Emergence of Time in the State–Relational–Entropy Framework
Year: 2026
Type: article
Abstract: In the State–Relational Entropy (SRE) framework the word "time" carries three distinct objects under one name — the external step count $t_{\rm ext}=n$, the memory-intrinsic time $\tau_{\rm mem}\propto\sqrt n$, and a third quantity defined here. We propose to read the expected amount of change in the world at step $n$ as the time worth of that step,and call $T_{\rm em}$ the emergent time. No new free real number is introduced; $\lambda$ and the step scale follow existing registrations and the evolution loop is unaltered. Seven verdicts follow. (1) $\Omega$ saturates monotonically toward $1$ (gap $0.1540$ at $n=999$; self-consistency $2.2\times10^{-16}$). (2) The approach obeys $1-\Omega=c\ln n/\sqrt n$ with $c=0.694157$, maximum relative residual $1.589\%$, versus $26.43\%$ for a pure power law — the logarithm is the fingerprint of the harmonic tail; the same law is predicted with zero free parameters from two laws of round 77, agreeing to $0.28\%$–$1.65\%$. (3) Relative step-to-step non-uniformity falls from $4.23\times10^{-3}$ to $6.10\times10^{-5}$ and matches the value differentiated from the saturation law within $[0.873,1.407]$ — uniformity is a derived property, not an assumption. (4) Locally it is not uniform: a single element's rewriting rate rises with its own age, from $0.058931$ (newborn) to $0.957750$ (oldest) at $n=999$. (5) Universality: sweeping $\lambda$ over a factor of $15$ does not move the limit, only the speed of approach — because the limit is a saturation value and a saturation value carries no coupling strength. (6) A phase transition lives in the *form* of the update rule, not in the value of parameters: squaring one denominator moves the limit from $1.0000$ to about $0.390$, the gap reaching $0.4629$ at $1099$ steps and still widening. (7) The price of emergence is that time is not unique: $\tau_{\rm mem}\propto\sqrt n$ and $T_{\rm em}\propto n$ diverge in ratio ($6.89\times$ and growing), so no rescaling makes both uniform. Three limits are stated explicitly. The emergent time has shape but no scale — "how many seconds is one step" remains external input. At attainable sizes time constancy holds only to about $22\%$ ($T_{\rm em}/n=0.7776$ at $n=999$). And two pre-registered guesses were refuted by measurement ($c\propto\lambda^{-1}$ gives slope $-0.577$; the variant kernel settles at a fixed point near $0.39$ instead of freezing) and are registered as open items, not evidence. The load-bearing separation is structure without memory: the joint distribution of the state matrix rejects full independence (row-sum variance $1.94$–$2.03\times$ the iid value, KS $p<10^{-4}$), yet two different histories become indistinguishable after continued evolution (exact permutation $p\ge0.3841$, none significant after Holm correction). The present state carries zero bits about its own past — which is exactly why time must be accumulated rather than read off. All numbers hold inside the SRE model only; comparisons with physical time are structural analogies, not numerical predictions. 在状态—关系熵(SRE)框架里,「时间」一词同时承载着三个互不相同的对象:外部步数 $t_{\rm ext}=n$、记忆内禀时间 $\tau_{\rm mem}\propto\sqrt n$,以及本文定义的第三个量。本文主张把第 $n$ 步世界的期望改变量读作该步所值的时间,并称 $T_{\rm em}$ 为涌现时间。本文不引入任何新的自由实数:$\lambda$ 与步数规模沿用既有登记,演化循环一字未改。 七项判决如下。(1)饱和:$\Omega$ 单调趋于 $1$($n=999$ 处缺口 $0.1540$,自洽校验 $2.2\times10^{-16}$)。(2)饱和律:$1-\Omega=c\ln n/\sqrt n$,$c=0.694157$,最大相对残差 $1.589\%$,而纯幂律残差达 $26.43\%$——$\ln n$ 是调和尾巴的指纹;该律由第 77 轮两条既有律零参数预测,与实测差 $0.28\%$–$1.65\%$。(3)渐近均匀:相对步进非均匀度从 $4.23\times10^{-3}$ 降到 $6.10\times10^{-5}$,与由饱和律微分而来的预测之比落在 $[0.873,1.407]$——均匀性是推论,不是假设。(4)内部不均匀:单个元素的改写率随自身年龄单调上升,$n=999$ 时从 $0.058931$(新生)到 $0.957750$(最老)。(5)普适性:$\lambda$ 扫过 $15$ 倍不改极限,只改逼近速度——因为极限是饱和值,而饱和值不携带耦合强度。(6)相变在规则形态、不在参数取值:把一处分母平方,极限由 $1.0000$ 移到约 $0.390$,$1099$ 步处差距已达 $0.4629$ 且持续扩大。(7)代价:时间不唯一。$\tau_{\rm mem}\propto\sqrt n$ 与 $T_{\rm em}\propto n$ 的比值发散($6.89$ 倍且继续),不存在使两者同时匀速的重标定。 三条限定必须写明。涌现时间只有形状、没有尺度——「一步等于多少秒」仍是外加输入;在可达规模上「时间恒定」只有约 $22\%$ 的精度($n=999$ 时 $T_{\rm em}/n=0.7776$);另有两条动笔之前的猜测被实测推翻($c\propto\lambda^{-1}$ 实测斜率 $-0.577$;变体核落到约 $0.39$ 的不动点而非失控冻结),按纪律登记为开放项,不作证据。 承重的一条分离是有结构、无记忆:状态矩阵的联合分布拒斥完全 iid(行和方差为 iid 的 $1.94$–$2.03$ 倍,KS $p<10^{-4}$),但两条互异历史在续跑后不可分辨(精确置换 $p\ge0.3841$,Holm 校正后无显著)。当前状态对自己的过去携带零比特——这正是「时间只能被累积、不能被读取」的确切含义。全部数值仅在 SRE 模型内部成立,与物理时间的对照一律为结构类比,非数值预言。
Author: Lu, Yue
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7219769583
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