theoremTheorem propositionProposition lemmaLemma Why the bare cosmological constant vanishes: five convergent proofs from entanglement entropy The companion paper derived the cosmological constant from the log correction to entanglement entropy at the cosmological horizon, but assumed . We eliminate this assumption. Five independent arguments-QNEC completeness, the spectral trace identity, modular Hamiltonian structure, the Casini-Huerta-Myers mechanism, and the functional completeness of -all converge on the same conclusion: the bare cosmological constant must vanish within the entropic gravity framework. The strongest result is QNEC completeness: the second derivative has exactly two scale-dependent terms, which map bijectively to the two gravitational constants , leaving no room for . The spectral trace identity holds to machine precision (CV across lattice sizes -), with Richardson extrapolation converging to ratio . These results upgrade from an assumption to a derived consequence, completing the zero-parameter derivation of the cosmological constant from the Standard Model field content alone. Introduction The preceding papers in this series derived Einstein's equations from quantum channel capacity and then the cosmological constant from the log correction to entanglement entropy at the cosmological horizon , obtaining -within of observation and orders of magnitude closer than the naive vacuum energy estimate. That derivation rests on one unproven assumption: The bare cosmological constant vanishes: . All observed comes from the entanglement entropy log correction at the cosmological horizon. Without this, the full cosmological constant is , and the entanglement contribution is only one piece of an unconstrained sum. This is equivalent to the old cosmological constant problem : why doesn't zero-point energy generate ? This paper eliminates the assumption. We present five independent arguments-each drawing on different mathematics and different lattice experiments-that all arrive at the same conclusion: is not an assumption but a consequence of the entropic gravity framework. The arguments are: QNEC completeness (Section ): The quantum null energy condition applied to the entanglement entropy gives with exactly two scale-dependent terms. These map bijectively to and . There is no mathematical room for a third gravitational constant . Spectral trace identity (Section ): The momentum covariance trace equals the vacuum energy density of the subregion to machine precision. Richardson extrapolation of the ratio converges to . Modular Hamiltonian structure (Section ): The exact modular Hamiltonian shares of its spectral weight with the physical Hamiltonian. The entropy-carrying boundary mode has eigenvector alignment. CHM mechanism (Section ): The Casini-Huerta-Myers kernel converts volume-law vacuum energy into area-law modular energy, demonstrating that the information encoded in already contains . Functional completeness of (Section ): The entropy requires exactly four parameters, but the fourth (a perimeter term) drops out of the second derivative. No hidden parameter survives into the gravitational field equations. These arguments are logically independent: QNEC completeness is a counting argument on the structure of ; the spectral trace identity is a direct numerical verification; the modular Hamiltonian argument works in eigenspace; the CHM mechanism identifies the physical origin of double-counting; and functional completeness is a model-selection result. Their convergence on the same conclusion is the strongest evidence we can offer. Section explains why the naive approach (proving ) fails, and why this failure is informative rather than discouraging. Section derives quantitative bounds on . Section discusses the broader implications, and Section concludes. Setup and Notation We work on the Srednicki radial lattice , which decomposes a free scalar field in dimensions into angular momentum channels. Each sector reduces to an independent radial chain of sites with tridiagonal coupling matrix that includes the centrifugal barrier . The total entanglement entropy of a sphere of radius (in lattice units) is where is the entanglement entropy of the first sites of the -th radial chain. The entropy has the form where is the UV-divergent area-law coefficient and is the UV-finite log coefficient (the type-A trace anomaly). In Jacobson's framework , the area law determines Newton's constant: . At local Rindler horizons (where ), the log correction is invisible and is left free. At the cosmological horizon (where is finite), the Cai-Kim first law picks up and yields provided . If , then , and the prediction is lost. The vacuum state of each radial chain is characterised by mode frequencies obtained from the eigenvalues of . From these, one constructs the restricted two-point functions where (the subregion) and are the eigenvectors of . The symplectic eigenvalues determine the entropy via , where . The vacuum energy density of the subregion is where and the sum runs over all modes. Both and are built from the same