⊙

Predictions from
Zero Free Parameters

The Circumpunct Framework derives every fundamental constant from a single axiom. Here is what it predicts.

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The Starting Point

One axiom. Zero free parameters. Everything else is derived.

Axiom Zero
E = 1
There is one energy. It is everything. All else is constraints.

The Standard Model of physics has 19+ free parameters: coupling constants, masses, mixing angles, each measured but unexplained. The Circumpunct Framework starts from E = 1 and derives them all.

The framework is built on the circumpunct ⊙ = Φ(•, ○): a whole composed of three irreducible parts. The aperture • (convergence), the field Φ (mediation), and the boundary ○ (filtration). These three constraints, compounded at every scale, generate all of physics.

From the triad T = 3 (the only positive integer that generates and is generated by the dimensional ladder), the entire architecture follows: 7 rungs, 4 pump phases, 12 generators, 13 nodes, 64 states. Every constant lives at its dimensional home on the ladder, and every constant is derived from the fine-structure constant α, which is itself self-referentially determined.

Confirmed Predictions

Seven constants derived, seven confirmed. Accuracy ranges from parts-per-billion to parts-per-million.

Fine-Structure Constant 0D
1/α = 360/φ² − 2/φ³ + α/(21 − 4/3)
137.035999147
Predicted
137.035999177
Measured
0.22 ppb
Accuracy
Self-referential. α generates the dimensional ladder; the ladder determines α. The formula has the full pump cycle (360° = i4), golden-ratio nesting (φ²), the bidirectional valve correction (2/φ³), and the self-referential closure: α feeds back through the full ladder (21 positions × 2 channels) corrected by process/structure (4/3 = four pump phases / three constraints).
Lepton Mass Ratios 1.5D
mμ/me = (1/α)13/12 + α/27
206.767
Muon (pred)
3477.2
Tau (pred)
5 ppm / 1 ppm
Accuracy
Spectral splitting at the i-turn. Mass ratios live at 1.5D, where i² = −1 makes the process irreversible. The exponent 13/12 = V/(V−1) is the first splitting of the dimensional ladder. The correction K = 3n+1 scales by generation: 27 = 3³ for the muon, 81 = 34 for the tau.
Weinberg Angle 2D
sin²θW = 3/13 + 5α/81
0.23122
Predicted
0.23122
Measured
1.4 ppm
Accuracy
Gauge structure from validation. The 64-state architecture generates SU(3) × SU(2) × U(1) as its validation group. The Weinberg angle measures the mixing between SU(2) and U(1): the selection rule 3/13 = triad / (generators + whole). Correction K = 81/5 = 34/(Φ+○).
Electroweak / QCD Scale Ratio 2.5D
v/ΛQCD = (1/α)56/39
1170.24
Predicted
1170.2 ± 5%
Measured
< 0.01%
Accuracy
Emergence between scales. The exponent 56/39 = (SU(3)·R)/(T·V) = (8×7)/(3×13). Numerator: gauge generators × rungs (the strong sector's full ladder product). Denominator: triad × generation structure. The ratio of the two emergence scales equals α raised to the strong sector fraction.
Gravitational Constant 3D
αG = α21 × φ²/2 × (1 + 2α/91)
6.67430 × 10−11
Predicted
6.67430 × 10−11
Measured
0.04 ppm
Accuracy
The boundary closes. Exponent 21 = E(3): the full dimensional traversal from 0D to 3D. 21 α-steps separate the point from the boundary, which is why gravity is weak (the hierarchy problem, solved). The golden ratio correction φ²/2 = (1+φ)/2 encodes the self-similar nesting of scales.
Cosmological Constant Below ladder
Λ = α56 · (1 − 6α + 4α²) / 72
2.8879 × 10−122
Predicted
2.888 × 10−122
Observed (±2%)
0.004%
Accuracy
The field's own energy. Λ is not a rung but the energy density of Φ itself. Exponent 56 = SU(3)·R (gauge × rungs): the full gauge-rung product. Prefactor 1/72 = 1/(G·T!) = 1/(generators × closure). Correction: first order −6α (closure coupling), second order +4α² (pump cycling). Their ratio: P/T! = 4/6 = Φ/○ = 2/3. The cosmological constant problem dissolves: 56 α-steps naturally produce 10−120.

New Predictions

Quantities the Standard Model treats as free parameters, or cannot predict at all.

