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 bare (tree-level) result, and it is the part that is locked: α₀⁻¹ = 137, unambiguous — 16 faces, the triangular number, plus 1.
The remaining 0.036 — the dressed value 137.035999084 that experiments measure — is where honesty matters. Getting from 137 to 137.036 is ordinary QED vacuum polarisation, and the framework does not derive it. The lattice count is charge-blind: it weights every face element equally, which is exactly why it gives a clean integer 137. The physical electromagnetic dressing, by contrast, is charge-squared-weighted — a different object entirely, so the clean count has nothing to say about the decimal. The status is therefore split cleanly in two: the bare 137 is derived and locked; the dressed 137.036 is an open problem, carried as standard external QED rather than a framework prediction. The framework stakes its claim on the clean integer, not the decimal.
2. The Weak Mixing Angle: A Ratio of Charge Traces
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 measured low-energy value, at the Z pole, is:
sin²θ_W ≈ 0.231
In the Standard Model, this is a free parameter. It is measured, not predicted. The theory provides no reason why it should be 0.231 rather than 0.3 or 0.1 or any other value between 0 and 1.
On the information lattice, the weak mixing angle is fixed by the register’s charge bookkeeping. Tracing the charge assignments through the eight-bit code and forming the standard ratio of traces gives
sin²θ_W = Tr(T₃ₗ²) / Tr(Q²) = 3/8
at the grand-unification scale — exactly the classic Georgi–Glashow GUT value. Under ordinary renormalisation-group running down to laboratory energies, 3/8 flows to the measured low-energy value (≈ 0.231 at the Z pole). The framework gets the standard number, by charge consistency: the weak sector is a success of charge-bookkeeping, not a free input.
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, which the proton mass fixes) and the infrared cutoff (the cosmological horizon, set by the Hubble rate H₀). The proton scale is the primary input; Newton’s constant, the Planck mass, and the Hubble rate emerge together as outputs.
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 ties the gravitational and cosmological scales together:
M²_P = 24π α² Λ³_QCD / (H₀ Ω_Λ)
With Λ_QCD fixed by the proton mass and Ω_Λ (the dark energy fraction) measured, this relation delivers the Planck mass and the Hubble rate together. The Planck mass it yields is:
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 fixed by the framework’s record clock. The substrate keeps its records on a 28-channel service ledger, whose spectral gap is Δ₁ = 1/28. That single number sets the slow drift of the vacuum away from a pure cosmological constant:
w₀ = −27/28 ≈ −0.964
The vacuum begins at w = −1 at the Big Bang and rises slowly along a thawing trajectory
w(a) = −1 + a/28, i.e. w_a = −1/28,
always staying just above −1 — there is no phantom crossing. The same 28-channel gap fixes the scalar spectral index of the primordial fluctuations, through the duality w₀ = −n_s, so the dark-energy and inflationary numbers come from one count.
The current DESI/Planck constraints favour an equation of state slightly above −1, consistent with −27/28. Future data from DESI Year 5, the Euclid satellite, and the Vera Rubin Observatory will sharpen this: if w₀ pins to exactly −1 (a pure cosmological constant), or the data establish a phantom crossing (w < −1 in the past), the prediction fails. The whole trajectory is pinned at zero free parameters by the 28-channel gap.
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: Bare α₀⁻¹ = 137 derived and locked (counting faces). The dressed 137.036 is an open problem, not a derivation — the face count is charge-blind, the physical dressing charge-squared-weighted; the precise low-energy value is standard external QED.
Weak mixing angle: Charge-forced to the GUT value sin²θ_W = 3/8 at unification, running to the measured low-energy value (standard, not a new constant).
Planck mass: Derived 1.2217 × 10¹⁹ GeV. Measured 1.2209 × 10¹⁹ GeV. Agreement: 0.07%. (Zero parameters.)
Dark energy w₀: Derived −27/28 ≈ −0.964 (the 28-channel spectral gap), with thawing slope w_a = −1/28. Consistent with current DESI/Planck constraints favouring w₀ slightly above −1. (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 dressed fine-structure constant remains an open problem — the framework locks the bare integer 137, not the decimal. 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 the bare α, charge-trace bookkeeping for the weak angle (the standard 3/8), the proton-primary ledger 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 charge-trace derivation of the weak angle, 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.