Angular Transport in Discrete Black-Hole Interiors: Process Accounts, Spin-Order Lattices, and the Rigid-Rotation Spectrum
离散黑洞内区的角向输运：过程账、旋向格与刚转谱

1 Introduction, Scope, and Dependency Matrix

This series studies discrete black-hole interiors within a pure-occupancy model class: earlier papers established the radial narrative of collapse [P1], the interior information and timing conventions [P2], and potential exclusion with stratified monotonicity [P3]. In these interiors the only degree of freedom is occupancy and the only dynamics is the legality of moves; the radial voice is in place, the angular voice is not. This paper supplies it. Within this pure-occupancy model class, angular net flow is first of all a transfer-level quantity; its cumulative real-valued account is in general not exact along closed histories. Finite state spaces can still carry orientation, phase classes, and discrete instantaneous-flow tiers; but a state extension that retains an unbounded rigid-rotation history while integrating the cumulative account exactly must be infinite. What this paper establishes is the definition of that account, the proof of its non-exactness, the rigidity of its single-tick spectrum and the conditions that break it, the layered structure of its observable compressions, and two negative results: the angular account does not alter the shape of the smooth core, and it does not supply radial support in this model.

Scope first. Every account in this paper is combinatorial: steps are counted by moves, layers by shells, and sums are taken with declared weights. Scope sentence: this paper establishes a combinatorial angular-transport account; the shell weights, time scale, and energy interface required to pass from this account to physical angular momentum are all left unestablished. Accordingly, four non-claims hold throughout. First, this paper does not claim that physical angular momentum fails to be a state quantity; what is proved is only the non-exactness of the combinatorial cumulative account within this model class. Second, this paper does not identify the unweighted edge-step account with Kerr angular momentum; between the two stand the three interfaces of shell weight, time, and energy. Third, this paper does not claim that all entropy accounts and release orders are blind to the angular sector; the blind zones established are named one by one and are finite in number. Fourth, this paper does not claim that after rigidity breaking the spectrum must fill the interval, and it never speaks of a continuous spectrum; every spectrum here is a discrete combinatorial one. Questions touching inner horizons, closed timelike curves, or negative-coordinate extensions lie outside the scope and are not adjudicated.

The paper delivers three positive results and one negative. Positive, first: the process account, comprising the agreement of three accounts, the underdetermination of instantaneous flow by occupancy, and the non-exactness of the cumulative account (Section 3). Second, spectral legislation: single-shell rigid quantization, the universal amplitude bound, the interface subgroup theorem with its degenerate shortcut, together the integer legislation of spectral lattice spacing (Sections 4 and 5). Third, observational compression: the four-layer observation maps and the nesting of their zero fibers (Section 6). Negative: invariance of the core shape and absence of radial support from annular configurations (Section 7). Section 8 corroborates the theorems with differential-rotation statistics under declared measures, Section 9 lists open problems and interface debts, and Section 10 concludes and restates the firewall.

Minimal dependency matrix. All principal results are conditional; their hypotheses are tabulated below and restated in full at each item; nothing is cited in place of writing. Tier notation: D marks definitions and model inputs; T1 marks conditional theorems with explicit hypotheses; T2 marks readings assembled from conditions; T3 marks speculative interface remarks that carry no load; C1 marks exact enumerations, with C1-R meaning independently recomputed and C1-P meaning pending independent rerun. All numerical values cited in this paper are C1-R.

Result	Timing semantics	Occupancy	Library	Weights	Tier
Three-account agreement (3.1)	synchronous	arbitrary	legal transfers	per-shell vector	T1
Non-exactness (3.3)	synchronous	saturated core	contains a prograde rotation	any positive weights	T1
Two budgets (3.5)	both	arbitrary	per branch	explicit $\alpha$	T1
Directed word length (3.6)	synchronous	saturated core	$\mathsf A_1$	edge-step	T1
Rigid quantization (4.1)	synchronous	saturated	occupancy-identity	per-shell	T1
Amplitude bound (4.3)	synchronous	arbitrary	legal transfers	edge-step	T1
Interface subgroup (5.2)	synchronous	saturated core	occupancy-identity	edge-step	T1
Spectral lattice (5.4)	synchronous	saturated core	full occupancy-identity	edge-step	T1
Observation layers (6.1)	synchronous	arbitrary	legal transfers	declared $\alpha$	T1
Core and annulus (7.1 to 7.3)	single-hop and synchronous	non-smooth states	per clause	none	T1
Differential-rotation statistics (8)	synchronous	saturated core	declared measures	$\alpha=\mathbf 1$	C1-R and T2

Guide: Section 2 fixes the model, timing, and conventions; Section 3 establishes the process account and its non-exactness, closing the distinction between process and state with the directed word length; Section 4 gives the single-shell spectrum and its sign symmetry; Section 5 the interface subgroup and rigidity breaking; Section 6 observational compression; Section 7 the negative results; Section 8 the statistics; Sections 9 and 10 close.

2 Model, Timing, and Conventions

Every item in this section is a definition or model input, tier D; only Lemma 2.4 is a truth-apt claim, tier T1.

Definition 2.1 (onion graph and occupancy). An onion graph $G$ consists of shells $j=0,\dots,R$, shell $j$ having capacity $C_j\ge1$ with cells $(j,i)$, $i\in\mathbb Z_{C_j}$. Tangential edges join ring neighbours within a shell: a cycle for $C_j\ge3$, a single edge for $C_j=2$, none for $C_j=1$. Radial edges join each cell $(j,i)$ with $j\ge1$ to its inward parents $\mathrm{par}(j,i)\subseteq\mathbb Z_{C_{j-1}}$, at least one. Occupancy is $n\in\{0,1\}$ per cell ($n_{\max}=1$); a configuration $x$ is an occupancy function with token number $N=\sum x$. Shells with $C_j\le2$ are called degenerate. The default parent construction, inherited from [P3], is primary-plus-successor; every result states its own hypotheses and none relies on the default.

Definition 2.2 (timing and legal transfers). The model carries two timing semantics. (a) Sequential single hop: one token moves along one incident edge; the move is legal iff the source cell is occupied and the target cell empty. (b) Synchronous composite: one tick applies a move set $\theta=\{a\to b\}$, legal iff four clauses hold: every move runs along an incident edge; each source emits at most one move; each target receives at most one move; every moved token starts from an occupied cell and the final occupancy of every cell remains in $\{0,1\}$. The move set of a legal synchronous transfer is a partial permutation on the graph, and its support decomposes into vertex-disjoint directed paths and closed chains; the occupancy-identity transfers are exactly the unions of closed chains (occupancy identity leaves a partial permutation no open path, so the support is purely closed chains). (c) Naming: a transfer with empty move set is the empty transfer; transfers that are occupancy-identity yet have nonempty move sets leave the state unchanged while the account may be nonzero, and their existence is the forerunner of the non-exactness of Section 3. Which semantics each result uses is recorded in the dependency matrix. This clause restates, self-containedly, the timing conventions of the earlier papers ([P3] Definition 3.0).

Definition 2.3 (sequential structure). A primary-parent selector picks one inward parent $p_j(i)\in\mathrm{par}(j,i)$ per cell. Interface $j$ is sequential if there is a non-decreasing lift $\tilde p_j:\mathbb Z\to\mathbb Z$ with $\tilde p_j(i+C_j)=\tilde p_j(i)+C_{j-1}$. Increments are $g_i=\tilde p_j(i{+}1)-\tilde p_j(i)\ge0$ with period sum $\sum_{i

Lemma 2.4 (transport of orientation). Under sequential interfaces, pick either ring direction on the innermost non-degenerate shell and recursively choose coordinates outward so that each $\tilde p_j$ is non-decreasing; the whole tail acquires an orientation, and reversing the initial choice reverses everything, so the orientation is unique up to a global flip. Proof: order preservation of the lift transports the direction interface by interface; the two initial choices are the two global flips, and no circularity of direction enters. $\blacksquare$ [T1; the hypothesis is the sequential structure itself.]

Definition 2.5 (orientation profile and projection). Fix the orientation profile $\omega$ by Lemma 2.4. The prograde successor map is defined only on shells with $C_j\ge3$; the orientation projection $\tau$ assigns $+1$ to prograde tangential moves, $-1$ to retrograde ones, and $0$ to radial moves and to tangential moves on degenerate shells.