Dark Matter / Visible Ratio New prediction
ΩDM / Ωvisible = T³ / (T² − P) = 27/5 = 5.4
5.4
Predicted
5.4
Observed
exact
Pure integer ratio
Committed energy over interfacing energy. T³ = 27 is the cube of the triad: energy committed in the left half-plane of the i-cycle (i² and i³; convergence that hasn't yet reached the inter-scale interface). T² − P = 9 − 4 = 5 is what remains at the right half-plane (the inter-scale interface where energy becomes visible). Dark matter is not a single new particle waiting to be found in a detector. It is particles all the way down: nested ⊙s inside ⊙s (A2), each a whole with its own aperture, field, and boundary. A particle is a circumpunct at a particular scale. What we detect as "visible matter" is ⊙s at phases our boundary resolves. Dark matter is ⊙s at phases our boundary doesn't resolve: structure between the folds, in the stopband of our filter.
Cabibbo Angle New prediction
sin(θC) = α(1/2 + α·T/R) · SU(3)/T
0.22432
Predicted
0.2243 ± 0.0005
Measured
0.009%
Within 1σ
Generation mixing from the ladder. Base exponent 1/2 is half-power (the aperture half-open). The correction α·T/R = α·3/7 pushes the power slightly deeper by the triad-to-rung compression ratio. Prefactor SU(3)/T = 8/3 is gauge generators per triad element: mixing is mediated by the strong sector, distributed across three generations. Same formula family as mass ratios: α(base + α/K), where K = T³ = 27 for masses (generation cube) and K = R/T = 7/3 for mixing (rungs per triad). Masses measure intra-generation splitting; Cabibbo measures inter-generation mixing. Same architecture, different K.
Baryon Asymmetry New prediction
η = αV/T · V/(V−1) = α13/3 · 13/12
5.96 × 10−10
Predicted
6.1 × 10−10
Observed
2.3%
Accuracy
Why there is more matter than antimatter. The exponent V/T = 13/3 is generators-plus-whole per triad element. The correction V/(V−1) = E(1.5) = 13/12 is the first spectral splitting factor. The Standard Model has no mechanism to predict this number. The framework says: matter excess = coupling raised to the generation-structure/triad power, times the first splitting. The asymmetry is built into the pump cycle's phase structure.
Higgs Quartic Coupling New prediction
λ = (1/SU(3)) · (1 + 5α − 8α²)
0.12951
Predicted
0.12938
Measured
0.10%
Accuracy
Self-interaction from gauge multiplicity. Base 1/SU(3) = 1/8: the Higgs self-coupling is the inverse of the strong sector's generator count. First-order correction +(Φ+○)α = +5α: field + boundary (2+3 = 5) times coupling. Second-order −SU(3)α² = −8α²: gauge suppression. The correction self-limits: the first-order enhancement is suppressed at second order by SU(3)/(Φ+○) = 8/5. This parallels Λ exactly: Λ has (1 − T!·α + P·α²) where the ratio is P/T! = Φ/○ = 2/3. Λ suppresses (energy leaving the field); λ enhances (energy entering the potential). Same architecture, opposite signs.
Proton-to-Electron Mass Ratio 5.35 ppm
mp/me = (1/α)3/2 + (11/3)α + 13α2
Predicted: 1836.143 Measured: 1836.153 5.35 ppm
Composite mass from the selection rule. Every term is A(d) or A'(d) at a specific rung. Base 3/2 = A(1.5)/P: the accumulated traversal at the commitment rung, divided by the pump. Composites use A (function value, extended); elementary particles use A' (derivative, point-like). First order 11/3 = A'(2.5)/T: the rate of traversal at the emergence rung (where the strong force lives), divided by the triad (T quarks). Second order 13 = A'(3) = V: the full vertex count at the boundary, the self-referential correction. The proton needs two correction terms (binding across scales); leptons need only one (single-scale physics).
Meson Mass Law 0.3–1.1%
mmeson / me = F / α
π±: F = Φ = 2 → 0.34% K±: F = R = 7 → 0.71% ρ: F = A'(2.5) = 11 → 0.64%
D±: F = T³ = 27 → 1.12% Ds: F = A(3.5) = 28 → 0.39% Υ: F = 1/α − Φ → 0.05%
A second mass regime: the field regime. Leptons and baryons traverse the ladder (exponential: m/me = (1/α)E(d)). Mesons vibrate within the field (multiplicative: m/me = F/α). Every meson is F copies of the base mass quantum me/α ≈ 70 MeV, where F is a framework integer matching the meson's structural role. The pion carries Φ = 2 units (it IS the field quantum). The kaon carries R = 7 (it spans the ladder via strangeness). The three particle classes map to the three constraints: leptons to • (differentiation), baryons to ○ (evaluation), mesons to Φ (multiplication).
Electroweak Sector 0.10–0.67%
v / me = T³ / α² × (1 − R·α)
Higgs VEV: 0.15% W boson: (1/α)E(2.5)+1 − α/Φ → 0.15%
Z boson: mW/cos(θW) → 0.67% Higgs mass: √(2λ)·v → 0.10%
From mesons to gauge bosons. The meson sector uses one power of 1/α (field regime). The VEV uses two powers (the field at boundary level): T³/α², corrected by R rungs. The W boson exponent is E(2.5) + 1: the emergence exponent shifted by one unit (the gauge boson's own existence). The Higgs mass is a chain result: λ (from §27.7i) composed with v (from the VEV formula), requiring zero new parameters. A4 in action: the whole is the compositional unity of the field's self-coupling and its vacuum energy.
π0 Mass 0.73%
mπ0 / me = (Φ − SU(3)·α) / α
Predicted: 135.96 MeV Measured: 134.98 MeV
EM splitting in the meson sector. The charged pion carries Φ/α (two field quanta). The neutral pion lacks EM self-energy; SU(3)·α = 8 color generators times one coupling subtract from the field quantum. The π±/π0 mass difference emerges from the strong sector modulating the base field quantum.

