Exclusion of Uniform Strict Potentials and Stratified Monotonicity in Discrete Black-Hole Interiors: Refuting the Same-Shell Bridge, Nested Observation Kernels, and the Release-Order Fork
离散黑洞内区的势定界与分层单调：同壳桥之否决、观测核之嵌套，与释放序之分岔
The predecessor paper left two openings exposed. The first is the main entrance: on what configuration space does there exist a bounded-below potential that descends strictly, by a uniform amount, along every transition. The second is an open bridge: does spherically symmetric zero circulation in the continuum entail acyclicity of the discrete same-shell directed subgraph. This paper closes both in the negative, and delivers the replacement and its structure.
For the first, a counter-model along the refinement sequence is constructed: a chequerboard family of disjoint oriented plaquettes, one unit out and one unit in at every covered cell, occupancy-identical, with Θ(ε⁻²) full-strength directed cycles at every level, whose first-order coarse-grained tangential flow pairs with every smooth test field at O(ε) and converges weakly to the zero field. Continuum first-order mean spherical symmetry therefore does not entail same-shell acyclicity (Kill 10). The counter-model's second moment converges weakly to the strictly SO(3)-invariant tangentially isotropic form ½P_tang, so lifting the symmetry reading to second order does not rescue the bridge either; and since the second moment does not vanish, no claim of total geometric blindness is made: what is killed is the bridge of symmetry, not the bridge of energy.
For the second, it is proved that a bounded-below uniformly strictly descending potential does not exist on any state space that covers the cyclic era and is forward-defined on it: a finite extension produces a cycle by pigeonhole and uniform descent on a cycle is impossible, while an acyclic extension must contain an infinite injective orbit along which uniform descent exhausts a bounded-below potential in finitely many steps. The machinery of this theorem belongs entirely to the predecessor's Theorem 4.3; what is new is the premise, the forward-definedness of the cyclic era. The predecessor's boxed question asks precisely for such a potential; on any state space that keeps the interior perpetually active, the answer is the empty set. That finitely many labels do not rescue it is strengthened from the predecessor's "radial potentials are label-blind" to "no such potential of any kind can be erected." What is excluded is exactly the conjunction of bounded below, uniform positive gap, and per-transition strict descent; non-strict stratified potentials are not excluded, and Φ_r itself is an instance: strictly descending on the transient region, flat on the recurrent class.
The replacement is stratified monotonicity: cumulative monotonicity of vacancies, uniqueness of the smooth sector as the closed recurrent class of each fixed-occupancy sector, and the three-clause closure of the saturated core, which together yield that every tracer label's radial coordinate is non-increasing, strictly decreasing with an audited step count before freezing, and radially conserved after. Occupancy freezing and label radial freezing are thereby demoted from rule inputs to conditional theorems, and the configuration multiplicity resides entirely in the skin.
Blind zones exist only relative to an observation map. The zero fibres of net edge flow, of occupancy, and of radial cut flux are universally nested, and strictly nested within this paper's full exchange transition repertoire; the strictness witnesses are precisely this paper's load-bearing objects: ring shifts lie in the occupancy kernel but not the net-flow kernel, and skin same-shell open chains lie in the radial kernel but not the occupancy kernel. Hence a threefold degeneracy of first-order silence: a dead core, a core of full-strength mutually cancelling circulation, and a core of radial mixing look alike at the occupancy layer and at the first-order coarse-grained flow layer, and silence observed there does not adjudicate the interior (Kill 12, jurisdiction stated). Discrimination requires ascending in moment or in layer: the second moment sees the anisotropy of radial pulsation, and tracer labels see the release order. Under the erosion protocol the cross-shell inversion fraction of the shell-cycling rung is identically zero, and this is a theorem; the mixing rung has an expectation plateau, conditional on exchangeability, given by a combinatorial closed form, and the multi-seed measurements agree with it within statistical error. The occupancy multiplicity is area-level on scaling families with C_skin ≍ A and filling fraction converging in (0,1), with lattice-dependent coefficients; the mechanism that pins the filling fraction is listed as an outstanding debt.
The labels of this paper are external tracers and are not counted among physical states; the paper claims nothing about what the interior actually is and does not adjudicate the choice of inheritance interface. What it delivers is two structural refutations and one observational-indiscernibility refutation, a replacement bridge, a nesting of observation kernels, and one model-internal discriminant.
Keywords: discrete black-hole interior; Lyapunov potential; same-shell circulation; observation kernel; tracer labels; area-level multiplicity; release order
Conventions and notation
The refutation convention follows the SAE ZFCρ series [4], §10, and the predecessor [1], §7.3; numbering continues from Kill 10. Three-domain separation, the labelled transition system 𝒯 = (𝒳, Θ, s, t, M), the configuration graph, and the occurrence unfolding all carry the predecessor's notation from its §3 and are not restated. H_occ, H_r and H_Lyap follow the predecessor's Definitions 5.1 through 5.3, and H_tang its Definition A.4. The net edge flow J_θ of a transition is the field of the flow decomposition in the predecessor's Proposition 6.9, and is used in §6.
Citations to prior work are given to section and proposition, not to the DOI alone. The commitment level of every result is collected in the master status table of §11; the body marks T1 (proved or computed), T2 (assembled hypothesis), and T3 (programmatic correspondence) in place.
Except where declared otherwise, lattice statements take n_max = 1 on the onion graph: shells j = 0, …, R, shell j a ring of C_j cells with same-shell adjacency along the ring available in both directions, and every cell possessing at least one inward parent on shell j − 1. Timing conventions are declared explicitly at every point of use, in the discipline of the predecessor's §3.4.
1 Introduction
1.1 The problem
The predecessor, by three-domain separation and the orbit dichotomy, showed that trappedness claims for a discrete interior must declare their bridges clause by clause, and closed with a boxed question (predecessor §7.5):
$$\boxed{\ \text{On what configuration space does there exist a bounded-below potential that descends strictly and uniformly along every transition?}\ }$$
It simultaneously left an open bridge (predecessor §7.1): the continuum circulation no-go of its Appendix A is a statement in the continuum limit, while H_tang is a statement about a finite graph, and between the two neither a bridge nor a counter-model was supplied. The predecessor's Corollary A.3 was explicit that all that could be asserted was the absence of a delivered bridge F5_continuum ⇒ H_tang, not that the implication had been refuted.
Two questions were therefore left hanging: whether that bridge is unbuilt or unbuildable, and whether that potential is unfound or non-existent. The difference is the difference between a work schedule and a structure. This paper proves that both are structure.
1.2 What this paper does
First, it delivers the counter-model to F5_continuum ⇒ H_tang (§2, Kill 10): full-strength discrete same-shell circulation can coexist with first-order coarse-grained mean spherical symmetry; what is killed is the first-order bridge, and the survival of the second moment is recorded alongside.
Second, it establishes stratified monotonicity (§3): the lemma of cumulative vacancy monotonicity, the proposition that the smooth sector is the unique closed recurrent class of each fixed-occupancy sector, and the three clauses of saturated-core closure. Occupancy freezing and label radial freezing thereby become conditional theorems, and configuration multiplicity resides entirely in the skin.
Third, it establishes the structure of the label layer (§4): the tracer status of labels, the permutation groups induced by legal transitions under the three radial conditions, and the label speed limit.
Fourth, it proves the exclusion of potentials (§5): a bounded-below uniformly strictly descending potential does not exist on any state space covering the cyclic era and forward-defined there. The predecessor's boxed question thereby receives its final answer.
Fifth, it establishes observation kernels and their nesting (§6): the inclusions hold universally and are strict within this paper's full exchange transition repertoire, together with the threefold degeneracy of first-order silence.
Sixth, it delivers the release-order discriminant (§7) and the scaling of area-level multiplicity (§8), and gathers the reading of all the pieces into a three-era picture (§9, conditional on flow inheritance, T2).
1.3 Nature and limits of this paper
This paper, like its predecessor, is an audit, with one additional layer of delivered replacement. It claims no material core, no interior actuality, touches no singularity, and gives no observational projection. The choice of inheritance interface (stepwise or flow, §6) is physical input; this paper writes the ledger on both sides and does not choose. The labels of this paper are external tracers not counted among physical states (§4); the consequences of physicalising them are listed in §10. The qualifications of lattice results (onion graph, n_max = 1, the declared timings and schedulers) are marked in place; unverified generalisations are listed in §10 and not claimed.
1.4 Organisation
§2 refutes the same-shell bridge. §3 erects stratified monotonicity. §4 erects the label layer. §5 proves the exclusion of potentials. §6 erects the nesting of observation kernels. §7 the release-order discriminant. §8 area-level multiplicity. §9 the three-era picture (T2). §10 discussion and the refutation table. §11 the master status table. Appendix A the details of the counter-model, Appendix B the lattice certifications and script hashes.
1.5 Dependency matrix
| Result of this paper | Predecessor items it rests on |
|---|---|
| Theorem 2.1 (Kill 10) | predecessor Definition A.4, Proposition A.2, Corollary A.3; the exchange semantics of Lemma 3.9 |
| Lemma 3.1 through Corollary 3.4 | predecessor Corollary 6.11; the flow decomposition of Proposition 6.9; Proposition 2.4 (the left end of inheritance) |
| Proposition 4.1 | predecessor Lemma 3.9; Corollary 6.11; Remark 6.12 |
| Theorem 5.2 | predecessor Theorem 4.3 Parts B and D; Definition 5.3 |
| Propositions 6.1 and 6.2 | predecessor Proposition 6.9's flow decomposition; the predecessor's §3 three-domain discipline |
| Corollary 6.4 | this paper's Theorem 2.1 with Lemmas 3.3 and 6.3 |
| §7 | this paper's Corollary 3.4 and Proposition 4.1 |
| §8 | this paper's Proposition 3.2; the input–output ledger of Paper 1 [2] §2.E and its earned bounds |
| §9 (T2) | predecessor Theorem 4.3 Parts A and C; predecessor Theorem 6.1; predecessor §5.5 exit matrix |
2 Refuting the Same-Shell Bridge (Kill 10)
2.1 The implication at stake and three readings
The predecessor's Appendix A proves zero circulation in the continuum limit (Theorem A.1), and its Proposition A.2 proves that H_tang is equivalent to acyclicity of the same-shell directed subgraph. What is to be refuted is the implication F5_continuum ⇒ H_tang: that spherically symmetric zero circulation in the continuum limit forces same-shell acyclicity at every finite level.
