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arXiv:2610.09747

A Minimal Bicomplex Extension of the Complex Scalar Algebra of Quantum Mechanics with an Ideal-Valued Sector

Ralf Otte

quant-phSubmitted Oct 7, 2026• Announced Oct 8, 2026
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Abstract

Standard quantum mechanics takes the complex numbers as its scalar algebra. We ask whether this complex structure is algebraically fundamental or may instead form a distinguished embedded sector of a larger commutative and associative scalar algebra. Requiring the ordinary complex sector to remain intact while admitting an additional stable complex-like analytic sector turns the latter into a proper ideal, makes zero divisors unavoidable in finite dimension, and implies a lower bound of four real dimensions. The bicomplex algebra realizes this minimum, and under an additional compatibility assumption the four-dimensional realization is unique up to isomorphism. The resulting algebra exhibits two distinct decompositions: the canonical decomposition into orthogonal idempotent ideals and a decomposition adapted to the embedded complex sector and the additional ideal-valued sector. The ordinary complex scalars therefore lie diagonally across the canonical idempotent decomposition. The ideal sector supports Taylor series, oscillatory exponentials, Fourier-type kernels, and Schrödinger-type expressions, and carries a formal ideal-valued eigenvalue structure. Restricting the Born rule to the embedded complex sector recovers its standard form. Whether this dual scalar architecture has physical significance beyond the algebraic framework remains an open question.

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46 pages

Related papers

The bicomplex algebra as a minimal extension of the scalar algebra of quantum mechanics

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Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.

•Mar 24, 2026•0 citations

Real-commutant phase selection from real Hilbert representations

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Physics Letters A•Aug 13, 2026•0 citations

Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle

Dean Kulik

Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle Driven by Dean Kulik July 2026 Abstract Standard quantum mechanics assumes complex amplitudes. Classical and quantum mechanics assume Hamilton's principle. Neither is derived; both are imported. This paper compiles both from the same five primitive constraints {C0, C1, C1a, C2, C5} established in Paper 1. Two main theorems are proved. First, the Division Algebra Theorem: the unique admissible scalar continuation algebra is the complex numbers C, established by reducing to the Frobenius candidates {R, C, H}, eliminating R via C5's dual degree-of-freedom requirement, and eliminating H via the simplicity of H as a real algebra combined with C1a's independence constraint. Second, the Extremal Theorem: a field configuration is locally admissible under the compiled constraints if and only if it extremizes the action functional delta S = 0. 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Conservation (L3) and locality (L4) are compiled consequences of this set, not primitives. The definitions are reproduced here for self-containment. C0 (Distinction exists): At least two states exist that are not identical. This is the minimum condition for any structure to be meaningful. C1 (Continuation exists): Every admissible state s admits at least one successor state s' with s' not equal to s under the transition algebra. Terminal states and stationary continuation are excluded. The identity element e of the composition algebra (empty composition, satisfying e composed with T = T) is required by the algebra but is not itself a physical continuation event (Lemma B, Section 4.2). C1a (Independence Preservation): Independent distinctions remain independently continuable. The continuation of A cannot depend on the presence or continuation of B when A and B are independent. C2 (History Preservation): Every continuation preserves the set of states reachable from prior states in the history. C2 forbids deletion of accessible states from the transition graph and establishes the ordering relation from reachability. C5 (Observable Event Primitive): An observable event is the minimum continuation that simultaneously carries two independent degrees of freedom: propagation (a continuous spatial datum) and residue (a distinct temporal datum). C5 bridges transition structure and observation, and forces dim_R(D) >= 2 for any admissible scalar algebra D (Lemma D, Section 4.2). §3 The Independence Constraint and Its Algebraic Form C1a is the primitive that does the most structural work in this paper. Its algebraic form is derived immediately. Let T_A and T_B be continuation operators acting on independent subsystems A and B. Their joint continuation must factor as T_AB = T_A ⊗ T_B. Factorization requires that the result is independent of which update was applied first. Otherwise the joint continuation depends on an ordering between independent events, which is additional structure not licensed by C0 to C2. Therefore: [T_A, T_B] = T_A T_B − T_B T_A = 0 for all independent A, B Non-commutativity of independent continuation operators is a direct C1a violation by definition, not a violation of an algebraic nicety. This single equation is the structural spine of the paper. Scope. This commutation requirement applies to the scalar continuation algebra — the algebra of weights on independent continuation amplitudes. Physical symmetry operators (angular momentum generators, Lorentz boosts, gauge generators) act at a different layer and are not required to commute. This restriction is explicit and essential throughout. §4 The Division Algebra Theorem §4.1 Frobenius Reduction [C] Theorem [C]. The admissible scalar continuation algebra D is a finite-dimensional division algebra over R. Proof. The compiled linear state space (C1 + C1a, Paper 1 §4.1) is a vector space over some scalar field D. Four properties of D follow from the primitives: No zero divisors (from C0 + C1): C0 requires nonzero states to remain distinguishable under scalar multiplication. If scalar d ≠ 0 maps nonzero state ψ to zero, two distinct states become indistinguishable — a C0 violation. Additionally, C1 requires that every continuation produces a nonzero successor amplitude. If two nonzero amplitudes a and b satisfy a·b = 0 in D, then applying amplitude a followed by b to a lawful continuation produces a terminated (zero) amplitude state — a C1 violation. Therefore D has no zero divisors from both C0 and C1. Unit element (from C1): C1 requires the composition algebra to have an empty composition (Lemma B). This is the unit element of D. Division algebra (from no zero divisors + unit + finite dimension): For finite-dimensional associative algebras over a field, the combination of no zero divisors and a unit element is equivalent to being a division algebra. Every nonzero element has a two-sided inverse. Over R and finite-dimensional (from C5 + M1): Real scaling is the minimal requirement for continuous amplitude variation. C5 forces dim_R(D) ≥ 2 (see Lemma D). The minimality principle (M1) selects the minimal structure with no unconstrained degrees of freedom: dim_R(D) = exactly 2. Infinite-dimensional scalar structures carry unconstrained degrees of freedom not licensed by any primitive and are excluded by M1. Frobenius (1877): The only finite-dimensional division algebras over R are R (dimension 1), C (dimension 2), and H (dimension 4). Therefore D ∈ {R, C, H}. □ §4.2 The Four Admissible Conditions Any admissible scalar continuation algebra must satisfy four conditions, each compiled from the primitives. Together with Frobenius, these identify C as the unique survivor. Lemma A (Associativity) [C] A physical continuation history is an ordered sequence of lawful continuations. Any algebra representing that history must assign the same aggregate continuation map regardless of how the sequence is parenthesized. If associativity failed, the parenthesization — a bookkeeping convention — rather than the physical history would determine the outcome, violating C0. Therefore D is associative. □ Lemma B (Identity) [C] Sequential composition requires a neutral element — the empty composition e satisfying e ∘ T = T ∘ e = T for all T. The empty composition is not a continuation event; it represents the absence of composition, a structural requirement of the algebra. C1 excludes stationary continuation (T(s) = s as a physical event). It does not exclude the neutral element required by the composition algebra. These apply at different layers: e is required by the algebra and excluded as a physical continuation by C1. No contradiction. Identity is correctly placed in the representational layer. □ Lemma C (Commutativity of the scalar algebra) [C] The scalar weight on a continuation amplitude is itself a continuation operation ψ ↦ a·ψ. For independent amplitudes ψ_A and ψ_B with independent weights a_A and a_B, applying the weights in either order must give the same result — otherwise the combined amplitude depends on ordering between independent operations, which is a direct C1a violation. Therefore a_A · a_B = a_B · a_A for all elements of D. D is commutative. □ Lemma D (Two-dimensional) [C + M1] C5 requires every observable event to simultaneously carry two independent degrees of freedom: propagation and residue. These are not the same degree of freedom. A one-dimensional algebra has only one real degree of freedom and its only intrinsic transformation generates Z_2, which is discrete and cannot encode continuous propagation. Therefore: C5 forces: dim_R(D) ≥ 2 (lower bound) M1 selects: dim_R(D) = 2 (minimality) Higher-dimensional structures (quaternions, octonions, and beyond) carry additional degrees of freedom not licensed by any primitive. They are excluded by M1, not by a physical prohibition. The paper answers 'why not four?' with: four is not forced. Two is forced, and M1 selects the minimum forced structure. □ §4.3