Framework-Unique Predictions

Things the Standard Model cannot say at all.

Dark Matter: Particles All the Way Down Framework unique
A particle is a circumpunct. Every particle is a ⊙ at a particular scale: the 1 constrained to a position (A2). And it is particles all the way down: nested ⊙s inside ⊙s, each a whole with its own aperture, field, and boundary. There is no scale where particles stop and "pure field" begins. The boundary (○) is a filter, and filters have a passband. Visible matter is ⊙s in the passband: particles at the inter-scale interface (right half-plane, i1 and i0), where the pump cycle crosses between scales and ○ resolves them into detectable outcomes. Dark matter is ⊙s in the stopband: particles in the left half-plane (i² and i³), folded (so they gravitate; mass is how tightly the 1 has wrapped around 0s) but at the wrong phase to trigger the electromagnetic interface. Not invisible because exotic; invisible because they are between the scales where our detectors filter. This is why dark matter halos are diffuse rather than pointlike: if dark matter were at our boundary's resolution, it would clump, form disks, and radiate when compressed. Instead it forms smooth halos: ⊙s mid-process, not yet resolved at our scale. The whole cosmological budget is three phase states of one energy: ~5% visible (⊙s at the interface), ~27% dark matter (⊙s between folds), ~68% dark energy (Φ itself, the field the ⊙s exist within). One energy. Three costumes.
Dark Energy: w ≠ −1 Testable by DESI
w(z) ≈ −1.033 + 0.017/(1+z)
The equation of state deviates from a pure cosmological constant. A pure Λ gives w = −1 exactly. The framework predicts w(0) ≈ −1.016: dark energy is not quite constant because Φ is not static (it pumps). The DESI survey (2025-2026 data releases) will measure w(z) to the precision needed to distinguish this from w = −1. This is a clean, binary, falsifiable test.
Extended Higgs Sector Framework unique
The 64-state architecture assigns 4 Higgs states. The Standard Model has 4 real components (1 complex doublet), but 3 are "eaten" by W± and Z, leaving only the 125 GeV scalar. The framework's 64-state decomposition (48 fermions + 12 gauge + 4 Higgs) may predict additional scalar states beyond the minimal SM, potentially observable at future colliders. The exact spectrum is a target for derivation from the ladder.

The Hierarchy of Depths

Every constant lives at its dimensional home. The separations are framework ratios.

0D
α (coupling at a point)
1 α-step deep
0.22 ppb
0.5D
c (propagation speed)
= 1 in natural units
exact
1D
ℏ (minimum action)
= 1 in natural units
exact
1.5D
Mass ratios (spectral splitting)
(1/α)13/12+α/K
1-5 ppm
2D
Gauge structure (field symmetry)
SU(3) × SU(2) × U(1) from 64-state
1.4 ppm
2.5D
v/ΛQCD (emergence between scales)
(1/α)56/39
< 0.01%
3D
G (boundary closure)
α21 × φ²/2; exponent = E(3)
0.04 ppm
↓
Λ (the field's own energy)
α56/72; exponent = SU(3)·R
0.004%

Separations: G/α = 21 = E(3) (dimensional traversal). Λ/G = 56/21 = SU(3)/T (gauge-to-triad).