Refutation first requires a criterion for F5_continuum along the refinement sequence, and there are three candidate readings. The weak reading: the discrete direction field, paired against every smooth test field, has the zero field as its limit. The strong reading: the field converges to zero strongly in L². The pointwise reading: pointwise convergence. This paper takes the weak reading, on the following grounds. The continuum macroscopic field just is the coarse-grained field, and coarse-graining just is pairing against smooth test fields; the predecessor's Theorem A.1 constrains smooth macroscopic fields, the discrete field is not smooth, and the theorem therefore acts on its weak limit and not on its pointwise values. The pointwise reading writes "no sub-grid microstructure" into the premise, and the conclusion goes vacuous. The strong reading is an additional energy-vanishing hypothesis: strong L² convergence demands that the kinetic energy of sub-grid motion vanish in the limit, and no substrate that keeps full-strength microscopic motion satisfies it (the second-moment survival of Remark 2.2). This paper's refutation does not act on that strengthened hypothesis; the jurisdiction of Kill 10 is the weak reading and its symmetry strengthening (Corollary 2.3). Hence the weak reading.
2.2 The counter-model and the theorem
Construction (disjoint oriented plaquette family). Take finitely many coordinate regions with pairwise disjoint interiors and piecewise smooth boundaries, forming a partition of the shell surface; in each region take a square subdivision in geodesic normal coordinates with mesh ε, vectors expressed in the local frame. Tile the interior of each region with disjoint 2 × 2 cell blocks up to a seam band of width O(ε) along the region boundary, the seam bands left untiled; orient the four edges of every block as one directed cycle, all with the same orientation. The permitted same-shell moves are the edges of these cycles; the covered fraction is 1 − O(ε).
Theorem 2.1 (Kill 10). The refinement sequence above satisfies: (i) the same-shell directed subgraph at every level contains Θ(ε⁻²) directed four-cycles, each of graph circulation (units transferred around the cycle) identically four, not decaying with ε; (ii) for every compactly supported smooth tangential test field φ, the pairing of the first-order coarse-grained flow is O(ε) and converges weakly to the zero field, which is exactly spherically symmetric with zero circulation, so that the weak reading of F5_continuum holds in the limit in its strongest form. Hence F5_continuum ⇏ H_tang.
Proof. (i) is by construction. (ii) The four unit tangent vectors of one closed cycle sum to zero in the frame at the block centre; parallel-transporting each vector to the block centre costs a curvature correction of O(ε²) per vector. A single block's pairing therefore satisfies
$$\Bigl|\epsilon^2\sum_{c\in\text{block}}u(c)\cdot\varphi(c)\Bigr|=\Bigl|\epsilon^2\sum_{c}u(c)\cdot\bigl(\varphi(c)-\varphi(x_{\text{block}})\bigr)\Bigr|+O(\epsilon^4)\le 4\epsilon^3\,\|\nabla\varphi\|_\infty+O(\epsilon^4)$$
The number of blocks does not exceed Cε⁻², so the total pairing is O(ε). There are O(1) seam bands, each containing O(ε⁻¹) untiled cells contributing zero to the pairing; the principal sum is the block sum, bounded as above. ∎
$$\boxed{\ \text{Full-strength micro-circulation can coexist with first-order coarse-grained mean spherical symmetry; symmetry does not forbid same-shell cycles.}\ }$$
Remark 2.2 (distinguishing four objects). Four things must be separated. First, the first-order coarse-grained flow: weakly convergent to zero, as proved. Second, vorticity in the distributional sense: zero as a consequence of the first. Third, per-cycle graph circulation: identically at full strength four; this is a micro-turnover quantity invisible to the first two. Fourth, the second moment u ⊗ u: its weak limit does not vanish. On the disjoint plaquette family, |u| = 1 at every covered cell, the per-component second moment is ½, and the mean of the outer products is exactly ½P_tang (P_tang = I − r̂ ⊗ r̂, the intrinsic tangential projector; Corollary 2.3). What is killed is therefore the first-order bridge; total geometric blindness is not claimed here, and this boundary is folded into the jurisdiction of Kill 12 (§6.4). What second order can see, however, is the presence of motion and its radial-versus-tangential anisotropy, not the order of orientation: the three legal plaquette families of distinct orientation (uniform, chequerboard, per-block random) have identical one-point moments term by term, and their difference lives only in lattice-scale two-point correlations (the comparison of §2.3), which coarse-graining erases by definition. To infer non-vanishing first-order coarse vorticity from the non-decay of per-cycle circulation is to confuse the third object with the second, an instance of the strength over-reading warned against in the predecessor's §7.3; the normalisation trap is in Appendix A.
Corollary 2.3 (second-order upgrade). For the whole-shell tiled family above, the second-moment field Q_ε = u_ε ⊗ u_ε satisfies, for every smooth second-order test tensor A, ⟨Q_ε, A⟩ → ½∫P_tang : A. Indeed the mean of the outer products in each block, in the block-centre frame, is exactly ½(e₁ ⊗ e₁ + e₂ ⊗ e₂) = ½P_tang(x_block), and P_tang is the intrinsic tangential projector, frame-independent; the per-block pairing error against A is O(ε³) over O(ε⁻²) blocks; on the seam bands Q_ε vanishes while the target field does not, with deficit |∫_seam ½P_tang : A| ≤ Cε‖A‖_∞. Altogether |⟨Q_ε − ½P_tang, A⟩| ≤ Cε(‖A‖_∞ + ‖∇A‖_∞). Hence Q_ε ⇀ ½P_tang, strictly SO(3)-invariant (purely tangentially isotropic, radial component zero). Lifting F5_continuum from the first moment to the conjunction of first and second (first-order weak zero and second-order invariant form) therefore still does not entail H_tang: this family satisfies both while cycles fill every level. Whoever would rescue the bridge must appeal to energy vanishing (the strong reading) or to the inheritance of lattice-scale two-point correlations (§2.3), both additional hypotheses beyond symmetry, with their own ledgers to carry. ∎
Remark 2.4 (realisability and the circulation gauge). The theorem's principal object is the orientation field on permitted edges, and H_tang's object is a finite directed graph, so the graph circulation of (i) is the proper gauge; if edges are weighted by continuum line integrals, a single cycle's physical circulation is O(ε) while the cycle density is Θ(ε⁻²), and the per-cell flow speed remains at full strength, one cell per tick, not decaying with ε. A strengthening: this orientation field is realisable on a saturated shell as a legal family of synchronous-exchange ring shifts (legality in the composite criterion of §3's synchronous-exchange semantics), one out and one in at every cell every tick, occupancy-identical, with exact integer conservation; if orientations alternate as a chequerboard, the pairing decays faster still: adjacent blocks carry opposite signs, and pairing a block pair cancels the zeroth and first Taylor terms simultaneously, so each block pair contributes O(ε⁴) rather than O(ε³), lowering the whole by one further order. The fully packed and real-amplitude variants are in Appendix A.
2.3 Numerical certification
The integer unit flow of the fully packed diamond variant is asserted divergence-free cell by cell at every level, and the cycle count is asserted level by level to match the closed form 2m² + 2m + 1 with m = ⌊0.3n⌋, the exact form of Θ(ε⁻²) (script killten.py, Appendix B):
| n | directed 4-cycles | per-cycle graph circulation | \ | ⟨u_n, φ⟩\ | segment order | |
|---|---|---|---|---|---|---|
| 32 | 181 | 4 | 7.682e-3 | |||
| 64 | 761 | 4 | 3.841e-3 | −1.000 | ||
| 128 | 2 965 | 4 | 1.847e-3 | −1.056 | ||
| 256 | 11 705 | 4 | 9.051e-4 | −1.029 | ||
| 512 | 47 125 | 4 | 4.526e-4 | −1.000 | ||
| 1024 | 189 113 | 4 | 2.263e-4 | −1.000 |
The order of the flow pairing matches the O(ε) bound of Theorem 2.1. The moments of the disjoint plaquette family are asserted as identities: block means ⟨u⟩ = (0,0) and ⟨u ⊗ u⟩ = diag(½, ½) (n = 128, 2812 cells, |u| = 1 at every cell), identical term by term across the three families of uniform, chequerboard, and per-block random orientation. The normalised two-point correlation C(2e) separates exactly what the one-point moments cannot: uniform +1.000 and chequerboard −1.000 are deterministic values; the per-block random one is a random variable, with 40-seed mean −0.007, standard deviation 0.040, range [−0.087, 0.084], consistent with its fluctuation scale O(blocks^{−1/2}) = 0.038. The blindness claimed is in the limit gauge: the three families share the same weak limits of first and second moments, while the finite-ε first-order pairing decay rate differs by family (uniform O(ε) is the bound of Theorem 2.1; chequerboard is faster by the adjacent-block sign cancellation, Remark 2.4), the same discipline as Corollary 6.4, and observers of finite resolution are outside the claim. Legality versus illegality (closure) needs no two-point data; the occupancy layer sees it directly. What the scripts certify are the planar local-block identities and two-point correlations; the whole-shell SO(3) weak limit is carried by the analytic proof (Corollary 2.3). Every object of Remark 2.2 is thus touchable here. The real-amplitude diamond variant has segment slopes flow −2.00 and vorticity −1.00, and the smooth-envelope variant decays super-algebraically; see Appendix A.
