Information is not in the carrier; it is in the definition. Communication does not transmit content; it only transmits a trigger. The Same-Origin World System is a carrier-independent formal theory. Its core quantity is the same-origin degree, its structure is the same-origin matrix, and its evolution is row-stochastic dynamics. Core chain: 1 = N → same-origin degree → same-origin matrix → dynamics → convergence → one controls all. 1 = N: The 1 selected from N has an internal structure that is the N of the next level. This recursion can continue indefinitely. Same-origin degree: D_same(i,j) = exp(-D_fine × Σ w_k × d_k²), describing the similarity between attributes. Same-origin matrix: The matrix of all pairwise same-origin degrees, normalized to be row-stochastic. Dynamics: x(t+1) = D × x(t). The spectral radius of a row-stochastic matrix is ρ(D)=1, and the system converges to the principal eigenvector. One controls all: Changing one attribute changes all attributes. Four-copy local verification: End A has A1 + B2, End B has B1 + A2. Rigid binding A1=A2, B1=B2. Local comparison equals cross-end comparison. No communication is needed to verify consistency. Pulsar triggering: Both ends observe the same pulsar. The N-th pulse triggers both ends simultaneously. Event occurrence + index selection. Non-local connection, carrier-independent. Quantum function correspondences: Superposition (the same-origin matrix contains all states simultaneously), interference (phase θ_ij), entanglement (shared trigger source), non-locality (pulsar triggering), measurement (triggering), collapse (state change after triggering), many-worlds (decision point selects matrix). Information geometry correspondences: Probability distribution (difference distribution), Fisher metric (relational distance d = -ln D_same), divergence (same-origin degree), e-connection (internal weight), m-connection (external weight), geodesic (minimum-cost path), curvature (non-uniformity of difference distribution), natural gradient (same-origin dynamics). Carrier independence: The core is mathematics, independent of the speed of light, Planck constant, or electron charge. It can be implemented in optics, acoustics, electronics, or quantum systems. 1 = N. Control one = control all. Information is definition, communication is trigger, verification is local, synchronization is pulsar. Carrier-independent; the core is mathematics.
Information is not in the carrier; it is in the definition. Communication does not transmit content; it only transmits a trigger. The Same-Origin World System is a carrier-independent formal theory. Its core quantity is the same-origin degree, its structure is the same-origin matrix, and its evolution is row-stochastic dynamics. Core chain: 1 = N → same-origin degree → same-origin matrix → dynamics → convergence → one controls all. 1 = N: The 1 selected from N has an internal structure that is the N of the next level. This recursion can continue indefinitely. Same-origin degree: D_same(i,j) = exp(-D_fine × Σ w_k × d_k²), describing the similarity between attributes. Same-origin matrix: The matrix of all pairwise same-origin degrees, normalized to be row-stochastic. Dynamics: x(t+1) = D × x(t). The spectral radius of a row-stochastic matrix is ρ(D)=1, and the system converges to the principal eigenvector. One controls all: Changing one attribute changes all attributes. Four-copy local verification: End A has A1 + B2, End B has B1 + A2. Rigid binding A1=A2, B1=B2. Local comparison equals cross-end comparison. No communication is needed to verify consistency. Pulsar triggering: Both ends observe the same pulsar. The N-th pulse triggers both ends simultaneously. Event occurrence + index selection. Non-local connection, carrier-independent. Quantum function correspondences: Superposition (the same-origin matrix contains all states simultaneously), interference (phase θ_ij), entanglement (shared trigger source), non-locality (pulsar triggering), measurement (triggering), collapse (state change after triggering), many-worlds (decision point selects matrix). Information geometry correspondences: Probability distribution (difference distribution), Fisher metric (relational distance d = -ln D_same), divergence (same-origin degree), e-connection (internal weight), m-connection (external weight), geodesic (minimum-cost path), curvature (non-uniformity of difference distribution), natural gradient (same-origin dynamics). Carrier independence: The core is mathematics, independent of the speed of light, Planck constant, or electron charge. It can be implemented in optics, acoustics, electronics, or quantum systems. 1 = N. Control one = control all. Information is definition, communication is trigger, verification is local, synchronization is pulsar. Carrier-independent; the core is mathematics.
Traditional interstellar communication depends on signal transmission across space. The farther the distance, the greater the delay. The starting point of this theory is the same-source constraint between two ends. Same-source means that the devices, chips, hardware, software, rules, content, and materials of the two ends are exactly identical. Not similar, not compatible, but exactly identical. When the two ends are exactly identical, the communication logic changes accordingly. Any operation that one end needs to perform, the other end also has the same operation. Any content that one end needs to display, the other end also has the same content. The content is already at both ends; it does not need to be transmitted from one end to the other. Timing does not need to be transmitted either; a single trigger signal activates both ends simultaneously. Both ends execute independently, and the results are necessarily identical. Because the constraint conditions are exactly the same; the same conditions and the same starting point necessarily yield the same result. Communication is decomposed into two parts: definition and trigger. Definition is the content, rules, and materials pre-loaded at both ends. Trigger is a signal that only says “start.” No content needs to be transmitted; no timing needs to be transmitted. Quantum entanglement does only one thing in this system: make the trigger signals at both ends occur simultaneously. It transmits no information, only the fact of “simultaneous occurrence.” Both ends evolve independently, and the results are not affected by spatial position. Distance does not affect the final result. Because there is no information carrier transmitting content through space, the time needed for both ends to reach the same result is determined only by how fast the local system settles down, and is independent of spatial distance. The same-source architecture offers a new paradigm for interstellar communication: not to spread information across the universe, but to make both ends identically built, letting information appear at both ends simultaneously. Information is not transported, but pre-placed at both ends and evoked by instruction.
