Meta-theory

The 7 architectural principles of PT

Not imposed rules. Emergent properties of the sieve.

When applying PT to a new domain (molecular chemistry, biology, geophysics), one observes that the same structures recur. Not by convention: by architectural necessity. These seven principles were identified in March 2026 during PTCS v0.6 (computational chemistry) development, then verified across other domains.

Each principle is an emergent property of the sieve at $mu^* = 15$, hence a consequence of T5 and T6. None is postulated. Their force comes from universality: they apply equally to $alpha_{ m EM}$ and to the chemical bonding of a diatomic molecule.

The seven

Seven universal patterns

Sequential cascade

The sieve filters in ORDER: p = 3 → p = 5 → p = 7. Each filter sees the survivors of the previous one.

Tree-level α_EM = ∏ sin²(θ_p) is CRT-orthogonal so order does not matter. But dressing corrections ARE sequential. CKM is an explicit cascade: V_us → V_cb → V_ub (powers of λ).

Ex: CKM: V_us = λ, V_cb = A·λ², V_ub = A·λ³·R

Shannon cap (D_KL ≤ 1 per channel)

Each CRT channel carries AT MOST 1 unit of information. sin²θ_p < 1 for every p.

The cap is AUTOMATIC for PT quantities. The resummation G_Fisher/cos²θ_3 = 5.12 sums ALL orders of the same channel (total throughput, not instantaneous).

Ex: C_K^tree = 4/sin²θ_3, NLO = 1/(p₁p₃)

Pythagoras = orthogonal CRT

Channels on DIFFERENT CRT circles combine via Pythagoras.

sin²θ_W = γ_7² / (γ_3² + γ_5² + γ_7²) = Pythagorean norm. In chemistry: E² = E_cov² + E_ion². Same branch / same role: direct addition (V_us = sin²_3 + sin²_5 on q⁻).

Ex: sin²θ_W = 0.2377 (tree), 0.23119 (corrected)

Bifurcation 3 regimes

The q⁺/q⁻ bifurcation gives 3 regimes, not 2.

Regime 1 (strong coupling, EA/IE ≥ s): full transfer = confinement. Regime 2 (weak coupling, EA/IE < s): partial transmission = QED. Regime 3 (endothermic, BH ≤ 0): quantum fluctuations = VP. The threshold s = 1/2 is exactly the γ_p > 1/2 threshold from T5.

Ex: Atomic IE: 3 regimes read directly

Holonomic rotation

Energy PROJECTS onto the sieve frame (Ry, m_e) via holonomic rotation.

Rotation amplitude depends on angular coverage (hybridisation in chemistry, Berry phase in physics). This is the MECHANISM by which D_KL converts into an observable. Chemistry: IE projects onto Ry via cos²θ_3 · (Ry − IE). Physics: masses = holonomic rotations around m_e.

Ex: Berry integral 3π = N_gen · π fixes m_μ/m_e

Geometric resummation

The effective coupling for fluctuations is G_Fisher/cos²θ_3 = 5.12.

Resummation of ALL perturbation orders on the p = 3 channel: G_Fisher · Σ sin²₃ⁿ = 4/(1−sin²₃) = 4/cos²₃. In physics, this is the running coupling. The ratio C_Koide / (G_Fisher/cos²₃) = cos²₃/sin²₃ = cot²θ_3 links the two branches.

Ex: Fisher-Koide identity: C_K = 4/sin²₃ + (1+5δ₃²/18)/21

Anti-double-counting (GFT)

log_2(m) = D_KL + H is an IDENTITY. No energy can be added that is already counted in D_KL.

The q⁺/q⁻ bifurcation is a PARTITION: sin²(q⁺) and sin²(q⁻) measure orthogonal projections of the same prime. No double-counting. Every PT coefficient respects this algebraic identity.

Ex: Ω_info + Ω_Λ + Ω_baryon = 1 (Clausius split)

Natural question

Why seven?

One might suspect the number 7 is itself chosen. It is not. Principles 1–3 (cascade, Shannon cap, Pythagoras) describe the sequencing of channels; principles 4–6 (bifurcation, rotation, resummation) describe the regimes in which each channel operates; principle 7 (anti-double-counting) is the consistency constraint binding them all.

Three sequencings + three regimes + one consistency = 7. And this is exactly the same $3 + 3 + 1$ structure as the main cascade: three active primaries $\{3, 5, 7\}$ + three dressing corrections + one GFT consistency.

Not a coincidence: the architectural principles are the cascade, read at a meta-level. Hence their universality.

Tested universality

Where these principles are verified

Quantum chemistry (PTCS)

IE Z = 1–118 at 0.06 % MAE, EA 0.98 %, exact periods. Empirical origin of the 7 principles.

Particle physics

43 SM observables, 0.316 % MAE, no continuously fitted parameter. Every correction obeys the 7 principles.

Cosmology

H₀, n_s, Ω_info, F_echo within < 0.2 % of Planck. q⁺/q⁻ bifurcation → dark sector.

Nuclear

408/413 PASS, magic numbers 2, 8, 20, 28, 50, 82, 126 derived. Resummation and bifurcation active.

Biology (PT_PROTEIN)

Universal q⁺/q⁻ bifurcation across 3 biopolymers; Ramachandran torus 15/15; RNA cos_therm at 90 ppm.

Perception (SieveColorSpace)

3 opponent channels forced by {3, 5, 7}; L-M weight = γ_3/Σγ_p (PT 0.385 vs measured 0.373); Pythagorean metric via CRT.

Going further