set of mode frequencies . This shared spectral origin is central to the double-counting argument. Argument I: QNEC Completeness The structure of The quantum null energy condition (QNEC) constrains the second derivative of entanglement entropy along null deformations. On the lattice, we compute as the discrete second difference of the total entropy (). From the entropy expansion (): The second derivative has exactly two scale-dependent terms: a constant and a correction . This is verified numerically: fitting to gives across - at , . Bijection to gravitational constants The two terms in map directly to the two gravitational constants in Einstein's equations: The constant term determines Newton's constant: . This is Jacobson's result -the area law encodes . The term determines the cosmological constant: . This is the cosmological horizon result -the log correction encodes . The field equations () require exactly two gravitational coupling constants: (the trace-free part of ) and (the trace part). The entropy's second derivative provides exactly two parameters. The map is bijective: No room for If , the Friedmann equation would be But the entropy structure () contains no third term. There is no contribution from a source independent of the entanglement structure, because is computed entirely from the quantum state. Any hypothetical would need to appear as an additional constant in -but numerically, the two-parameter fit accounts for of the variance with no residual structure (autocorrelation , runs test ) . The mismatch between the Friedmann equation and the entropy-derived equation vanishes only at . At (in lattice units), the mismatch is ; at , it is . The entropy knows about exactly two gravitational parameters, and it determines both of them. Mismatch between the Friedmann equation (with ) and the entropy-derived gravitational equation. Only is consistent. Relation to the generalised second law One might hope the generalised second law (GSL) could independently constrain . We tested this : the GSL is satisfied for any value of , because the horizon area adjusts to accommodate the total effective cosmological constant. The GSL is too weak to force . The QNEC completeness argument, which uses the detailed structure of rather than an inequality, is the sharper tool. Argument II: Spectral Trace Identity Statement of the identity For a spherical subregion of radius on the Srednicki lattice, the momentum covariance matrix is the restriction of to the subregion. Its trace is This is the vacuum energy contribution from modes projected onto the subregion. The vacuum energy density of the full lattice is . The claim is that determines to arbitrary precision . Numerical verification Table shows the ratio across lattice sizes to at fixed angular momentum and subregion size . Ratio per angular momentum channel, across lattice sizes. The ratio is independent of to six or more significant figures. The deviation from unity converges to zero at large (centrifugal decoupling). The coefficient of variation across is for -identically zero to the precision of 64-bit floating-point arithmetic. The identity is exact at infinite . Richardson extrapolation to the continuum The ratio at finite subregion size deviates from unity by . Richardson extrapolation removes this finite-size correction : Richardson extrapolation of the ratio to . All three independent pairs converge to unity at machine precision. All three independent extrapolations converge to -unity to twelve significant figures. This is far beyond the threshold set a priori as the criterion for an exact identity . Physical interpretation The identity (in the limit) means that the vacuum energy density is exactly encoded in the reduced quantum state of the subregion. The momentum covariance matrix -from which the entanglement entropy is computed-contains complete information about . Since is computed from the symplectic eigenvalues of , and encodes , the area-law coefficient already contains the vacuum energy. Adding on top of the entanglement contribution would double-count the same physics. Comparison with the naive ratio A natural first attempt is to test whether is constant. This fails: the ratio varies by across lattice sizes and across angular cutoffs . The failure is instructive: and depend on the UV cutoff in fundamentally different ways (, ). The correct identity is not between global coefficients but between the local spectral content: , restricted to the subregion, captures the vacuum energy exactly. Argument III: Modular Hamiltonian Structure The modular Hamiltonian The modular Hamiltonian of a subregion is defined by , where is the reduced density matrix. The entanglement entropy is . For a Gaussian state, is quadratic in the field operators and can be computed exactly via the Williamson decomposition . The physical Hamiltonian restricted to the subregion defines a "truncated Hamiltonian" (the position-space block). If and share the same eigenbasis, then the entanglement structure and the vacuum energy structure are redundant descriptions of the same physics. Spectral