Communication = Pre-installed + Trigger. Both ends are pre-loaded with exactly the same set of rules and content. What A wants, B already knows. No need to transfer content—only a trigger signal, saying "start" simultaneously. The trigger signal is the simplest signal possible: happen, or not happen. It carries no content. Both ends independently run by the same set of rules. These rules contain no distance. So how long it takes depends only on how fast its own internal reaction is, not on how far apart they are. Distance no longer determines delay. There are five triggering methods, choose any one: Quantum entanglement: touch one side, the other side knows instantly. Pulsar: both ends watch the same star, count to the same pulse. Cosmic background radiation: both ends point to the same direction in the universe. Multiple pulsars combined: same as above, higher precision. Atomic clock: both ends synchronize watches, act at the same time. For interstellar distances, quantum entanglement is preferred. If quantum fails, switch to the next one in order. In the worst case, even if there is a delay in trigger time between the two ends, as long as both are started, the final result is the same. Information is not carried over; it is pre-installed, and started by instruction.
宣言 Manifesto 标准量子力学的数学基础是结合代数:算子乘法满足 (AB)C = A(BC),态空间是希尔伯特空间(正定内积空间),演化由幺正群 U(N) 描述。 The mathematical foundation of standard quantum mechanics is associative algebra: operator multiplication satisfies (AB)C = A(BC), the state space is a Hilbert space (positive-definite inner product space), and evolution is described by the unitary group U(N). 然而,自然界存在深刻的非结合结构:从八元数在弦论中的角色,到 Jordan 代数在量子逻辑中的起源,非结合性不是病态,而是更一般的对称性破缺。中道积分理论已经证明:非结合代数上的围道积分可以通过算术平均唯一地定义,结合子精确量化每条分支的偏差。 However, nature harbors profound non-associative structures: from octonions in string theory to Jordan algebras in quantum logic, non-associativity is not pathological but a more general symmetry breaking. The Mid-Path Integral theory has proven: contour integrals on non-associative algebras can be uniquely defined via arithmetic averaging, with the associator precisely quantifying each branch’s deviation. 本文提出一个根本性的转向:不再把非结合代数塞进希尔伯特空间的模子里,而是直接从复化三维数(ℂ³)的内生结构出发,重建量子力学的全部公理。标准量子力学将是本文框架在“结合性极限”(δ → 0)和“标量子空间限制”(ψ₁ = ψ₂ = 0)下的子理论。 This paper proposes a fundamental pivot: instead of forcing non-associative algebras into the Hilbert-space mold, we reconstruct all axioms of quantum mechanics directly from the endogenous structure of the complexified ternary numbers (ℂ³). Standard quantum mechanics will be a sub-theory of this framework in the “associative limit” (δ → 0) and the “scalar subspace restriction” (ψ₁ = ψ₂ = 0). 预备知识:ℂ³ 代数结构 Preliminaries: ℂ³ Algebraic Structure 乘法表 Multiplication Table 基为 {1, e₁, e₂},乘法由以下关系定义: Basis {1, e₁, e₂}, multiplication defined by: ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ 基本性质 Basic Properties • 交换性:X·Y = Y·X(乘法表对称) Commutativity: X·Y = Y·X (symmetric multiplication table) • 分配律:X·(Y+Z) = X·Y + X·Z(严格成立) Distributivity: X·(Y+Z) = X·Y + X·Z (strictly holds) • 非结合性:一般地 (X·Y)·Z ≠ X·(Y·Z) Non-associativity: generally (X·Y)·Z ≠ X·(Y·Z) • 结合子:Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) Associator: Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) • 严格定理:Δ(X,X,X) ≡ 0(幂结合性在自乘时成立) Strict theorem: Δ(X,X,X) ≡ 0 (power-associativity holds for self-multiplication) • 严格定理:Δ(X,X,Y) ≠ 0 当 Y ≠ X(混合非结合性) Strict theorem: Δ(X,X,Y) ≠ 0 when Y ≠ X (mixed non-associativity) • 定理 2.5(非结合激活条件):当 x₀=y₀=z₀=0 时 Δ(X,Y,Z) ≡ 0。非结合性只在至少一个元素的标量分量非零时显现。 Theorem 2.5 (Non-associative activation): Δ(X,Y,Z) ≡ 0 when x₀=y₀=z₀=0. Non-associativity only manifests when at least one element has non-zero scalar component.
