Paper release: r1578 Released: July 25, 2026

Keywords: de Sitter horizon, shock wave, static patch, finite screen, generalized entropy, icosahedral graph

The sign problem

Chen, Stanford, Tang, and Yang study a static-patch proposal in which a Euclidean gravitational path integral with an observer worldline computes a Hilbert-space trace . Their shock-wave calculation includes observer recoil and gravitational backreaction. The resulting out-of-time-ordered correlator conflicts with cyclicity and positivity: its leading regularized contribution increases. Geometrically, the negative shock advances the signal.

The finite-screen calculation below isolates a capacity mechanism for this sign. Three levels remain separate throughout:

1. exact entropy and graph identities for a finite sector model;

2. an analytic relaxation of integer sector dimensions for stating gradients and curvatures;

3. the physical dictionary identifying screen capacity transfer with a gravitational horizon shock.

The first two levels are mathematical statements. The third is a conditional attachment. In particular, a finite capacity premise does not evade the trace obstruction in Ref. . Finite-dimensionality is part of the premise tested there.

The smooth de Sitter normalization

Write the horizon shock equation in spacetime dimension \(d\) as \[ \begin{equation} \left(-\nabla^2_{S^{d-2}}-\mu^2\right)X^+ =8\pi G\,T^-, \qquad \mu^2=\frac{d-2}{2}|f'(r_c)|r_c=(d-2)\kappa r_c . \label{eq:shock} \end{equation} \] For pure de Sitter space of radius \(L\), \[ f(r)=1-\frac{r^2}{L^2}, \qquad r_c=L, \qquad \kappa=\frac1L. \]

The smooth \(\ell=1\) shock is generated by a de Sitter isometry and is pure gauge, with the corresponding source component removed by the gravitational constraint . This explains the zero of the operator in Eq. [eq:shock]. Transferring that normalization to a discrete screen requires a separate premise, stated in Section 4.

Finite entropy and the capacity-transfer direction

The exact entropy maximum

Let a finite screen have positive integer sector dimensions \(d_1,\ldots,d_n\), and let \(p_i\) be a probability distribution over the sectors. Set \[ M=\sum_{i=1}^n d_i, \qquad S_{\mathrm{gen}}(p,d) =-\sum_{i=1}^n p_i\log p_i+\sum_{i=1}^n p_i\log d_i. \]

Thus a logarithmic capacity \(N=\log M_0\) agrees exactly with the extremal generalized entropy when the closed screen carries total dimension \(M_0\). This is a maximization over \(p\), with the sector dimensions held fixed. It does not classify extrema with respect to the \(d_i\).

Exact gradient and Hessian

At the maximizing distribution, the expectation of the logarithmic sector operator is \[ \begin{equation} \mathcal A(d) :=\sum_{i=1}^n\frac{d_i}{M}\log d_i . \label{eq:areaobservable} \end{equation} \] The dimensions are integers in the finite model. Extend them to positive real variables solely for the following differential calculation.

Two consequences prevent a false extremum statement. First, \(\nabla\mathcal A=\mathbf 1/(nd)\) at the symmetric point, so the point is not stationary in the full positive orthant. Second, a fixed-horizon-capacity variation obeys \(\mathbf 1^T\delta d=0\), and hence \[ \delta d^T(\operatorname{Hess}\mathcal A)\delta d =\frac{\|\delta d\|^2}{nd^2}>0 \] for every nonzero tangent variation. The symmetric point is locally minimizing along the fixed-\(M\) redistribution directions. The negative eigenvalue belongs to the homogeneous direction, which changes \(M\).

The one-sided budget maximum

The de Sitter sign concerns transfer between the horizon and the observer, rather than redistribution among horizon sectors. Let the closed total capacity be \(M_0\). Assign \(c_{\mathrm{obs}}\geq0\) units in the finite capacity ledger to the observer, leaving \[ M_h=M_0-c_{\mathrm{obs}} \] on the horizon. Suppose the transfer depletes every horizon sector by the same factor \(1-f\), where \(f=c_{\mathrm{obs}}/M_0\). For integer dimensions, restrict to values for which the depleted dimensions remain integers. Then \[ \begin{equation} S_{\mathrm{gen},h}^{\max}(f)-S_{\mathrm{gen},h}^{\max}(0) =\log(1-f), \label{eq:entropydrop} \end{equation} \] and \[ \begin{equation} \mathcal A((1-f)d)-\mathcal A(d) =\log(1-f). \label{eq:areadrop} \end{equation} \] Both relations are exact.

For transfers of \(2\%\), \(5\%\), and \(10\%\), \[ \log(0.98)=-0.020203\ldots,\qquad \log(0.95)=-0.051293\ldots,\qquad \log(0.90)=-0.105361\ldots. \] The exact finite order and its analytic interpolation are the budget statements relevant to the shock sign. Proposition 3 supplies compatible negative homogeneous curvature, while also showing why the transfer result cannot be described as an interior maximum at fixed horizon \(M\).

Conditional gravitational interpretation

In the horizon coordinates used in Ref. , the constant term of the shock operator carries the sign of \(B'(0)/B(0)=2r'(0)/r_c\). For a cosmological horizon, the transverse area decreases from the horizon toward the observer, and the shock has the time-advance sign. Under the two physical identifications \[ \text{closed screen capacity}=\text{fixed horizon--observer budget}, \qquad \mathcal A=\frac{A_{\mathrm{horizon}}}{4G}, \] Theorem 4 supplies the corresponding conditional sign mechanism. Identifying the ledger coordinate \(c_{\mathrm{obs}}\) with the observer mass used in the gravitational shock calculation is an additional physical dictionary. The finite entropy theorem and transfer law establish none of these identifications.

