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

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===== We construct a deterministic four-dimensional phase space with coordinates (p,τ0,n,τ)(p,\tau_0,n,\tau)(p,τ0,n,τ), where ppp denotes the precision index, τ0\tau_0τ0 the initial structural tolerance, nnn the cumulative experiment count, and τ\tauτ the effective tolerance after refinement and accumulation. Precision refinement is modeled as multiplicative contraction by a factor 2−p2^{-p}2−p, representing systematic increases in measurement resolution. =====

Experiment accumulation enters through a logarithmic erosion factor

κ(n)=1−αlog⁡ ⁣(1+nn0),\kappa(n)=1-\alpha\log\!\left(1+\frac{n}{n_0}\right),κ(n)=1−αlog(1+n0n),
with fixed parameters α∈(0,1)\alpha\in(0,1)α∈(0,1) and n0>0n_0>0n0>0. This choice ensures sublinear tightening with experiment count and a finite exhaustion horizon. For naive scaling, the effective tolerance is

τnaive(p,τ0,n)=τ0 2−p κ(n),\tau_{\mathrm{naive}}(p,\tau_0,n)=\tau_0\,2^{-p}\,\kappa(n),τnaive(p,τ0,n)=τ02−pκ(n),
which defines a family of smooth tolerance surfaces indexed by discrete nnn.

Collapse is defined deterministically by comparison to a fixed residual bound ∣r∣max⁡|r|_{\max}∣r∣max. The collapse condition τ<∣r∣max⁡\tau<|r|_{\max}τ<∣r∣max corresponds geometrically to intersection with a horizontal absorbing boundary plane in (p,τ0,τ)(p,\tau_0,\tau)(p,τ0,τ) space. For each experiment count nnn, this intersection occurs at a finite precision index

p\''(τ0,n)=log⁡2 ⁣(τ0 κ(n)∣r∣max⁡),p^\''(\tau_0,n)=\log_2\!\left(\frac{\tau_0\,\kappa(n)}{|r|_{\max}}\right),p\*(τ0,n)=log2(∣r∣maxτ0κ(n)),
which decreases monotonically with increasing nnn, yielding a finite experiment horizon beyond which collapse is unavoidable for any nonzero precision.

For comparison, an infrared fixed-point scaling is defined by

τIR(p,τ0)=τ\''+(τ0−τ\'')2−p,τ\''≥∣r∣max⁡,\tau_{\mathrm{IR}}(p,\tau_0)=\tau^\''+(\tau_0-\tau^\*)2^{-p},
\qquad \tau^\''\ge |r|_{\max},τIR(p,τ0)=τ\''+(τ0−τ\'')2−p,τ\''≥∣r∣max,
which is independent of nnn and never intersects the collapse boundary. This surface forms a time-invariant, non-collapsing region of the phase space.

The rendered 4D projection embeds (p,τ0,τ)(p,\tau_0,\tau)(p,τ0,τ) explicitly and represents the fourth dimension nnn as discrete stacked surfaces. All quantities are computed analytically; no stochastic elements, fitting procedures, or optimization objectives are introduced. The geometry therefore provides an exact, visual representation of the theory’s collapse criteria and stopping rules.

© Robert R. Frost
2026-01-03