Whitepaper

Quantum Mirror Gravity: The Critical Line as a Ubiquitous Horizon

Gravity as resolution symmetry; particles as reflective outputs; antiparticles as conjugate reflections. General Relativity in the infrared; soft, testable quantum structure in the ultraviolet.

Abstract

We propose Quantum Mirror Gravity (QMG): a minimal, angle-preserving quantum interface— the critical line—that is ubiquitous at every resolution point (node/atom). Each local interaction with the mirror resolves two linked outputs: a particle on our side and the charge-conjugate antiparticle across the line. The mirror acts through a scalar, frequency-dependent response $A(\kappa)$ and phase $\phi(\kappa)$ with $\kappa=\omega\,\ell_{\rm res}(x)$, where $\ell_{\rm res}(x)$ is a local resolution scale. Because the action is diagonal in angular harmonics, spin/tensor structure is preserved; hence General Relativity (GR) is recovered in the infrared: propagators and geodesics agree with GR when $A(0)=1$. At high frequency, $A(\kappa)\to 0$, providing controlled UV softness in loops and near horizons, while maintaining chirality and CPT. Gravity emerges as resolution symmetry in action: coarse-graining the local mirror rule yields spacetime curvature and geodesics. The framework is zeta-agnostic (any admissible $A$ works) but zeta-capable (a critical-line, zeta-structured $A$ adds fine, testable spectral dents).

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  • Derivations (A–E)
  • From SSC to QMG: step-by-step mapping

    • 1. ELI5 intuition

    Imagine reality woven from a single, invisible mirror that is everywhere—the critical line. Every tiny place (each atom/node) touches this mirror. When a place “looks” into it, the mirror resolves a particle on our side and a matching antiparticle behind the mirror. The mirror keeps angles and shapes intact—so GR stays true—but gently tames extreme, high-energy wiggles (a frequency-aware “muffler”). Add up all these local interactions and you get the smooth curves of spacetime: that is gravity as the macroscopic shape of a microscopic, angle-preserving rule.

    2. Core objects & definitions

    2.1 The critical line (the mirror)

    The critical line is an operational quantum interface present at all points. It acts in spectral space with dimensionless argument $\kappa=\omega\,\ell_{\rm res}(x)$, where $\omega$ is a mode frequency and $\ell_{\rm res}(x)>0$ is a local resolution scale.

    2.2 Local reflection map

    $$ \psi^{\mathrm{out}}_{+} = e^{i\phi(\kappa)}\,A(\kappa)\,\psi^{\mathrm{in}}, \qquad \psi^{\mathrm{out}}_{-} = \mathsf C\!\left[\psi^{\mathrm{out}}_{+}\right], $$

    where $A(\kappa)\in(0,1]$ with $A(0)=1$ and $A(\kappa)\to 0$ as $\kappa\to\infty$, and $\phi(\kappa)\in\mathbb R$. The action is diagonal in spherical harmonics $Y_{\ell m}$ (it multiplies each by the same scalar), hence angularity—and so spin/tensor content—are preserved.

    2.3 Resolution state

    The field $\ell_{\rm res}(x)$ encodes how finely the mirror resolves at $x$. Heuristically, larger energy/curvature tightens resolution (stronger suppression at fixed $\kappa$); flat regions are gentle.

    2.4 Reflection vs projection (terminology)

    In earlier SSC language, matter arose by projection from an internal space via a kernel. Here we use the equivalent but more physical picture: reflection from the ubiquitous mirror. “Angularity” replaces the role of basis harmonics: it is conserved by the mirror and is the seed of particle properties.

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    2.6 Time–space–light–structure as mirror properties

    In QMG, time, space, light, and structure are emergent aspects of the same angle-preserving reflection. Clock rates and distances arise from how the local scale ℓ_res(x) and response A(κ) weight paths and fields; light follows the angle-preserving limit (null geodesics) where A(0)=1. Because (A,φ) admit infinite variability within admissibility (analytic/causal, bounded), the substrate provides effectively unbounded configuration freedom—a physical basis for open-ended dynamics and macroscopic agency—while remaining deterministic in its GR limit.

