PRH | Essay | 7.5 • Gravity as Local Information, Quantum as Blur

Gravity as Local Information, Quantum as Blur Black Holes, Speed Limits, and the Two Poles of Reality Aleksandar Perišić September 2025 Abstract We propose a unified, operational bridge between gravity and quantum theory based on an information–energetics principle: no free information. Gravity is treated as a local constraint on accessible information density, while quantum theory supplies the irreducible blur that gravity cannot resolve. Attempts to extract arbitrarily precise predictive structure within a finite region require energy that gravitates; beyond a threshold, horizons form, sealing the information behind black holes. In this view, black holes do not break the light-speed limit—they enforce it: you may “know” more locally at the price of losing the ability to communicate it outward. We formalize the tradeoffs with clean bounds (Bekenstein-type information capacity [ 1 , 2 , 3 , 4 ], Schwarzschild collapse criterion, Landauer erasure cost [ 7 , 8 ], quantum speed limits [ 9 , 10 ]), introduce a two-pole envelope for scale (quantum micro-pole and gravitational macro-pole), and state an operational Bridge Principle: you cannot learn inward more finely than information can flow outward. The middle of the scale-space is effectively Gaussian; the extremes act as poles of ignorance. This article can be read as a gravity/quantum case study within a broader blur–based epistemic framework developed in [19]. 1 Program in one page P1. Gravity ≡ local information budget. For a laboratory of radius R and total energy E inside, the accessible information Iobeys the Bekenstein-type capacity bound S≤2πkB ℏcR E, I :=S kBln 2 ≤2π ln 2 R E ℏc,(1) while compressing Einto Rpushes the Schwarzschild radius rs=2GE c4=2GM c2,(E=Mc2)(2) towards R. Crossing rs=Rforms a horizon: outward channels are severed. P2. Black holes enforce c .At a horizon, the entropy saturates an area law SBH = kBA 4ℓ2 P with ℓ2 P = Gℏ c3 [ 1 , 5 , 6 ]; no information escapes at a finite rate to infinity. Attempts to “read the future” by concentrating resources cannot create communicable superluminal advantage: the price is isolation. P3. Quantum as irreducible blur. To refine predictions you must pay Landauer (energy per erased bit) and obey quantum speed limits (time to create orthogonal states). Gravity only sees coarse stress-energy; the unrevealed fine structure is the quantum blur. In the language of the general blur framework [ 19 ], quantum behaviour is the micro-pole where this irreducible blur is codified by ℏ. 1 2 Axioms and bounds: one law, three faces Fix a spherical region of radius Rcontaining total energy Eat temperature T. Capacity / Bekenstein-type S≤2πkB ℏcRE, (3a) I:=S kBln 2 ≤2π ln 2 RE ℏc,(3b) Collapse / Schwarzschild rs=2GE c4,(3c) E≥c4 2GR=⇒horizon,(3d) Landauer erasure (min. energy to erase ∆Ibits at temperature T) ∆E≥kBTln 2 ·∆I, (3e) Quantum speed limits (Mandelstam–Tamm; Margolus–Levitin) τ≥maxπℏ 2 ∆E,πℏ 2 (⟨H⟩−E0).(3f) Principle 2.1 (No free information).For fixed ( R, T ), the outward communicable information rate ˙ Iout is bounded by a function of ( R, E, T )that saturates at horizon formation. Increasing inward resolution (pushing blur scale σ↓ 0) at fixed ( R, E )eventually decreases ˙ Iout by triggering collapse. 3 Gravity as a local informational field Proposition 3.1 (Information–gravity tradeoff).Let R be fixed, and let P denote the average dissipated power over a time window τ . Any protocol that achieves inward information gain ∆ I within time τinside the lab obeys ∆I≤min2π ln 2 RE ℏc,P τ kBTln 2,with E≤c4 2GR, (4) else a horizon forms. Thus, pushing ∆ I at fixed R forces E upward until communication to infinity is cut. Idea. Combine the capacity bound (1) – (3b) and Landauer (3e) with the collapse threshold (3d) . The minimum of the two information ceilings applies prior to collapse; beyond E = ( c4/ 2 G ) R the achievable outward rate approaches zero. 4 Black holes as speed-limit enforcers Lemma 4.1 (No fast-forward without isolation).If a procedure attempts to outpace causal signaling by increasing local energy density (e.g. to forecast many bits of the “future” of a system), then, as E↗c4 2GR and rs↗R , the asymptotic redshift diverges and the achievable outward information rate to infinity tends to zero. The agent may store more local bits (area law), but cannot transmit them to the exterior at finite rate. Idea. Near rs→R , null rays from the interior suffer unbounded redshift/time dilation for distant observers; operational outward capacity collapses (cf. (3b) , (3d) , and black-hole thermodynamics [1,5,6,14]). 