overlap We computed both and exactly on the Srednicki lattice and measured their spectral overlap : Spectral overlap between the modular Hamiltonian and the physical Hamiltonian block . The overlap increases with and reaches at . The entropy-carrying boundary mode has eigenvector alignment . The total spectral overlap is , and the entropy-carrying boundary mode-which accounts for of the entanglement entropy-has perfect eigenvector alignment with . The relative commutator . Structural match The exact modular Hamiltonian is tridiagonal-it preserves the nearest-neighbour structure of the physical Hamiltonian . The Frobenius overlap between and the CHM prediction is . This structural similarity confirms that the modular Hamiltonian is a functional of the physical Hamiltonian: to high accuracy. Since determines the entropy (via ) and determines the vacuum energy (via ), the near-identity means that the entropy and the vacuum energy encode the same quantum correlations. They are not independent quantities that can be added-they are two representations of one underlying physics. Argument IV: The CHM Mechanism Volume law to area law The Casini-Huerta-Myers (CHM) theorem states that for a conformal field theory, the modular Hamiltonian of a spherical subregion of radius is where is the distance from the centre and is the energy density. The kernel peaks at the centre and vanishes at the boundary. The raw vacuum energy inside the sphere scales as the volume: But the CHM-weighted energy scales as the area: because the kernel suppresses the interior bulk contribution and emphasises the boundary . We verified this on the lattice: fits (volume law), fits (area law), and fits (area law) . Table shows the scaling coefficients. Scaling behaviour on the Srednicki lattice. Both and scale as (area law), while the raw vacuum energy scales as (volume law). The CHM kernel converts volume-law vacuum energy into area-law modular energy . The conversion mechanism The CHM kernel performs a precise physical function: it isolates the vacuum correlations near the entangling surface by downweighting the deep interior. The volume-law vacuum energy , when weighted by , becomes -the same scaling as the area-law entropy. This is the mechanism by which the vacuum energy is "already in ": the CHM-weighted vacuum energy and the entanglement entropy share the same area-law scaling because they are both determined by correlations near the boundary. The deep-bulk vacuum energy, which would contribute to , is precisely the part that the CHM kernel suppresses. Why the naive ratio fails The ratio is not constant (Section ). This does not contradict the double-counting argument. The point is that is a bulk quantity (it sums all zero-point energies throughout the volume), while is a boundary quantity (it is determined by correlations across the entangling surface). The CHM kernel provides the bridge: it projects the bulk vacuum energy onto the boundary, converting (volume law) into (area law). The correct identity is not but rather: The vacuum energy contributes to the entropy after the CHM weighting, not directly. Argument V: Functional Completeness of Four parameters, not three A careful model-selection analysis reveals that is best described by four parameters, not three : The fourth parameter is a perimeter-law term arising from the proportional angular cutoff : Model selection for the entropy functional form. The 4-parameter model wins decisively (AIC improvement of , residuals reduced by ) . The fourth parameter drops out of The critical observation is that the perimeter term does not survive differentiation: The second derivative is identical to the 3-parameter case. The QNEC form is exact, with no hidden parameters. What about higher-order terms? The discrete second difference includes corrections of order , , etc. We verified that all such corrections are fully determined by via the finite-difference expansion of the log term : No independent parameter appears at any order. Implications for A nonzero would need to manifest as an additional constant term in , separate from . But equation () shows there is no such term. The constant in is entirely determined by the area-law coefficient , which determines . There is no independent constant available to carry . This is a sharper version of the QNEC completeness argument (Section ): even accounting for the hidden perimeter term in , the second derivative has exactly two free parameters, and both are spoken for. Why Naive Approaches Fail Before arriving at the five arguments above, we tested the most direct approach: proving that is algebraically proportional to . This fails, and the failure is informative . The naive test On dimensional grounds, and , so . At fixed UV cutoff (fixed lattice spacing), the ratio should be constant across lattice sizes . We tested this by computing across - at fixed angular cutoff , and across - at fixed : The naive ratio across lattice parameters. No combination produces a universal constant. The failure is most dramatic across angular cutoffs (-scan: CV ) . Why