2.4 Consequences
The second half of the predecessor's §5.2 gap sought to exclude same-shell transfer by H_tang, and H_tang has no source in first-order symmetry: the row F5_continuum ⇒ H_tang in the predecessor's status table changes from open to refuted (in the gauge of the weak reading and its second-order symmetric strengthening, Corollary 2.3). Whoever wants same-shell acyclicity must posit it separately and carry its ledger; §5 will show that even obtaining it does not rescue the uniform potential.
3 Stratified Monotonicity
This section erects three items at the occupancy layer, which together form the replacement bridge. Legality splits by timing semantics, in the discipline of the predecessor's §3.4. Under sequential-hop semantics, an atomic transfer x ⇝ y is legal iff, at the start of that step, n_x > 0 and n_y < n_max. Under synchronous-exchange semantics, atomic transfers are not judged one by one: a complete transition θ is legal iff each source's total outflow does not exceed its initial occupancy, the result t(θ) of simultaneous application satisfies 0 ≤ n ≤ n_max everywhere, and θ decomposes into a union of unit paths and closed chains (a composite of partial matchings and ring shifts), so that label trajectories are defined and neither split nor merge. Thus a single hop requires a pre-existing vacancy, a saturated closed chain shifts synchronously and legally, and an open chain requires capacity only at its terminal end. The per-edge radial condition requires each permitted edge to satisfy r(y) ≤ r(x) (non-increasing) or r(y) < r(x) (strictly inward), applied to every atomic transfer under the semantics in use; the status of this condition is §6's subject. Transfers conserve the total occupancy N, so the transition graph is layered by N, and all statements below live within a fixed-N sector.
Definition 3.0 (the three rungs). Rung one (strictly inward): every atomic transfer satisfies r(y) < r(x). Rung two (per-edge non-increasing): every atomic transfer satisfies r(y) ≤ r(x). Rung three (net constraint only): only the complete transition is required to satisfy ΔΦ_r(θ) ≤ 0, with no per-edge condition on atomic transfers. All three rungs conserve total occupancy. Idle transitions (empty transfer set, M(θ) = ∅) are not counted in the transition set; occupancy-identical transitions with non-empty transfer (ring shifts, swaps) are in it, and the three names — idle transition, occupancy-identical transition, self-loop of an extended state — are kept distinct. The exclusion of idles applies to the pure occupancy-transfer systems of this section through §4; an extended state space X′ may possess internal transitions with M = ∅ and a changed extended state (phase advance, record change), which Theorem 5.2 does not exclude and whose exclusion its proof does not need (§5.3). Exchange semantics and the schedulable edge set are declared at each point of use.
3.1 Cumulative monotonicity of vacancies
Lemma 3.1. Under per-edge radial non-increase, for every ρ the cumulative occupancy N(ρ) = Σ_{r(x)≤ρ} n_x is non-decreasing along orbits; a strictly inward step strictly increases N(ρ) for ρ ∈ [r(y), r(x)). Equivalently, the total number of vacancies inside any radius is non-increasing: vacancies flow only outward.
Proof. One transfer decreases n_x by one and increases n_y by one; N(ρ) changes only when r(y) ≤ ρ < r(x), and then by +1. ∎
Corollary 3.1a (ledger identity). On the onion graph with integer shell coordinates and inward parents exactly one shell lower, the number of inward atomic transfers (not the tick count of composite transitions) equals the total decrease of Φ_r; moreover, when the total vacancy count H_tot > 0, the mean vacancy radius r̄_vac = (n_max Σ_x r(x) − Φ_r)/H_tot is non-decreasing.
3.2 Smoothing: uniqueness of the closed recurrent class in a fixed sector
Proposition 3.2. On the onion graph (shells connected, same-shell edges available both ways, every cell with at least one inward parent), fix the total occupancy N and write uniquely N = Q_k + m with Q_k = Σ_{j≤k} C_j and 0 ≤ m < C_{k+1}. Then in the transition graph of that sector under per-edge non-increasing sequential hops, the unique closed recurrent class (sink strongly connected component) is the smooth sector 𝒮_N: shells 0 through k full, exactly m units on shell k + 1, all empty above; its size is C(C_{k+1}, m). "Transient" is here graph-theoretic: not in the sink SCC, independent of scheduling. The endpoints N = 0 and N = V are the all-empty and all-full singleton sectors (convention Q_{−1} = 0); the proposition is stated for 0 < N < V, the endpoints being trivial.
Proof. First, every non-smooth state has a finite escape route that strictly lowers Φ_r. Let j₀ be the lowest shell containing a vacancy (all shells below it full). If no cell above shell j₀ is occupied, the state is already in 𝒮_N (with k = j₀ − 1). Otherwise take the lowest occupied cell v above shell j₀; two cases exhaust. If r(v) > j₀ + 1, then shell r(v) − 1 is entirely empty (v being lowest), so an inward parent of v is empty; drop inward one step and Φ_r strictly decreases. If r(v) = j₀ + 1, shell j₀ contains a vacancy; since shells are connected, push the vacancy along the ring toward an inward parent of v: if the neighbouring cell toward the target is occupied, that unit hops into the vacancy (same shell, Δr = 0, legal) and the ring distance decreases by one; if it is empty, chase that nearer vacancy instead, and the ring distance again strictly decreases; after finitely many steps an inward parent is empty, v drops, and Φ_r strictly decreases. Degenerate shells cause no trouble: if C_{j₀} = 1, containing a vacancy means being empty, the inward parent of v is that very cell, and the drop is immediate with no ring walk. Since transfers never increase Φ_r and every non-smooth state has a finite strictly decreasing path out, no non-smooth state lies in any closed recurrent class. Second, closure: in the smooth sector every inward target is full, and the only legal moves are same-shell single hops on shell k + 1 (when 0 < m < C_{k+1}), which stay inside. Third, strong connectivity: token sliding around the skin ring with a vacancy makes all m-arrangements mutually reachable. In the degenerate cases m = 0, or the top shell exactly consumed, the sector is a single point with no legal sequential hop; for synchronous exchange see §3.3. ∎
3.3 Closure of the saturated core
Lemma 3.3. Suppose shells 0 through k are saturated and all vacancies lie outside. Under per-edge radial non-increase, for every legal transition (including composite synchronous exchanges): (i) core occupancy is invariant: a core cell at n_max cannot net-increase; if it net-decreased, the flow decomposition of the predecessor's Proposition 6.9 yields a path from that cell to some net-increasing cell, which must have spare capacity and hence lies outside the core at larger radius, and a per-edge non-increasing path cannot climb; contradiction. (ii) Every transfer touching the core is same-shell: an open chain entering the core cannot exit, since exit needs an outward edge, so open chains never touch the core; only closed chains touch it, and along a closed chain the Δr sum to zero with each term non-positive, hence each term zero (an application of the predecessor's Corollary 6.11). (iii) Therefore the radial coordinate of every tracer label inside the core is constant at all times, while tangential motion remains. Under strict inwardness the statement is stronger: closed chains do not exist (predecessor §5.3) and the core has no events at all. ∎
Remark 3.3a (what is frozen). What the lemma establishes is occupancy freezing and label radial freezing; tangential events are not frozen (iii). Degrees of freedom outside the model (phase, momentum, stress) are not this paper's objects and are not in the freezing claim. "Freezing" carries this meaning throughout.
3.4 The stratified bridge
Corollary 3.4 (stratified monotonicity). Under per-edge radial non-increase, every tracer label's radial coordinate is non-increasing; its strictly decreasing atomic steps are audited (Corollary 3.1a); and inside the saturated core its radial coordinate is constant (Lemma 3.3(iii)).
This is the inwardness inherited through the predecessor's Proposition 2.4, lifted to what the label layer can actually build: descent exhausts itself in finitely many steps, while events need not exhaust with it. Its condition is the per-edge radial condition itself, whose provenance is the interface choice (§6), a load-bearing physical input, not a theorem of this paper. Read through erosion, Corollary 3.4 yields the conservation of release order; the status of that reading is §7's subject.
3.5 Numerical certification, part one
The headline items follow; full tables and script in Appendix B (arealaw.py).
First, exhaustive certification. On three onions, from the outermost-packed initial state, the reachable set is exactly the whole sector C(V, N) (Lemma 3.1's constraint is vacuous at this initial state, now landed as an assertion): 286, 480 700, 27 132; within them the closed recurrent class passes the three assertions of closure, strong connectivity and absorption, and equals exactly the C(C_skin, m) smooth arrangements (10, 36, 28), all else transient.
Second, the ledger identity. On the middle onion V = 309, N = 261: saturation used 668 atomic steps, of which the 227 inward steps equal ΔΦ_r exactly; cumulative monotonicity and mean-vacancy-radius monotonicity pass at every checkpoint; four hundred thousand steps after saturation touch exactly the 96 cells of the full skin shell, not one step beyond.
Third, the two radial conditions contrasted. Under strict inwardness the same initial state terminates on a face containing two partially occupied shells, and history dependence appears: on the onion [1,4,6,8] with N = 13, only 27 of the 28 smooth arrangements are reachable as terminal states, and only 278 of the 290 heaps. Non-increase erases history through the recurrent class. Terminal states are characterised as heaps (every occupied cell's inward parents occupied); heap counts and smooth counts both scale linearly with the skin over the test range, with per-skin-cell entropies 0.499 to 0.648 and 0.499 to 0.581 respectively; the heap coefficient is still drifting over the test range, and no asymptotic claim is made.
4 The Label Layer
Ontological status of labels. The labels of this paper are external tracers: they track transfer history and release order, like tracer isotopes, and label permutations are not counted among physical states; the physical state is the occupancy configuration. Hence the multiplicity of §8 is a quantity of the occupancy quotient, while the groups of this section characterise the tracer permutations that the dynamics can induce, that is, transport structure, not state multiplicity. If labels were physical degrees of freedom, the total entropy would be counted differently and would in general not be area-level (log Π_j C_j! is volume-times-log for three-dimensional families); that choice and its consequences are listed in §10 and belong to the information-theory series interface.
