Gravity / Horizons
A horizon hides the postings that cross it.
A black hole of horizon area A carries the entropy A/(4ℓP²). On the ledger, what a horizon hides is a set of postings, one from each of its records. Give each crossing one bit and the coefficient of the area depends on the horizon’s orientation: 0.108, 0.325 or 0.1875 at the identified spacing, none of them the quarter. The ledger’s own count, and a lattice black hole itself, are open.
The counts below: provedSpacing and phase premise: identifiedThe ledger’s count, a lattice black hole: open
- The record p
- Records on the boundary of its past
Live · exact count, Proposition 4.3
6
- Counted
- 74records on the cut
- 2m² + 2
- 74the proposition’s count
- Area term
- 72A/(√3 b²), with A = 2√3 m² b²
- Plus
- 2the Euler characteristic of a sphere
The boundary of the past of a record p is the records below p that agree with p on at least one counter. Its cut m ranks below p is the surface of a tetrahedron with equilateral faces of side √2 mb, for every κ. A count of records on such a closed surface is exactly an area term plus a topological one.
Black-hole paper, Proposition 4.3; it reads the posting order as the physical causal order, an identification
With JavaScript on, this figure counts the records on the boundary of a record’s past live. m ranks below, they number exactly 2m² + 2: an area term 2m² and the Euler characteristic 2.
The entropy
One bit per crossing does not give the quarter.
A horizon of the posting order hides one posting from each of its records. Give each hidden posting one bit, convert with the phase premise, and the entropy is S = CA/ℓP². The coefficient C depends on how the cut lies in the lattice, so a count that gives every record the same content is not a function of area alone.
- Records on the cut
- Crossing postings
- The quarter
Live · exact, Table 2 and Proposition 4.6
Kind of cut
Spacing
1
- Cut
- observer facet
- Per record
- √3 b²of area, 1 crossing per record
- C
- 0.1083below the quarter
What it rests on. C = k n λrec²/(4√D b²) for k bits per crossing, n crossings per record and one record per √D b². The phase premise gives ℓP² = λrec²/(4 ln 2) (an identification); one bit per crossing is a model; the identified spacing takes κ = 1 and one record length per rank, so λrec² = ¾ b². None of the four is the quarter there. No whole number of states per record gives it either: a facet or null cut would need 24/√3, between 4 and 5, an axis cut 28/3, between 6 and 7.
The long-wave factor. With the measured coupling Z times the tree-level one, every coefficient is divided by Z, so a suitable Z brings one kind of cut to the quarter, the inward axis cut at Z = ¾; the ratios stay, so no Z gives it on two. At b = λrec the inward axis cut is exactly ¼, and the facet and null cuts are not.
Black-hole paper, Table 2 and Proposition 4.6
The numbers
What a horizon holds.
The quarter
2πas
The entropy law S = A/(4Gℏ) holds at every area exactly when the density is 2πas per unit area, with G = 1/(8πs) and ℏ = 1/a; under the phase premise that is one bit per face of area λrec².
Black-hole paper, Proposition 4.1
By orientation
0.108
One bit per crossing on an observer facet; 0.325 on a null plane, 0.1875 on an axis cut, at the identified spacing and tree level. None is 1/4.
Black-hole paper, Proposition 4.6
Topology
+2
Records on the cut of a record’s past number A/(√3 b²) + 2: the 2 is the Euler characteristic of the sphere, as 0 is that of a torus.
Black-hole paper, Propositions 4.2 and 4.3
A field-like ledger
1 bit
If the ledger is one state at each record, an observer horizon hides exactly one bit, whatever its area: no area law, and at most a perimeter law on any horizon.
Black-hole paper, Proposition 4.5, the field reading (a model)
How it works
Horizons on the ledger.
1 · The posting order
The posting order holds no black hole.
In the posting order no nonempty future-closed set is spatially bounded. An observer that posts on every counter without bound has the whole lattice in its past; one whose postings on one counter stop at M has the event horizon xa = M. A trapped region needs spacelike postings, weak fields trap nothing, every trapped region leaks along postings, and the path sum’s own configurations hold no black hole. A strong-field lattice black hole is open; with a regular far field it needs an action exactly symmetric under vertex moves (an inference).
Black-hole paper, Section 3 and Propositions 3.2 and 3.8
2 · What it hides
A horizon hides one posting from each record.
The only way out of the past of an observer whose postings on counter a stop at M is a posting of counter a from a record with xa = M, and once it has landed, everything after it stays outside. What the horizon hides is that set of postings, one from each of its records; the same holds for every past set. The side of a posting, debit or credit, is the one datum a reader of records cannot see.
Black-hole paper, Propositions 4.4 and 4.5; the posting order as the observer’s causal order is an identification
3 · The count
A quarter on every horizon needs records that are correlated.
In the history reading every posting across a horizon is a separate event, k crossings carry 2k patterns of sides, and the count grows with area; one bit each then gives the coefficients above, which depend on orientation. A quarter that is the same on every horizon needs a count that correlates the records, with one density on every kind of cut. That count is the paper’s open target.
Black-hole paper, Section 4 and Table 1
4 · Radiation and information
The radiation is preliminary; the information question is open.
On test backgrounds that are not lattice solutions, the lattice scalar’s computed in-out occupation is a Planck spectrum that tends to the Hawking temperature as the spacing shrinks: preliminary numerical evidence, read as emission only by inference. Where the information goes depends on a unitary evolution in record time, which the identified Regge path sum lacks: at κ = 1 it has a growing mode in record time.
Black-hole paper, Sections 5 and 6 and Table 1
5 · A static exterior
A static vacuum metric is not yet a lattice vacuum.
The lattice’s equations act on squared edges, and the fifteen edge classes of a cell carry the ten components of a metric and five combinations more, which can carry a curvature source where Einstein’s tensor vanishes. For a uniform tidal field an exact correction cancels that source at a record, for every κ above ½. At the next order one route fails: a static cubic field at κ = 1 that solves every linearized equation has second-order Ward defect 756√3, so no second-order correction makes it solve the thirty equations at a record. That excludes one route, not every static exterior. A lattice black hole must still solve the exterior equations beyond that order, pass to cells of the causal types it needs, and show a horizon.
Static-exteriors paper, Theorems 2.1 and 3.1 and Section 6
Sources: the black-hole paper is The Recognition Ledger of a Black Hole: Horizon Records, Entropy Counts and Lattice Hawking Radiation, version 1.2; the static-exteriors paper is Static Exteriors and the Black-Hole Problem on the Recognition Lattice.
Words on this page, in plain terms: record · posting · record time · phase premise.
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