2 5 Quantum as blur: two poles with a Gaussian middle Gravity responds to coarse stress-energy; phases and fine entanglement are largely invisible to classical curvature. We model the accessible scale profile with a two-pole envelope on radius r∈[0, R∗]: wα,β(r) = 1 B(α, β)r R∗α−11−r R∗β−1,0< α, β < 1,(5) where B is the Euler beta function. The mesoscopic regime is captured by a Gaussian kernel Gσ(r)centered at r0: K(r) = wα,β(r)·Gσ(r), Gσ(r) = 1 √2πσ exp−(r−r0)2/(2σ2).(6) Remark 5.1 (Interpretation).Near r0 (our everyday scales), effective field theory is well-behaved (Gaussian core). Approaching either pole, the kernel weight rises while control vanishes: at r↓ 0 quantum indeterminacy dominates; at r↑R∗ gravitational backreaction dominates. Horizons are the operational boundary where inward probing cannot exceed outward flow. In the abstract blur language of [ 19 ], this two-pole picture is one concrete instance of a general meta-phenomenon: for well-posed flows on compact regions, each fixed blur scale admits only finitely many effective histories (finite blur-families), and the poles are where such blur-descriptions necessarily break down. 6 Bridge Principle (operational) Theorem 6.1 (Inward learning ⇒ outward cost).For any protocol Πoperating within radius R and energy E over time τ , with blur scale σ (smaller σ means finer inward resolution), the outward communicable information throughput satisfies ˙ Iout(Π) ≤Φ R, E, T, τ, σ,Φ(R, E, T, τ, σ)−−−→ σ↓00whenever E↗c4 2GR. (7) Equivalently: you cannot learn inward (reduce σ ) beyond the rate at which information can flow outward; attempts to do so convert communicability into isolation. Sketch. Bound the attainable ∆ I by (4) , convert to a rate via τ , and account for channel capacity degradation as redshift grows near rs→R . The limit Φ → 0reflects the vanishing exterior lightlike channels in the horizon limit. 7 Thought experiment: forecasting by energy concentration An agent inside radius R builds a device to “read the future” by (i) increasing measurement bandwidth and (ii) accelerating computation. Each additional bit of reliable prediction costs at least kBTln 2in energy and Ω( ℏ/ ∆ E )in time, while increasing E curves spacetime. If the agent pushes far enough, a horizon forms: they may know more, but nobody outside can ever learn that they know. Thus, black holes are not loopholes to c; they are the mechanism that preserves it. 8 Consequences and tests (qualitative) • Design bound. Any architecture promising unbounded predictive throughput from a bounded region must either (a) radiate waste heat (raising T ) and slow down (QSL), or (b) increase Eand self-isolate (horizon tendency). 3 • Dual edges. Phenomena at ultra-small and ultra-large scales will remain governed by pole-like ignorance (quantum randomness / gravitational opacity). Mid-scale Gaussian models remain robust. • Holographic budgeting. Protocols that appear to surpass blur limits must reveal either hidden reservoirs (larger effective R or T ) or trade outward communicability for inward storage; cf. Bekenstein/holographic bounds and QNEC [2,12,4,13,3]. 8.1 Time as Causal Residue Between Two Uncertainty Principles We sit between two operational “uncertainties”: the small-scale (quantum) limit ∆E∆t≳ℏ 2(heuristic energy–time relation; operative bounds via QSLs (3f)),(8) and the large-scale (gravitational/causal) limit that caps communicable information from a bounded region of radius Rand power P: ˙ I≤min     P kBTln 2 | {z } Landauer-limited erasure rate ,2πR ℏcln 2 P | {z } Bekenstein envelope      , E≤c4 2GR(no-collapse at fixed R; hence over a window τ,P≤E/τ). (9) These relations formalize that causality is not a free lunch: sustaining any nonzero outward rate ˙ I requires finite energy flux P over a nonzero dwell time τ , and shrinking τ↓ 0at fixed R either demands P→∞ (which backreacts up to collapse) or forces ˙ I→ 0. In this sense, time is the residual budget over which dynamics unfolds: the slack variable that reconciles quantum kinematics with gravitational causality. Operational corollary (time cost per bit). Let τbit be the minimal latency to communicate one reliable bit from the region to infinity. Combining (3f) and (9), τbit ≥maxπℏ 2 ∆E,kBTln 2 P,ℏcln 2 2πR 1 P,(10) with P itself bounded by the no-collapse condition at radius R . Thus any attempt to “speed up causality” by reducing τbit must either raise ∆ E (quantum-limited) or P (gravity-limited); pushing too far isolates the system (horizon formation), while starving the budgets sends ˙ I→ 0. The T→0limit (frozen channel, coherent blur). In the idealized cryogenic limit, T→0+, P bounded,∆Ebounded,(11) a thermal erasure channel is unavailable and relaxation times diverge (third-law phenomenology); quantum speed limits impose a finite lower bound on state-change times. Consequently, for any realizable protocol with bounded resources, ˙ I−−−−→ T→0+0,(12) even though the microscopic state retains coherent structure (vacuum/ground-state correlations). The vacuum is thus not empty—it harbors fluctuations and entanglement—yet, absent carriers and budgets, it is operationally informationless to the outside. 