Primitives, Definitions, Axioms
Primitives: Fundamental geometric entities (points/relations), a connectivity structure, and a measure assigning invariant intervals. No background manifold is assumed a priori.
Definitions:
- State: equivalence class of configurations under symmetry group G.
- Event: minimal update consistent with constraints C.
- Observer: a congruence of events admitting a monotonic parameter (proper-like).
- Distance/Time: operational measures derived from invariant intervals.
Axioms:
1) Geometric Sufficiency: physical laws emerge from geometric relations among primitives.
2) Relational Invariance: observable quantities are invariant under G.
3) Local Consistency: updates respect local conservation constraints C.
4) Minimality: among consistent evolutions, choose the one minimizing action-like functional A.
5) Correspondence: in appropriate limits, recover standard relativistic and FRW-like behavior.
Notes: These axioms are intentionally compact; formal statements and proofs are given on the Mathematics pages.
Definitions:
- State: equivalence class of configurations under symmetry group G.
- Event: minimal update consistent with constraints C.
- Observer: a congruence of events admitting a monotonic parameter (proper-like).
- Distance/Time: operational measures derived from invariant intervals.
Axioms:
1) Geometric Sufficiency: physical laws emerge from geometric relations among primitives.
2) Relational Invariance: observable quantities are invariant under G.
3) Local Consistency: updates respect local conservation constraints C.
4) Minimality: among consistent evolutions, choose the one minimizing action-like functional A.
5) Correspondence: in appropriate limits, recover standard relativistic and FRW-like behavior.
Notes: These axioms are intentionally compact; formal statements and proofs are given on the Mathematics pages.
Motivation: The primitives/axioms are chosen to be observationally anchored yet mathematically lean, enabling derivations without introducing extra fields unless required by data.
System Dynamics
Evolution Law: dS/dλ = argmin_{Δ} A[Δ] subject to constraints C and invariants of G. In continuum limit, this yields field equations resembling curvature–content balance. Discrete updates correspond to event additions respecting causality-like partial ordering.
Conservation & Fluxes: Noether-like quantities appear from symmetries in G; continuity relations emerge in the coarse-grained limit.
Scaling Regimes: (i) Local regime approximates relativistic dynamics; (ii) Cosmological regime admits homogeneous/isotropic solutions with scale factor a(λ).
Initial/Boundary Conditions: Minimal-action seeds with low-complexity priors; sensitivity explored in Derivations.
Conservation & Fluxes: Noether-like quantities appear from symmetries in G; continuity relations emerge in the coarse-grained limit.
Scaling Regimes: (i) Local regime approximates relativistic dynamics; (ii) Cosmological regime admits homogeneous/isotropic solutions with scale factor a(λ).
Initial/Boundary Conditions: Minimal-action seeds with low-complexity priors; sensitivity explored in Derivations.
Hint: The Euler–Lagrange form appears when A is local and λ is continuous; discrete formulations use variational updates on graphs.
Parameters & Constraints
Core Parameters:
- κ: geometric coupling setting curvature–content relation.
- α_i: symmetry-breaking coefficients governing departures from perfect homogeneity/isotropy.
- ξ: discretization scale or coarse-grain cutoff.
- Λ_g: geometric vacuum term emerging from A (not assumed a priori).
Priors & Bounds:
- κ > 0 from local stability; ξ below smallest resolved observational scale; |α_i| << 1 for late-time isotropy.
Constraints Used:
- Invariance under G; locality of updates; positivity of effective densities; causal ordering of events.
Estimation Strategy: Fit a(λ) and distance relations jointly to standard probes (SNe, BAO, CMB peaks) while allowing α_i perturbations; cross-validate with growth data.
- κ: geometric coupling setting curvature–content relation.
- α_i: symmetry-breaking coefficients governing departures from perfect homogeneity/isotropy.
- ξ: discretization scale or coarse-grain cutoff.
- Λ_g: geometric vacuum term emerging from A (not assumed a priori).
Priors & Bounds:
- κ > 0 from local stability; ξ below smallest resolved observational scale; |α_i| << 1 for late-time isotropy.
Constraints Used:
- Invariance under G; locality of updates; positivity of effective densities; causal ordering of events.
Estimation Strategy: Fit a(λ) and distance relations jointly to standard probes (SNe, BAO, CMB peaks) while allowing α_i perturbations; cross-validate with growth data.
