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, admits α (the electron aperture's cross-scale coupling strength, |•electron|) as one measured input to the κ nesting operator, and derives the other constants from α plus a single small pool of framework integers.
The framework is built on the circumpunct ⊙ = (✹ ∘ i ∘ ⊛)(Φ(•, —, ○)): a whole composed of four structural dimensions unified by the pump cycle. The aperture • (convergence, 0D), the line — (commitment, 1D), the field Φ (mediation, 2D), and the boundary ○ (filtration, 3D). These four constraints, compounded at every scale, generate all of physics.
From the triad T = 3 (forced by seven independent routes, including Route 6: (T−1)! = Φ from P! = G·Φ, and Route 7: S = PT = P³ iff T = 3), 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. α = |•electron| is admitted as input to κ0,0 (the representation layer); its CODATA value additionally admits a structurally motivated closed form at 0.22 ppb (the auxiliary-claim layer).
A common objection, fair and worth answering up front: "Some of the numbers you derive have units; those aren't fundamental constants." The framework's position on this is explicit.
1. The fundamental predictions are dimensionless. Every entry on this page reduces to a dimensionless ratio: α itself, mass ratios like mμ/me, mixing angles like sin²θW, the gravitational coupling αG = α21·φ²/2·(1+2α/91), the cosmological constant Λ in Planck units, the dark matter to visible ratio. Where SI values appear (G shown as 6.67 × 10−11, the cosmological constant in Planck units, W/Z/Higgs masses in MeV), they are the dimensionless prediction times a unit scaffold assembled from measured me, ℏ, c. The prediction lives in the dimensionless ratio; the SI form is the ratio re-expressed in the reader's unit system.
2. α is admitted as measured input, not derived from the pump cycle. The four-beat pump F is trace-preserving (it cannot set a coupling scale); κ (the nesting operator carrying ⊂[α]) is where α lives, as κ0,0 = |•electron|, the strength of the electron aperture's bond to the 0D aperture of the greater whole, with the 2D electromagnetic field as mediator. Everything else on this page is a structural composition around α; the claim is not "α is derivable from axioms," it is "given α, the rest follows from T = 3 plus the framework integer pool." This is the representation layer.
3. The α closed form is an auxiliary claim, separable from the representation layer. The CODATA value of α additionally admits the structurally motivated expression 1/α = 360/φ² − 2/φ³ + α/(59/3) = 137.035999147 (0.22 ppb from CODATA), where every factor is framework-native (360 = P!·T·(Φ+○) = 24·3·5 via the P! = G·Φ identity at T = 3, 59/3 = (P·V+R)/T = (S−Φ−T)/T via Route 7 of the
octave-wrap lemma, φ-exponents fixed by A3 + dimensional analysis). No factor-level residuals remain as of 2026-04-22, and the three-term
shape is pinned by a station-content argument (α is a scalar 0D-to-0D coupling κ
0,0 with a 2D mediator; content at exactly three stations: 2D mediator, 3D closure, 0D self-return; no 1D term because scalarity from equidistance of • forces directional content to zero; see
alpha_derivation.html §5). The residual convention is bookkeeping ("one term per station with content"), consistent with A(d)'s one-integer-per-rung bookkeeping throughout the framework. If future precision pushes 1/α outside 0.22 ppb of this value, the auxiliary claim fails, and α reverts to input-only at κ
0,0; the representation layer is untouched. Full audit:
α derivation.
4. "Zero free parameters" is a claim about the compositional shape, not about the measurement count. Given α, each card below composes one measured constant from framework integers (T, P, Φ, R, V, SU(3), A(3)) without per-formula tuning. The restrictive grammar (the integer pool is small and its operations are fixed by the ladder) means not every combination lands on a known value; the fact that the right combinations exist and produce sub-percent to sub-ppb accuracy is what's on the record. What remains open: whether the grammar is independently predictive (whether it reproduces a not-yet-measured constant without post-hoc fitting). The pre-registered predictions section below stakes that claim on specific future measurements.
