Theoretical Structure

The Intrinsic Response Framework

A constitutive metric response as a candidate identity for cold dark matter — ontologically agnostic, empirically grounded, and explicitly falsifiable.

Conceptual Foundation

Role vs. Identity: A Formal Distinction

The ΛCDM model stands as the standard paradigm of cosmology, yet after decades of engagement, every attempt to directly identify the foundational particle of cold dark matter has returned a null result. This motivates a formal distinction between the role of dark matter and its identity.

Defining the CDM Role

In the language of General Relativity, the CDM role is the assertion that there exists an effective stress-energy tensor sector T^X_μν such that the Einstein Field Equations are satisfied:

G_μν = 8πG ( T̄_μν + T^X_μν )

This combined tensor must yield a metric that reproduces the full suite of geodesic phenomenology: timelike geodesics (galaxy dynamics), null geodesics (gravitational lensing), and cosmological perturbations (CMB power spectrum).

The Intrinsic Response as a Candidate Identity

The intrinsic response framework proposes that the extra sector is not an independent entity with its own initial conditions, but is instead a constitutive response functional of the baryonic sector itself:

T^resp_μν = F( T̄_μν, ∇T̄_μν, … ; θ )

The "darkness" is simply that this response is non-luminous and need not have electromagnetic couplings. The null detection record is reframed not as a failure to find dark matter, but as a success in defining what it is not.

Mechanism

Boundary-Layer Closure

The response sector is activated near a baryonic boundary layer — the transition radius R_t where the baryonic acceleration equals the critical scale a₀. Outside this activation zone, an approximate flux-conservation law constrains the response to a 1/R acceleration tail, matching the classic halo profile but derived rather than assumed.

Transition Radius

g_bar(R_t) ≈ a₀ a₀ ≈ 1.2 × 10⁻¹⁰ m s⁻²

R_t is determined deterministically from the baryonic mass model — no free parameter. The product cH₀ ~ a₀ motivates a possible cosmological origin.

Asymptotic Tail

g_resp ~ Q/R (R → ∞) ρ_resp(R) ∝ R⁻²

The effective density scales as R⁻², yielding an asymptotic g ∝ 1/R tail — matching the isothermal halo profile but as a derived consequence.

The Single Free Parameter

The entire galaxy-domain closure is described by a single free parameter Q per galaxy — the asymptotic extra v² contribution. Two independent estimators are computed:

Q_best: χ² minimisation over all data points

Best-fit amplitude; sensitive to inner kinematics.

Q_est: Huber M-estimator outer subset (R ≥ 0.6 R_max)

Model-free; Spearman ρ(Q_est, Q_best) = 0.972 across 175 galaxies.

Key Prediction

Oscillatory Metric Response

A striking prediction of the framework is that 46 of 175 SPARC galaxies exhibit sign changes in Δ(R) = V²_obs − V²_bar — an oscillatory pattern consistent with boundary-layer standing waves. The spatial period scales with galaxy size rather than a cosmological constant.

λ ≈ 45.6 kpc

Median wavelength

Across 46 sign-change galaxies

λ/2 ≈ R_max

Half-wavelength

Key empirical scaling result

Δ(R) = 0 radii

Nodal prediction

Morphological ring features

Φ_resp(r) = A cos(κr + δ) e^(−r/L) κ = 2π/λ (galaxy-size dependent, not cosmological)

Lensing Discriminant

Branch A vs. Branch B

A crucial theoretical decision is the default branch for the galaxy regime, which determines the relationship between the two Newtonian-gauge scalar potentials:

Default

Branch A: Metric-Equivalent

Φ = Ψ (η = 1)

Scalar potentials are equal — no gravitational slip. Lensing follows dynamics in the standard way. This is the conservative default assumption.

Testable

Branch B: Slip-Allowed

η(R) ≡ Ψ/Φ ≠ 1 (boundary-layer zones)

Potentials differ in activation zones, predicting a specific lensing-dynamics mismatch. Weak lensing surveys can adjudicate between branches.

Theoretical Context

Positioning: Not MOND, Not Standard CDM

The intrinsic response framework occupies a distinct position in the theoretical landscape. It is not a competitor to dark matter — it is a candidate for its identity.

Framework	Dark Matter	GR	Cluster Scale	Cosmological Scale
ΛCDM	Particle fluid	Standard	✓ Explained	✓ Explained
MOND / TeVeS	Modified dynamics	Modified	⚠ Tension	⚠ Tension
Intrinsic Response	Constitutive response	Standard	? Rung 3 test	? Rung 5 test

The framework avoids MOND's primary vulnerability by seeking to be the dark component rather than explain it away. The cluster and cosmological scales are the highest rungs of the discriminants ladder — open empirical questions, not refutations.

Theoretical Extension

Spacecymatics: The Spectral DNA of the Response Operator

A formal analogy exists between the eigen-spectrum of the IRT response operator and the nodal geometry of classical Chladni cymatics. Wherever the response sector is governed by a linearized operator L_resp, its eigenpairs are the "spectral DNA" of the response geometry — the galactic analogs of Chladni mode shapes and nodal lines.

The sign-alternating lobes of Δ(R) correspond to nodal annuli of these eigenmodes. The characteristic ring spacing and decay envelope are set by spectral parameters (κ, screening length L) selected by activation/quenching "boundary" conditions near R_t and at large scales. This framework enables a spectral taxonomy of galaxy response morphologies — classifying galaxies by mode index, damping Q-factor, and phase alignment, exactly as eigen-spectra classify vibrating plates.

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Explore the Full Symbol Glossary

Complete reference for all mathematical notation used in the framework.

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