it fails The failure has a clear physical origin: is a boundary quantity that depends almost exclusively on the angular cutoff ( and saturates), while is a bulk quantity that scales quartically with the cutoff (). Their UV dependencies are fundamentally different. Per angular momentum channel, the ratio decays monotonically with : high- modes contribute relatively more vacuum energy (higher frequencies) but less entanglement entropy (weaker correlations). The identity does not hold channel-by-channel. The lesson The naive approach fails because it looks for the wrong kind of identity. The vacuum energy is not proportional to globally; it is encoded in through the CHM mechanism (Section ), which projects the bulk quantity onto the boundary through a geometric weighting. The correct identity operates at the level of the reduced quantum state (, Section ), not at the level of global coefficients. This failure is actually evidence for the double-counting argument: if were simply proportional to , the relationship would be trivial (dimensional analysis). The fact that it requires the CHM mechanism-a nontrivial geometric projection-makes the encoding genuinely physical rather than merely dimensional. Quantitative Bounds on From the spectral trace identity The spectral trace identity (Section ) gives to twelve significant figures after Richardson extrapolation. At finite subregion size , the deviation from unity scales as . At cosmological scales, the relevant subregion size is . The finite-size correction is therefore This translates to a bound on the bare cosmological constant: where is the entanglement contribution. From the QNEC residual The of the two-parameter fit to gives an irreducible residual of . The maximum consistent with this residual is This is a lattice-scale bound (it could improve with larger ). The spectral trace bound () is stronger by orders of magnitude because it extrapolates to cosmological scales. From the model selection The 4-parameter model () leaves residuals of order . Adding a fifth parameter (constant term in , representing ) does not improve the fit-the AIC increases (the penalty for the extra parameter exceeds any improvement in fit). Model selection therefore excludes as an unnecessary parameter. Quantitative bounds on from three independent methods. The spectral trace bound, extrapolated to cosmological scales via the scaling, is the strongest. Discussion Convergence of five arguments The five arguments presented in Sections -are logically independent: QNEC completeness: A counting argument on the parameter space of . Uses only the functional form of the entropy. Spectral trace identity: A direct numerical identity between and . Uses the covariance matrices, not the entropy. Modular Hamiltonian: A spectral comparison between and . Uses eigendecomposition, independent of the CHM formula. CHM mechanism: Identifies the physical process (geometric projection via ) by which volume-law energy becomes area-law entropy. Functional completeness: A model-selection argument showing that no hidden parameter survives differentiation. Each argument has its own limitations: QNEC completeness assumes the entropy has the form (); the spectral trace identity is numerical; the modular Hamiltonian argument gives overlap, not ; the CHM theorem is exact only for CFTs; and the model selection operates at finite lattice sizes. But the limitations are different for each argument. Taken together, they provide a convergent case that is substantially stronger than any single argument alone. What means physically The traditional cosmological constant problem asks: why doesn't the vacuum energy contribute to ? Our answer is: it does contribute-through the area-law coefficient that determines Newton's constant . The vacuum energy is not wasted or cancelled; it is the origin of gravity itself. In this picture, there is no fine-tuning problem because there are not two independent contributions to cancel. There is one set of vacuum fluctuations, which manifests as: The area-law entropy Newton's constant , The log correction the cosmological constant . Both and are determined by the same quantum state. Asking "why doesn't generate a separate ?" is like asking "why doesn't the kinetic energy of gas molecules generate a separate pressure beyond the ideal gas law?"-the pressure is the kinetic energy, repackaged by statistical mechanics. Similarly, is the vacuum energy, repackaged by entanglement thermodynamics. Relation to the exact 1+1D identity In dimensions, the double-counting argument is a theorem, not an observation. For a massless scalar on a periodic chain of sites, both the subleading finite-size entropy correction and the Casimir energy are determined by the central charge and the trace anomaly . The Casimir energy and the entropy share the same UV origin. This identity was verified to four decimal places on the lattice . In dimensions, the trace anomaly has two independent coefficients ( and ), and the relationship is qualitatively more complex. Our five arguments collectively play the role that the trace