    3. Gravity as connective geometry

    Because the mirror’s action is a scalar multiplier in angular space, GR’s tensor structure is preserved. In momentum space,

    $$ D^{\mathrm{eff}}(k;x) = D^{\mathrm{GR}}(k)\,A\!\big(k\,\ell_{\rm res}(x)\big), \qquad A(0)=1. $$

    Worldlines extremize a weighted length, producing GR geodesics in the infrared and tiny, controlled softening at high $\kappa$:

    $$ S[x]=\int A\!\big(\kappa(x)\big)\,ds,\qquad ds=\sqrt{g_{\mu\nu}\,dx^\mu dx^\nu}, $$ $$ \nabla_u u^\mu + \big(\delta^\mu_{\ \nu}-u^\mu u_\nu\big)\,\nabla^\nu \ln A = 0, \quad (\nabla A=0\Rightarrow \nabla_u u^\mu=0). $$

    4. Angularity & emergence of particles and mass

    The mirror preserves the angular labels $(\ell,m)$, so particle properties tied to spin and angular structure are stable. Particles and antiparticles are paired reflective outputs (with chiral phases $\pm\phi$), while the effective mass scale of fields is shaped by how $A(\kappa)$ suppresses ultraviolet modes and how $\ell_{\rm res}(x)$ varies—giving the macroscopic appearance of mass/curvature.

    5. GR & SM equivalence; the QM link via the critical line

    GR (deterministic geometry)

    • Angle-preserving action \Rightarrow GR’s spin-2 tensor structure intact.
    • Infrared limit $A(0)=1$ \Rightarrow GR propagators/geodesics recovered; PPN $\gamma=\beta=1$.
    • Horizons: impose small, frequency-dependent reflectivity $\mathcal R(\omega)\propto A(\omega\ell_{\rm res})$ \Rightarrow tiny QNM shifts, late-time echoes.

    SM (local quantum dynamics)

    • Chirality/CPT respected via opposite phases for $L/R$ with the same $A$.
    • Loops: the same $A$ softens UV contributions; e.g., Higgs one-loop bound $|\delta m_H^2|\le C/(16\pi^2\ell_{\rm res}^2)$.
    • Low-energy fits unchanged at $\kappa\to 0$; high-$\kappa$ tails tamed without new light fields.
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    Derivation links. GR preservation follows from angle-diagonal action (Derivation A) and weighted geodesics (Derivation B). SM consistency and UV softness follow from the shared spectral response in loops (Derivation C). Near-horizon phenomenology uses the same map (Derivation D). Chirality/CPT are preserved (Derivation E).

    6. Predictions & tests

    6+. Global predictions of the Uniform Reflection Principle (URP)

    Each item states (i) the claim, (ii) an operational prediction, and (iii) how to test/falsify it.

    P1. Universality of the mirror law (single origin)

    Claim. All local reflections share the same functional form of A(κ), φ(κ); only the scale ℓ_res(x) varies.

    Prediction. A single family A(κ) and a single parameter field ℓ_res jointly fit: GW ringdown echoes (horizon reflectivity), the Higgs one-loop bound, and high-freq EHT ring modulations.

    Falsifier. No common A and ℓ_res explain these datasets simultaneously.

    P2. Indistinguishability of quantum instances

    Claim. Any local reflection is indistinguishable after rescaling frequency by ℓ_res.

    Prediction. Echo spectra from different black holes collapse to a master curve when plotted vs. ω·ℓ_res.

    Falsifier. Source-dependent echo shapes that cannot be reconciled by rescaling ℓ_res.

    P3. Gravity & light are the “vectors” of quantum computation

    Claim. Light fixes null cones (conformal structure); gravity (free fall) fixes the connection (projective structure). The mirror computes by transporting states along these structures.

    Prediction. No extra GW polarizations; no birefringent/tensor-mixing effects. Only scalar, frequency-dependent amplitude shaping via A(κ).

    Falsifier. Detection of new GW polarizations or polarization-dependent propagation beyond GR bounds.

    P4. Infinite variety bound to a single quantum state

    Claim. The mirror’s admissible (A,φ) supply unbounded configuration freedom while remaining one universal interface.

    Prediction. The Higgs loop obeys |δm_H^2| ≤ C/(16π^2 ℓ_res^2) with 0.01 ≲ C ≲ 0.5 (model-family dependent), and the same ℓ_res matches GW/EHT fits.

    Falsifier. No consistent ℓ_res across loops and horizons, or empirical need for A(0)≠1 (which would break GR in the IR).

