Space and Time May Grow from Simple Links

In the paper's words

The prevailing view treats spacetime as a pre‑existing arena, a stance that blocks unified descriptions of gravitation and quantum phenomena and incurs large research expenditures. A concrete quantitative gap exists because existing emergent‑spacetime proposals lack computable, falsifiable measures of the geometric structure produced by a relational substrate. We adopt a purely analytical approach that (i) maps the autaxic relational effort metric onto the universal rotation‑curve relation, (ii) models the substrate as a d‑dimensional lattice of process‑events, and (iii) derives three explicit scaling results.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. For $d=3$, $r=10$, $L=100$ the correction factor is $$ 1-\frac{10}{100^{3}} = 1-\frac{10}{1\,000\,000}=0.99999, $$ so $P_{\text{corr}}(10)\approx9{,}765{,}625\times0.99999\approx9{,}765{,}527$, a negligible change at this scale but a growing effect for larger $r$ or smaller $L$.

   from 3. Path‑count approximation and its limits

2. Assuming reuse of existing survey data and modest computational resources (a few CPU‑core weeks), the incremental cost is on the order of $ 10⁴ USD for data processing, personnel time, and simulation development.

   from 8. Practical Implications and Engineering Pathways

How it is done

1. We adopt a purely analytical approach that (i) maps the autaxic relational effort metric onto the universal rotation‑curve relation, (ii) models the substrate as a $d$‑dimensional lattice of process‑events, and (iii) derives three explicit scaling results.

   from Abstract

2. Cohomological proof of Cartan conjugacy – [2] provides a rigorous algebraic method for classifying structures, informing our use of cohomological tools to identify invariant relational patterns within autaxic dynamics.

   from 2. Background and Related Work

3. We treat autaxic process‑patterns as discrete events linked by a relational “effort” metric $E$, defined as the number of fundamental relational updates required to propagate influence between two events.

   from 3. Methods

4. Starting from $\Phi(r)=-M/r$ (A1) we compute the radial derivative: $$ \frac{\mathrm{d}\Phi}{\mathrm{d}r}= \frac{\mathrm{d}}{\mathrm{d}r}\!\left(-\frac{M}{r}\right)=\frac{M}{r^{2}} . $$ Insert into the rotation‑curve relation (A3): $$ V^{2}= r \times \frac{M}{r^{2}} = \frac{M}{r}. $$ Set $M=10$ and $r=5$: $$ \frac{M}{r}= \frac{10}{5}=2,\qquad V^{2}=2,\qquad V=\sqrt{2}\approx1.41421356. $$

   from 4.1 Galactic rotation derivation

What would falsify it

1. The main galactic result would be falsified if high‑precision rotation‑curve measurements at radii where $M/r$ is well constrained systematically deviate from $V=\sqrt{M/r}$ by more than observational uncertainties, and the deviation cannot be attributed to baryonic modelling.

   from Falsifiability

2. The lattice scaling results would be falsified if simulations of a $d$‑dimensional process‑event lattice produce boundary‑touching frequencies or path counts that differ from the analytic predictions beyond statistical fluctuations.

   from Falsifiability

3. A concrete quantitative gap exists because existing emergent‑spacetime proposals lack computable, falsifiable measures of the geometric structure produced by a relational substrate.

   from Abstract

Where the claim stops

1. The galactic‑rotation derivation assumes a Newtonian potential (A1) and neglects relativistic corrections and baryonic disk structure.

   from Limitations

2. The path‑count approximation $(2d-1)^{r}$ ignores self‑avoidance constraints that become significant for large $r$ on finite lattices.

   from Limitations

3. Future work will aim to derive the proportionality constant $\alpha$ from autaxic dynamics, extend the analysis to relativistic regimes, and validate the lattice predictions through dedicated simulations.

   from 9. Conclusion

Where to read

1. Abstract158 words · 1 min
2. 1. Introduction212 words · 1 min
3. 2. Background and Related Work62 words · 1 min
4. 3. Methods232 words · 1 min
5. 4. Analysis244 words · 1 min
6. 5. Results53 words · 1 min
7. 6. Extended Derivations and Worked Examples1,036 words · 5 min
8. 7. Discussion208 words · 1 min
9. 8. Practical Implications and Engineering Pathways231 words · 1 min
10. 9. Conclusion106 words · 1 min
11. Version History47 words · 1 min
12. Provenance and AI Involvement225 words · 1 min