Definition 2.6 (three quantities and initial values). Per transfer, the single-tick shell net flow is $\ell_j(\theta)=\sum\tau(m)$ over tangential moves in shell $j$; the cumulative account is $\Lambda_j(T)=\sum_{t\le T}\ell_j(\theta_t)$ with $\Lambda_j(0)=0$; the tracer unwound angular step account $w_{a,j}$ counts, per token, its signed tangential steps in shell $j$ in edge-step units, with $w_{a,j}(0)=0$; only on closed sojourn segments, where $w_{a,j}/C_j\in\mathbb Z$, is it called a winding number. Physical angular momentum $J_{\rm phys}$ is not a quantity of this paper; it belongs to the interface.

Definition 2.7 (vector account and explicit weights). The per-shell vector account $\boldsymbol\ell=(\ell_j)$ is the primary object; scalar totals exist only with explicit positive weights, $\ell_\alpha=\sum_j\alpha_j\ell_j$, $\alpha_j>0$. Three gauges: the edge-step gauge $\alpha=\mathbf 1$, combinatorially privileged since every move counts an integer; the turn gauge $\alpha_j=1/C_j$; the physical gauge $\alpha^{\rm phys}$, unestablished, deferred to the interface. A shared orientation grants only sign comparability across shells; magnitude comparison requires a declared weight.

Definition 2.8 (spin-order lattices). Four names: $\mathsf A_1$, prograde tangential moves only, radial and retrograde forbidden; $\mathsf A_2$, every atomic move has $\tau(m)\ge0$; $\mathsf A_3^\alpha$, every transfer has $\ell_\alpha(\theta)\ge0$; $\mathsf A_0^\alpha$, every transfer has $\ell_\alpha(\theta)=0$. $\mathsf A_0^\alpha$ is the zero fiber of $\mathsf A_3^\alpha$, listed as the neutral sector, not an independent fourth clamp.

Definition 2.9 (four observation layers). Four deterministic compression maps: the gross move record $\mathcal O_{\rm gross}(\theta)=M(\theta)$; the tangential net edge-current field $J^\omega_\theta$, defined on tangential edges of non-degenerate shells with the positive direction from the orientation profile, $J^\omega_\theta(e)=\#(e_+)-\#(e_-)$; the shell net flow $\ell_j(\theta)=\sum_{e\subset\text{shell }j}J^\omega_\theta(e)$; the weighted total $\ell_\alpha(\theta)$. The nesting and strictness of the zero fibers are the subject of Section 6; the kernel semantics is inherited from [P3] Section 6.

Definition 2.10 (generalized defect and the sequential trichotomy). On interface $j$, any inward parent $q$ of cell $i$ carries the offset $\delta_q(i)=(q-p_j(i))\bmod C_{j-1}$ and the generalized defect $D(i,q)=i-\tilde p_j(i)-\delta_q(i)$; at the primary parent $\delta=0$. Sequential trichotomy: if the primary map is non-constant the lift is unique and the increment vector well defined; if constant, a sequential lift exists but is not unique, the increment formula does not apply, and all cells share one parent; if the period sum is a proper multiple of $C_{j-1}$ the map is non-sequential (crossing).

3 Process Accounts, Non-Exactness, and the Directed Word Length

This section defines the account on legal transfers, proves that occupancy does not determine the instantaneous flow, proves that the cumulative account is not exact, and closes the distinction between process and state quantities with the directed word length. Hypotheses are restated at each item.

Proposition 3.1 (three accounts agree). Hypotheses: synchronous composite legality (Definition 2.2 b); the tracer rule that each move is charged to the token it carries; the orientation projection vanishes on both sides of degenerate shells (Definition 2.5). Then per tick, $$\sum_{e\subset\text{shell }j}J^\omega_{\theta_t}(e)=\ell_j(\theta_t)=\sum_a\bigl(w_{a,j}(t)-w_{a,j}(t-1)\bigr)\qquad(\forall j),$$ that is, the net edge-current account, the shell account, and the summed tracer unwound accounts take the same value each tick; the first equality is by Definition 2.9, the second is this proposition. Proof. Each tangential move carries exactly one token, and its $\tau$ enters simultaneously the net current of its edge, the net flow of shell $j$, and the unwound account of its token; the three sums are one finite sum in different orders, and exchanging the order gives the claim. Radial moves contribute zero to all three accounts, so tokens crossing shells create no boundary terms. $\blacksquare$ [T1]

Lemma 3.2 (bare occupancy does not determine flow). There exist a configuration $x$ and two legal transfers $\theta,\theta'$ from $x$ with $\boldsymbol\ell(\theta)\neq\boldsymbol\ell(\theta')$. Proof. On a non-degenerate saturated shell, the prograde and retrograde full-shell rotations are both legal occupancy-identity transfers (closed chains, Definition 2.2), with accounts $+C_j$ and $-C_j$. Hence the single-tick flow is not a function of the occupancy state. $\blacksquare$ [T1]

Theorem 3.3 (non-exactness). Hypotheses: the library is the occupancy-identity class of saturated transfers (Definition 2.2 c), containing at least one prograde rigid rotation of a non-degenerate saturated shell available to scheduling; the weight $\alpha$ is any positive weight. (i) Occupancy layer: no real-valued function $\Psi$ of occupancy states satisfies $\ell_\alpha(\theta)=\Psi(x_{\rm fin})-\Psi(x_{\rm init})$ for all library transfers, since the library contains occupancy-identity transfers with account $\alpha_jC_j\neq0$ (rotations), for which the right side is identically zero. (ii) Labelled layer: no such function of labelled states exists either, since $C_j$ prograde rotations form a closed history restoring all labels while accumulating $C_j\cdot\alpha_jC_j\neq0$, and any labelled-state function has zero net change along a closed history. (iii) Finite extensions: any finite state extension $E$ admitting an infinite lift of the rotation history has no exact primitive, since the lifted orbit inside finite $E$ is eventually periodic, one period accumulates a positive multiple of $\alpha_jC_j$, and any function on $E$ changes by zero over a period. $\blacksquare$ [T1]

Corollary 3.4 (sharpening). Finite states can carry orientation, phase classes modulo $C_j$, and discrete flow tiers; what the theorem excludes is only the global existence of a real-valued exact primitive; any extension that integrates the unbounded real-valued cumulative account exactly must be infinite. [T1]

Proposition 3.5 (two budgets). Radial side: the budget of inward atomic moves is finite and equals the potential drop $\Delta\Phi_r$, inherited from the ledger identity, Corollary 3.1a of [P3], together with its potential exclusion theorems; not reproved here. Tangential side, each branch conditional on the existence of a repeatable transfer: if the unclamped occupancy-identity library contains both rotations of some non-degenerate saturated shell, the cumulative account can run to $\pm\infty$; if the $\mathsf A_3^\alpha$ library contains a repeatable $\theta_+$ with $\ell_\alpha(\theta_+)>0$, it runs one-sidedly unbounded from zero; if the $\mathsf A_0^\alpha$ library contains a repeatable $v\in\mathrm{Spec}(\boldsymbol\ell)\setminus\{0\}$ with $\alpha\cdot v=0$, per-shell accounts oscillate unboundedly; absent such a vector no conclusion is drawn on that branch. Main sentence: radial descent has a finite countdown; the angular cumulative account has no countdown supplied by $\Phi_r$. [T1; the reading "the hole can rotate forever but cannot fall forever" is an assembled conditional reading, tier T2, valid where the branch hypotheses hold.]