•Jul 26, 2026•0 citations

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Standard quantum mechanics takes the complex numbers as its scalar algebra. We ask whether this complex structure is algebraically fundamental or may instead form a distinguished embedded sector of a larger commutative and associative scalar algebra. Requiring the ordinary complex sector to remain intact while admitting an additional stable complex-like analytic sector turns the latter into a proper ideal, makes zero divisors unavoidable in finite dimension, and implies a lower bound of four real dimensions. The bicomplex algebra realizes this minimum, and under an additional compatibility assumption the four-dimensional realization is unique up to isomorphism. The resulting algebra exhibits two distinct decompositions: the canonical decomposition into orthogonal idempotent ideals and a decomposition adapted to the embedded complex sector and the additional ideal-valued sector. The ordinary complex scalars therefore lie diagonally across the canonical idempotent decomposition. The ideal sector supports Taylor series, oscillatory exponentials, Fourier-type kernels, and Schrödinger-type expressions, and carries a formal ideal-valued eigenvalue structure. Restricting the Born rule to the embedded complex sector recovers its standard form. Whether this dual scalar architecture has physical significance beyond the algebraic framework remains an open question.
Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.
Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle Driven by Dean Kulik July 2026 Abstract Standard quantum mechanics assumes complex amplitudes. Classical and quantum mechanics assume Hamilton's principle. Neither is derived; both are imported. This paper compiles both from the same five primitive constraints {C0, C1, C1a, C2, C5} established in Paper 1. Two main theorems are proved. First, the Division Algebra Theorem: the unique admissible scalar continuation algebra is the complex numbers C, established by reducing to the Frobenius candidates {R, C, H}, eliminating R via C5's dual degree-of-freedom requirement, and eliminating H via the simplicity of H as a real algebra combined with C1a's independence constraint. Second, the Extremal Theorem: a field configuration is locally admissible under the compiled constraints if and only if it extremizes the action functional delta S = 0. Hamilton's principle is the global expression of local C1-consistency. A compiled Measurement Theorem follows from the channel decomposition forced by C5: collapse is a C2 history-update, not a C1 transition. The independence constraint [T_A, T_B] = 0 serves as the structural spine: it compiles conservation (C3 to L3), forces the complex scalar field (Division Algebra Theorem), and will constrain gauge sector factorization. One axiom. Three compiled results. §1 Introduction Two of the deepest assumptions in physics are that quantum amplitudes are complex-valued and that physical trajectories extremize the action. Both are usually treated as foundational inputs. The goal of this paper is to show both are compiled consequences of a primitive constraint set that mentions neither. The companion paper (Paper 1) compiled: conservation, locality, the free-field Lagrangian L, and the wave equation, from the five primitive constraints {C0, C1, C1a, C2, C5}. The present paper derives two further results from the same set, and establishes the structural connection between them. The two theorems share the same spine. The independence constraint C1a — that independent distinctions continue independently — has the algebraic form [T_A, T_B] = 0 for independent operators. This single condition eliminates quaternionic scalar algebras (Division Algebra Theorem), blocks coordinated creation-destruction channels (compiling conservation), and will factorize the gauge group (T5, T6). Section 7 develops this as the Independence Spine Theorem. Paper scope. Both theorems concern the free field (Paper 1, C8). Interactions require C9 (equivalence class coupling) and are addressed in Paper 4. The Measurement Theorem (Section 5) is fully compiled. The Born rule requires one additional input (C7 geometry; see Section 5.3 and open problem Omega_B). §2 The Primitive Constraints The irreducible primitive set established in Paper 1 is {C0, C1, C1a, C2, C5}. Conservation (L3) and locality (L4) are compiled consequences of this set, not primitives. The definitions are reproduced here for self-containment. C0 (Distinction exists): At least two states exist that are not identical. This is the minimum condition for any structure to be meaningful. C1 (Continuation exists): Every admissible state s admits at least one successor state s' with s' not equal to s under the transition algebra. Terminal states and stationary continuation are excluded. The identity element e of the composition algebra (empty composition, satisfying e composed with T = T) is required by the algebra but is not itself a physical continuation event (Lemma B, Section 4.2). C1a (Independence Preservation): Independent distinctions remain independently continuable. The continuation of A cannot depend on the presence or continuation of B when A and B are independent. C2 (History Preservation): Every continuation preserves the set of states reachable from prior states in the history. C2 forbids deletion of accessible states from the transition graph and establishes the ordering relation from reachability. C5 (Observable Event Primitive): An observable event is the minimum continuation that simultaneously carries two independent degrees of freedom: propagation (a continuous spatial datum) and residue (a distinct temporal datum). C5 bridges transition structure and observation, and forces dim_R(D) >= 2 for any admissible scalar algebra D (Lemma D, Section 4.2). §3 The Independence Constraint and Its Algebraic Form C1a is the primitive that does the most structural work in this paper. Its algebraic form is derived immediately. Let T_A and T_B be continuation operators acting on independent subsystems A and B. Their joint continuation must factor as T_AB = T_A ⊗ T_B. Factorization requires that the result is independent of which update was applied first. Otherwise the joint continuation depends on an ordering between independent events, which