Theory — Dimensional Coherence Theory

Theory

The canonical action, polytope topology, McKay correspondence, dimensional axis, and anti-dimension cosmology underlying Dimensional Coherence Theory.

The framework is a Brans–Dicke scalar–tensor theory in which a scalar field P (the "tie field") crystallises via Gross–Pitaevskii dynamics on a 600-cell lattice. The icosahedral primitives V=12, E=30, F=20, |2I|=120, χ=2, n_irreps=9, φ=(1+√5)/2 plus the geometric P₀ = cos(36°)/cos(18°) = 0.850651 force every numerical prediction. Zero free parameters.

Canonical action

The framework is a Brans–Dicke scalar–tensor theory in which a scalar field P (the "tie field") crystallises via Gross–Pitaevskii dynamics on a 600-cell lattice. The action is

$$ S = \frac{1}{16\pi G}\!\int\! d^4x \sqrt{-g}\left[P\,R - \frac{\omega(P)}{P}(\partial P)^2 - 2V(P)\right] + S_{\rm matter}[\psi, P\!\cdot\! g] + S_{\rm DM}[{\rm disformal}] + S_\theta $$

with the field-dependent coupling $\omega(P) = (c_{BD} P^2 - 3)/2$ and $c_{BD} = 138{,}189$. The master identity $2\omega_0 + 3 = c_{BD}\,P_0^2 = 100{,}077$ closes exactly for the equilibrium values P₀ = 0.850651 (geometric: cos(36°)/cos(18°); 0.041% from the calibrated 0.851), ω₀ = 50,037. General Relativity is recovered in the limit P → 1, ω → ∞. Full derivation: DCT-FND-V2.

600-cell topology

The 600-cell is the unique convex regular 4-polytope with 120 vertices, 720 edges, 1200 faces, and 600 cells, with binary-icosahedral symmetry 2I. Vertex coordination z = 12, vertex-figure face count f_v = 20, giving the GP three-body / two-body coupling ratio β = f_v/z = 5/3. The Avrami susceptibility χ_Avr = 1 − P₀² = 0.276 (0.284 with LHY) is independently confirmed by four derivation paths.

Spectral data on the adjacency Laplacian: 9 distinct eigenvalues with multiplicities matching the 9 irreducible representations of 2I, Σd² = 120. Casimir spectral identities Σ = f_v + z − 1 = 31 and Σ_{C·d} = 154 = 2·7·11 are exact integers; G_LHY = 3701/6300 with 3701 prime is exact rational; spectral gap μ_min = (3 − √5)/4. Derivation and chance-probability test: DCT-SPI-01.

McKay correspondence

Each finite subgroup of SU(2) is associated by McKay (1980) and Slodowy (1980) with an affine Dynkin diagram of ADE type. The case relevant to DCT is 2I → E₈. The McKay matrix satisfies A_McKay d = 2d for d = (1,2,3,4,5,6,4,3,2); the affine 𝐸̃₈ Dynkin spectral radius equals 2.0. The branching 248 = (78,1) ⊕ (1,8) ⊕ (27,3) ⊕ (27̄,3̄) is exact. Three fermion generations are forced by the (27,3) representation; the ℤ₃ coset partition of the 600-cell vertex set produces CKM mixing in the Yukawa sector. Standard-Model derivation chain: DCT-SM-01.

Dimensional axis (DCT-AXS-01)

The paper DCT-AXS-01 re-derives P₀ from icosahedral midradius/circumradius geometry: P₀ = cos(36°)/cos(18°) = 0.850651. This derives P₀ from icosahedral geometry and re-tags 77 observables DERIVED_FROM_PRIMITIVES. The dimensional axis carries the novel forward predictions for photon dispersion (η = 1/12, n = 2), GW echo delay ((3/5)·r_s/c·ln(r_s/l_P)), the BEC breathing-mode boson (m* = ℏH₀/c² ≈ 1.44×10⁻³³ eV/c²), and gravitational decoherence (Γ = (3/5)·GM²/(ℏd)).