Proposition 3.6 (directed word length; same group, different account). Hypotheses: saturated core; the $\mathsf A_1$ library, whose nonempty transfers are exactly disjoint unions of prograde rigid rotations of non-degenerate shells. Writing $\mathbf k=(k_j)\in\prod_j\mathbb Z_{C_j}$, $0\le k_jEqual reachable groups do not imply equal process cost: the distinction between process and state quantities closes here. [T1]

4 Rigid-Rotation Spectrum, Amplitude Bound, and Sign Symmetry

Theorem 4.1 (single-shell rigid quantization). Hypotheses: shell $j$ non-degenerate ($C_j\ge3$) and saturated; the transfer $\theta$ is a legal synchronous transfer whose moves lie entirely on the tangential edges of shell $j$ and which is occupancy-identity. Then every nonempty connected component of the move graph of $\theta$ is a directed cycle on the ring, and on a chordless ring the only directed cycles are the two-cycle on one edge (adjacent swap), the full prograde rotation, and the full retrograde rotation. Hence $$\ell_j(\theta)\in\{0,\pm C_j\},$$ with $\pm C_j$ attained iff $\theta$ contains the corresponding full-shell rigid rotation. Proof. Saturation and occupancy identity make the restriction a permutation of the ring cells; the partial-permutation clauses decompose it into directed cycles; on a chordless ring a cycle's support is one edge or the whole ring. $\blacksquare$ [T1]

Proposition 4.2 (single-tick spectrum of the radial lattice). Hypotheses: all shells saturated; the library is the disjoint union of intra-shell tangential occupancy-identity transfers (no radial moves). The vector spectrum is the product $\prod_{C_j\ge3}\{0,\pm C_j\}$ with degenerate components identically zero, and the weighted total spectrum is the signed subset-sum set $\{\sum_j\varepsilon_j\alpha_jC_j:\varepsilon_j\in\{-1,0,1\}\}$. Proof: shells do not interact; apply Theorem 4.1 shellwise. $\blacksquare$ Note: holes in the total spectrum depend on $\{C_j\}$ and $\alpha$; in the edge-step gauge the two-shell tower with capacities $(3,4)$ misses $\pm2,\pm5,\pm6$; the exact enumeration (C1-R) is in Appendix C. Spectra with holes are combinatorial spectra, not energy spectra. [T1]

Proposition 4.3 (universal amplitude bound and equality). Hypotheses: any legal synchronous transfer, arbitrary occupancy, open chains with vacancies included. In the edge-step gauge, $$|\ell_{\mathbf 1}(\theta)|\le\sum_{C_j\ge3}C_j=:M.$$ Equality holds iff the restriction of $\theta$ to every non-degenerate shell is a full-shell rigid rotation of one and the same direction; the degenerate sector may carry disjoint zero-account activity besides. Proof. A partial permutation lets each cell emit at most one move; only tangential moves on non-degenerate shells are charged, one unit each, so the bound is the count of non-degenerate cells. Equality forces every such cell to emit a same-sign tangential move; under the occupancy constraint this means, shell by shell, "every cell moves to its prograde (retrograde) neighbour", i.e. a full rigid rotation, and conversely. Degenerate moves are uncharged and do not obstruct equality. $\blacksquare$ [T1]

Definition 4.4 (sign involution). On occupancy-identity transfers (unions of closed chains, Definition 2.2 c), set $\rho(\theta)=\{b\to a:a\to b\in\theta\}$, reversing every move. Then $\rho(\theta)$ is again a legal occupancy-identity transfer and $\boldsymbol\ell(\rho\theta)=-\boldsymbol\ell(\theta)$. [D]

Lemma 4.5 (involution count and fixed points). On the occupancy-identity library $\rho$ is an involution, so the enumeration satisfies $$N_{\rm all}=N_0+2N_+,$$ with $N_0$ the number of zero-account transfers and $N_+$ the positive ones, positives and negatives paired. Fixed points: $\mathrm{Fix}(\rho)$ is exactly the unions of swaps; in the radial-move-free library $\mathrm{Fix}(\rho)=\{\boldsymbol\ell=\mathbf 0\}$ by Theorem 4.1; in general only $\mathrm{Fix}(\rho)\subseteq\{\boldsymbol\ell=\mathbf 0\}$, strict iff the library contains a compensating transfer with $\rho(\theta)\neq\theta$ and $\boldsymbol\ell(\theta)=\mathbf 0$. Proof: involution and sign reversal are immediate movewise; the count is orbit length one or two; fixed points are movewise self-reverse, i.e. swaps. $\blacksquare$ Note: for the onion of capacities $(1,3,4)$, the full library gives $1995=521+2\times737$ and the radial-free library $53=27+2\times13$; exact enumeration (C1-R) in Appendix C. [T1]

Lemma 4.6 (amplitude bound for a single shell with vacancies). Hypotheses: shell $j$ non-degenerate with $v\ge1$ vacancies; a legal synchronous transfer whose moves lie entirely on the tangential edges of shell $j$. Then $|\ell_j(\theta)|\le C_j-v$, with equality attained by the synchronous shift of all $C_j-v$ tokens. Proof. Each source emits at most one move and must be occupied, so the tangential step count is at most the token number $C_j-v$, each step $\pm1$, giving the bound; for equality let every token move to its prograde neighbour: each target receives at most one move and the final occupancy is the shifted configuration, still in $\{0,1\}$, hence legal, with account $C_j-v$. $\blacksquare$ [T1]

Three end-of-section clarifications. First, the quantization of Theorem 4.1 rests on saturation, shell-confined moves, and the chordless ring; remove any one and it can fail, and how it fails is the subject of the next section. Second, total spectra depend on the weight: the edge-step and turn gauges shape the same library differently, and magnitude comparisons must declare their gauge. Third, all spectra here are discrete combinatorial spectra; any bridge from combinatorial to physical spectra belongs to the interface and is not built here.

5 Interface Subgroups, Degenerate Shortcuts, and Rigidity Breaking

Closed-chain interface accounts are pinned to an integer subgroup, and its algebraic lattice spacing is legislated by capacities, increments, and parent offsets. This is the section's headline and the paper's second principal result. With the single-shell spectrum rigid (Theorem 4.1), once cross-shell moves enter the library the question is not how wide the library opens but which numbers the interface generates: on which lattice the new account values fall and with what spacing. The section first exhibits three families of primitive cycles as generators, then proves the local subgroup theorem and the global spacing, and closes with the spectral-lattice corollary and the ladder of spacings.

Lemma 5.1 (construction of primitive cycles). Setting: interface $j$ with the notation of Definitions 2.3 and 2.10; the library is the legal occupancy-identity class (Definition 2.2 c); all shells saturated. (i) Shared-parent three-cycle: if ring neighbours $i,i{+}1$ share an inward parent $q$ (through any parent), the closed chain $(j,i)\to(j,i{+}1)\to(j{-}1,q)\to(j,i)$ is legal with account $+1$; reversed, $-1$. (ii) Increment cycle: if the interface is sequential with non-constant primary map and both shells non-degenerate, then for each column $i$ the closed chain $\gamma_i$, descending from column $i{+}1$ to $p(i)+g_i$, walking $g_i$ retrograde steps below to $p(i)$, ascending to column $i$, and taking one prograde step above, has $g_i+3$ edges and account $1-g_i$; $g_i=0$ recovers the primary form of (i), and $g_i=2$ gives a five-edge cycle of account $-1$. (iii) Secondary-parent cycle: if cell $i$ has a secondary parent $q$ with offset $\delta=\delta_q(i)\ge1$, the chain descending from $i$ through $q$, walking $\delta$ retrograde steps below to $p(i)$, and ascending back to $i$ has account $-\delta$; $\delta=1$ gives a three-cycle. (iv) Degenerate clause: if the lower shell has $C_{j-1}\le2$ and the upper is non-degenerate, the orientation projection vanishes below and the constructed cycle has account $+1$ regardless of $g_i$. Proof: each construction is a simple cycle, hence under saturation a permutation of its cell set, satisfying the synchronous clauses; accounts are read off from the projection stepwise. $\blacksquare$ [T1]