Data Note: estimation uses public datasets; details and code are linked from Derivations and Evidence & Tests pages.
Correspondence with Standard Cosmology
Mapping Overview:
- FRW Limit: homogeneous/isotropic solutions reproduce an FRW-like metric with effective scale factor a(λ) and curvature term k_eff determined by κ and Λ_g.
- Redshift Relation: 1+z ≈ a(λ_0)/a(λ_e) with small corrections O(α_i).
- Distance Measures: Luminosity and angular-diameter distances match standard forms in the correspondence limit; deviations parameterized by α_i and ξ.
- Early-Time Behavior: Alternatives to inflation arise via geometric smoothing without extra fields, yet standard successes (horizon/flatness) are retained in limit cases.
Where It Differs: Growth of structure and late-time acceleration can diverge measurably for certain parameter ranges.
- FRW Limit: homogeneous/isotropic solutions reproduce an FRW-like metric with effective scale factor a(λ) and curvature term k_eff determined by κ and Λ_g.
- Redshift Relation: 1+z ≈ a(λ_0)/a(λ_e) with small corrections O(α_i).
- Distance Measures: Luminosity and angular-diameter distances match standard forms in the correspondence limit; deviations parameterized by α_i and ξ.
- Early-Time Behavior: Alternatives to inflation arise via geometric smoothing without extra fields, yet standard successes (horizon/flatness) are retained in limit cases.
Where It Differs: Growth of structure and late-time acceleration can diverge measurably for certain parameter ranges.
Further Reading: See side-by-side analyses and case studies comparing to ΛCDM, including fit quality and residuals.
Unique, Testable Predictions
1) Low‑z Distance Modulation: A percent‑level, scale‑dependent deviation in D_L(z) around z≈0.1–0.3 linked to α_i; sign and amplitude fixed by fitted κ, ξ.
2) Growth Anomaly: fσ8(z) tilt differs at z<1 without invoking new dark sector microphysics; predicts a specific redshift of maximum deviation.
3) CMB Lensing Slope: Slight change in lensing power at high‑ℓ consistent with ξ; cross‑correlates with galaxy lensing.
4) Standard Sirens: GW distance–redshift curve shows the same modulation as EM standard candles, breaking degeneracies.
5) Early‑Time Smoothness: BAO phase shift pattern consistent with geometric smoothing, testable with next‑gen surveys.
Each item includes falsifiable ranges; see Evidence & Tests for datasets and thresholds.
2) Growth Anomaly: fσ8(z) tilt differs at z<1 without invoking new dark sector microphysics; predicts a specific redshift of maximum deviation.
3) CMB Lensing Slope: Slight change in lensing power at high‑ℓ consistent with ξ; cross‑correlates with galaxy lensing.
4) Standard Sirens: GW distance–redshift curve shows the same modulation as EM standard candles, breaking degeneracies.
5) Early‑Time Smoothness: BAO phase shift pattern consistent with geometric smoothing, testable with next‑gen surveys.
Each item includes falsifiable ranges; see Evidence & Tests for datasets and thresholds.
Status: Predictions 1–2 under active analysis; 3–5 scheduled pending data releases. Replication links provided on Evidence & Tests.
Assumptions & Limitations
Core Assumptions:
- Symmetry group G captures all observationally relevant invariances at targeted scales.
- Action-like functional A is local (or weakly nonlocal) and bounded below.
- Coarse-graining to continuum is well-defined for the regimes considered.
Known Limitations:
- Microphysical interpretation of Λ_g remains open.
- Precise mapping between λ and cosmological time requires calibration.
- High‑curvature/early‑epoch behavior needs further mathematical control.
Next Steps: Formalize the λ↔t mapping and extend proofs of stability; see Mathematics→Foundations for ongoing work.
- Symmetry group G captures all observationally relevant invariances at targeted scales.
- Action-like functional A is local (or weakly nonlocal) and bounded below.
- Coarse-graining to continuum is well-defined for the regimes considered.
Known Limitations:
- Microphysical interpretation of Λ_g remains open.
- Precise mapping between λ and cosmological time requires calibration.
- High‑curvature/early‑epoch behavior needs further mathematical control.
Next Steps: Formalize the λ↔t mapping and extend proofs of stability; see Mathematics→Foundations for ongoing work.