CMB BIREFRINGENCE & S⁷ CHIRALITY

THE PREDICTION

This prediction is derived from two independent primitives that appear on the math page:

• S_classical = 5/7 comes from the S⁷ path geometry — five rotatable paths operational in the classical state. This is a state-space count, not a fit.

• p_c = 0.3116 comes from 3D site percolation theory (Stauffer & Aharony) — a dimensionless lattice constant, taken as input.

The ratio (5/7)/p_c = 2.2923 is therefore not circular and not fitted — it is built from two independent frameworks (S⁷ path geometry and 3D percolation theory) combined in one formula.

PRE-REGISTRATION EVIDENCE

The Δβ formula has been in TSO since at least v7.2, well before ACT DR6's 2025 announcement. Documentary evidence:

• TSO v7.2 t-shirt design prints the formula β = S/p_c and labels the phase-transition region Δβ. The shirt is tagged "TSO v7.2" and exists as a physical artifact from the v7.2 era. A photograph is archived with the project.

• β derivation suite notebook — a runnable Colab notebook that recomputes the 2.2923 value from the primitives and tests candidate S⁷ angular conversion factors. β derivation suite (Colab).

• The current predictions page lists β > 0 as a formally tracked pre-registered prediction (entry 9), added May 11, 2026 — the prediction itself is older, but the formal tracker entry dates from the session in which the predictions page was built. The underlying formula and sign prediction are both older than that entry.

OBSERVATIONAL STATUS

Both measurements are consistent with the TSO pre-registration on sign and order of magnitude. Neither measurement is consistent with the Standard Model prediction (β = 0 exactly).

The TSO prediction is not a pre-registered exact numerical match. At v7.2 and since, the framework has consistently predicted β_dim = 2.2923 and a positive sign, but the conversion from dimensionless to degrees has remained open. So the correct framing is: sign and order-of-magnitude confirmed, exact value pending.

THE SIGN: S⁷ OCTONION HANDEDNESS

The positive sign (β > 0 rather than β < 0) is the feature that distinguishes the TSO prediction from "TSO predicts β ≠ 0 but has no sign preference." The sign argument is as follows.

TSO's path structure {X₁, X₂, x, y, z, T, ∅} is seven elements, and the five rotatable paths form an SO(5) rotation group. The state space these paths live in is S⁷ — the seven-sphere — because the continuous family of path assignments with unit normalization naturally inhabits the unit sphere in eight-dimensional path-amplitude space.

S⁷ is not an arbitrary seven-sphere. It is the norm-one sphere of the octonions, the unique non-associative normed division algebra over the reals. S¹, S³, and S⁷ are the only parallelizable spheres; S⁷ is the one where parallel transport fails to commute in a specific way, captured by the associator [a,b,c] = (ab)c − a(bc). The associator is non-zero, and it has a preferred orientation: octonion multiplication is chiral in a way that the commutative and associative algebras (reals, complexes, quaternions) are not.

When the first spanning cluster crystallized at cosmic percolation and selected a chirality — the seed-crystal model described on roof-antimatter — the selection was not a coin flip between two equally-probable outcomes. S⁷'s preferred orientation biases the seed selection toward one handedness. That preferred handedness is what produces β > 0 rather than β < 0 in the CMB polarization rotation.

This argument is currently asserted in the framework rather than demonstrated notebook-style. The demonstration task would be: derive the associator's orientation explicitly, show how it maps onto the percolation seed-crystal model, and show that the resulting chirality produces a positive β in the standard CMB convention. This is on roof-open as an open theory task.

THE OPEN PROBLEM: EXACT CONVERSION TO DEGREES

The dimensionless value β_dim = 2.2923 is cleanly derived. Converting it to degrees requires the natural angular unit of S⁷, which has not yet been cleanly derived from the octonion geometry.

The β derivation suite notebook tested ten candidate S⁷ angular factors derived from surface area (π⁴/3), volume (π⁴/24), Hopf fibration, and several combinations. All candidates fall in the correct order of magnitude (roughly 0.1°–0.6°), but none falls within 1σ of the observed 0.264°. The closest tested candidate was (2π/7) in degrees / 100, which gives 1.179° — off by roughly 16σ.

The most likely source of the correct conversion is the octonion associator angle — the natural angular scale associated with the failure of associativity. This has a specific value that can in principle be computed from octonion geometry, but it was not tested in the current notebook suite, and computing it correctly is not trivial. This is the primary open theory task for this page.

Status: if the associator angle computation returns a conversion factor that lands β_dim = 2.2923 near 0.264°, the derivation is complete and the prediction upgrades from "sign + order of magnitude" to "exact value." If the computation gives a clearly different number, either (a) the S⁷ geometric framing needs revision or (b) there is an additional geometric factor that the current derivation is missing.

κ = 4/3 CROSS-SCALE UNIVERSALITY

Independent of the β value, TSO predicts κ = 4/3 (2D percolation universality) appears across multiple scales:

• IBM quantum hardware (below-threshold exponential fit exponent)

• CMB temperature cluster fractal dimension: D_f = 1.85 vs 91/48 = 1.896 at 1.15σ

• CMB polarization handedness (this page)

• Rydberg sigmoid prediction (κ = 4/3 specifically, to be tested in the pending Rydberg experiment)

Whether the same κ appearing across these scales is a deep result or a coincidence of the universality class is an open question. It is consistent with the TSO claim that all phase transitions across scales share the same d_eff ≈ 2 universality class at threshold, but this is not independently sharp enough to constitute a dedicated test.

CONNECTION TO BARYON ASYMMETRY

The CMB birefringence and the matter-antimatter asymmetry are proposed to be two consequences of the same event: the first crystallization of the cosmic spanning cluster, which broke chirality symmetry. See roof-antimatter for the seed-crystal model and the baryon asymmetry calculation.

Quantitatively connecting β and η from the same set of lattice primitives — such that the observed values of both are predicted from one formula — is an open task. Currently the baryon asymmetry calculation is a retrodiction (applied Walton-Chalmers / Avrami metallurgical math to TSO parameters and got η within 19%), and the CMB β is a pre-registered sign + order-of-magnitude prediction. Unifying them into a single formula that predicts both would be a significant strengthening.

WHAT'S STRONG, WHAT'S OPEN

FALSIFICATION

This prediction fails if any of the following is confirmed:

• Improved precision measurements find β consistent with zero (falsifies the seed-crystal chirality-breaking mechanism entirely).

• Improved precision measurements find β definitively negative (falsifies the S⁷ octonion handedness sign argument — would require re-examining whether octonion parity maps to positive or negative rotation in the standard CMB convention).

• The octonion associator angle, when computed correctly, converts β_dim = 2.2923 to a value clearly outside the 0.1°–0.6° range (would indicate the framework has an error in the β_dim derivation or the conversion assumption).

NOTEBOOKS

• β derivation suite (Colab) — recomputes β_dim from primitives, tests candidate S⁷ conversion factors

• Baryon asymmetry (Walton-Chalmers / Avrami) — the seed-crystal model this page depends on