Theorem 5.2 (interface subgroup theorem). Hypotheses: both shells non-degenerate; primary map sequential and non-constant; $\theta$ a legal occupancy-identity transfer supported on the shell pair $\{j{-}1,j\}$. Then its edge-step account lies in $$G_j\mathbb Z,\qquad G_j=\gcd\bigl(C_j,\;C_{j-1},\;\{|1-g_i|\}_i,\;\{\delta_q(i)\}_{i,q}\bigr),$$ and $G_j$ is exactly the greatest common divisor of the primitive simple-cycle accounts of the pair. Proof. Let $d$ divide all listed generators. The generalized defect satisfies the two congruences $$D(i{+}1,p(i{+}1))-D(i,p(i))=1-g_i\equiv0,\qquad D(i,q)-D(i,p(i))=-\delta_q(i)\equiv0\pmod d,$$ and its period drift $C_j-C_{j-1}\equiv0$, so all radial endpoints share a residue $c_j$ of $D$ modulo $d$. Define the $d$-valued potential $\psi(j{-}1,x)=x$ and $\psi(j,i)=i-c_j$, both modulo $d$, columns taken in the shared orientation of Definition 2.5. Then along each move: a tangential step changes $\psi$ by exactly its $\tau=\pm1$, the wrap being harmless since $d\mid C$; a radial step is uncharged and changes $\psi$ by $q-(i-c_j)\equiv c_j-D(i,q)\equiv0\pmod d$. The transfer decomposes into vertex-disjoint cycles (Definition 2.2), each of which returns $\psi$ to its start, so the total account is $\equiv0\pmod d$. Generation: $C_j$ and $C_{j-1}$ are realized by full rotations, $|1-g_i|$ by the increment cycles, $\delta_q(i)$ by the secondary cycles, all primitive simple cycles; hence the cycle-account gcd divides the generators and is no coarser than $d$; with the containment this gives equality. $\blacksquare$ [T1]

Corollary 5.3 (global spacing and degenerate shortcuts). Tail clause: if $\theta$ is supported on the non-degenerate tail and every interface satisfies the hypotheses of Theorem 5.2, set $d=\gcd_jG_j$, take the per-interface residues $c_j$, define recursively $s_j-s_{j-1}=c_j\pmod d$ with $s_{j_*}=0$ at the innermost tail shell, and put $\psi(j,i)=i-s_j$ modulo $d$; every radial edge changes $\psi$ by zero, every tangential edge by $\tau$, and every closed chain has account $\equiv0\pmod d$. Global clause: any occupancy-identity legal transfer on the whole graph has account in $G^{\deg}\mathbb Z$ with $$G^{\deg}=\gcd\bigl(d,\;\{1:\text{interfaces with degenerate lower and non-degenerate upper shell}\}\bigr),$$ degenerate interfaces contributing the generator $1$ through Lemma 5.1 (iv), so the containment is trivial there while the generation is real; without degenerate interfaces the tail clause applies. The degenerate core is a unit-generating shortcut: in the fully saturated occupancy-identity closed-chain library with at least one non-degenerate shell, any onion with a one- or two-cell inner shell has global algebraic lattice spacing one, and account quantization survives only on tails away from degenerate hubs. In-class witness: capacities $(2,12,4)$ give tail spacing two, measured tail cycle gcd two, yet a legal shortcut three-cycle of account $+1$ and a legal odd-account transfer $+5$; exact enumeration (C1-R) in Appendix D. $\blacksquare$ [T1]

Corollary 5.4 (spectral lattice). Hypotheses in full: fully saturated core; occupancy identity; synchronous partial permutation; the complete occupancy-identity library; every non-degenerate interface satisfying the hypotheses of Theorem 5.2. Then $$\mathrm{Spec}(\ell_{\mathbf 1})\subseteq G^{\deg}\mathbb Z\cap[-M,M].$$ Registered falsity outside the domain: a single same-shell hop into a vacancy has account $\pm1$ and escapes the lattice already on a tower of global spacing four; the lattice statement holds only for the closed-chain library and not for spectra with vacancies. Proof: Corollary 5.3 with Proposition 4.3. $\blacksquare$ [T1]

Ladder of spacings and the rigidity-breaking criterion. Theorem 5.2 splits the landscape in three. Spacing $G_j=C$ (equal capacities, single parent, increments all one): every interface closed-chain account is winding number times $C$, and the rigidity of Theorem 4.1 survives the cross-shell library. Spacing $1unit spacing implies neither a single-tick $\pm1$ nor a hole-free spectrum: there is a single-parent interface with $G=1$ whose primitive cycle accounts contain no $\pm1$ and whose full spectrum is riddled with holes (C1-R, Appendix D); the bridge from subgroup to packing is an open problem, deferred to Section 9.

Table 5.1 (four interface examples). Proofs and exact enumerations in Appendix D.

Interface structure	Generator profile	$G_j$	Spectral clamp
Equal-capacity single-parent uniform tower	increments all one, no offsets	$C$	spectrum $\subseteq C\mathbb Z$, rigidity survives
One-cell centre	degenerate shortcut	$1$	all cells share the centre, unit tier
Congruent single-parent tower	increments in one residue class	intermediate	spectrum $\subseteq d\mathbb Z$, even-lattice witness on record
Default double parents	offsets all one	$1$	breaking everywhere, confluent with the shared-parent language

6 Observational Compression and Zero Fibers

This short section establishes one thing: angular information degrades along a four-layer deterministic compression, the shape of the loss being exactly a nesting of zero fibers; net flow is the shallowest visible layer, not the only one. The four maps are those of Definition 2.9.

Proposition 6.1 (nesting and strictness of zero fibers). Hypotheses: legal synchronous transfers, arbitrary occupancy, any positive weight. Universal containments $$\ker\mathcal O_{\rm gross}\subseteq\ker J^\omega\subseteq\ker\boldsymbol\ell\subseteq\ker\ell_\alpha,$$ where $\ker$ denotes the class of transfers on which the layer vanishes and $\ker\mathcal O_{\rm gross}$ is the empty transfer. Each strictness is conditional on a witness. First, gross-visible with vanishing edge current: a same-edge adjacent swap (the two directions cancel on one edge) and a purely radial swap are both nonempty with $J^\omega\equiv0$. Second, current-visible with vanishing shell account: on one non-degenerate shell, two disjoint edges carrying one prograde and one retrograde legal hop in the same tick, e.g. capacity five, tokens at columns zero and three, vacancies at one, two, four, moves $0\to1$ and $3\to2$; the two edge currents are $+1$ and $-1$ while the shell account is zero. Third, shell-visible with vanishing total: neutral compensation, conditional on library and weight; in the turn gauge any counter-rotation of two non-degenerate shells, in the edge-step gauge only equal-capacity pairs; libraries without such a vector make this containment non-strict. Proof: containments hold since each layer is a function of the previous; the witnesses are computed layer by layer. $\blacksquare$ [T1]

Proposition 6.2 (three named blind zones). The invisibility of the angular account is confined to the following three observables, each named and finite. First, the occupancy projection is blind to the account: on the saturated core, the all-prograde and all-retrograde rotation schedules have identical occupancy trajectories tick by tick (the saturated constant) while their cumulative accounts diverge with opposite signs; enumerated witness (C1-R) in Appendix C. Second, the occupancy-quotient entropy is blind: the entropy count of [P3] is defined on the occupancy quotient and is invariant under every occupancy-identity transfer. These two follow because their domains are functions of occupancy or its quotient, while occupancy-identity transfers with nonzero account are in the library (Definition 2.2 c, Theorem 3.3). Third, the cross-shell release statistics of whole-shell synchronized ablation are blind to the account, by a separate argument: this statistic is not a pure occupancy function, reading each label's birth shell and release round; yet purely tangential scheduling never changes any label's shell membership, and whole-shell synchronized ablation releases strictly in shell order, so the cross-shell inversion statistics are invariant under the insertion of any tangential schedule; the rotation-blind inversion witness (C1-R) is in Appendix C. $\blacksquare$ [T1]

In sum: a non-vanishing coarse layer forces non-vanishing at every finer upstream layer; a non-vanishing fine layer may vanish under further compression. Visibility of the full move record does not guarantee visibility of the edge current; edge-current visibility does not guarantee shell-account visibility; shell-account visibility does not guarantee weighted-total visibility. That net flow is the shallowest visible layer, and not the only one, means precisely this: whatever leaves a trace at the net-flow layer leaves traces at all upstream layers, while upstream traces need not reach the net flow. This is also the positive reading of the third non-claim of Section 1: the paper does not call all observation blind to the angular sector; the blindness established is the three names of Proposition 6.2, and the upstream edge-current layer sees more.

7 The Angular Account Neither Alters the Core Shape Nor Supplies Radial Support

This section establishes the paper's negative results. Hypotheses are restated per item; both timing semantics appear, each declared.