4.1 The permutation groups of the three rungs
Proposition 4.1. On the saturated core, suppose every adjacent transposition on a same-shell ring, and (on the net-constraint rung) every edge transposition, can enter a legal schedule, and tracers are distinguishable. Then the tracer permutation group induced by legal exchange transitions switches rung by rung with the radial condition (Definition 3.0): under strict inwardness it is trivial, since inward edges strictly decrease shell number, no cycles hence no ring shifts; under per-edge non-increase it is Π_j Sym(shell_j), the upper bound by the radial invariance of Lemma 3.3(iii), the lower bound because same-shell adjacent transpositions generate each shell's full symmetric group; under net constraint only (ring shifts indifferent to geometry, since the configuration is unchanged and ΔΦ_r = 0, predecessor Lemma 3.9) it is Sym of the whole core, since transpositions along the edges of a connected graph generate the full symmetric group (take a spanning tree; conjugates of tree-edge transpositions give all transpositions). ∎
Remark 4.1a. The three-way split inherits the predecessor's Remark 6.12: where ring shifts live is decided by the radial condition, not by geometry. The proposition lifts this to a characterisation of groups: what transport structure the interior dynamics can remember switches rung by rung with the radial condition.
4.2 The label speed limit
Lemma 4.2. Under exchange semantics one transfer traverses one adjacency edge, and one label rides at most one transfer per tick, hence moves at most one edge per tick; within t ticks a label's displacement does not exceed graph distance t.
Corollary 4.3. The time to realise a permutation π satisfies T(π) ≥ max_x d_G(x, π(x)); taking a pair realising the diameter and a permutation exchanging them gives the worst case max_π T(π) ≥ diam(G). The identity and adjacent transpositions carry no such bound; the lower bound is per permutation. On local lattices diam ∼ V^{1/d}, so the worst-case permutation time is bounded below by a power law Ω(V^{1/d}); a necessary condition for scrambling time ∼ log V is adjacency with diam ∼ log V.
4.3 Numerical certification, part two
Headlines; full tables in Appendix B (labels.py).
First, exhaustive groups. BFS on the onion [1,3,4]: the non-increasing rung reaches 144 = Π C_j! with every label on its birth shell in every state; the net-constraint rung reaches 40320 = 8!; the strictly inward rung's 11 inward edges all strictly decrease shell number.
Second, the dynamic face of radial invariance. V = 166 with skin vacancies, twenty thousand ticks of synchronous exchange: label shells unchanged and inward moves never legal, assertions passing at every checkpoint.
Third, transport numerics. Seeds are on record per experiment (local 101 through 103, matched-edge 201 through 203), local RNGs isolated, independent of call order; medians are standard sample medians (even samples take the mean of the central two), bootstrap on the same statistic. Under random-matching scheduling on the net-constraint rung, the central label's first-passage median to the outermost shell, with bootstrap 95% intervals: R = 12 gives 68 [52, 105], R = 20 gives 280.5 [226.5, 470], R = 28 gives 804.5 [431, 2067]; t_med/R² is 0.47, 0.70, 1.03, with few trials and wide intervals, so only compatibility with diffusive scaling is claimed and no exponent is fitted. The numerical support for diffusive scaling is in the mean squared displacement: at R = 20 the radial MSD grows near-linearly from 0.59 at t = 1 to 14.18 at t = 32 (regression slope about 0.44 to 0.46 per tick), reaching 43.58 at t = 256; compatibility is read with interval propagation, the R = 12 interval times (20/12)² giving [144, 292], which intersects the R = 20 interval [226.5, 470]. Adding one layer of random long-range perfect matching collapses the medians to 19 [13.5, 26.5], 28 [10, 53], 62 [25.5, 116], a reduction of roughly four to thirteen times over local (bootstrap intervals conditional on the listed matching seeds; instance-to-instance graph fluctuation is not included); what this certifies is that transport is no longer limited by the local diameter, not a measured logarithm, and not that the resulting graph is an expander.
4.4 Correspondence with fast scrambling (T3)
The T1 inside the fence is Corollary 4.3's per-permutation lower bound and the measured diffusion. For the black-hole fast-scrambling conjecture [6] and information mirrors [7] with t* ∼ log S: within this model class the necessary condition is log-diameter adjacency; bounded-degree expanders are the natural sufficient candidate, while log diameter is neither equivalent to expansion nor by itself a guarantee of fast mixing (two expanders joined by one bridge edge is the counterexample). The correspondence is T3, programmatic; its deepening belongs to the information-theory series interface and is not developed here.
5 Exclusion of Potentials
5.1 The premise: forward-definedness of the cyclic era
Call a dynamical sector a cyclic era if occupancy no longer decreases while transitions never cease. §3 and §4 establish its existence: under sequential hops the smooth sector is a closed recurrent class with a perpetually live skin (0 < m < C_skin, Proposition 3.2); under synchronous exchange, ring shifts inside the saturated core are always legal (the same-shell closed chains of Lemma 3.3(ii)). Hence any state space covering the cyclic era has its transition relation forward-defined on that sector: every state has at least one successor. The sector is moreover forward-closed in X′: transitions project to the occupancy layer, the smooth sector is closed (Proposition 3.2) and the saturated core is closed (Lemma 3.3), so the preimage is forward-closed, and the cycle test and orbit test below act on this subgraph. The degenerate case (the full sector with m = 0 under sequential hops) has no successor, is of terminating type, and does not enter the premise.
5.2 The two-cycle lemma
Lemma 5.1. If there exist states x ≠ y with both x → y and y → x legal, then H_Lyap fails: Ψ(x) ≤ Ψ(y) − δ ≤ Ψ(x) − 2δ. ∎
Remark 5.1a. Under exchange semantics the two-cell swap is a legal transition and is its own inverse; this is an extension of the predecessor's k ≥ 3 ring-shift convention, not a restatement, and is recorded here. Whenever the swap acts with finite order on the extended state space, a finite cycle appears at once and the lemma applies directly. Even without swaps, applying a same-shell ring rotation C_j times returns to the start, again yielding a finite cycle; the lemma is not the theorem's only dependency.
5.3 The exclusion theorem
Theorem 5.2. Let X′ be any state space whose transitions project to occupancy transitions or occupancy identities, covering the cyclic era and forward-defined on it. Then H_Lyap (a bounded-below Ψ with uniform δ > 0 strictly decreasing along every transition, in the predecessor's Definition 5.3) fails on X′.
Proof. The preimage of the sector in X′ is forward-closed (§5.1). Its transition graph either contains a cycle or is acyclic. With a cycle: every edge on the cycle has ΔΨ ≤ −δ while the telescoping sum vanishes; contradiction. This is the predecessor's Theorem 4.3 Part D applied. Acyclic: forward-definedness gives an infinite injective orbit from any state, along which Ψ drops by at least δ per step while bounded below, exhausted within ⌊(Ψ(x₀) − inf Ψ)/δ⌋ steps; this is the predecessor's Part B, which needed neither finiteness nor determinism. Contradiction with infinitude. ∎
The configuration-layer transition set contains no idle transitions (Definition 3.0); internal transitions of the extended layer (M = ∅ with a changed extended state) are not excluded and need not be, and if genuine self-loops were present, H_Lyap would fail by them at once, beyond rescue. The theorem therefore does not lean on self-loops of empty occurrence. Occupancy-identical transitions with non-empty transfer remain legal: taking X′ = 𝒳 (no extension), a ring shift is a self-loop at the configuration layer and Lemma 5.1 applies on the spot; taking a tracer-bearing X′, ring shifts are not self-loops and the cycle comes from swaps or from C_j-fold ring rotation (Remark 5.1a). Both routes go through. The attribution must be straight: the machinery of this theorem belongs entirely to Parts B and D of the predecessor's Theorem 4.3. What is new here is the premise, the forward-definedness of the cyclic era (§5.1), and the target of application.
5.4 Corollaries and their limits
Corollary 5.3 (final answer to the boxed question). The predecessor's §7.5 asks precisely for such a potential: bounded below and uniformly strictly descending along every transition. On any state space that keeps the interior perpetually active, the answer is the empty set. H_Lyap entails that every maximal orbit is finite, so it can occur only in terminating models; on finite or finitely branching terminating graphs the potential can indeed be constructed (rank by longest remaining path). The freedom in "what state space" only selects the manner of death within terminating models. Within the chosen transition system, a certificate of a uniformly strictly descending potential is a certificate of combinatorial termination, not a physical death certificate.
Corollary 5.4 (strengthening for finite extensions). The predecessor's §5.4 proved radial potentials label-blind; here is the stronger statement: on any finite extension, no bounded-below uniformly strictly descending potential of any kind can be erected. Finite and forward-defined, the pigeonhole yields a cycle, and the cycle kills every such Ψ. To break the cycle one must make the occurrence history free of exact recurrence, whose typical realisation is a monotone counter, a quantity strictly increasing along transitions, such as ticks, accumulated action phase [5], or growing records; but not the only one. Whatever the variable, an acyclic forward-defined system has infinitely injective orbits, and the acyclic branch of Theorem 5.2 kills it regardless of realisation. The tension the predecessor recorded, that the more complete the state space the harder finiteness is to prove, here closes without a needle's eye: the finite must cycle; the acyclic has infinite orbits, and on an infinite orbit uniform-gap strict descent and a lower bound cannot both be kept. No seam on either side.
Remark 5.4a (what is and is not excluded). Excluded is exactly the conjunction of bounded below, uniform positive gap, and per-edge strict descent. Non-strict Lyapunov functions, potentials flat on recurrent classes, state-dependent decrements tending to zero, ordinal ranks, and local potentials on the transient sector are all outside the exclusion; and Φ_r itself is the instance of a stratified potential: strictly descending on the transient region, flat on the recurrent class. This is precisely the mathematical content of §3's stratified monotonicity: the active recurrent sector excludes a global uniform strict potential while admitting a stratified one, strict on transients and flat on recurrence.