The axis paper introduces the anti-dimension dual coupling β_up = 3/5 = V/F, the reciprocal of β = 5/3 = F/V. The pair {β, β_up} drives the framework: β governs BEC condensation (downward), β_up governs anti-dimension dissolution (upward).

Anti-dimension cosmology (DCT-ADC-01)

The companion v9 paper DCT-ADC-01 derives:

Ω_DM/Ω_b = 9·β_up = 27/5 = 5.400 — first geometric derivation of the dark-matter / baryon ratio. 0.10σ vs Planck PR3 (5.394 ± 0.065).
Λ ~ (φ²+φ⁻²)·10⁻⁽¹²⁰⁺χ⁾ ≈ 3×10⁻¹²² — order-of-magnitude derivation of the cosmological constant from the icosahedral primitives. The exponent |2I|+χ = 122 in log₁₀ is exact.
Splashback R_sp/R_200m = P₀^{β_up} = 0.851^{3/5} = 0.9075 — 0.95σ vs More+2015.
Apparent w_a = −Δγ/0.21 = −0.714 — growth–geometry signature on a stable ΛCDM background, reproducing the DESI Y3 apparent w_a at 0.20σ.
EW hierarchy m_W²/M_Pl² ~ 10⁻³⁴ — geometric suppression chain.
η_B = (2/|2I|)·exp(−Σ′/n_irreps) = 6.174×10⁻¹⁰ — baryon asymmetry. 0.95σ vs PDG 6.10×10⁻¹⁰.

SM mass spectrum (DCT-MSS-01)

The paper DCT-MSS-01 derives the complete Standard-Model mass spectrum and couplings from icosahedral primitives. Thirty-five predictions, 34 HIT + 1 SOFT.

Mass	Geometric formula	Predicted	Measured	Match
m_c (charm)	(V/n_irreps)·m_p = (4/3)·938.3	1251 MeV	1270 ± 20 MeV	0.9σ
m_b (bottom)	(n_irreps/2)·m_p = (9/2)·938.3	4222 MeV	4180 ± 30 MeV	1.4σ
m_s (strange)	(χ/F)·m_p = (1/10)·938.3	93.8 MeV	93.4 ± 0.8 MeV	0.5σ
m_t (top)	(Σ′+Σ_d)·m_p	172.64 GeV	172.57 ± 0.29 GeV	0.18σ
m_u/m_d	9/19	0.4737	0.474 ± 0.056	0.01σ
m_s/m_d	F = 20	20.0	20.2 ± 1.5	0.13σ

Key DCT constants

DCT is a Brans–Dicke scalar–tensor theory in which scalar field P (the tie field) crystallises via Gross–Pitaevskii dynamics on a 600-cell lattice. Two equilibrium values, P₀ and ω₀, related by the master identity through the coupling normalisation c_{BD} — these three numbers are the structural anchors of the framework.

Symbol	Name	Value	Derivation
P₀	Tie-field equilibrium amplitude	cos(36°)/cos(18°) = 0.850651 (geometric); 0.851 (calibrated, 0.041%); 171/200 = 0.855 (topological, 0.47%)	Brans–Dicke condensate equilibrium; icosahedral midradius/circumradius (DCT-AXS-01).
ω₀	Brans–Dicke coupling at equilibrium	50,037	Solar-system Cassini bound ω > 40,000 satisfied with margin.
c_{BD}	BD coupling normalisation (NOT speed of light)	138,189	Renormalisation-group fixed point; Δc/c ~ 10⁻⁸ over 60 e-folds.
f_v	Vertex-figure face count of icosahedron (= F)	20	Topological invariant of the 600-cell vertex figure.
z	600-cell coordination number (= V)	12	Each vertex of the 600-cell has 12 nearest neighbours.
β	GP three-body / two-body coupling ratio	f_v/z = 5/3	Ratio of icosahedral face count to 600-cell coordination.
β_up	Anti-dimension dissolving coupling	z/f_v = V/F = 3/5	Reciprocal of β. Drives DM ratio, Λ, splashback, apparent w_a.
\|2I\|	Binary icosahedral group order	120	= V·E·F/χ topological invariant.
Σ′	Angular-momentum Casimir sum of 2I irreps	154 = 2·7·11	Spectral sum over irrep dims [1,2,2,3,3,4,4,5,6].
n_irreps	Number of 2I irreducible representations	9	McKay correspondence: 2I → affine E₈.
χ_Avr	Avrami susceptibility	0.276 (= 1 − P₀²); 0.284 with LHY	Four independent derivation paths.
Master identity	Connecting P₀, ω₀, c_{BD}	$2\omega_0 + 3 = c_{BD}\,P_0^2 = 100{,}077$	EXACT for n = 2.