Lemma 7.1 (mutual reachability under one-way sliding). Hypotheses: a non-degenerate shell $C\ge3$ with fixed occupancy $0

Theorem 7.2 (invariance of the core shape). Hypotheses: fixed token number $N$; sequential single-hop semantics; the free hop library clamped to prograde ($\mathsf A_2$ at the single-hop level, where $\mathsf A_3$ coincides with it since a single move's net is its step). Then the unique smooth sink strongly connected class established for fixed $N$ in [P3], Proposition 3.2, keeps all three of its properties. First, every non-smooth state still has a finite path strictly decreasing $\Phi_r$, since the legality of an inward drop depends on occupancy only, never on orientation. Second, the smooth sector remains closed. Third, the skin shell remains strongly connected under prograde hops, by Lemma 7.1; for shells with $C\le2$ the lemma does not apply and one cites instead the original vacancy-pushing argument in the proof of [P3] Proposition 3.2, valid for all $C\ge2$. Hence a tangential spin-order clamp does not change the shape of the core: existence, uniqueness, and closedness of the smooth sink are untouched. $\blacksquare$ [T1]

Proposition 7.3 (annular configurations supply no radial support). Call $x$ annular if $N>0$, shells $0$ through $h{-}1$ are empty ($h\ge1$), and all occupancy lies in shells $h$ through $H$. Hypotheses: synchronous semantics; the unclamped legal library; the annulus contains at least one legal tangential move (a rigid rotation of a saturated shell $C\ge3$, or a single hop next to a vacancy on an unsaturated shell). Three clauses.

First, channel and budget: the inward channel is always open in the sense that the lowest occupied shell has parents in empty shells; the number of inward atomic moves needed to reach the smooth sector is exactly $\Delta\Phi_r$ (the ledger identity, [P3] Corollary 3.1a) and is unchanged by any tangential interleaving, while the number of ticks can be stretched arbitrarily. What is invariant is the step count, not the tick count.

Second, persistence and account: every annulus with a legal tangential move admits an infinite purely tangential schedule that never schedules a radial step, so the cavity persists by scheduling refusal alone. The zero-account branch is an additional condition: if the library also contains a repeatable nontrivial zero-account transfer (a saturated-shell adjacent swap, a swap of two occupied neighbours, or any other nonempty element of $\ker\boldsymbol\ell$), the same cavity persists under tick-by-tick zero account. This is not universal; the precise condition is that either some other shell carries a repeatable nonzero counter-signed tangential step, or the shell itself already has a zero-account transfer; the registered counterexample is an annulus consisting of this one shell alone, one token and two vacancies at $C=3$, where every nonempty tangential step has $|\ell|=1$, no swap is available, and no tick-by-tick zero-account schedule exists. The account value is therefore no switch on the radial channel; rotation is purely concurrent.

Third, transience: every nonempty annulus, having an empty inner layer, is non-smooth, hence transient in the dynamics of Theorem 7.2 and outside the closed recurrent class.

Joint statement: the shape can be maintained indefinitely by scheduling, while nothing in the model lets the angular account enter radial legality or the radial budget; the lifetime of an annulus is the restraint of a scheduler or the fiat of a clamp, never a work of spin. The name "no centrifugal term" means exactly this sentence and makes no universal physical negation. $\blacksquare$ [T1]

8 Differential-Rotation Statistics Under Conditional Measures

This section corroborates the theorems by sampling statistics under declared measures. Every conclusion is relative to its measure, no measure is natural, and nothing here is claimed as a law; all values are C1-R (Section 1), with full protocols, seeds, exact moments, and complete tables in Appendix E and recomputation material in Appendix F.

Measures and protocols. The component universe comprises both rigid rotations of every shell, all swaps, and all simple cycles of length at most six taken in both orientations, quotiented by support and account vector, each element passing the legality clauses of Definition 2.2. The three families are the default-double-parent onion of capacities $(1,3,4,5,6)$, the congruent single-parent tower $(12,4)$, and the equal-capacity single-parent uniform tower $(4,4,4)$, with coordinates given by their global algebraic lattice spacing (Corollary 5.3). The free measure is uniform on the universe with independent ticks; the clamped runs use $\mathsf A_2$ per Definition 2.8, movewise; the compensation runs draw compensating pairs at declared mixing rates; the neutral runs use half zero-account single components and half disjoint zero-sum pairs; the skin run is a random-cell sequential sweep on a single ring, a token possibly moving twice per tick. Wherever exact moments exist, the exact value from direct summation over the universe is canonical and the empirical value is a check.

Table 8.1 (summary of differential-rotation statistics; pure combinatorial-measure sampling, not time evolution).

Object	Setting	Highlights
Normalized shell drifts, equal or not	$\mathsf A_2$ uniform component measure, three families	congruent tower equal ($0.164$ vs $0.166$); uniform tower mid-shell double ($0.200,0.372,0.197$); default family fast inside, slow outside ($0.071,0.152,0.116,0.036$)
Sign of neighbour-shell covariance	free uniform component measure	congruent tower strongly anticorrelated $-0.750$; uniform tower strongly correlated $+0.351$; default family mixed ($-0.004,+0.023,-0.067$)
Two forms of compensation	$\mathsf A_3$ mixture, rates $0,\tfrac14,\tfrac12$	intra-component base rate $0.440$; pairwise compensation rising to $0.968$; total account always ascending
Neutral internal flow strength	zero-account mixture	mean $2.608$ (interval $2.582$ to $2.635$), total account identically zero
Per-tick increment variance rate	same	rising outward, $0.661$ to $1.259$
Mobility-curve peak	single ring $C=12$, random-cell sweep	peak at half filling ($v=6$, mobility $0.356$), both ends declining (Appendix E)

Four readings, each prefixed by its measure, all tier T2.

First. Under the length-six, support-and-account-quotiented, uniform component measure with $\alpha=\mathbf 1$, the sign of the neighbour-shell covariance is a scale weighing the truncated cycle census: compensating cycles dominating gives anticorrelation, same-sign long cycles dominating gives correlation, and the two in balance give mixed signs; the three families of Table 8.1 are its three readings. No claim of a natural coupling constant for general graph families is made.

Second. Under the same measure clamped by $\mathsf A_2$ of Definition 2.8, when the only positive-account components are rigid rotations, the normalized shell drifts are equal; once all-prograde mixed cycles enter the library, the profile is generically differential. The equality comes from the combinatorial symmetry of library and sampling measure, not from any dynamical equation introduced here.

Third. Under the declared $\mathsf A_3$ mixture at $\alpha=\mathbf 1$, compensation splits into two forms: intra-component (a nonnegative single component with negative shell entries) and pairwise (disjoint positive and negative components in one tick); local retrogression is financed by global gain, and the clamp forbids total retreat, never differential rotation. This is the statistical quantification of the one-sided branch of Proposition 3.5; changing the weight changes the sufficiency criterion.

Fourth. Under the half-zero-single, half-zero-pair measure, a static total does not imply static shells: the total account is identically zero every tick while the internal flow strength exceeds two per tick and per-shell accounts diffuse linearly. This quantifies the neutral branch of Proposition 3.5 and does not extrapolate to all $\mathsf A_0$ schedules.

9 Open Mathematical Problems and Physical Interface Debts

Problem 9.1 (packing). A natural conjecture says that after rigidity breaking the occupancy-identity spectrum must fill the interval $[-M,M]$. The conjecture is false: unit lattice spacing does not imply a hole-free spectrum, the boundary marker of Section 5 being on record, and the claim is withdrawn. The defensible form narrows to: for default-double-parent growing onions and for single-parent floor growing onions, the occupancy-identity spectrum is the full segment $[-M,M]$. Both families have global spacing one: the former by the secondary-parent offsets (Table 5.1); the latter likewise, since floor growth makes capacities strictly increase across interfaces, the increments of the non-decreasing lift sum to $C_{j-1}$ over $C_j$ terms, forcing a zero increment, hence a shared-parent pair and a $\pm1$ cycle (Lemma 5.1 (i)). Exhaustion of three families and constructive certificates for one large family support the conjecture (C1-R, Appendix D); a general packing proof is not established. Tier: Open, no load carried.

Problem 9.2 (complete characterization of primitive $\pm1$ cycles). The four constructions of Lemma 5.1 are sufficient, the subgroup of Theorem 5.2 is the obstruction; the intermediate band is real: there is a single-parent interface with spacing one whose primitive cycle accounts contain no $\pm1$ (Appendix D). The exact characterization of when a primitive $\pm1$ cycle exists is open.