Remark 5.5 (the gap upgraded). The predecessor distinguished remainder from work schedule: a structural remainder versus a bridge merely unbuilt. Here H_Lyap ascends from unbuilt to unbuildable (conditional on a live interior), and the gap becomes structural in grade: the occupancy-identical sector is missed by every function of occupancy and by every such potential, and is caught by the tracer layer; the tracer layer then generates its own new remainder, the openness of finiteness and the release-order fork (§7); and below the tracer there is no zero-response remainder left to catch (Remark 6.2b, under the same conditions: complete gross-layer recording, no hidden variables beyond M), so the descent of zero-response remainders terminates, within this model class, at the tracer layer, after which only the fork of finite versus infinite remains. This passage is a framework-level reading, T2, supported by the T1 of Theorem 5.2 and Remark 6.2b.
6 Observation Kernels and Stratified Blind Zones
6.1 Two inheritance interfaces
Continuum trappedness (predecessor Proposition 2.4) constrains worldlines and events; its inheritance by the discrete layer requires an interface. Stepwise inheritance demands every transfer strictly inward; this is the realisation of the predecessor's §5.3 item three, H_r holds by definition, the predecessor's Part C collects on the spot, forward totality is lost, and the core dies. Flow inheritance demands the coarse-grained occupancy flow inward or zero; under it same-shell transfer and occupancy-identical exchange are both licensed.
6.2 Observation maps and their nested kernels
Blind zones exist only relative to an observation map. For any transition θ, with J_θ its net edge flow (the field of the predecessor's Proposition 6.9 flow decomposition), define four observations: 𝒪_cur(θ) = J_θ; 𝒪_occ(θ) = Δn_θ; 𝒪_rad(θ) = {F_ρ(θ)}_ρ, the family of net fluxes through radial cuts; and 𝒪_gross(θ) = M(θ), the gross transfer multiset. "Kernel" below means the zero fibre, ker 𝒪 := 𝒪⁻¹(0) (Θ is not a linear space).
Proposition 6.1 (characterising occupancy identity). The following are equivalent: (i) s(θ) = t(θ); (ii) the net transfer field has zero net flux through every cut, including single-cell cuts; (iii) θ is invisible to occupancy and to every function of occupancy.
Proof. The net flux through the single-cell cut {x} is Δn_x, so (ii) is equivalent to cell-by-cell balance, i.e. (i); (i) and (iii) entail each other since functions of occupancy see only occupancy. ∎
Remark 6.1a (weakness of radial cuts, sufficiency inside the core). Zero flux through the radial-cut family alone pins only the per-shell sums, not the within-shell distribution; the skin same-shell single hop is the witness: all F_ρ = 0 while Δn ≠ 0. Inside the saturated core, however, radial cuts suffice: a same-shell single hop needs spare capacity at its target, illegal at saturation, so the legal transitions touching the core are closed chains only (Lemma 3.3(ii)), and closed chains are occupancy-identical.
Proposition 6.2 (nesting). In any transition system, ker 𝒪_cur ⊆ ker 𝒪_occ ⊆ ker 𝒪_rad; if the transition repertoire contains ring shifts of k ≥ 3, the first inclusion is strict; if it contains skin same-shell open chains, the second is strict; if it contains two-cell swaps, then ker 𝒪_cur ≠ ∅. This paper's exchange repertoire contains all three, so the chain is strict.
Proof. Δn = −div J, so J = 0 entails Δn = 0 entails invariance of all shell sums; both inclusions hold in any system. Strictness by witnesses: a ring shift of k ≥ 3 has J ≠ 0 with div J = 0; a skin same-shell open chain has Δn ≠ 0 with all F_ρ = 0; a two-cell swap has M ≠ ∅ with J = 0. ∎
Remark 6.2a (granularity). A two-cell swap's net edge flow is zero (one unit each way on one edge, cancelling), so it lies in the first kernel; the gross-transfer layer is finer, resolves the swap, and the second moment is taken at the gross layer (§6.4). This paper's chain stops at the net layer; the finer subdivision of the gross layer is used in §6.4.
Remark 6.2b (the zero-response chain has a bottom; conditions stated). Suppose the gross-layer observation records the transfer multiset completely; idle transitions are expelled (Definition 3.0); and transitions are determined by endpoints and M, with no hidden variables beyond M. Then ker 𝒪_gross = ∅ and the chain of zero-response blind zones stops after four layers: ker 𝒪_rad ⊋ ker 𝒪_occ ⊋ ker 𝒪_cur ⊋ ker 𝒪_gross = ∅. What is bounded is the emptiness of the zero fibre, not injectivity of the observation: confusion between distinct transitions sharing the same M is not thereby excluded. The regress in which every layer breeds its own zero-response blind zone fails here; the chain has a bottom. Extensions carrying metadata beyond M (phase, internal records) are outside this clause and their bottom must be proved separately. The remaining question is not whether the ladder is inexhaustible, but how deep continuum observation can descend (§10 openings).
6.3 Four layers of visibility
The nested kernels give four layers of visibility (strictness within this paper's full exchange repertoire, Proposition 6.2), and each layer is exactly one of this paper's load-bearing objects:
| layer | criterion | what lives there | what probe sees it |
|---|---|---|---|
| 0 radially visible | 𝒪_rad ≠ 0 | inward transfer; the descent era's ledger | radial cut flux |
| 1 radially blind, occupancy-visible | 𝒪_rad = 0 and Δn ≠ 0 | skin same-shell rearrangement; §8's C(C_skin, m) multiplicity | occupancy snapshots |
| 2 occupancy-blind, net-flow-visible | Δn = 0 and J ≠ 0 | ring shifts of k ≥ 3 | net edge flow |
| 3 net-flow-blind | J = 0 (transfer non-empty) | two-cell swaps | gross layer: full tracer trajectories and gross transfer records (release order one of their protocol compressions, §7); second moments (§6.4) |
$$\boxed{\ \text{Blind zones exist only relative to an observation map; there is no unconditional blind zone.}\ }$$
ΔΦ_r is a scalar compression of 𝒪_rad: under per-edge sign uniformity (rung two) it is a sufficient probe, and the ledger of Corollary 3.1a is this compression; in general 𝒪_rad ≠ 0 with ΔΦ_r = 0 can coexist (inward and outward transfers at different radii cancelling), so the table's criterion is 𝒪_rad. §8's area-level multiplicity lives in layer one, and §4 and §7's label structure in layers two and three; the sections are thereby rungs of one ladder, and the chain has a bottom (Remark 6.2b).
6.4 The threefold degeneracy of first-order silence
Lemma 6.3 (core occupancy identity on rung three). In the smooth sector (vacancies all in the skin), under net constraint only, every transition changing core occupancy is illegal. A core cell cannot net-increase (it sits at n_max); a net decrease sends its unit to spare capacity, which lies wholly on shell k + 1 or beyond, an ascent of at least one; the compensating descent would need a higher unit and a lower vacancy, and the smooth sector has neither, so net ΔΦ_r > 0, violating the net constraint. ∎
Corollary 6.4 (threefold degeneracy). The three interiors of a frozen core, the dead core (stepwise inheritance), the shell-cycling core (rung two), and the radially mixing core (rung three), look alike as silence at the occupancy layer and at the specified coarse limit of the first-order flow layer: the dead core has no transitions; the core transitions of rungs two and three are all occupancy-identical (Lemma 3.3(i) and Lemma 6.3), so occupancy snapshots are constant; and rung two may run a full-strength mutually cancelling plaquette family (Theorem 2.1) whose first-order coarse flow vanishes weakly, while rung three's swaps have exactly zero net edge flow. First-order silence is therefore consistent with all three, and observing it does not adjudicate the interior. The three silences have distinct identities, and the qualifier cannot be dropped: the dead core's flow is strictly zero level by level; the cancelling family of rung two vanishes only in the weak ε → 0 limit; the swaps of rung three have exactly zero net edge flow. At finite scale, without the limit gauge, the likeness does not hold.
Jurisdiction note. What is refuted is certifying the interior from first-order silence, not the uselessness of the first-order layer: coherent bulk rotation on rung two (a non-cancelling family) is visible at the net-flow layer. Discrimination requires ascending in moment or in layer. The gross-layer second moment is a candidate: rung three's radial swaps show in the rr component while the dead core gives zero (an existence witness, Appendix B); but rung three's transition set contains rung two's, it may run only same-shell exchange, activity may be intermittent, so becoming a classifier requires a definition over transition distributions or time averages, its formalisation listed among §10's openings; this paper does not treat it as a finished discriminator. The tracer layer is §7's subject, and separates mixing from non-mixing.
6.5 The interface choice is an input
Each interface carries its ledger: the stepwise dead core was priced by the predecessor; the flow-inherited cycling core is given its shape by §3 through §5 here. Which is the correct reading of inheritance is physical input, not theorem; this paper writes both ledgers and does not choose. The gap does not vanish but narrows to a single coin; that is its final form.
7 The Release-Order Discriminant
7.1 Protocol and quantity
The erosion protocol: peel the saturated onion's outermost occupied shell round by round, with τ ticks of autonomous dynamics between rounds. The discriminant is the cross-shell inversion fraction: among label pairs of distinct birth shells, the fraction whose release-round order inverts their birth-depth order. This is a model-internal diagnostic, protocol-dependent, not a black-hole observational prediction.
7.2 The shell-cycling rung: identically zero
A direct consequence of Corollary 3.4: labels never cross shells, the release round is determined by the birth shell, and the deeper is always later out. This is a theorem, not a measurement; in the numerics it lands as an assertion, passing for all seeds and intervals (Appendix B).