Conditional icosahedral spectrum

Consider the graph Laplacian of the twelve vertices of the icosahedron. Its eigenspaces decompose under \(A_5\) as \[ \mathbb R^{12}\simeq\mathbf1\oplus\mathbf3\oplus\mathbf5\oplus\mathbf3', \] with raw Laplacian eigenvalues \[ 0,\qquad 5-\sqrt5,\qquad 6,\qquad 5+\sqrt5, \] respectively.

These two assumptions are independent. Assumption 5 fixes the scale by placing the \(\mathbf3\) at the smooth value \(2\). Assumption 6 selects the finite operator whose remaining eigenvalues are being compared.

The value \(-2\) carries no enhancement relative to the continuum invariant mode. Any different comparison obtained by leaving the graph scale unfixed is a normalization artifact.

Ports and edge sectors

The logarithmic sector operator naturally suggests an edge carrier. The low-lying spectrum is unchanged by this choice.

For the icosahedron, \(k=5\), \(n=12\), and \(m=30\). The edge-sector spectrum is therefore \[ \{0^1,(5-\sqrt5)^3,6^5,(5+\sqrt5)^3,10^{18}\}. \] It contains the full port spectrum plus a high block at \(10\). In particular, the normalization-free ratio between the \(\mathbf5\) and rotation-triplet Laplacian eigenvalues is \[ \frac{\lambda_{\mathbf5}}{\lambda_{\mathbf3}} =\frac6{5-\sqrt5} =\frac{3(5+\sqrt5)}{10} =2.170820\ldots \] on both carriers. The dodecahedral face graph gives \(\varphi^2=2.618033\ldots\), while the smooth \(\lambda_{\ell=2}/\lambda_{\ell=1}\) ratio is \(3\).

Lemma 8 closes the port-versus-edge ambiguity once the nearest-neighbour Laplacian premise is accepted. It does not establish that premise.

The repair generator has a different domain

Let \(X_r\) be the configuration space at fixed sector structure, with measure \(\pi_r\), and let \(P_v\) be the single-site heat-bath conditional expectation. The repair generator \[ L^{\mathrm{rep}}=\sum_v(I-P_v) \] acts on \[ K_r=L^2(X_r,\pi_r). \] Since each \(P_v\) is an orthogonal projection, \[ \langle f,L^{\mathrm{rep}}f\rangle =\sum_v\|(I-P_v)f\|^2. \] Consequently, \[ \ker L^{\mathrm{rep}}=\bigcap_v\operatorname{Ran}P_v. \] Reducing this intersection to the constants requires a separate irreducibility or intersection proof and is unnecessary for the scope statement here.

A horizon shock changes the sector dimensions \(d_\alpha\) in \[ L_C=\sum_\alpha(\log d_\alpha)P_\alpha . \] Such a change deforms the configuration space and lies outside \(L^2(X_r,\pi_r)\) at fixed sector structure. The repair generator therefore does not act on the sector-dimension shock. Its spectral gap neither excludes nor supplies the time-advance mode. The capacity-transfer law in Theorem 4 is a response input on the enlarged sector-dimension space; a dynamical generator on that space is open.

Claim boundary

The coefficient multiplying the shock response requires a generalized-entropy normalization and a physical horizon-area dictionary on one common domain. The finite calculation fixes neither. It also gives no Hilbert-space trace whose correlators satisfy the conditions tested in Ref. .

Closed tested classes and surviving alternatives

The external trace obstruction concerns the proposal tested in Ref. : a Euclidean static-patch path integral with an observer worldline interpreted as a positive cyclic Hilbert-space trace for the stated correlators. Horizon-screen descriptions without that trace interpretation remain outside the tested class.

The repair-domain statement concerns the declared fixed-sector \(L^{\mathrm{rep}}\) on \(L^2(X_r,\pi_r)\). A separately constructed dimension-changing generator remains possible. The graph-spectrum statement concerns the scaled nearest-neighbour icosahedral port Laplacian and its line graph under exact gauge transport. Weighted, longer-range, refinement-limit, and other source-derived kinetic operators remain possible and require their own spectra.

None of these boundaries disproves Observer-Patch Holography. They delimit one trace interpretation, one fixed-sector generator, and one conditional finite kinetic operator. The exact finite entropy, transfer, and line-graph identities survive independently of those physical attachments.

Conclusion

A closed capacity budget has a distinguished transfer direction. Moving capacity from a horizon screen to its observer changes the extremal entropy and the uniformly normalized logarithmic area observable by \(\log(1-f)\). The sign is negative for every allowed positive finite transfer and throughout the positive-real interpolation, so the zero-observer allocation is a one-sided boundary maximum. The analytic Hessian supports a negative homogeneous curvature and a positive fixed-horizon-capacity tangent curvature. Keeping both facts visible avoids an incorrect interior-maximum argument.

The pure de Sitter normalization \(\mu^2=\lambda_{\ell=1}\) is exact. An icosahedral spectrum follows only after exact discrete gauge transport and a nearest-neighbour kinetic operator are assumed. The regular line-graph identity then makes its low spectrum independent of choosing ports or edge sectors. These results supply a conditional finite-screen mechanism for the time-advance sign, with the ledger-to-mass map, coefficient, gravitational attachment, and sector-dimension dynamics left as separate obligations. The exact identities remain valid if those attachments fail; such a failure would reject the corresponding physical interpretation inside its stated class.