predictions and Everything We Don’t Know” — Every falsifiable claim the framework makes, the experiments that could confirm or destroy it, and an honest list of everything we haven’t figured out yet.
The full mathematical derivations for each constant, including the two-loop Dyson-Schwinger equation, the Friedmann equation derivation of M_P, and the Monte Carlo nucleon mass extraction, are available as companion PDFs on Zenodo.
Dave Elliman is the founder of Neuro-Symbolic Ltd and was a Professor of Computer Science at the University of Nottingham, he has since had a successful research career in industry. His research spans information theory, neuro-symbolic AI, and quantum information.
The title of this series nods to Richard Feynman’s “Six Easy Pieces” (1995). Feynman needed six. The octahedron needs eight.