7.3 The mixing rung: the exchangeable-limit expectation plateau
Proposition 7.1 (expectation plateau in the between-round exchangeable limit). Suppose the first round peels the not-yet-mixed outermost shell (C_R cells), the remaining N′ = N − C_R labels mix sufficiently between rounds so that release batch and birth shell are exchangeable, and the erosion batches are b₁, …, b_k. Then the tie fraction is Σ_i C(b_i, 2)/C(N′, 2), and the expected cross-shell inversion fraction is
$$\mathbb E[I]=\frac{P_{\rm cross}-C_R(N-C_R)}{P_{\rm cross}}\cdot\frac{1-\text{tie}}{2}$$
Proof. First-round pairs are concordant by construction; the remaining cross pairs, under exchangeability, invert and concord with equal probability, each (1 − tie)/2; a tie holds iff the pair is released in the same batch, whose probability under exchangeability is the within-batch pair fraction. ∎
This is an expectation, not a per-realisation bound: single realisations fluctuate on both sides, and the measured ranges already cross it (τ = 288 up to 0.332, τ = 2048 up to 0.344), which corroborates rather than contradicts. For the instance used (C_j = 3j, R = 12): cross-shell pairs 24 687, first-round concordant pairs 7164, tie fraction 2178/19701 = 0.1106, expectation plateau 0.3157 (script ceiling.py). Exchangeability is the condition, and what the numerics test is whether mixing is sufficient: multi-seed measurements (erosion_seeds.py, Appendix B) give τ = 24 mean 0.2720 (standard error 0.0041), τ = 288 mean 0.3134 (0.0026), τ = 2048 mean 0.3156 (0.0062); the last differs from the plateau by 10⁻⁴, within the Monte Carlo uncertainty (0.02 standard errors).
7.4 Limits: the discriminant's jurisdiction
The converse of "inversions identically zero" fails: under the stepwise dead core labels do not move, and inversions are likewise zero. This discriminant therefore separates mixing from non-mixing (splitting rung three out of Corollary 6.4's threefold degeneracy), not the live pinned shell from the dead core; the candidate discriminator for the interface choice is the gross-layer second moment (the jurisdiction note of §6.4), whose formalisation is among §10's openings.
7.5 Status
Release order is a tracer-layer quantity, consistent with §6.3: the fork at layers two and three is adjudicable only by observation beyond occupancy and radial flux. This paper delivers one tracer-layer item; one second-moment item is listed as an opening (§6.4). That the shell-cycling rung's order is incorruptible is a theorem; that the mixing rung's order is saturated is a measurement under the protocol. Its correspondence to the information release order of evaporation radiation is T3, at the information-theory series interface.
8 Area-Level Multiplicity
8.1 The pinned entropy
Corollary 8.1. The occupancy multiplicity of the smooth sector is C(C_skin, m) (Proposition 3.2). Writing p = m/C_skin, ln C(C_skin, m) = C_skin H(p) + O(ln C_skin) (the Stirling asymptotic with p bounded away from 0 and 1, H the binary entropy), and the exact bound ln C(C_skin, m) ≤ C_skin ln 2 holds always; the degenerate end m = O(1) gives only m ln C_skin + O(1), logarithmic. The scripts compute the exact ln C(C, m) throughout; the asymptotic form is used only in the text. This is a quantity of the occupancy quotient (§4 opening).
8.2 Scaling
$$\boxed{\ \text{On scaling families with }C_{\rm skin}\asymp A\text{ and }m/C_{\rm skin}\to p\in(0,1)\text{, }S_{\rm occ}\asymp A\text{; strengthened to }C_{\rm skin}/A\to\kappa>0\text{, }S_{\rm occ}\sim\kappa A\,H(p)\text{; the area-level bound }S\le C_{\rm skin}\ln2\text{ is a theorem; the mechanism pinning }p\text{ is not erected here; coefficients depend on the lattice.}\ }$$
The two filling regimes contrasted (scripts scaling2.py and ceiling.py): in the near-saturated regime the three-dimensional family's pinned entropy fits S ∼ R^2.03 against free counting S ∼ R^2.30, area versus area-times-log, one logarithmic factor apart; in the half-filled regime the pinned entropy's upper bound is core face times ln 2, with core faces fitting R^1.97 (3D) and R^0.98 (2D), while free counting for the 3D family fits S ∼ R^2.90, volume-level. The half-filled pointwise pinned entropy swings with m/C between zero and C ln 2 (at C_skin = 96, per-cell entropy from 0.048 at m = 1 to 0.667 at m = 48, a fourteen-fold span), so only the bound's scaling is reported and no swing-contaminated fit. In evaporation N decreases while m sweeps the whole range, so a "law" requires a mechanism pinning p, listed as §10's debt; coefficients depend on lattice and filling fraction, and only the scaling is universal, in the discipline of the predecessor's Appendix B.
8.3 The ledger against Paper 1
Paper 1 [2], §2.E, lists inputs and outputs face up, with the freezing rule among the inputs. This paper demotes occupancy freezing and label radial freezing from rule inputs to conditional theorems: given the onion graph, a saturated initial state, and per-edge radial non-increase, the frozen core and the area-level skin multiplicity are theorems; what remains as input is the radial condition itself, that is, §6's interface, load-bearing and underived. The hierarchy reads: interface (input); freezing and area-level multiplicity (theorems).
Whose area this is must also be said plainly. This paper's area is the skin cell count of the frozen core; its relation to the horizon area is not erected here, and the predecessor has already moved the algebraic extrapolation crossing core radius with horizon radius out of the trusted domain (the removal list of its §7.4), which this paper does not re-cite. The correspondence with Paper 1 is therefore a structural correspondence of shared area-level scaling, not a shared underlying mechanism of one same area; the objects also differ, there the radiation entropy and min-cut bound of a stabiliser network, here the multiplicity of occupancy configurations. Marked T2.
9 The Three-Era Picture (T2, conditional on flow inheritance)
This section is model-internal assembly throughout: the parts are T1 and the assembly is hypothesis, conditional moreover on the flow-inheritance interface: under stepwise inheritance the core dies, no cyclic era exists, and the three eras degenerate to descent and termination. Its correspondence to the life of physical black holes is T3, not claimed. The predecessor's §5.5 exit matrix reads its rows as mutually exclusive model options; this section reads them instead as successive stages of one object's life.
The descent era. During collapse strictly inward edges abound, Φ_r descends, the predecessor's Part B countdown runs, atomic steps are audited (Corollary 3.1a); vacancies are expelled shell by shell (Lemma 3.1) and the interior saturates from the inside out.
The frozen cycling era. Descent bottoms out, uniform strict descent no longer exists, the predecessor's Part C no longer bites, and forward completeness survives. Under finite autonomy Part A sends orbits into the periodic branch or finite recurrent wandering; by the predecessor's Theorem 6.1, net outflow on a period is zero, so the autonomous frozen core does not radiate, and radiation must be a non-autonomous event of the skin. This era's core dynamics retreats wholly inside the occupancy kernel (Corollary 6.4); Theorem 2.1 is precisely the existence proof of its coexistence with first-order macroscopic spherical symmetry: microscopically at full turnover, first-order silent.
The evaporative truncation era. Skin emissions strip cell by cell, the state space contracts with them, the time-resolved version of the exit matrix's non-autonomous evaporation row; under the shell-cycling rung the release order is the depth order (§7).
Compatibility. With Kill 6: freezing is a theorem of the dynamics, not blockage inferred from saturation. With Kill 7: recurrence comes from finiteness and determinism, not from being trapped. With Kill 9: what is forbidden is net outflow on a period; transient radiation on the approach to the cyclic era is outside the prohibition. The timing dependence is stated: live core events need exchange-type timing; under sequential hops a dead core with a live skin is an equally coherent picture, the dead-core rung of §4's three.
This section claims nothing about the interior's actuality; the three eras are a reading, whose use is to let the proved parts see one another, and to turn "what is the interior" into the askable "which rung of the three."
10 Discussion and Conclusion
10.1 What stands
The paper's arc closes in five links: first-order symmetry cannot save the same-shell bridge (Kill 10); in models with flow inheritance that keep the cyclic era forward-active, same-shell recurrent structure cannot be excluded by symmetry, and stratified monotonicity becomes the buildable bridge (§3 through §4); on state spaces covering the cyclic era and preserving forward-definedness, a uniformly strictly descending potential cannot be erected (Theorem 5.2), closing the boxed question; the place where it cannot be erected is precisely the occupancy-kernel sector, where first-order observation is silent and the silence is threefold degenerate (§6); breaking the degeneracy requires ascending in moment or layer, and the tracer release order adjudicates one branch (§7). The question the predecessor's coda left hanging, whether the inwardness of events and worldlines can be lifted to uniform strict descent, has, on state spaces covering the cyclic era and preserving forward-definedness, the answer no, and the no is a theorem. The two ledgers no longer cancel against each other but are filed each in its own book: inwardness in the tracer's book, forward-definedness in the event's book, the type discipline making the two books lawfully coexist.
10.2 This paper's refutations
By the criterion of the predecessor's §7.3.1, this paper records three.
| # | hypothesis | how refuted | source |
|---|---|---|---|
| Kill 10 | continuum spherically symmetric zero circulation (weak reading) entails discrete same-shell acyclicity | disjoint oriented plaquette family: full-strength cycles at every level, first-order coarse flow O(ε) weakly to zero | Theorem 2.1 |
| Kill 11 | some extended state space covering the cyclic era and forward-defined (labels, phases, records, or other internal freedoms) makes H_Lyap hold | the finite yields cycles by pigeonhole and cycles kill uniform descent; the acyclic yields infinite injective orbits and Part B kills bounded-below uniform descent | Theorem 5.2 |
| Kill 12 | silence at the occupancy layer and the first-order coarse-flow layer can certify the frozen core's interior (dead, shell-cycling, radially mixing) | the three look alike in the specified first-order coarse limit: occupancy identity, with first-order flow strictly zero, weakly vanishing, or net zero; silence is consistent with all three | Corollary 6.4 |
Jurisdiction note on Kill 12: what is refuted is certifying the interior from first-order silence; the second-moment layer and the tracer layer are outside the refutation and in principle can adjudicate (§6.4, §7); coherent bulk rotation of rung two is visible at the net-flow layer and likewise outside.