Problem 9.3 (non-sequential account theory). Crossing primary maps admit no lift and the increment language fails; small examples show rich cycle accounts. An account theory over non-sequential parent structures is open.

The angular voice of the three eras (T2). The three eras of the earlier papers were told radially; their angular voice can be assembled sentence by sentence from the theorems above, each sentence citing its part. Infall era: the tangential bias of falling matter writes the account tick by tick while radial steps are uncharged, the two ledgers running concurrently (Definition 2.5, Proposition 3.1). The saturation tick: once a shell fills, single-hop semantics dies and closed chains become the only carrier (Definition 2.2); a single shell with $v\ge1$ vacancies has single-tick account bounded by $\pm(C-v)$ (Lemma 4.6); at the saturation tick the belly of the spectrum dies as its extremes are born, the interior values all extinguished and $\pm C$ first reachable (Theorem 4.1). Tracer shell coordinates freeze from then on (the label radial lock, [P3] Lemma 3.3), and the occupancy history decouples completely from the account history (Proposition 6.2, first zone). Frozen-cycling era: rigid rotation can be kept resident by scheduling, the single-shell spectrum is $\{0,\pm C_j\}$, and cross-shell spacing is legislated by the interfaces (Theorems 4.1 and 5.2). Evaporation era: the cross-shell release statistics of whole-shell synchronized ablation inherit the order of [P3] and are blind to the purely tangential account (Proposition 6.2, third zone); the single-ring mobility curve shows the fully occupied end and the depleted end as the two low-activity wings of one combinatorial mobility curve (Table 8.1, last row; C1-R and T2). Naming these two wings collapse and evaporation is a toy-level interface label only, tier T3, carrying no load.

Interface conjecture (T3, the whole paragraph load-free). The momentum layer of this model is truncated by convention and deferred to the interface, following the division of [P2]. From this one may conjecture that the four absences called no frame dragging, no ergoregion provenance, no inheritance law, and no centrifugal term share this single truncation as their root. The boundary of what is actually proved here, item by item: no centrifugal term has a T1 exact meaning, namely the joint statement of Proposition 7.3 that nothing in the model lets the angular account enter radial legality or the radial budget; the in-model form of no dragging follows from the same proposition while the physical name is not established; the provenance of ergoregions and the inheritance of angular momentum are not established at all and remain interface-level observations, since $J_{\rm phys}$ itself is unestablished and the $J$ of collapse has nowhere in this model to be delivered. That the four share one root is a conjecture, not a theorem of this paper.

10 Conclusion and Firewall

Three concluding sentences. First, angular net flow lives at the transfer level, and its cumulative account is in general not exact. Second, the rigid-rotation spectrum and its interface lattice spacing are legislated jointly by the local graph structure and the radial clauses. Third, angular activity neither alters the shape of the smooth core nor supplies radial support in this model.

Four interface debts, listed singly. Shell weights $\alpha^{\rm phys}$: the paper declares only the two combinatorial gauges, edge-step and turn; the physical gauge is unestablished. Time scale: no bridge from ticks to physical time is built, and every rate here is a per-tick rate. Energy interface: no link from accounts to energy is established, and spectra with holes are combinatorial, not energetic. Delivery of $J_{\rm phys}$: physical angular momentum is not a quantity of this paper; its state character, conservation, and inheritance lie outside the scope.

The scope sentence, restated: this paper establishes a combinatorial angular-transport account; the shell weights, time scale, and energy interface required to pass from this account to physical angular momentum are all left unestablished. The four non-claims of Section 1 hold throughout as at the start.

Appendix A Primitive-Cycle Constructions and Legality

Constructions are given in coordinate lists. Legality is checked clause by clause against Definition 2.2 (b): at most one move out per source, at most one in per target, final occupancy unchanged; under saturation each construction is a permutation of its cell set, so the clauses hold automatically.

(a) Shared-parent three-cycle (Lemma 5.1 (i)): moves $(j,i)\to(j,i{+}1)$, $(j,i{+}1)\to(j{-}1,q)$, $(j{-}1,q)\to(j,i)$; projections $+1,0,0$; account $+1$; reversal gives $-1$. (b) Increment cycle $\gamma_i$ (ii): $(j,i{+}1)\to(j{-}1,p(i){+}g_i)$; $g_i$ retrograde steps below to $(j{-}1,p(i))$; $(j{-}1,p(i))\to(j,i)$; $(j,i)\to(j,i{+}1)$; $g_i{+}3$ edges, projections $0,(-1)^{\times g_i},0,+1$, account $1-g_i$; $g_i=0$ reduces to (a) in primary form, $g_i=2$ is a five-edge cycle of account $-1$. (c) Secondary cycle (iii): $(j,i)\to(j{-}1,q)$; $\delta_q(i)$ retrograde steps below to $(j{-}1,p(i))$; $(j{-}1,p(i))\to(j,i)$; account $-\delta_q(i)$; $\delta=1$ a three-cycle. (d) Degenerate clause (iv): with $C_{j-1}\le2$ the projection vanishes below, so the account of (b) is $+1$ regardless of $g_i$. All cycles are simple with pairwise distinct support cells, hence primitive.

Appendix B Full Proof of the Global Recursive Potential

Full proof of the tail clause of Corollary 5.3. Suppose every tail interface satisfies the hypotheses of Theorem 5.2 and set $d=\gcd_jG_j$. First, existence of per-interface residues: by the two congruences in the proof of Theorem 5.2, all generalized defects at radial endpoints of interface $j$ are congruent modulo $d$; call the value $c_j$; it is independent of the lift representative since the period drift $C_j-C_{j-1}\equiv0\pmod d$. Second, recursion of shell constants: put $s_{j_*}=0$ at the innermost tail shell and $s_j\equiv s_{j-1}+c_j\pmod d$; well defined, one constant per interface. Third, the potential: $\psi(j,i)=i-s_j\bmod d$ with columns in the shared orientation of Definition 2.5; ring wrap-around is harmless since $d\mid C_j$. Fourth, movewise increments: a tangential step in shell $j$ changes $\psi$ by $\pm1$, exactly its projection and its account; a radial step $(j,i)\to(j{-}1,q)$ is uncharged and changes $\psi$ by $q-s_{j-1}-(i-s_j)\equiv q-i+c_j\equiv c_j-D(i,q)\equiv0\pmod d$, ascents symmetrically. Fifth, closure: an occupancy-identity legal transfer decomposes into vertex-disjoint closed chains (Definition 2.2), each returning every cell and hence $\psi$ to its start, so the total account is $\equiv0\pmod d$. $\blacksquare$ The global clause follows by adjoining the generator $1$ of the degenerate interfaces (Appendix A (d)), containment trivial and generation real.

Appendix C Exact Enumeration Checks for the Analytic Theorems (C1-R)

All numbers come from one standard run of the scripts listed in Appendix F and have passed independent recomputation (the meaning of C1-R, Section 1). Onion of capacities $(1,3,4)$: occupancy-identity single components $642$ (swaps $18$, directed cycles $624$), legal composites $1995$; involution counts $1995=521+2\times737$ and, radial-free, $53=27+2\times13$ (witnesses of Lemma 4.5); full spectrum the segment $[-7,7]$; minimal positive closed-path account $9$ for the directed word length (instance of Proposition 3.6). In the edge-step gauge the two-shell radial-lattice spectrum of capacities $(3,4)$ misses $\pm2,\pm5,\pm6$ (note to Proposition 4.2). Capacities $(1,3,4,5)$: full spectrum $[-12,12]$, $25$ points. Capacities $(1,3,4,5,6)$: amplitude $M=18$, constructive certificates for all values $0$ to $18$. Capacities $(1,2,8)$: degenerate-shell projection verified zero, full spectrum the segment of amplitude $8$. Interface-subgroup double computation: cycle-account gcds of the six interfaces of the standard growing families all equal the analytic formula of Theorem 5.2, all $1$; default double-parent equal towers $C=3$ to $6$ likewise. Blind-zone witnesses: the pair of schedules with identical occupancy trajectories and diverging accounts, and the rotation-blind ablation inversion order, are both on record (witnesses of Proposition 6.2).

