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Consistency Bound on the Emergent Gravitational Scale in SPU

Abstract

In this document we derive a consistency bound on the emergent gravitational scale $\Lambda_{\mathrm{SP}}$ in the SPU framework. The bound follows from basic requirements of locality, universality, and RG decoupling between gauge and gravitational sectors. The result constrains the ratio $\Lambda_{\mathrm{SP}} / M_{\mathrm{GUT}}$ to a narrow and physically meaningful range, ensuring internal coherence of the theory without introducing tunable parameters.

1. Why a Consistency Bound Is Needed

In SPU, gravity emerges as a collective phenomenon characterized by a stiffness scale:

$$\Lambda_{\mathrm{SP}}$$

Gauge interactions unify dynamically at:

$$M_{\mathrm{GUT}} \approx 10^{16},\mathrm{GeV}$$

For the theory to be consistent, these two scales cannot be arbitrarily related. A bound is required to guarantee:

  • decoupling of gravity from gauge RG flow,
  • universality of gravitational coupling,
  • absence of strong-gravity effects below unification.

2. Lower Bound: Decoupling from Gauge Dynamics

Gravity must not affect gauge running below $M_{\mathrm{GUT}}$. This requires:

$$\Lambda_{\mathrm{SP}} > M_{\mathrm{GUT}}$$

If $\Lambda_{\mathrm{SP}}$ were comparable to or smaller than $M_{\mathrm{GUT}}$:

  • gravitational fluctuations would enter gauge beta functions,
  • unification would be destabilized,
  • gauge-sector predictions would lose robustness.

Therefore, a strict lower bound exists:

$$\frac{\Lambda_{\mathrm{SP}}}{M_{\mathrm{GUT}}} > 1$$

3. Upper Bound: Collective Nature of Gravity

Gravity in SPU arises from collective averaging over fermionic degrees of freedom.

Let:

$$N_f^{\mathrm{eff}} = 128 - \delta$$

If $\Lambda_{\mathrm{SP}}$ were parametrically larger than the collective enhancement scale:

$$\Lambda_{\mathrm{SP}} \gg \sqrt{N_f^{\mathrm{eff}}} \times M_{\mathrm{GUT}}$$

then:

  • gravity would effectively decouple entirely,
  • no observable long-range interaction would remain,
  • the equivalence principle would be lost.

This yields an upper bound:

$$\frac{\Lambda_{\mathrm{SP}}}{M_{\mathrm{GUT}}} \lesssim \sqrt{N_f^{\mathrm{eff}}}$$

4. Combined Consistency Window

Combining the two constraints:

$$1 < \frac{\Lambda_{\mathrm{SP}}}{M_{\mathrm{GUT}}} \lesssim \sqrt{N_f^{\mathrm{eff}}}$$

Using SPU values:

$$N_f^{\mathrm{eff}} \approx 127$$

gives:

$$1 < \frac{\Lambda_{\mathrm{SP}}}{M_{\mathrm{GUT}}} \lesssim 11$$

This is a narrow window, not an arbitrary range.

5. Natural Saturation of the Bound

From the semi-analytic estimate (previous document):

$$\Lambda_{\mathrm{SP}} \approx \sqrt{N_f^{\mathrm{eff}}} \times M_{\mathrm{GUT}}$$

SPU naturally saturates the upper edge of the allowed window.

This explains:

  • the extreme weakness of gravity,
  • the absence of deviations at accessible energies,
  • the numerical proximity to the observed Planck scale.

No fine tuning is involved.

6. Independence from δ Fine Details

Importantly:

  • small changes in $\delta$ modify $\Lambda_{\mathrm{SP}}$ only logarithmically,
  • the bound survives even if $\delta$ varies by $O(10%)$.

Thus, the consistency window is structural, not accidental.

7. What This Bound Does Not Assume

This derivation does not rely on:

  • quantum gravity models,
  • string theory,
  • extra dimensions,
  • supersymmetry,
  • specific UV completions.

It follows solely from SPU principles.

8. Falsifiability Criteria

The SPU gravitational sector would be falsified if:

  1. $\Lambda_{\mathrm{SP}} &lt; M_{\mathrm{GUT}}$ were required phenomenologically,
  2. gravity affected gauge RG flow below $M_{\mathrm{GUT}}$,
  3. long-range gravity were absent,
  4. equivalence principle violations were detected.

Any of these would invalidate the construction.

9. Summary

  • SPU predicts an emergent gravitational scale $\Lambda_{\mathrm{SP}}$.
  • Consistency imposes a narrow allowed window.
  • The observed hierarchy lies naturally within it.
  • Gravity is weak because it is collective and constrained.

This completes the consistency analysis of the gravitational scale in SPU.

End of document.