Not counted as refutations: the three-era picture is T2 assembly; stratified monotonicity and the shell-cycling LIFO are positive results; random long-range perfect-matching edge addition is a display of possibility, not a narrowing of the space.
10.3 Moved out, firewalls, and openings
Between the predecessor's Theorem 6.1 and the information-theory series' [3] claims on gravitational-wave outflow, this paper supplies one key without choosing: what Theorem 6.1 forbids is net outflow on a period, transients are outside the prohibition; the choice belongs to the information-theory series. The deepening of the fast-scrambling correspondence, and the unfolding of release order against radiation information order, go to the same interface. Kerr, three-dimensional shell networks, n_max > 1, non-uniform descent variants (state-dependent δ, quasi-Lyapunov functions, supermartingales), and the statement of the exclusion under randomised parallel and block-synchronous timings, are listed as sequels and not presumed portable. The full state space of phase and momentum (predecessor §7.1 item three) belongs to the quantum-mechanics series interface. The SAE framework-level record-chain reading, beyond Remark 5.5's remainder upgrade, does not enter this paper.
Four openings of this paper's own. First, the mechanism pinning the filling fraction p (§8's debt). Second, the formalisation of the discriminators: the candidate for adjudicating the interface choice is the gross-layer second moment (zero for the dead core, non-zero tangential pulsation for the live pinned shell, §6.4), which becomes a discriminator only under a definition over transition distributions and time averages; and the adjudicability of order lies one layer deeper, since the three legal families of distinct orientation have identical one-point moments term by term, the distinction living only in lattice-scale two-point correlations (§2.3), so the real question is which lattice-scale two-point correlations a continuum field theory retains; the closure test is already carried by the occupancy layer (non-closing rearrangements change occupancy and live in layer one), and only the orientation-coherence branch hangs; if it is not retained, orientation order is in principle unadjudicable by limit-gauge continuum observation (the finite-ε decay-rate differences are in §2.3). Third, the ladder question half-answered: within this model class the chain of four layers has a bottom (Remark 6.2b), and the regress of every layer breeding its own blind zone is false; what remains is the descent depth of continuum observation, first order stopping outside layer zero (Kill 10 with Corollary 2.3), second order touching layer three's anisotropy but not its order, tracers having no continuum counterpart. Fourth, the total entropy under physicalised labels (Route B of §4's opening), at the information-theory series interface.
10.4 Coda
The mathematics of this paper is elementary in the extreme: the vectors of a closed cycle sum to zero, pigeonholes make cycles, and what is bounded below cannot descend uniformly forever. Nothing specific to black holes lives in this elementary combinatorics; it lives in one point of measure: the impossibility of a uniformly strictly descending potential is not a failure of the model, but the way an interior's perpetual activity necessarily leaves a remainder on the configuration ledger. Where the remainder lives, the nesting of observation kernels carves; tracers catch it; the fork erects it; and the fork is judged by order.
11 Master Status Table
| item | level |
|---|---|
| Refutations | |
| Theorem 2.1 (Kill 10) | proved, conditional theorem (weak-reading gauge; local frames, seam bands, circulation gauge all listed), with numerical certification |
| Theorem 5.2 (Kill 11) | proved; machinery from predecessor Theorem 4.3 Parts B and D, premise the cyclic era's forward-definedness |
| Corollary 6.4 (Kill 12) | proved; restricted to the first-order coarse limit, jurisdiction limited to silence's certifying power |
| Same shell and second order | |
| Remark 2.2 and Corollary 2.3 | the four-object separation is formal discipline; the second-order weak limit ½P_tang on the whole shell (SO(3)-invariant) is T1; the orientation-coherence two-point comparison is computed (three legal families; the random family's multi-seed fluctuation on record) |
| Stratified monotonicity | |
| Lemma 3.1 and Corollary 3.1a | proved (atomic-step gauge, H_tot > 0) |
| Proposition 3.2 smoothing | proved, fixed-N sector; conditional on shell connectivity, two-way same-shell edges, at least one inward parent per cell; endpoints N ∈ {0, V} trivial |
| Lemma 3.3 three-clause core closure | proved, on predecessor Proposition 6.9 and Corollary 6.11; what freezes is occupancy and label radial coordinates |
| Corollary 3.4 the stratified bridge | proved, conditional on the per-edge radial condition (the interface choice, load-bearing input) |
| The label layer | |
| Status of labels | tracers, not counted among states; the physicalising choice listed among openings |
| Proposition 4.1 three-rung groups | proved, conditional on schedulable adjacent and edge transpositions and tracer distinguishability; BFS-certified |
| Lemma 4.2 and Corollary 4.3 | proved (per-permutation lower bound; worst case bounded below by a power law) |
| Transport numerics | computed; seeds on record, standard medians with bootstrap intervals; compatible with diffusive scaling, small samples fit no exponent; log diameter necessary, expanders sufficient candidate (T3) |
| Exclusion and observation kernels | |
| Corollary 5.3 the boxed question's final answer | proved (necessity direction; constructible on finitely branching terminating graphs) |
| Corollary 5.4 finite-extension strengthening | proved (gauge: bounded-below uniformly strictly descending potentials) |
| Remark 5.4a excluded and not excluded | formal discipline; Φ_r the instance of a stratified potential |
| Remark 5.5 the remainder upgraded | T2 reading, supported by Theorem 5.2 |
| Propositions 6.1 and 6.2, Lemma 6.3 | proved; inclusions universal, strictness conditional on the repertoire (present here), witnesses landed as assertions; zero-response chain bottom stated with conditions (Remark 6.2b, no injectivity claim) |
| The interface choice | input, not adjudicated here |
| Release order | |
| Shell-cycling inversions identically zero | proved (consequence of Corollary 3.4), landed multi-seed as assertions |
| Proposition 7.1 expectation plateau | proved, conditional on between-round exchangeability; an expectation, not a per-realisation bound; multi-seed measurements agree within statistical error |
| The discriminant's jurisdiction | separates mixing from non-mixing; dead core and live pinned shell look alike, interface-adjudicating observation listed among openings |
| Area-level multiplicity | |
| Corollary 8.1 | proved, an occupancy-quotient quantity; leading term with O(ln C) remainder |
| Scaling | computed, test-range statements; conditional box; the p-pinning mechanism the debt |
| Correspondence with Paper 1 | T2, structural grade; this paper's area is the core skin, its relation to the horizon not erected |
| The three-era picture | T2, conditional on flow inheritance; the physical correspondence T3, not claimed |
| Not treated here | Kerr; three-dimensional shell networks; n_max > 1; non-uniform descent variants; randomised parallel timing; the full phase–momentum state space; the Info-4 choice; the record-chain reading's unfolding; Route B total entropy's adjudication |
Appendix A: Details of the Counter-Model
The fully packed variant
On the corners of the square subdivision place a signed stream function Ξ(v) = σ(v) a(x_v), with σ = (−1)^{i+j} the chequerboard parity and a an envelope; a cell edge's flow is the difference of the two corner values, and taking signs gives an integer unit flow. Under the diamond envelope (boundary along level lines of i + j) the unit flow is exactly divergence-free cell by cell; every interior corner is an elementary vortex, adjacent vortices alternate in orientation, and every edge is shared by two vortices: this is the full packing. Its directed four-cycle count has the closed form 2m² + 2m + 1 with m = ⌊0.3n⌋, the lattice-point count of the diamond, asserted level by level. The disjoint plaquette family of Theorem 2.1 is its sparsification, with a shorter proof and the same shape of conclusion.
The legality contrast
The choice of envelope is not style but conservation: after taking signs, the smooth bump and the sharp-edged disc both leak ±1 around one boundary ring (64 and 256 cells at n = 64 and n = 256), and only the diamond is identically zero, because its cut lines coincide with the level lines of the chequerboard parity. The legal counter-models for integer occupancy dynamics are therefore the diamond full packing and the disjoint plaquette family; with real amplitudes all three envelopes are usable, at different convergence rates.
Moments and two-point correlations
The cell-averaged first-order field: the disjoint family's block means are zero; the fully packed variant is identically zero cell by cell in the bulk, the cancellation happening at cell scale, its energy residing at the edge and gross layers. Second moments are taken on the disjoint family: |u| = 1 at every covered cell, per-component second moments ½, ⟨u ⊗ u⟩ = diag(½, ½) asserted (n = 128, 2812 cells), and identical term by term across the uniform, chequerboard, and per-block random orientation families. Two-point correlations: normalised C(2e) is +1.000 for uniform and −1.000 for chequerboard (deterministic, 2664 pairs); the per-block random family over 40 seeds (7 through 46) has mean −0.007, standard deviation 0.040, range [−0.087, 0.084], the fluctuation scale O(blocks^{−1/2}) with 703 blocks; one-point moments agree at every order, and orientation order lives only in lattice-scale two-point correlations. This is the numerical face of Remark 2.2 and Corollary 2.3.
Decay of the real-amplitude variants
Segment slopes of the real-amplitude diamond variant: flow −2.000 to −2.002, vorticity −0.999 to −1.000 (n from 24 to 384); the smooth-envelope variant's pairing decays super-algebraically, segment slopes falling from −4 to −16 into numerical floor. The cancellation is no edge effect.