5. Each row of the accuracy table is a compositional test, not a calibration fit. No parameters are tuned per entry. When a formula lands at sub-percent accuracy, the claim is that the framework's integer pool contains the structural content of that constant. When it lands at several percent, either the framework is missing a correction or the pool is too coarse at that rung. Errors are reported honestly; no entries are suppressed.
The rest of this page is a walk through the compositions, rung by rung.
Three categories. Entries are classified as K (known-fit: structurally motivated composition of an already-measured value; post-hoc but not tuned, since pool integers and α-exponents are forced by the ladder), P (pre-registered prediction: value committed before measurement lands; pass/fail/inconclusive), or D (derived identity: integer relation with no free parameters at all, e.g., conservation of traversal, Route 7, S = P³ iff T = 3). K entries show the grammar is restrictive; P entries show whether it is predictive; D entries show what is true by construction.
Structural compositions of already-measured values. Pool integers and α-exponents are forced by the ladder (not tuned), so each K is a restrictiveness test: could the grammar have landed elsewhere? Median 2-3 integers per formula; typical accuracy sub-percent to sub-ppb.
Committed values awaiting measurement: P1 (1/α next decimal, ±5 ppb), P2 (X≡X/X−X length ratio, ±1%), P3 (mycelial Murray's Law exponent, ±0.15), P4 (five-virtue sequence effect size, d ≥ 0.5). None settled yet. Status tracked in the history page.
Integer relations with no free parameters. Conservation of traversal (0+1+2=3), Route 6 ((T−1)! = Φ from P! = G·Φ at T = 3), Route 7 (S = P³ iff T = 3), seven independent T = 3 self-determinations. True by construction of the pool, not fits.
What this page can and cannot claim. Can claim: the integer pool is sparse (14 symbols: T, P, Φ, R, G, V, S, A(d), A'(d), SU(3), φ, π, ◐, and α as input), the operations are fixed by the ladder (A(d), A'(d), compositional products, α-powers at structural rungs), and given α as input, 44 known dimensionless constants land inside their measurement uncertainty by structurally forced compositions.
Cannot claim: that the grammar is independently predictive (the 4 pre-registered P entries are the first live test; no P has settled). If 2 of 4 P entries fail their bands, the
scoreboard flags "predictiveness in doubt" and piece #3 of the frontier plan reopens with new targets.
Retraction protocol. When a precision measurement shifts a K fit outside its originally claimed accuracy band, the old formula gets a
Revision Notice block in circumpunct_framework.md (not silently deleted), the retraction is logged in the
history page, and if the retraction is factor-level, the relevant rung is marked "pool choice under review." When a P prediction lands outside its pre-committed band, it demotes from P to R (retracted); it is
not re-fit to the new value. No silent retraction; every change between measurements is logged. Post-hoc refits of a failed P into a new K are flagged "re-fit after failure" in the Brier score. Full protocol in
plans/predictions_scoreboard.md §4.
Companion documents. Full audit across all 52 entries with pool-integer counts and uniqueness notes:
plans/predictions_scoreboard.md. Running log of settled/retracted entries as they accumulate:
predictions_history.html. The walking cards below are the eight headline K entries plus the four P entries; the scoreboard carries the complete set.
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).
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.
1/α = 360/φ² − 2/φ³ + α/(59/3) = 137.035999147
137.035999147
Framework value
5σ
Current Cs/Rb disagreement
± 5 ppb
Falsification band
The 0D rung as test. Current precision measurements disagree: Berkeley-Cs (2018) gives 137.035999046(27), LKB-Rb (2020) gives 137.035999206(11), a 5σ split the Standard Model does not adjudicate. The framework's pool-native closed form (360 = P! · T · (Φ+ˆ), with the φ³ sign balance-forced and 59/3 emerging from the
octave-wrap lemma) lands at 137.035999147, within 1 ppb of the midpoint. The full audit (each symbol justified, the representation layer separated from the auxiliary closed-form claim, no factor-level residuals remaining as of 2026-04-22 with the φ-exponent rule closed by A3 + dimensional analysis) is in the
α derivation. Pre-registered: as atom interferometry tightens (next-generation Cs, Rb, and independent Sr at 10−100 ppt precision), the converged value will land at 137.035999147 ± 5 ppb. Outside this band, the closed-form auxiliary claim fails; α reverts to input-only at κ
0,0 and the 0D rung loses its structural name (the ladder composition survives, the name does not).