Full parameter register: /constants/. Source-of-truth:.

Notation collisions

Each entry below is a symbol that means at least two different things. Always disambiguate explicitly.

c: Speed of light vs c_{BD} = 138,189 (BD coupling normalisation). Always use c_{BD} for the coupling, plain c only for speed of light.
P (capital): The tie field (coherence amplitude) — the Brans–Dicke scalar. NOT pressure, NOT probability.
n: Three different n's: n_ω (ω(P) exponent = 2), n_Avr (Avrami exponent = 1), n_holo (holographic = 2). Always specify.
m: Yukawa mass m_Y ≈ 0.023 h/Mpc; GP boson m* ≈ 1.44×10⁻³³ eV; proton mass m_p. Always disambiguate.
α: PPN α; fine-structure α_EM; bump coefficient α_bump = 0.405; LHY α_LHY. Always specify.
β: GP coupling β = f_v/z = 5/3 vs PPN β. Always specify.
B_s: Disformal coupling value 5.46×10⁷ (a number) vs B(P) the general disformal function. B_s = value; B(P) = function.
g_μν vs g̃_μν: Einstein-frame metric g vs Jordan-frame metric g̃ = P·g. The relation c_GW = c is exact in the Einstein frame.
(2ω₀ + 3) ≈ 100,077: BD stiffness, NOT 2ω₀ ≈ 100,074. The +3 comes from conformal coupling to R.

Frame conventions

DCT is a scalar–tensor theory. The same observable can have different values in different frames, and consistent comparison requires fixing the frame in which the measurement is interpreted.

Frame	Definition
Einstein frame	Standard GR metric g_μν. c_GW = c exactly. The action is canonical.
Jordan / physical frame	Conformally rescaled metric g̃_μν = P·g_μν. This is what laboratory rulers and clocks 'see'. Atomic clock rates depend on P. H_phys = H_E/√P₀ = 73.019 km/s/Mpc.
SPS frame	Stress-Path-Selected frame used for cosmological observables. Cosmic-chronometer fits compare H_obs(z)/H_ΛCDM(z) — see /observables/cc-hz/.

Common terms

DCT: Dimensional Coherence Theory. The current name of the framework.
Tie field: The Brans–Dicke scalar field P. The "tie" name is descriptive — it ties the geometry of the 600-cell to physical observables.
Anti-dimension: The β_up = 3/5 dual sector. Where β = 5/3 governs BEC condensation (downward), β_up governs the dissolving / anti-condensing dynamics (upward) that produce dark-matter abundance, the cosmological constant, and the splashback radius.
600-cell: Unique convex regular 4-polytope: 120 vertices, 720 edges, 1200 faces, 600 cells. Binary-icosahedral symmetry 2I.
2I: Binary icosahedral group, order 120. Discrete subgroup of SU(2). McKay correspondence: 2I → affine E₈.
Master identity: 2ω₀ + 3 = c_{BD}·P₀² = 100,077. The single closed identity that ties all PPN, BAO, SBD, SM derivations to the icosahedral primitives.
Conformal-wall theorem: S_YM[P·g] = S_YM[g] in 4D. Standard Model gauge physics is invariant under g → P·g. DCT-CMB-01.
Empirical isolator: An observable whose match uniquely distinguishes DCT from competing frameworks. Tier classification: POTENTIALLY_EXCLUSIVE / SHARED_VALUE / STRUCTURAL_THEOREM / ANTI-PREDICTION.