4 Archimedean view. Time here plays the role of an “Archimedean background”: locally a small cost per causal step, but integrated globally an enormous resource enabling histories to exist. Decreasing that residue (starving the budgets) freezes dynamics; increasing it (funding ∆ E or P ) restores flow until gravity pushes back. In between lies the effective, Gaussian-like middle where our physics lives. 9 Two apparent realities at a horizon (operational picture) At a black-hole horizon there are two equally valid operational descriptions, tied to where the observer’s outward channel terminates. (i) Infaller’s local frame (smoothness). By the equivalence principle, a freely falling traveler who survives tidal forces experiences no sharp drama at the horizon: local physics remains crisp and low-blur. Proper times and thermometer readings along the worldline behave regularly; any breakdown is deferred to deep interior scales. (ii) Distant observer’s frame (opacity). For an exterior observer, the infaller’s outward channel collapses: signals redshift and time-dilate toward zero rate as null rays skim the horizon. Operationally, the traveler’s state becomes information-theoretically blurred—not because local microstructure vanishes, but because its communicable information to infinity tends to zero within finite exterior time (cf. Principle 2.1). In this sense “disappearance” means loss of extractable information, not necessarily destruction of local structure. Inside: a gradient to an inner blur radius Within the horizon, outward-directed null rays still move inward; causal cones tip so that every timelike worldline ends at the interior pole. Communication among nearby layers remains possible for a while, but operational capacity degrades rapidly. We formalize this with an adjacent-layer capacity. Definition 9.1 (Adjacent-layer capacity).Let Br and Br−δℓ be concentric shells of proper separation δℓ inside a Schwarzschild black hole. Over a window τ, define Cadj(r;δℓ, τ):= sup protocols I(Br→Br−δℓ;τ) τ, the maximal achievable classical information rate from Brto Br−δℓ. Definition 9.2 (Operational inner blur radius).Fix small thresholds ε, ℓc> 0. The blur radius rblur is the smallest radius for which Cadj(r;δℓ, τ)< ε/τ for all 0< δℓ ≤ℓc. Beyond rblur , even immediately adjacent layers are, for practical purposes, causally decoupled at any finite resource level—operationally a “soup of randomness” with no recoverable channel structure. Order-of-magnitude estimate (classical GR + Planck cutoff). For a Schwarzschild mass M, the curvature invariant K(r) = RabcdRabcd =48 G2M2 c4r6 5 reaches the Planck scale when K∼ℓ−4 Pwith ℓ2 P=Gℏ c3. Solving gives a quantum-gravity radius rqg(M)≈(48)1/62−1/3 | {z } ≈1.5rs1/3ℓ2/3 P, rs=2GM c2.(13) Thus for macroscopic M , rqg ≪rs : any fundamental breakdown (the “pole” of total indetermination) is far inside the horizon. In our framework one may identify rblur ∈[ 0, rqg(M) ], with the precise value controlled by microscopic scrambling/mixing that throttles Cadj before Planckian curvature is reached. Equation (13) supplies a conservative inner bound tied to curvature alone. Operational summary. From the infaller’s view the horizon is mundane; from infinity it is an information barrier. Deeper in, the gradient of communicability steepens until, at rblur , even nearest-neighbor layers fail to sustain a usable channel. Whether this effective blur pole sits exactly at the geometric center ( r = 0) or at a finite rblur > 0is a question of microphysics; either way it lies well inside the known radius rs . This picture meshes with the No-Free-Information law: pushing inward resolution beyond what can be communicated outward inevitably converts knowledge into isolation. 10 Refutation Probe and a Runnable Testbed Aim. Provide (i) an internal consistency probe that stress-tests the No-Free-Information (NFI) principle against standard physics, and (ii) a runnable testbed (toy experiment/simulation) that a reader can implement to seek a violation. We close with a crisp falsifiability checklist and our verdict. A. Internal Consistency Probe (desk check) We examine plausible “escape hatches” and how known physics responds. 1. Reversible computing vs. Landauer. Reversible gates avoid dissipation per logical step, but reliable prediction requires error correction at nonzero noise. Stabilization introduces entropy export; at fixed R this either (a) raises E (pushing rs↑R ) or (b) slows the net inference rate by cycle time and redundancy. Together with quantum speed limits, this preserves the NFI envelope [7,8,9,10]. 