The normalisation trap
The unnormalised corner-circulation weighted sum Σ_v c(v)ψ(v) tends on the fully packed variant to a non-zero constant (about 0.331 for the test function and weights used), which is ε⁻¹ times ⟨ω, ψ⟩. To cite its non-vanishing as evidence of surviving first-order vorticity is to mistake a normalisation; correctly normalised, the pairing is O(ε), consistent with the weak zero of the flow. The survival of second-order quantities (above) is unrelated to this: that is a genuine survival, this a normalisation error.
Appendix B: Lattice Certifications and Scripts
Exhaustive certification (§3.5 item one)
| onion | N | reachable = C(V, N) | smooth | transient | strictly inward terminal | heaps | smooth among terminal |
|---|---|---|---|---|---|---|---|
| [1,3,4,5] | 10 | 286 | 10 = C(5,2) | 276 | 12 | 12 | 10 |
| [1,3,5,7,9] | 18 | 480 700 | 36 = C(9,2) | 480 664 | 278 | 290 | 36 |
| [1,4,6,8] | 13 | 27 132 | 28 = C(8,2) | 27 104 | 100 | 115 | 27 |
The three assertions of closure, strong connectivity and absorption pass in every case; from the outermost-packed initial state the reachable set is exactly the full sector. Heap counts are computed independently by shellwise dynamic programming over inward-closed sets, and the assertion that all terminal states are heaps passes.
Entropy per skin cell (§3.5 item three)
| R | heaps | smooth |
|---|---|---|
| 3 | 0.499 | 0.499 |
| 4 | 0.557 | 0.531 |
| 5 | 0.598 | 0.553 |
| 6 | 0.627 | 0.569 |
| 7 | 0.648 | 0.581 |
Test-range statement; the heap coefficient is still drifting, no asymptotic claim.
Dynamical checks (§3.5 item two)
Middle onion V = 309, N = 261, core shells 0 through 7, skin m = 48: saturation used 668 atomic steps, with 227 inward steps equal to ΔΦ_r = 227; cumulative monotonicity, the ledger identity, and mean-vacancy-radius monotonicity pass at every checkpoint; four hundred thousand post-saturation steps touch exactly the 96 cells of the skin shell. The strictly inward control terminates on a face with two partially occupied shells.
The label layer (§4.3)
Exhaustive groups: non-increasing rung 144, net-constraint rung 40 320, asserted. Radial invariance: V = 166, N = 151, twenty thousand ticks, asserted. Transport (local seeds on record; standard median, bootstrap 95%, trials, range): local, R = 12 seed 101 gives 68 [52, 105] (12 trials, 33 to 294); R = 20 seed 102 gives 280.5 [226.5, 470] (12 trials, 120 to 764); R = 28 seed 103 gives 804.5 [431, 2067] (8 trials, 395 to 2405); t_med/R² is 0.47, 0.70, 1.03. Radial MSD (R = 20): 0.59 at t = 1, 14.18 at t = 32, 43.58 at t = 256. Random long-range perfect-matching edge addition: seeds 201 through 203, giving 19 [13.5, 26.5], 28 [10, 53], 62 [25.5, 116].
Release order (§7)
Multi-seed statistics (erosion_seeds.py, seeds from 1000):
| τ | seeds | rung-3 inversion mean | s.d. | range |
|---|---|---|---|---|
| 24 | 20 | 0.2720 | 0.0184 | [0.231, 0.301] |
| 288 | 20 | 0.3134 | 0.0116 | [0.289, 0.332] |
| 2048 | 10 | 0.3156 | 0.0197 | [0.281, 0.344] |
The expectation plateau is 0.3157 (the exchangeable-limit expectation, not a per-realisation bound), with components: cross-shell pairs 24 687, first-round concordant pairs 7164, tie fraction 2178/19701 = 0.1106; the τ = 2048 mean differs from the plateau by 10⁻⁴, within 0.02 standard errors, while the range tops of 0.332 and 0.344 exceed the plateau, corroborating that it is an expectation. At τ = 0 there is no dynamics and inversions are identically zero, deterministically. Rung two at τ ∈ {24, 288}, five seeds each, passes the identically-zero assertion case by case.
Micro-examples (§5 and §6)
The explicit two-cycle on the three-cell ring; two hundred random finite forward-defined labelled extensions all exhibit reachable cycles (reachable state counts 36 to 42, bound 42); nine occupancy-identical transitions with zero net flux through every cut; the second-moment rr component: ring shift 0 against radial swap 2 (gross layer); the two nesting witnesses landed as assertions: a ring shift with J ≠ 0 and div J = 0, a same-shell open chain with Δn ≠ 0 and all radial fluxes zero.
Scaling (§8)
Three-dimensional family C_j = 2j²: near-saturated pinned S from 47.54 to 1415.53, fit R^2.03; free S from 88.13 to 4177.52, fit R^2.30. Half-filled core faces C: 50, 128, 288, 512, 882, 1352, fit R^1.97; free S fit R^2.90, volume V fit R^2.89. Two-dimensional family C_j = 3j: near-saturated pinned R^1.03, free R^1.23; half-filled core face R^0.98, free R^1.98, volume R^1.97.
Declaration of timings and schedulers
Exhaustive and smoothing certifications use sequential hops; the dynamic radial-invariance run uses synchronous exchange with same-shell adjacent swaps, vacancy hops, and whole-ring rotations; transport and release order use random-matching scheduling. Invariance assertions are scheduler-independent (their coverage is the theorems' business); diffusion constants, first-passage distributions and protocol statistics depend on scheduler and protocol, and are measurements under the declared protocols. Seeds of the random experiments are on record item by item (local RNGs, independent of call order); reported medians are standard sample medians with bootstrap on the same statistic, resampling seed the experiment seed plus 7777.
Scripts and hashes
| script | corresponds to | SHA-256 |
|---|---|---|
tangential.py | Appendix A real-amplitude variants and constructors | d101fc4da6027e722e087eb4b7b8797857b43ec972dc398d851646255609f79c |
killten.py | §2.3 table, cycle closed form, moments and two-point correlations, Appendix A legality contrast | 63d963c1d2927065c7baf2c3db16ae649975cb52c2534d53621ae036a6e3f79e |
arealaw.py | §3.5 and Appendix B exhaustive runs, dynamics, heaps | 39ca2ab687935caa69dfa1c831db59f05899fa064cd40b954968e7f128347c07 |
scaling2.py | §8.2 two-regime scaling | c5736476ed1977893179e710e4dd6415590f8705fc05e6565089fff67263140e |
labels.py | §4.3 and Appendix B label-layer tables and erosion machinery | 50c349c7f0eeb3fd44907577677937486788dfd88b3c1ba69f0e26788a37bd41 |
ceiling.py | §7.3 expectation plateau and §8.2 core-face fits | 354c265c0f45e5d0bf9b8ffa7d40a17fb4b81ec9b020f21d894299ae74f7926e |
erosion_seeds.py | §7.3 and Appendix B multi-seed statistics | e4d12332835d14d706d8b28555047e93d1b5b180859e6ff4437571985e4f443e |
microcert.py | §5 and §6 micro-examples, nesting witnesses, second-moment rr | 085fb3b35d1586549a1fd2568ed80b8d879509b3354927ad6d8110fa51d12d2e |
README.md | package guide (environment, order, seeds, resources) | 8c2824fdf297317a7b5b93c32d54c854dfb63d300d2b0cf3684d5b8e10255e53 |
No dependencies beyond NumPy. The recomputability commitment covers the scripts of the table above, that is, everything the numbers of the body and appendices actually rest on; the scripts and README ship packaged with this paper, item hashes as in the table, the package's SHA-256 recorded in the metadata.
Acknowledgments
The propositions of this paper passed through several rounds of independent review and full-script recomputation before this version, and several key corrections came from the reviewers' findings, gratefully acknowledged here. Among them: the cut-family gauge of the occupancy-identity characterisation, the distinction between an expectation plateau and a per-realisation bound, the separation of legality under the two timing semantics, the definition of the median statistic, the explicit seam-band deficit, and the second-order symmetry upgrade with the narrowed gauge of its two-point comparison, were all clarified or earned through review. Remaining errors are the author's.
Metadata
Title: Exclusion of Uniform Strict Potentials and Stratified Monotonicity in Discrete Black-Hole Interiors: Refuting the Same-Shell Bridge, Nested Observation Kernels, and the Release-Order Fork Author: Han Qin (秦汉), ORCID 0009-0009-9583-0018 Series: SAE Black Hole Series, Paper 3 License: CC BY 4.0 Concept DOI: 10.5281/zenodo.21521760 Language: English (independent rewrite; the Chinese original is a separate file of the same record) Script package: BH-P3-scripts-final.tar.gz, SHA-256: bbab97f34035ee037df059c9dabe01761a108c560cebb82a13a705bf407aeeaa The SHA-256 of this text is registered on the Zenodo record page (a file cannot contain its own hash), forming a double anchor with the package hash.
References
[1] Han Qin. Update Incompatibility in Discrete Black-Hole Interiors: Three-Domain Separation, Orbit Dichotomy, and the Bridge Gap in Radial-Potential Lifting. SAE Black Hole Series, Paper 2. DOI: 10.5281/zenodo.21499053.
[2] Han Qin. Area-Level Plateau and Self-Sufficient Window in a Frozen-Ledger Model. SAE Black Hole Series, Paper 1. DOI: 10.5281/zenodo.20913514.
[3] Han Qin. SAE Information Theory Series, Paper 4. DOI: 10.5281/zenodo.19880112.
[4] Han Qin. ZFCρ 68. DOI: 10.5281/zenodo.19739811.
[5] Han Qin. ℏ as the L₁↔L₂ Symplectic-Conjugation Closure Signature. SAE Quantum Mechanics Series, Paper 3. DOI: 10.5281/zenodo.20340595.
[6] Y. Sekino, L. Susskind. Fast Scramblers. JHEP 0810:065, 2008.
[7] P. Hayden, J. Preskill. Black holes as mirrors: quantum information in random subsystems. JHEP 0709:120, 2007.