is additional structure not licensed by C0 to C2. Therefore: [T_A, T_B] = T_A T_B − T_B T_A = 0 for all independent A, B Non-commutativity of independent continuation operators is a direct C1a violation by definition, not a violation of an algebraic nicety. This single equation is the structural spine of the paper. Scope. This commutation requirement applies to the scalar continuation algebra — the algebra of weights on independent continuation amplitudes. Physical symmetry operators (angular momentum generators, Lorentz boosts, gauge generators) act at a different layer and are not required to commute. This restriction is explicit and essential throughout. §4 The Division Algebra Theorem §4.1 Frobenius Reduction [C] Theorem [C]. The admissible scalar continuation algebra D is a finite-dimensional division algebra over R. Proof. The compiled linear state space (C1 + C1a, Paper 1 §4.1) is a vector space over some scalar field D. Four properties of D follow from the primitives: No zero divisors (from C0 + C1): C0 requires nonzero states to remain distinguishable under scalar multiplication. If scalar d ≠ 0 maps nonzero state ψ to zero, two distinct states become indistinguishable — a C0 violation. Additionally, C1 requires that every continuation produces a nonzero successor amplitude. If two nonzero amplitudes a and b satisfy a·b = 0 in D, then applying amplitude a followed by b to a lawful continuation produces a terminated (zero) amplitude state — a C1 violation. Therefore D has no zero divisors from both C0 and C1. Unit element (from C1): C1 requires the composition algebra to have an empty composition (Lemma B). This is the unit element of D. Division algebra (from no zero divisors + unit + finite dimension): For finite-dimensional associative algebras over a field, the combination of no zero divisors and a unit element is equivalent to being a division algebra. Every nonzero element has a two-sided inverse. Over R and finite-dimensional (from C5 + M1): Real scaling is the minimal requirement for continuous amplitude variation. C5 forces dim_R(D) ≥ 2 (see Lemma D). The minimality principle (M1) selects the minimal structure with no unconstrained degrees of freedom: dim_R(D) = exactly 2. Infinite-dimensional scalar structures carry unconstrained degrees of freedom not licensed by any primitive and are excluded by M1. Frobenius (1877): The only finite-dimensional division algebras over R are R (dimension 1), C (dimension 2), and H (dimension 4). Therefore D ∈ {R, C, H}. □ §4.2 The Four Admissible Conditions Any admissible scalar continuation algebra must satisfy four conditions, each compiled from the primitives. Together with Frobenius, these identify C as the unique survivor. Lemma A (Associativity) [C] A physical continuation history is an ordered sequence of lawful continuations. Any algebra representing that history must assign the same aggregate continuation map regardless of how the sequence is parenthesized. If associativity failed, the parenthesization — a bookkeeping convention — rather than the physical history would determine the outcome, violating C0. Therefore D is associative. □ Lemma B (Identity) [C] Sequential composition requires a neutral element — the empty composition e satisfying e ∘ T = T ∘ e = T for all T. The empty composition is not a continuation event; it represents the absence of composition, a structural requirement of the algebra. C1 excludes stationary continuation (T(s) = s as a physical event). It does not exclude the neutral element required by the composition algebra. These apply at different layers: e is required by the algebra and excluded as a physical continuation by C1. No contradiction. Identity is correctly placed in the representational layer. □ Lemma C (Commutativity of the scalar algebra) [C] The scalar weight on a continuation amplitude is itself a continuation operation ψ ↦ a·ψ. For independent amplitudes ψ_A and ψ_B with independent weights a_A and a_B, applying the weights in either order must give the same result — otherwise the combined amplitude depends on ordering between independent operations, which is a direct C1a violation. Therefore a_A · a_B = a_B · a_A for all elements of D. D is commutative. □ Lemma D (Two-dimensional) [C + M1] C5 requires every observable event to simultaneously carry two independent degrees of freedom: propagation and residue. These are not the same degree of freedom. A one-dimensional algebra has only one real degree of freedom and its only intrinsic transformation generates Z_2, which is discrete and cannot encode continuous propagation. Therefore: C5 forces: dim_R(D) ≥ 2 (lower bound) M1 selects: dim_R(D) = 2 (minimality) Higher-dimensional structures (quaternions, octonions, and beyond) carry additional degrees of freedom not licensed by any primitive. They are excluded by M1, not by a physical prohibition. The paper answers 'why not four?' with: four is not forced. Two is forced, and M1 selects the minimum forced structure. □ §4.3
The bicomplex algebra B = C ⊗R C has recently been identified as a minimal commutative and associative extension of the complex scalar algebra underlying quantum mechanics. This structure contains a proper ideal J ∼= C that admits a complex-like exponential representation and leads to an additive phase relation between the complex and ideal sectors. The present work explores the consequences of this algebraic phase coupling under the hypothesis that it may become physically effective in suitable experimental conditions. In particular, we consider an optical setup involving a nonlinear medium, where mixed terms between complex and ideal components may arise. Within this framework, a specific, testable prediction is derived: a lateral displacement of the source in a double-slit experiment could lead to an amplified fringe shift, whose magnitude depends on the geometric parameters of the setup. The predicted scaling differs significantly from the standard geometric expectation of conventional wave optics. The analysis is explicitly conditional on the assumed physical realization of the bicomplex phase coupling. The proposed experiment therefore provides a direct way to test whether the additional algebraic structure has observable consequences or remains purely formal.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.