Information-Theoretic Deniability and the Forest Analogy This paper establishes information-theoretic deniability through a precise structural argument: the Forest Analogy. The central result is a symmetry property showing that an attacker who possesses a valid alibi chain stands in exactly the same position as an attacker who knows only the public parameter N. The verification function cannot distinguish the real authentication chain from any alibi chain. This property holds without cryptographic assumptions — it is information-theoretic, not computational. The authentication chain is built on the linear Diophantine system N = 19A + 9B, in which every number N has approximately N/171 valid representations per layer. A three-layer Matryoshka structure produces a multiplicatively structured search space of approximately 10¹¹² operations. The key insight is that this search space is symmetric: it is equally large and equally unnavigable from outside the system as from within it. Uniform Sampling and the Generation Algorithm Uniform sampling means here that every valid chain — consisting of A, B, a1, b1, a2, b2 satisfying 19A + 9B = N, 19a1 + 9b1 = A, 19a2 + 9b2 = a1, and the digit-root conditions — has exactly equal probability of being generated as the real authentication chain. The generation algorithm achieves this through a hierarchical weighted CDF sampler (cumulative distribution function), operating as follows: 1. The distribution over all valid A-values is computed, where the weight assigned to a given A equals the exact number of valid chains on layers 1 and 2 that can be extended from that A. 2. A is sampled according to this weighted distribution. 3. Given A, the value a1 is sampled in the same manner, with weights determined by the exact number of valid (a2, b2)-pairs beneath that a1. 4. Finally, (a2, b2) is chosen uniformly at random from all solutions to 19a2 + 9b2 = a1. All weights are computed exactly. This hierarchical weighted sampling procedure therefore induces a provably uniform distribution over all complete chains, without requiring the generator to enumerate or store them explicitly. Proposition (Uniform Chain Distribution): Let N be a fixed public parameter. Let Omega(N) denote the set of all valid chains C = (A, B, a1, b1, a2, b2) satisfying: - 19A + 9B = N - 19a1 + 9b1 = A - 19a2 + 9b2 = a1 - and the digit-root conditions on each layer. Then the hierarchical weighted CDF sampler generates each chain C in Omega(N) with probability exactly 1 / |Omega(N)|. Proof. Define the following counting functions: - w2(a1) = |{(a2, b2) : 19a2 + 9b2 = a1, digit-root conditions satisfied}| - w1(A) = sum over all valid a1 of w2(a1), where the sum runs over all a1 satisfying 19a1 + 9b1 = A for some valid b1 - w0 = sum over all valid A of w1(A) = |Omega(N)| The sampler proceeds in three steps: Step 1. A is drawn with probability P(A) = w1(A) / w0. Step 2. Given A, a1 is drawn with probability P(a1 | A) = w2(a1) / w1(A). Step 3. Given a1, the pair (a2, b2) is drawn uniformly with probability P(a2, b2 | a1) = 1 / w2(a1). The joint probability of generating the complete chain C = (A, B, a1, b1, a2, b2) is: P(C) = P(A) * P(a1 | A) * P(a2, b2 | a1) = w1(A) / w0 * w2(a1) / w1(A) * 1 / w2(a1) = 1 / w0 = 1 / |Omega(N)| Since this holds for every C in Omega(N), the sampler induces the uniform distribution over Omega(N). "Remark" B and b1 are uniquely determined once A, a1, a2, and b2 are fixed, via the Diophantine equations. They do not need to be sampled separately. The uniformity therefore holds over all complete chains, including these values. Theorem (Forest Symmetry): Let N be a public parameter and let C = (A, B, a1, b1, a2, b2) be any valid verification chain. Then for every valid alibi chain C': P(C is authentic | verification succeeds, C') = P(C is authentic | verification succeeds) Proof: The verification function evaluates solely against N. By Proposition (Uniform Chain Distribution), every valid chain is generated with identical probability 1 / |Omega(N)|. The alibi C' therefore carries no information about which chain is authentic. Formally, C' and the event "C is authentic" are conditionally independent given that verification succeeds. The posterior probability is thus unaffected by knowledge of C'. The Forest Analogy: The Forest Analogy makes the symmetry precise. The public parameter N is the forest. Each representation (A, B) on layer zero is a tree. Each sub-representation on layer one is a branch. Each leaf is a layer-two representation. All leaves are identical — every chain verifies correctly against N. An attacker holding a valid alibi leaf cannot determine which leaf is the real one, because the verification function provides no distinguishing information. *This work is licensed under CC BY-SA 4.0. Commercial licensing (without ShareAlike) available on request: elissa_oui@outlook.com*
We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the simultaneous eigenstates |ψ)= A1', A2',...), (ψ|C|ψ)=0. This CSCO criterion may be considered as an extension of the Heisenberg uncertainty principle to quantum systems with either two (or more) particles or multi-degrees of freedom (MDF).