A Binary-Ternary Meta-Orbit PIR Framework Projective Conjugacy, Nested Residue Layers, Prime-Compatible Classes, Rotation Blocks, and Dual Limit Structure

This work reformulates the PIR framework as a mathematical study of how discrete arithmetic operations, harmonic trigonometric structure, and the small-step limit of calculus can be connected through a common transformation system. The starting point is the abstract PIR structure PIR = (P, I, R), where P represents partition or splitting, I represents identity or continuation, and R represents reference, return, or stopping. PIR is not identified directly with particular numbers, trigonometric functions, or matrices. Instead, these mathematical objects appear as different representations of an underlying operational structure. The arithmetic sector is generated by three elementary transformations: S₂ : n → 2n C : n → n − 1 N₃ : n → n / 3 where division by three is admissible only when the current integer is divisible by three. A sequence of these transformations forms an admissible operation word w. Every such word has the affine form F_w(n) = [2^{p(w)} n − a(w)] / 3^{d(w)}. Here, p(w) counts doubling operations, d(w) counts divisions by three, and a(w) records the accumulated history of center shifts. Thus a complicated discrete operation can be summarized by the three parameters (p, d, a). The next step introduces the harmonic angular coordinate θ = 2π / n. This is the point at which π enters the framework. The role of π is geometric rather than prime-arithmetic: it appears because the integer index is represented as an angular subdivision of a complete periodic cycle. Under the transformation Θ(n) = 2π / n, the arithmetic affine map is transported into a Möbius transformation. The central projective relation is M_w = J A_w J⁻¹, or equivalently, Θ ∘ F_w = F̂_w ∘ Θ. Thus the arithmetic transformation and the harmonic transformation are not merely evaluated using the same integer. They are projectively conjugate descriptions of the same operation word. The exact angular transformation is F̂_w(θ) = [2π · 3^{d(w)} · θ] / [2^{p(w)+1}π − a(w)θ]. At the continuum boundary θ → 0, differentiation gives the tangent character χ(w) = 3^{d(w)} / 2^{p(w)}. This is one of the central results of the framework. The same quantity can be interpreted in three ways: Arithmetic sector:χ_A(w) = 3^{d(w)} / 2^{p(w)} Harmonic sector:χ_R(w) = F̂′_w(0) Small-step sector:χ_D(w) = lim_{h→0} F̂_w(h) / h Therefore, χ_A(w) = χ_R(w) = χ_D(w). This establishes a common first-order scale character across the arithmetic, harmonic, and small-step representations. Because h = 2π / n, the limit n → ∞ corresponds to h → 0. This is the bridge from the discrete index system toward the small-step language of calculus. The framework also explains what happens to discrete operational history in this limit. The parameter a(w), which remembers the placement and ordering of center shifts, does not affect the first-order tangent character. It first appears at second order. Therefore, two operation words with the same p and d may have different finite structures while becoming identical to first order as θ approaches zero. In this sense, discrete ordering information can survive at finite scale but become suppressed in the continuum approximation. Layered self-reference enters through center extraction and normalization by three. A center shift produces N → N − 1. A factor of three is exposed exactly when N ≡ 1 mod 3. The depth of repeated normalization is measured by v₃(N − 1). For example, 22 = 1 + 3 × 7. The corresponding operation chain may be written 11 → 22 → 21 → 7, representing doubling, center shift, and division by three. However, the mechanism is not caused by the primality of 11 or 7. The exact arithmetic gate is the congruence condition 22 ≡ 1 mod 3. The same mechanism also occurs for composite inputs. Prime numbers therefore participate in interesting PIR operation orbits, but primality itself is not the cause of the layered normalization. The trigonometric sector is defined through θ_n = 2π / n, with the rotation weights N_P(n) = sin²(2π / n) and N_R(n) = cos²(2π / n). The exact special-angle condition is N_P(n) > N_R(n) when 8/3 < n < 8. For integer n, the P-dominant set is therefore {3, 4, 5, 6, 7}, with balance at n = 8. This result contains both prime and composite integers. Therefore, primality neither characterizes nor causes the special-angle regime. The important conclusion is that prime arithmetic, trigonometric special angles, π, and the calculus limit should not be connected merely through numerical coincidence. Their meaningful relationship is operational. The structural flow of the framework is PIR operation words→ arithmetic affine transformations→ harmonic Möbius transformations→ tangent character→ small-step limit. The common tangent character is χ(w) = 3^{d(w)} / 2^{p(w)}. Prime and modular arithmetic belong primarily to the discrete arithmetic representation. π and the special-angle trigonometric structure belong to the harmonic representation. The tangent limit connects the transported operation to the small-step scale associated with calculus. The framework therefore establishes an exact arithmetic-to-harmonic projective conjugacy and an exact scale-level tangent character. It does not yet prove full finite-h PIR operator intertwining, a unique physical R-return law, or a complete three-sector operator unification. Those remain open problems. The main contribution is therefore an explicit algebraic backbone connecting discrete arithmetic transformations, layered self-reference, Möbius geometry, harmonic trigonometry, π, and the continuum tangent limit within a single falsifiable PIR framework.