    P5. Angle-preserving, polarization-blind high-κ effects

    Claim. The mirror multiplies harmonics by a scalar; it cannot mix polarizations/spins.

    Prediction. Any deviations at high frequency are polarization-independent (GW & EM). EHT ring modulations, if present, do not rotate polarization bases.

    Falsifier. Observed birefringence or spin-mixing not attributable to plasma or instrument systematics.

    P6. Mass/curvature trend in near-horizon reflectivity

    Claim. Stronger curvature effectively tightens resolution: A(ωℓ_res) suppresses more for heavier, more compact objects.

    Prediction. Systematic trend: heavier BHs show weaker high-ω echo content after normalisation.

    Falsifier. No correlation of echo high-frequency suppression with mass/compactness.

    P7. Weighted-geodesic correction in extreme fields

    Claim. Worldlines obey ∇_u u^μ + (δ^μ_ν − u^μ u_ν)∇^ν ln A = 0 (Derivation B).

    Prediction. Tiny frequency-dependent Shapiro-delay/trajectory drifts near Sgr A* consistent with ∇ln A; null in the IR.

    Falsifier. Frequency-dependent path effects that imply angle-non-preserving dynamics.

    P8. Lab analogue universality

    Claim. QMG is zeta-agnostic: any admissible A(κ) works; universality is the key.

    Prediction. Engineering the same A(κ) in distinct analogue platforms (optical, acoustic) yields the same echo master curve after ω·ℓ_res normalization.

    Falsifier. Irreconcilable A shapes across platforms once normalized.

    7. Minimal axioms & constraints

    1. Ubiquity: each point has $\ell_{\rm res}(x)>0$.
    2. Angle preservation: action is diagonal in $Y_{\ell m}$.
    3. Admissibility: $A(0)=1$, $0<A\le 1$, $A\to 0$ as $\kappa\to\infty$; $A,\phi$ analytic/causal.
    4. Parity pairing: antiparticle output is charge-conjugate with opposite chiral phase.
    5. Classical limit: coarse-graining yields GR in the IR.

    8. FAQ & clarifications

    Is this modifying GR?

    No. GR’s tensor structure and infrared predictions remain intact; the mirror only adds a scalar, angle-preserving response.

    Do particles literally sit on a number-theory line?

    We use “critical line” operationally: a quantum interface with a scalar spectral response. A zeta-structured version is testable but not required.

    Where does mass come from here?

    From how $A(\kappa)$ softens high-frequency content and how $\ell_{\rm res}(x)$ varies; after coarse-graining, this appears as curvature and effective mass scales consistent with GR/SM in the IR.

    9. Uniform Reflection Principle & Proof Program

    Statement

    URP. At every node/point, the critical line applies the same angle-preserving map M[Y_{ℓm}] = A(κ)e^{iφ(κ)}Y_{ℓm} with κ = ω·ℓ_res(x), where ℓ_res(x) > 0 is a local resolution scale. The rule is uniform (same functional form of A,φ) across space and time; only ℓ_res varies.

    Inside-model theorem (sketch)

    1. Angle preservation ⇒ GR tensors in the IR. Because M is scalar in angular harmonics, it commutes with the spin-2 projector, so the graviton two-point function is D^eff = A(k·ℓ_res)·D^GR with A(0)=1 ⇒ GR is recovered at low frequency (Derivation A).
    2. Geodesics as weighted extremals. Worldlines extremize S=∫A(κ(x)) ds, yielding ∇_u u^μ + (δ^μ_ν − u^μ u_ν)∇^ν ln A = 0. If ∇A=0 (IR), free fall is GR geodesic (Derivation B). Hence “gravity as resolution symmetry in action.”
    3. Same filter in quantum loops. Using the same A in one-loop integrals gives the finite bound |δm_H^2| ≤ C/(16π^2 ℓ_res^2) (Derivation C), tying the quantum scale ℓ_res to macroscopic inferences (e.g., horizon echoes, Derivation D).

    Why light + gravity suffice (external foundations)

    In classical spacetime, light rays determine the conformal structure (null cones), while free-fall worldlines determine the projective structure. The Ehlers–Pirani–Schild (EPS) program shows that compatibility of these two structures reconstructs a (Levi-Civita) metric geometry, i.e., GR’s kinematics. Maxwell theory in 3+1D is conformally invariant, so “light” pins down the null cones; “gravity” (free fall) pins down the connection; together they fix the classical geometry.

