FROM CORRELATIONS TO CURVATURE: Projective Correlation Theory (V60)

We present Projective Correlation Theory (PCT), a discriminator-first framework: instead of building a complete fundamental theory and hoping it matches reality, PCT starts with a declared pregeometric correlation rule and asks a narrower question—if this correlation rule were true, what concrete, falsifiable patterns should appear in data?

PCT is built around a fixed analysis pipeline (the same steps every time). The pipeline takes primitive correlation data, applies projection operators (ways of compressing or “viewing” the correlations) and scope gates (BC1–BC5; explicit boundaries on what counts as a valid inference), and then outputs a small set of robust observables. “Robust” here means the results are designed to be insensitive to arbitrary analysis choices (the “θ” settings) and come with pre-registered decision rules plus null tests to control false positives.

PCT focuses on three main diagnostic families:

1. Spectral-dimension behaviour across scale
   PCT predicts that an effective “dimension” inferred from correlations can show a non-smooth, step-like feature rather than changing gently. The step is characterised by a dimensionless ratio
   κ ≡ ℓ*/r_H,
   where ℓ* is the characteristic scale at which the change happens and r_H is the relevant horizon scale for the system being tested. The core question is: does the data prefer a sharp change or a smooth trend?

2. A gravitational-wave ringdown change-point test
   In gravitational-wave ringdown data, PCT defines a change-point detector: it looks for a statistically defensible “moment when behaviour changes.” The key proposed regularity is a universal scaling of the change time:
   t_c / M should behave consistently across systems (M is the mass scale).
   This is formulated with explicit null controls (so noise and standard modelling artefacts have clear ways to fail the test).

3. Cross-channel consistency relations (GW ↔ CMB)
   PCT also predicts that if the same underlying correlation structure is responsible, then quantities inferred from gravitational waves (GW) and the cosmic microwave background (CMB) should not vary independently: there should be linked consistency relations across channels (e.g., κ showing “order-unity” agreement rather than being wildly different in different contexts).

What has been run so far (capsule-level confrontations)

Four “capsules” (small, fully specified confrontations) have been executed at Level 2:

1. Toy spectral-dimension test (1D chain):
   peak d_s = 1.723 at ℓ = 1.468, with a refinement diagnostic max|Δd_s| = 0.453.

2. Synthetic GW change-point detection:
   detected a change at t_c = 60 (timestamped 2020-01-01) with Δμ = 2.12 (a 21% step), confirming the detector behaves as intended on controlled data.

3. Planck 2018 ΛCDM + running inference:
   inferred running α_s = −0.0037 ± 0.0069; compared to a PCT-motivated expectation (−0.012 ± 0.005), giving 0.97σ tension (i.e., not a meaningful conflict yet).

4. LVK ringdown pipeline on GW150914:
   returned the correct NULL (no false detection under that test configuration), which is a required sanity check.

Concrete predictions PCT makes (as currently stated)

• Negative running of the scalar spectral index:
  α_s ∈ [−0.02, −0.005].

• Order-unity κ across channels:
  κ should come out broadly comparable in different observational arenas (GW and CMB), not tuned or wildly inconsistent.

• Step-vs-smooth preference in spectral dimension:
  the data should prefer a non-smooth feature in d_s(ℓ), not just a gentle drift.

What PCT explicitly does not claim

PCT does not claim:

• a complete UV theory of quantum gravity,

• a derivation of the Standard Model,

• or empirical confirmation at present.

It claims something narrower: a reproducible, scope-gated method for stress-testing “emergent geometry” signatures with clear controls against overfitting and confirmation bias.

What is new in v60 (as described)

This version also maps PCT’s structures onto several recent directions in quantum gravity and quantum tests of gravity (e.g., entanglement–geometry ideas, echo/quantised-QNM themes, topological fingerprints of spacetime, gravity-mediated entanglement experiments), and from that mapping it states seven new falsifiable predictions (NP-1 to NP-7) intended to tighten experimental contact.

Finally, the matter sector is extended to a toy |𝕀| = 3 construction with an SU(2) gauge structure, plus a stated upgrade path for anomaly checks.

Declared next upgrade target

A Level ≥ 3 confrontation on public GWTC data is identified as the next step.