Appendix D Boundary Examples and Counterexamples

(1) Equal-capacity single-parent uniform tower (Table 5.1, row 1): capacities $(4,4,4)$, identity primary map, increments all one, no offsets; Theorem 5.2 gives $G=\gcd(4,4,0)=4$; enumeration confirms all cycle accounts in $4\mathbb Z$ and the full spectrum likewise; rigidity survives the cross-shell library, the account being winding times capacity. (2) One-cell centre (row 2): $C_0=1$ makes all shell-one cells share the centre; shared-parent three-cycles give $\pm1$ and global spacing one. (3) Congruent single-parent tower (row 3): capacities $(12,4)$, primary $p(i)=3i$, increments all $3$, $G=\gcd(4,12,2)=2$; enumeration confirms the full spectrum in $2\mathbb Z$; the two-interface tower $(36,12,4)$ gives $2$ per interface with all bounded-length cycle accounts even. (4) Default double parents (row 4): offsets all one enter the generators, $G=1$; the secondary three-cycles of capacities $(4,4)$ with accounts $\mp1$ are the witness, confluent with the shared-parent language. (5) Intermediate band (Problem 9.2): capacities $(7,5)$, single parent, increments $(3,1,1,1,1)$, with neither a shared ring-neighbour parent nor an increment in $\{0,2\}$; $G=1$ while the primitive cycle accounts are $\{0,\pm2,\pm5,\pm7\}$ without $\pm1$; the full spectrum is $\{0,\pm2,\pm5,\pm7,\pm12\}$, amplitude $12$ with sixteen holes: the boundary marker that unit spacing implies no packing (Problem 9.1). (6) In-class witness of the degenerate shortcut (Corollary 5.3): capacities $(2,12,4)$, tail interface spacing $2$ with measured tail cycle gcd $2$, yet a legal shortcut three-cycle of account $+1$, a legal transfer of odd account $+5$, and full-graph cycle gcd $1$. (7) Sequential trichotomy (Definition 2.10): the constant-parent $(3,3)$ example has a non-unique lift, all cells share one parent, and $G=1$ follows directly; the crossing $(3,3)$ example with $p=(0,2,1)$ admits no lift, cycle accounts $\{\pm1,\pm2,\pm3,\pm4\}$, registered as material for Problem 9.3.

Appendix E Protocols and Full Tables of the Differential-Rotation Statistics

Protocols. Component universe: both rigid rotations of every non-degenerate shell, all swaps, and all simple cycles of length at most six in both orientations, quotiented by support and account vector, each element passing the legality clauses of Definition 2.2. Families and coordinates as in Section 8. Sampling: independent local random generators per block with registered seeds, free $501$, clamped $503$, compensation $507$, neutral $509$, skin $511+v$; independent ticks, $T=20000$ (four thousand per $v$ in the skin block). Environment Python 3.12.3, NumPy 2.4.4; deterministic blocks reproduce bitwise, stochastic blocks bitwise in the declared environment. Bootstrap: independent resampling, $B=400$, quantiles $2.5$ and $97.5$. Wherever exact moments exist, the exact value from direct summation over the universe is canonical and empirical values are checks.

Full tables. Universe: components $1398/32/114$ with rotations $8/4/6$; global spacings $1/2/4$, divisibility of all universe accounts asserted per family. Free block: per-shell drifts exactly zero, empirical agreeing; neighbour-shell covariances (exact) default family $(1,2)\,{-}0.004$, $(2,3)\,{+}0.023$, $(3,4)\,{-}0.067$, congruent tower $(0,1)\,{-}0.750$, uniform tower $(0,1)$ and $(1,2)$ both $+0.351$; rotation shares exact $0.006/0.125/0.053$. Clamped block: $\mathsf A_2$ components $537/6/35$, zero-account shares $0.201/0.670/0.231$; normalized drifts default family $(0.0709,0.1519,0.1155,0.0363)$, congruent tower $(0.1641,0.1655)$, uniform tower $(0.1997,0.3720,0.1972)$. Compensation block (rates $0/0.25/0.5$): compensation rates $0.000/0.247/0.500$, per-tick retrograde flow $0.440/0.698/0.968$, per-tick total gain $0.988/0.947/0.892$. Neutral block: internal flow mean $2.608$, median $2.0$, interval $(2.582,2.635)$; per-tick increment variance rates $(0.661,1.143,1.236,1.259)$. Skin block ($C=12$):

$v$	1	2	3	4	5	6	7	8	9	10	11
mobility	.096	.179	.246	.297	.331	.356	.350	.331	.295	.226	.133
increment variance	1.26	2.39	3.34	4.15	4.48	4.69	4.62	4.28	3.76	2.75	1.60

Appendix F Recomputation Material and Hashes

The numerical content of this paper comes from one standard run of four scripts and one dependency from the earlier series; per-file SHA-256 values follow, and the package hash of the published bundle is registered in the publication-record metadata following the series double-anchor convention.

spin.py: 7a60653de726d98b6ddd00f0152f9922bcad051e3528316666f45f7d6b96278a. angular.py: b200a17d7497cbf352aad8d2082ab5673ecacf5251b6f313be3dfe86d8bd7faa. families.py: fb4bd9611d68177fdafb0621b6b8f3f5b7ac50ef31a984140f415a292efc2509. diffrot.py: b605ccbe7f89250a27d56c5f275550ca34b6274378d7efddbae709adbc3aaf64. Dependency arealaw.py belongs to the published package of [P3], SHA-256 39ca2ab687935caa69dfa1c831db59f05899fa064cd40b954968e7f128347c07. Script package SHA-256: 24312a41889bae7fbd455e8127c81ad69f42fec9eec354cb65d08eb35b95cae6. Standard run: in one directory execute python3 spin.py, python3 families.py, python3 diffrot.py (the latter two importing angular.py and arealaw.py); this reproduces all numbers of Appendices C and E; environment Python 3.12.3 with NumPy 2.4.4; deterministic blocks reproduce bitwise, stochastic blocks bitwise in the declared environment; all numbers passed independent recomputation, hence C1-R.

Appendix G Complete Status Table

D (definitions and model inputs): Definitions 2.1, 2.2, 2.3, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 4.4; the component universe and sampling measures; the three gauges. T1 (conditional theorems, hypotheses restated in the text): Lemma 2.4; Proposition 3.1; Lemma 3.2; Theorem 3.3; Corollary 3.4; the main sentence of Proposition 3.5; Proposition 3.6; Theorem 4.1; Proposition 4.2; Proposition 4.3; Lemma 4.5; Lemma 4.6; Lemma 5.1; Theorem 5.2; Corollary 5.3; Corollary 5.4; Proposition 6.1; Proposition 6.2; Lemma 7.1; Theorem 7.2; Proposition 7.3. T2 (assembled conditional readings): the reading "rotate forever, fall never" of Proposition 3.5; the four readings of Section 8; the angular voice of the three eras and the two-wing combinatorial reading in Section 9. T3 and Open: Problems 9.1 (Open), 9.2, 9.3; the interface-conjecture paragraph; the physical naming of the two wings; shell weights, time scale, energy interface, and the delivery of $J_{\rm phys}$. C1-R: all numbers of Appendices C and E.

Acknowledgements

The author thanks several large-language-model systems for assistance in derivation checking, adversarial review, and independent numerical recomputation; all claims and all errors are the author's own.

Funding and competing interests: this research received no external funding; the author declares no competing interests.

References

[P1] Qin, Han. Area-Level Plateau and Self-Sufficient Window in a Frozen-Ledger Model: A SAE black-hole information-flow-topology toy model (Chinese and English editions). Zenodo. DOI: 10.5281/zenodo.20913514.

[P2] Qin, Han. Update Incompatibility in Discrete Black-Hole Interiors: Three-Domain Separation, Orbit Dichotomy, and the Bridge Gap in Radial-Potential Lifting (Chinese and English editions). Zenodo. DOI: 10.5281/zenodo.21499053.

[P3] Qin, Han. 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 (Chinese and English editions). Zenodo. DOI: 10.5281/zenodo.21521760.

