A Scalar-Tensor Companion to Radial Dimensionality Theory: A Chameleon Model Reproducing the Density Law, with Gravitational-Wave Predictions

This is a companion paper to the Radial Dimensionality Theory (RDT) series (Papers I–V), not part of its load-bearing chain. The series is carried by a single canonical mechanism — a modification of the gravitational divergence operator (Papers I–II), promoted to general relativity in Papers III–IV and shown to leave the vacuum sector general-relativistic in Paper V — under which RDT contracts compact objects. This paper explores a different construction: a chameleon-type conformal scalar-tensor model that reproduces the RDT density law rather than deriving it. We are explicit that the scalar is a distinct (fifth-force) mechanism, not the divergence operator; the genuine field-theoretic completion of that operator — an action whose stellar-regime limit is the modified Gauss law — remains the series' principal open problem, which this paper does not solve. We identify the effective spatial dimensionality with a radial metric warp, d_eff = 3 + rb′/b — a kinematic definition, not a claim that dimension "emerges" from the action. The warp is built from the dimensionless conformal exponent, b ≈ 1 + β₀φ/M_pl (correcting an earlier dimensionally-inconsistent b = 1 + φ), with dynamics governed by a chameleon-type scalar field. In dense regions where the scalar gradient is negative, d_eff < 3 — reproducing the contraction sign of the series, whose amplitude the revised series bounds rather than detects (A < 0.0087 at 95% confidence from solar data, Paper I, with percent-level contraction ceilings through white dwarfs, neutron stars, and tidal deformability in Papers II–IV). The model automatically satisfies the GW170817 constraint (gravitational-wave speed equals c exactly), provides natural chameleon screening, and is ghost-free, tachyon-free, and causal. The minimal chameleon equilibrium reproduces the series' saturating dimensional law with density exponent α = 1 — a value outside the canonical range α ∈ [0.02, 0.5] scanned in Paper I. Treated as an extension point (not part of the load-bearing parameter space), rerunning the Paper I analysis there yields the separate bound A < 0.012 (95% confidence), a saturated dimensional deficit below 0.036 (saturation floor d_eff > 2.964, distinct from the canonical floor 2.974), which caps the model's parameter combination. The crossover density follows the corrected relation ρ_* ~ m²M_pl²/β₀ (placing the crossover at the series reference density requires a scalar mass m ~ 3×10⁻¹⁷ eV). A central structural result is the tradeoff between two observationally distinct parameter regimes: a stellar-relevant regime (scalar mass m ≳ 10⁻⁶ eV) in which scalar effects are localized at stellar scales — where the solar-bounded phenomenology lives — but gravitational-wave dipole radiation is Yukawa-suppressed below detectability; and a GW-loud regime (m ≲ 10⁻¹² eV) producing observable dipole radiation in neutron star–black hole binaries, with different density-crossover behavior. The differential signature — NS–BH systems showing scalar effects while BH–BH systems match general relativity exactly, via the scalar no-hair theorem — provides a robust observational discriminant, consistent with Paper V's vacuum nulling: black-hole observables remain exactly general-relativistic even when the scalar is long-range. Constraints from binary pulsar timing, Solar System tests (Cassini), and current gravitational-wave observations map the allowed parameter space, with detection strategies outlined for Einstein Telescope and Cosmic Explorer. The quantitative result is reproducible from the linked repository, merrill2025_neutronstar_code — the same repository that accompanies Paper III, which hosts the series' shared, gate-validated stellar machinery. The α = 1 extension-point bound is computed by companion_alpha1_bound.py (formerly paper6_alpha1_bound.py), which reruns the Paper I solar analysis at α = 1 via an included verbatim copy of the Paper I model module and reproduces the A < 0.012 bound quoted here. This paper does not complete the theoretical foundation of RDT — it provides a scalar-tensor companion that reproduces the phenomenology by a different mechanism, with an action, field equations, stability proofs, a predicted (extension) exponent, and quantitative observational targets. The genuine completion of the divergence operator remains open, and the stellar-fit / gravitational-wave tradeoff is identified honestly as a central scientific result rather than a flaw. Version 3 Update (June 2026): Erratum on versioning: the previous synopsis was labeled "Version 3 Update" in error — it in fact described the Version 2 manuscript. The present synopsis describes and supersedes that as the actual Version 3. This version demotes the paper off the load-bearing chain and recasts it as a companion. Earlier versions presented it as "Paper VI: A Geometric Completion" — the field-theoretic origin of the series. That claim is withdrawn: the conformal scalar is a different mechanism (a density-dependent fifth force) that reproduces the RDT phenomenology, not a derivation of the canonical divergence operator of Papers I–II (its effect on stellar structure is force-like, and the paper does not show it reduces to the operator). Accordingly: the title, abstract, and framing now label it a scalar-tensor companion that reproduces (not completes) the series, with the genuine completion stated as the open problem; "emerges/derives" language is downgraded to "is identified with"; the predicted exponent α = 1 and its bound A < 0.012 (ε_sat < 0.036, floor 2.964) are flagged as an extension point outside the canonical scan (α ∈ [0.02, 0.5], A < 0.0087, floor 2.974), reported separately; two dimensional defects are corrected — the warp is built from the dimensionless conformal exponent b ≈ 1 + β₀φ/M_pl (not 1 + φ), and the crossover density is ρ_* ~ m²M_pl²/β₀ (dimension [mass]⁴, m ~ 3×10⁻¹⁷ eV at the reference density); the support script is renamed companion_alpha1_bound.py; and the series bibliography is updated to Papers III V8 and IV V6. Prior changes (Version 2, retained): the paper was aligned with the revised series (no detections; solar bound A < 0.0087; percent-level downstream ceilings), the abandoned logarithmic phenomenological form was dropped, the α = 1 solar bound was added (computed with the Paper I version-3 machinery, released in merrill2025_neutronstar_code), the regime was renamed "stellar-matched" → "stellar-relevant," and the bibliography was updated. No theoretical content is withdrawn: the action, stability results, c_GW = c, scalar charges, the differential signature, and the tradeoff are unchanged.