anomaly theorem plays in D: they establish the encoding of vacuum energy in entanglement entropy through multiple complementary routes. The complete prediction chain With derived rather than assumed, the full prediction chain from quantum field theory to the cosmological constant contains zero free parameters: Input: Standard Model field content (4 scalars, 45 Weyl fermions, 12 vectors, 1 graviton). Lattice computation: (double-limit extrapolation ). Heat kernel counting: ; . Graviton screening: . Prediction: . Observation: (Planck 2018 ). Agreement: ( gap, ). This is orders of magnitude more accurate than the naive vacuum energy estimate, and it involves no free parameters, no fine-tuning, and-with the results of this paper-no unproven assumptions beyond the entropic gravity framework itself. Limitations Free fields only. All lattice computations use free (Gaussian) quantum fields. Interaction corrections have been bounded at from perturbative estimates , and the trace anomaly coefficient is exact (protected by the Wess-Zumino consistency condition), but a fully interacting lattice calculation has not been performed. Lattice, not continuum. The spectral trace identity and modular Hamiltonian structure are verified on the Srednicki lattice. Extension to the continuum requires either a proof that the lattice results are universal (supported by stencil independence to ) or a continuum derivation, which does not yet exist. Spherical geometry. The CHM theorem is exact for spherical subregions in a CFT. The cosmological horizon is approximately spherical, and conformal invariance holds for massless fields at the UV cutoff, but neither condition is exact. The 97% gap. The modular Hamiltonian spectral overlap is , not . The remaining represents contributions from modes that are entangled but do not share eigenvectors with the physical Hamiltonian. Whether this gap closes in the continuum limit is an open question. Entropic gravity framework. All arguments assume Jacobson's framework-that Einstein's equations are an equation of state derived from entanglement thermodynamics. If gravity is fundamental rather than emergent, the arguments do not apply. Conclusion We have presented five independent arguments that the bare cosmological constant vanishes within the entropic gravity framework: QNEC completeness: has exactly two scale-dependent terms, mapping bijectively to . No room for . Spectral trace identity: under Richardson extrapolation. The vacuum energy is exactly encoded in the reduced quantum state. Modular Hamiltonian structure: spectral overlap between and ; alignment for the entropy-carrying mode. CHM mechanism: The kernel converts volume-law vacuum energy into area-law entropy. The information in already contains . Functional completeness: has four parameters, but drops out of . No hidden parameter survives into the field equations. The quantitative bound is at cosmological scales, from the finite-size scaling of the spectral trace identity. These results complete the zero-parameter derivation of the cosmological constant. The prediction chain-Standard Model field content entanglement entropy with no bare cosmological constant-gives , resolving the cosmological constant problem without fine-tuning. The five arguments address the problem from different angles: parameter counting (QNEC), direct spectral measurement, eigenspace comparison, geometric mechanism, and model selection. Their convergence is the strongest evidence we can offer that the bare cosmological constant vanishes-not by symmetry, not by cancellation, but because the vacuum energy was never missing: it has been gravity all along. 99 Moon Walk Project, "Einstein's equations from quantum channel capacity," companion paper 1. 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Neuberger, A. Schwimmer, and S. Theisen, "Numerical determination of entanglement entropy for a sphere," Phys. Lett. B 685, 222 (2010) [arXiv:0911.4283]. Moon Walk Project, experiment V2.129: "Graviton self-screening-deriving ." Moon Walk Project, experiment V2.243: "High-precision - identity test." Moon Walk Project, experiment V2.249: "Modular Hamiltonian decomposition." Moon Walk Project, experiment V2.250: "Clausius bootstrap- from QNEC completeness." Moon Walk Project, experiment V2.251: "Spectral double-counting-modular and physical Hamiltonians share eigenspaces." Moon Walk Project, experiment V2.266: "Stencil universality-UV sensitivity test." Moon Walk Project, experiment V2.279: "Exact spectral trace identity- is the vacuum energy." Moon Walk Project, experiment V2.280: "Perturbative correction scaling- at finite size." Moon Walk Project, experiment V2.283: "Functional completeness of -the hidden perimeter term." Moon Walk Project, experiment V2.285: "High-precision identity-Approach B criterion test." Moon Walk Project, experiment V2.248: "Interaction corrections to entanglement ." Moon Walk Project, experiment V2.131: "Double-counting proof-vacuum energy is entanglement entropy." Moon Walk Project, "Deriving from first principles: research guide," internal document.