2. Quantum metrology and squeezed states. Heisenberg scaling raises Fisher information F with photon number/energy; at fixed R , Bekenstein and collapse bounds cap usable E , and detector saturation plus backaction throttle outward capacity Cout [2,3]. 3. Traversable wormholes / warp drives. Known traversable constructions require tuned negative-energy fluxes and are constrained by quantum inequalities [ 18 ]; sustained, broadband, superluminal-capacity channels are not known to be possible. 4. Black hole evaporation and information return. Unitary Page-curve scenarios allow eventual information recovery via Hawking radiation, but practical decoding is conjectured to be computationally intractable (Harlow–Hayden) and outward rates are tiny (greybody/redshift) [ 14 , 16 , 15 , 17 ]. NFI concerns finite-time, finite-resource communication; no contradiction arises. 6 Interim verdict. Within established physics (GR, QFT, thermodynamics, quantum info), no known mechanism simultaneously increases inward precision (shrinks blur σ ) and maintains or increases outward channel capacity Cout at fixed ( R, E, T )without triggering backreaction that reduces Cout. The principle is coherent; outstanding caveats are noted in Remark 10.1. B. Runnable Testbed: Lieb–Robinson Lab Protocol A tabletop analogue with a built-in “speed of light” provides a direct stress test. Model. A 1D spin chain of length N with nearest-neighbor coupling, local Hamiltonian ˆ H = Pihi,i+1 of bounded norm. The Lieb–Robinson velocity vLR bounds information propagation [ 11 ]. Split the chain into: •region A: contiguous block of length R(the “lab”), •region B: a detector block far away at distance D≫R. Knobs (hold geometry fixed). 1. Blur σ :tune continuous-measurement strength γ in A (weak → strong), or tighten POVMs to increase local Fisher information about a hidden parameter in A . Smaller σ corresponds to larger γ. 2. Energy/temperature: bound total energy in A by fixing drive amplitude and duty cycle; record local energy density εA. Measured quantities. 1. Inward precision: estimate classical Fisher information F ( σ )from outcomes in A about the hidden parameter. 2. Outward capacity: prepare codewords in A and estimate an achievable classical capacity to B over time window τ via the Holevo quantity χτ (or mutual information from tomography). Define Cout(σ) := χτ/τ. Prediction (finite-resource speed limit). For fixed ( R, εA, τ )with D > vLRτ and weak leakage to B: ∂Cout ∂σ ≤0,∂Cout ∂σ <0once F(σ)exceeds the shot-noise scale.(14) Operationally: pushing σ↓ 0(stronger local readout/processing in A ) reduces coherent signal reaching B within τ because (i) measurement backaction and control pulses decohere carriers, (ii) finite vLR caps usable modes, and (iii) fixed energy budget is reallocated from communication to metrology. How to run. Superconducting qubits or cold atoms suffice. 1. Calibrate vLR by light-cone tomography. 2. Choose R, D, τ so that D≳vLRτ. 3. Sweep γ to vary σ ; at each setting, (i) estimate F ( σ )in A , (ii) estimate Cout ( σ )to B via repeated codeword transmission and tomography. 4. Plot Cout vs. F (or σ ). NFI predicts a descending tradeoff curve and a knee where stronger metrology penalizes comms. 7 C. Falsifiability Checklist A single empirical/theoretical counterexample refutes NFI: 1. At fixed ( R, E, T )and fixed geometry/couplings, demonstrate a regime where decreasing σ (increasing F )increases or leaves unchanged Cout over fixed τ to distant B , without hidden resource injection or long-range couplings. 2. Exhibit a sustainable superluminal channel or a capacity that grows as energy density approaches a collapse threshold once backreaction is included. Remark 10.1 (Caveats & scope).(i) Bekenstein-type bounds are widely supported but not proven in full generality; we use them as an envelope (see also Casini’s relative-entropy proof and covariant bounds [ 3 , 4 ]). (ii) Hawking radiation supports the blur picture: emission is effectively sourced in an extended near-horizon region, so the quantum blur does not terminate at a sharp surface. Energy—and, in unitary accounts, correlations/information—can leak outward, but the rate is extremely small (greybody filtering, redshift) and decoding is believed hard [ 14 , 15 , 16 , 17 ]. Our capacity notion concerns finite-time, finite-resource communication; thus no contradiction with NFI arises. (iii) Exotic negative-energy effects are constrained by quantum inequalities; sustained violations needed to raise Cout are not known to be possible [18]. Remark 10.2 (Computational corollary: link to P vs. NP ).In a physically local model with finite signal speed, “verification” can be local while “discovery” must respect light-cone propagation. High blur can make P look like NP on noisy instances, but unblurring inevitably incurs energy/time/causal costs. 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