Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

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Figure X.
Four-dimensional precision–tolerance phase map with experiment accumulation. Surfaces show the effective tolerance τ(p,τ0,n)\tau(p,\tau_0,n)τ(p,τ0,n) as a function of precision ppp and initial tolerance τ0\tau_0τ0 for discrete experiment counts nnn. The horizontal plane τ=∣r∣max⁡\tau=|r|_{\max}τ=∣r∣max is the collapse boundary. Naive scaling surfaces descend with increasing nnn and intersect the collapse plane at finite precision, while the infrared fixed-point surface remains invariant and never intersects the boundary.

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The figure depicts a projection of a four-dimensional deterministic structure with coordinates (p,τ0,n,τ)(p,\tau_0,n,\tau)(p,τ0,n,τ), where ppp is the precision index, τ0\tau_0τ0 the initial structural tolerance, nnn the cumulative experiment count, and τ\tauτ the induced tolerance. Each translucent surface corresponds to a fixed experiment count nnn under logarithmic experiment accumulation, τnaive(p,τ0,n)=τ02−pκ(n)\tau_{\mathrm{naive}}(p,\tau_0,n)=\tau_0 2^{-p}\kappa(n)τnaive(p,τ0,n)=τ02−pκ(n). Increasing nnn monotonically lowers the surface, reducing the viable precision horizon. The horizontal plane τ=∣r∣max⁡\tau=|r|_{\max}τ=∣r∣max defines an absorbing collapse boundary; intersection with this plane induces collapse deterministically. The infrared fixed-point surface τIR(p,τ0)\tau_{\mathrm{IR}}(p,\tau_0)τIR(p,τ0) is independent of nnn and forms a time-invariant foliation that never intersects the collapse boundary, illustrating robustness to unlimited experiment accumulation.

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The theory induces a four-dimensional phase space with coordinates (p,τ0,n,τ)(p,\tau_0,n,\tau)(p,τ0,n,τ), where ppp indexes precision refinement, τ0\tau_0τ0 is the baseline tolerance, nnn is the cumulative experiment count, and τ\tauτ is the effective tolerance. Experiment accumulation enters through a logarithmic erosion factor κ(n)=1−αlog⁡(1+n/n0)\kappa(n)=1-\alpha\log(1+n/n_0)κ(n)=1−αlog(1+n/n0). For naive scaling, τ(p,τ0,n)=τ02−pκ(n)\tau(p,\tau_0,n)=\tau_0 2^{-p}\kappa(n)τ(p,τ0,n)=τ02−pκ(n), producing a family of tolerance surfaces indexed by nnn. Collapse occurs deterministically when τ<∣r∣max⁡\tau<|r|_{\max}τ<∣r∣max, corresponding to intersection with an absorbing boundary plane. As nnn increases, the collapse index p\''(τ0,n)p^\''(\tau_0,n)p\''(τ0,n) decreases monotonically, yielding a finite experiment horizon. In contrast, infrared fixed-point scaling τIR(p,τ0)=τ\''+(τ0−τ\'')2−p\tau_{\mathrm{IR}}(p,\tau_0)=\tau^\''+(\tau_0-\tau^\'')2^{-p}τIR(p,τ0)=τ\''+(τ0−τ\*)2−p is independent of nnn and never intersects the collapse boundary, demonstrating asymptotic robustness under unlimited experiment accumulation.

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The phase space is a stratified subset

M⊂R4=⋃n∈N0Mn,\mathcal{M} \subset \mathbb{R}^4 = \bigcup_{n\in\mathbb{N}_0} \mathcal{M}_n,M⊂R4=n∈N0⋃Mn,
where each stratum

Mn={(p,τ0,τnaive(p,τ0,n))}\mathcal{M}_n = \{(p,\tau_0,\tau_{\mathrm{naive}}(p,\tau_0,n))\}Mn={(p,τ0,τnaive(p,τ0,n))}
is a smooth embedded surface. The collapse set

C={(p,τ0,n,τ):τ=∣r∣max⁡}\mathcal{C}=\{(p,\tau_0,n,\tau): \tau=|r|_{\max}\}C={(p,τ0,n,τ):τ=∣r∣max}
is a codimension-one absorbing boundary. Flow in the nnn-direction induces monotone downward motion of strata toward C\mathcal{C}C, yielding finite-time absorption for naive scaling. The infrared fixed-point surface defines an invariant foliation disjoint from C\mathcal{C}C. Thus, collapse corresponds to absorption in a stratified manifold with a sharp, deterministic boundary.

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© Robert R. Frost
2026-01-03