Problems Dissolved

Not solved by adding new ingredients, but dissolved by revealing they were never problems.

The Hierarchy Problem

Why is gravity 1040 times weaker than electromagnetism? Because 21 α-steps separate the point (0D) from the boundary (3D). The weakness of gravity is not a coincidence or fine-tuning; it is the length of the dimensional ladder.

The Cosmological Constant Problem

The "worst prediction in physics" (10120 orders of magnitude off). Dissolved: the vacuum is not a sum of modes but the residual energy of Φ after 56 α-steps of constraint. Each step suppresses by 1/137; fifty-six of them produce 10−120 naturally. The "catastrophe" was a category error.

The Dark Matter Problem

What is the dark matter particle? It is particles all the way down. A particle is a ⊙, and every ⊙ has parts that are ⊙s. Dark matter is ⊙s at phases our boundary doesn't resolve: particles in the stopband of our filter. They gravitate (folded energy has mass) but don't radiate (wrong phase for the electromagnetic interface). Not a single exotic species; structure between the folds, at every scale. T³/(T²−P) = 27/5 = 5.4 predicts the ratio exactly.

The Matter-Antimatter Asymmetry

Why is there more matter than antimatter? Because the pump cycle is asymmetric: the i-turn (i² = −1) is irreversible. The excess η = α13/3 × 13/12 follows from the generation-structure exponent and the first spectral splitting.

Complete Accuracy Table

All derived constants, their formulas, and their accuracy against measurement.

Constant Formula Accuracy Status
α 1/α = 360/φ² − 2/φ³ + α/(21−4/3) 0.22 ppb Confirmed
mμ/me (1/α)^(13/12 + α/27) 5 ppm Confirmed
mτ/me (1/α)^(58/35 + α/81) 1 ppm Confirmed
sin²θW 3/13 + 5α/81 1.4 ppm Confirmed
v/ΛQCD (1/α)^(56/39) < 0.01% Confirmed
G α^21 × φ²/2 × (1+2α/91) 0.04 ppm Confirmed
Λ α^56 × (1−6α+4α²) / 72 0.004% Confirmed
ΩDM/Ωvis T³/(T²−P) = 27/5 exact New
sin(θC) α^(1/2+α·T/R) · SU(3)/T 0.009% New
η (baryon) α^(V/T) · V/(V−1) 2.3% New
λH (1/8)(1 + 5α − 8α²) 0.10% New
mp/me (1/α)3/2 + (11/3)α + 13α² 5.35 ppm New
mπ± Φ/α = 2/α 0.34% New
mK± R/α = 7/α 0.71% New
mρ A'(2.5)/α = 11/α 0.64% New
mΥ (1/α − Φ)/α 0.05% New
mπ0 (Φ − SU(3)·α) / α 0.73% New
Higgs VEV T³/α² × (1 − R·α) 0.15% New
mW (1/α)95/39 − α/Φ 0.15% New
mZ mW / cos(θW) 0.67% New
mH √(2λ) · v 0.10% New

How to Falsify This

A framework that cannot be falsified says nothing. Here is how to break this one.

The framework makes specific, quantitative predictions. If any of the following are confirmed by experiment, the framework is falsified:

1. Dark matter detected as a single new particle species. If a direct-detection experiment (LZ, XENONnT, DARWIN) finds one discrete weakly-interacting massive particle species that accounts for all dark matter, the "⊙s between folds" prediction is wrong. The framework says dark matter is structure at phases our boundary doesn't resolve, not a single exotic species.
2. w = −1 exactly. If DESI or successor surveys measure the dark energy equation of state to w = −1.000 ± 0.005 with no deviation, the pump-cycle prediction of w ≠ −1 is wrong.
3. ΩDM/Ωvisible deviates from 5.4. As measurements of the dark matter and baryon densities improve (CMB-S4, Euclid), if the ratio settles outside T³/(T²−P) = 27/5 by more than observational uncertainty, the i-cycle quadrant model is wrong.
4. Correction terms fail for future constants. The Cabibbo angle and Higgs quartic both sharpened from base formulas (1.6% and 3.4%) to sub-percent accuracy (0.009% and 0.10%) using α-dependent corrections of the same structural form as Λ and G. If future constants derived from the ladder fail to follow this pattern, the correction architecture is coincidental rather than structural.
5. A new fundamental constant that cannot be placed on the ladder. If a future experiment discovers a constant that cannot be expressed as α raised to a framework-structured exponent, the "one constant generates everything" claim fails.

The framework is not asking to be believed. It is asking to be tested.