    Empirical proof program

    10. Derivations (A–E)

    Derivation A — Angle-preserving response preserves GR tensors

    Claim. If the mirror acts diagonally in spherical harmonics (multiplying each Y_{ℓm} by the same scalar A(κ)·e^{iφ(κ)}) then GR’s spin‑2 tensor structure is unchanged in the IR, and the effective propagator is D^eff(k;x)=D^GR(k)·A(k·ℓ_res(x)).

    M[Y_{ℓ m}] = A(κ) e^{i φ(κ)} Y_{ℓ m},   κ = ω · ℓ_res(x)
⇒ D^eff(k;x) = D^GR(k) · A(k · ℓ_res(x)), with A(0)=1.

    Sketch. The action is a scalar on the SO(3) harmonic basis, hence it commutes with rotations and leaves spin content intact; the graviton projector commutes with the map, so only a scalar factor appears.

    Derivation B — Weighted geodesic equation

    Define the weighted length functional S[x]=∫ A(κ(x)) ds, with ds=√(g_{μν} dx^μ dx^ν) and κ(x)=ω·ℓ_res(x). Varying S yields

    ∇_u u^μ + (δ^μ_ν − u^μ u_ν) ∇^ν ln A(κ(x)) = 0,    u^μ = dx^μ/ds
(If ∇A=0 ⇒ ∇_u u^μ = 0, the GR geodesic.)

    The correction is orthogonal to u, so no superluminality is introduced.

    Derivation C — One-loop Higgs mass correction bound

    Use the same spectral response A(k·ℓ_res) in Euclidean one-loop integrals:

    δ m_H^2 = (1/8π^2) ∫_0^∞ dk k^3 [ 6λ/(k^2+m_H^2) + (9/4)g^2/(k^2+m_W^2)
                                 + (3/4)g'^2/(k^2+m_Z^2) − 6 y_t^2/(k^2+m_t^2) ] · A(k·ℓ_res)
⇒ |δ m_H^2| ≤ C / (16 π^2 ℓ_res^2)

    Here C is a dimensionless angular average set by A; this yields a finite, testable relation between mirror scale and Higgs sensitivity.

    Derivation D — Horizon reflectivity and echoes

    Impose a small, frequency-dependent reflectivity at the would‑be horizon:

    ℛ(ω) = ℛ_0 · A(ω · ℓ_res) · e^{i φ(ω · ℓ_res)},   |ℛ_0| ≪ 1
Echo spacing ~ 2 |r_*|; nth echo envelope ∝ |ℛ(ω)|^n.

    The same A that bounds the Higgs loop shapes late-time GW residuals, enabling joint inference of ℓ_res.

    Derivation E — Chirality and CPT across the mirror

    For Weyl components, choose opposite phases while sharing the same A:

    (ψ_L)^out_- = e^{+i φ} · A · (ψ_L)^in,    (ψ_R)^out_- = e^{−i φ} · A · (ψ_R)^in.

    This realizes particle–antiparticle pairing with symmetric parity, preserves SM chiral structure, and leaves low‑energy amplitudes unchanged as A(0)=1.

    11. From SSC projection to QMG reflection — step-by-step mapping

    1. Kernel → Mirror: SSC’s projection kernel W(x,σ) maps to a local, angle‑preserving scalar response A(κ) with phase φ(κ), evaluated at κ=ω·ℓ_res(x).
    2. Resolution field: SSC’s resolution/constraint sector yields the positive field ℓ_res(x) that sets local spectral scale; coarse‑graining recovers GR in the IR.
    3. Angularity preserved: SSC’s harmonic structure (on the internal sphere) becomes the mirror’s angular‑preserving action on Y_{ℓm}, fixing spin/tensor content.
    4. Loops & horizons: SSC’s UV softness (“flexure”) becomes the shared spectral filter A in loop integrals (Deriv. C) and near‑horizon reflectivity (Deriv. D).
    5. Parity pairing: SSC’s neutrality/constraint logic corresponds to particle–antiparticle pairing with opposite chiral phases (Deriv. E) under a single interface.
    6. Empirics: SSC consistency checks (PPN, c_GW=c) are inherited via A(0)=1; novel QMG predictions enter through the small‑departure structure of A,φ at high κ.