(X≡X) / (X−X) → R/T² = 7/9 ≈ 0.778
0.779
Measured for C (0.19%)
Bond length at the 1.5D rung. Carbon's C≡C / C−C ratio measures 120/154 = 0.779, matching R/T² = 7/9 to 0.19%; the same ratio appears in the π1/σ energy ratio. The framework says this is not a carbon fact: the 1.5D rung (the i-turn, spectral splitting) sets a scale-invariant triple-to-single ratio for any homonuclear main-group bond. Pre-registered: when X≡X and X−X bond lengths are resolved to sub-percent precision for silicon (disilyne analogues Ar-Si≡Si-Ar'), germanium, phosphorus (diphosphorus derivatives), and arsenic, every ratio will land at 0.778 ± 1%. A clean 0.82 for Si (implying stronger multiplicity contraction than the ladder permits) or a clean 0.74 for P would falsify the "R/T² is universal at 1.5D" claim.
rparentn = Σ rchildn, nmycelia = 2.5
n = 2.5
Framework exponent
n = 3 / n = 2
Vascular (3D) / leaf (2D)
± 0.15
Falsification band
Dimension drives the exponent. Murray's Law conserves rn where n equals the effective transport dimension. Animal vasculature (3D bulk fluid) gives n = T = 3; leaf venation (2D flat lamina) gives n = Φ = 2 (confirmed). Mycelial networks sit at 2.5D: branching wood-decay fungi propagate through a porous substrate that is neither 3D bulk nor 2D plane, and the emergence rung (✹ at 2.5D) is literally where this half-integer transport lives. Pre-registered: when Murray's Law exponents are measured precisely in Armillaria, Phanerochaete chrysosporium, or similar branching mycelia at hyphal-network resolution, the exponent will land at 2.5 ± 0.15. A clean 3.0 (the network behaves as bulk) or a clean 2.0 (it behaves as a plane) would falsify the "transport dimension = ladder rung" reading.
GOOD → RIGHT → FAITHFUL → TRUE → AGREEMENT
Cohen's d ≥ 0.5
Predicted effect size
12 months
Durability window
Pre-registered design
Falsification: d < 0.3
Required sequence, not one pillar among many. The framework claims GOOD (boundary, ˆ), RIGHT (field, Φ), FAITHFUL (line, —), TRUE (aperture, •), AGREEMENT (whole, ⊙) must be walked in order: you cannot assess faithfulness until the space is open (RIGHT), you cannot confirm truth until the history of commitment has been checked (FAITHFUL), and AGREEMENT is the compositional closure (D5) of the prior four. Pre-registered: in a controlled longitudinal study of conflict resolution (workplace disputes, long-partnership mediation, or negotiated policy settlements), groups trained in the explicit five-virtue sequence will show Cohen's d ≥ 0.5 on twelve-month agreement durability versus unsequenced controls matched for initial signed-agreement rate. If d < 0.3, the "required sequence" claim reduces to "any virtue set works equivalently" and the ethical architecture is decorative rather than structural.
These are the bolts still to land. A framework that fits every measured value already in the books is a grammar; a framework whose pre-registered predictions land at the band it named becomes a theorem.
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 dimensionless coupling, mixing angle, or mass ratio that cannot be expressed as α raised to a framework-structured exponent (times a small-integer prefactor from the pool), the "α plus integer pool composes everything" claim fails. Note: this targets the representation layer. The auxiliary α closed form is a separate claim with its own falsification band (see the Physics pre-registered prediction, ± 5 ppb).
The framework is not asking to be believed. It is asking to be tested.