1 引言、辖区与依赖矩阵

本系列于纯占据模型类内考察离散黑洞内区:前篇已立坍缩之径向叙事〔P1〕、内区之信息与时序体例〔P2〕、以及势定界与分层单调〔P3〕。诸篇之内区,自由度唯占据之有无,动力学唯搬运之合法;其径向声部已备,而角向声部未立。本文补之:于此纯占据模型类,角向净流首先是转移层之量;其累计实值账沿闭历史一般非正合。有限状态仍可承载旋向、相位模类与离散瞬时流档位,但若要在保留无穷刚转史的同时精确积分无界累计账,状态扩展必须无限。本文所立者,即此账之定义、其非正合性之证明、其单拍谱之刚性与破刚之条件、其可观测压缩之层级,以及两条负结果:角向账不改光滑核之形,亦不在本模型中提供径向支撑。

辖区先立。本文所称之账,皆组合量:以搬运计步,以壳计层,以声明之权求和。正名句:本篇先立组合角向输运账,自该账至物理角动量之壳权、时间尺度与能量接口均未立。由此,四句判否,全篇恪守:其一,本文不主张物理角动量非态量,所证者唯组合累计账于本模型类之非正合;其二,本文不以无权之边步账认同 Kerr 角动量,二者之间尚隔壳权、时间与能量三重接口;其三,本文不称一切熵账与释放序皆盲于角向,所立之盲区逐条具名而有限;其四,本文不主张破刚之后谱必普遍填满,亦不用连续谱之语,一切谱皆离散组合谱。凡涉内视界、闭类时曲线与负坐标延拓之问,不属本文辖区,不审。

本文交付三正一负。正结果其一,过程账之立:三账相符、占据态不定瞬时流、累计账之非正合(第 3 节);其二,谱之立法:单壳刚转量子化、幅值上界、界面子群定理与退化捷径,合为谱格距之整数立法(第 4 至 5 节);其三,观测之压缩:四层观测映射与其零纤维之嵌套(第 6 节)。负结果:核形不变与环带无支撑(第 7 节)。第 8 节以声明测度下之差旋统计佐诸定理,第 9 节列开放问题与接口债,第 10 节结论并复诵防火墙。

最小依赖矩阵。 本文诸主结果皆条件式,其前件逐条列于下表;正文逐节复录所需前件,不以引代书。层级记号:D 为定义与模型输入;T1 为条件式定理,前件明书;T2 为条件拼装之读法;T3 为思辨接口,不承重;C1 为精确枚举之数,其中 C1-R 谓已经独立复算通过,C1-P 谓待独立复跑。本文所引数值皆 C1-R。

结果	时序语义	占据条件	库	权规范	层级
三账相符(3.1)	同步复合	任意	合法转移	逐壳向量	T1
非正合性(3.3)	同步复合	饱和核	含顺行刚转	任意正权	T1
预算之别(3.5)	二者	任意	逐支明书	显式 $\alpha$	T1
有向词长(3.6)	同步复合	饱和核	$\mathsf A_1$	边步	T1
刚转量子化(4.1)	同步复合	饱和	占据恒等	逐壳	T1
幅值上界(4.3)	同步复合	任意	合法转移	边步	T1
界面子群(5.2)	同步复合	饱和核	占据恒等	边步	T1
谱格点(5.4)	同步复合	饱和核	占据恒等全库	边步	T1
观测四层(6.1)	同步复合	任意	合法转移	声明 $\alpha$	T1
核形与环带(7.1 至 7.3)	逐次与同步	非光滑态	逐款明书	无涉	T1
差旋统计(8)	同步复合	饱和核	声明测度	$\alpha=\mathbf 1$	C1-R 加 T2

导览:第 2 节立模型、时序与约定;第 3 节立过程账并证其非正合,以有向词长收束过程量对态量之辨;第 4 节立单壳谱与反号对称;第 5 节立界面子群与破刚;第 6 节立观测压缩;第 7 节立负结果;第 8 节报差旋统计;第 9 至 10 节收束。

2 模型、时序与约定

本节诸件皆定义与模型输入,层级 D;唯引理 2.4 为可真可假之断言,层级 T1。

定义 2.1(洋葱图与占据)。 洋葱图 $G$ 由壳 $j=0,\dots,R$ 组成,壳 $j$ 具容量 $C_j\ge1$,其胞记 $(j,i)$,$i\in\mathbb Z_{C_j}$。切向边连同壳环邻:$C_j\ge3$ 时成环,$C_j=2$ 时唯一边,$C_j=1$ 时无。径向边连胞与其内亲:每胞 $(j,i)$($j\ge1$)配内亲集 $\mathrm{par}(j,i)\subseteq\mathbb Z_{C_{j-1}}$,至少一元。占据 $n\in\{0,1\}$ 逐胞($n_{\max}=1$);位形 $x$ 为占据函数,令牌数 $N=\sum x$。称 $C_j\le2$ 之壳为退化壳。本文体例承〔P3〕:内亲之缺省构造为主亲加后继之双亲;一切结果之前件逐条另书,不依缺省。

定义 2.2(时序与合法转移)。 本模型有二时序语义。甲,逐次单跳:一步迁一令牌沿一邻接边,合法当且仅当源胞占据而目标胞空。乙,同步复合:一拍施搬运集 $\theta=\{a\to b\}$,合法当且仅当四款俱备:每一搬运沿邻接边;逐源至多一出;逐靶至多一入;每被迁令牌之源占据,且终占据逐胞仍属 $\{0,1\}$。合法同步转移之搬运集为图上之 partial permutation,其支撑分解为点不交之有向路径与闭链;占据恒等之转移恰为闭链之并(占据恒等使 partial permutation 无开路径,故支撑纯为闭链)。丙,分名:搬运集空者称空转移;占据恒等而搬运非空者,态不变而账可非零,此类转移之存在为第 3 节非正合性之先声。二语义之施用逐结果明书于依赖矩阵。本款承前篇之时序体例(〔P3〕定义 3.0)而自含重述。

定义 2.3(序贯结构)。 主亲 selector 逐胞择一内亲 $p_j(i)\in\mathrm{par}(j,i)$。称界面 $j$ 序贯,若存在非降提升 $\tilde p_j:\mathbb Z\to\mathbb Z$,$\tilde p_j(i+C_j)=\tilde p_j(i)+C_{j-1}$。增量 $g_i=\tilde p_j(i{+}1)-\tilde p_j(i)\ge0$,周期和 $\sum_{i

引理 2.4(定向之传输)。 序贯诸界面下,任取尾段起始非退化壳环向之一,递归择其外诸壳坐标之向使各 $\tilde p_j$ 非降,则全尾段得一定向;反转起始向即整体反转,故定向恰差一全局。证:逐界面以提升之保序性传输一壳之向;起始之二择即全局反转之二,方向之先后无循环。$\blacksquare$〔T1,条件即序贯结构本身。〕

定义 2.5(定向廓与取向投影)。 定向廓 $\omega$ 依引理 2.4 取定。顺行后继映射唯于 $C_j\ge3$ 之壳有定义;取向投影 $\tau$ 逐搬运赋值:顺行切向步 $+1$,逆行切向步 $-1$,径向步与退化壳切向步俱 $0$。

定义 2.6(三量与初值)。 逐转移之单拍净流 $\ell_j(\theta)=\sum\tau(m)$,和取壳 $j$ 之切向搬运;累计账 $\Lambda_j(T)=\sum_{t\le T}\ell_j(\theta_t)$,$\Lambda_j(0)=0$;示踪解卷角步账 $w_{a,j}$,逐令牌以边步计其于壳 $j$ 之带号切向步,$w_{a,j}(0)=0$;唯闭合留居段上 $w_{a,j}/C_j\in\mathbb Z$ 方称绕数。物理角动量 $J_{\rm phys}$ 非本文之量,归接口。

定义 2.7(向量账与显式权)。 逐壳向量账 $\boldsymbol\ell=(\ell_j)$ 为本文之主对象;标量总账唯以显式正权立:$\ell_\alpha=\sum_j\alpha_j\ell_j$,$\alpha_j>0$。三规范:边步规范 $\alpha=\mathbf 1$,具组合特权,逐搬运整数计;转数规范 $\alpha_j=1/C_j$;物理规范 $\alpha^{\rm phys}$ 未立,归接口。共定向唯予跨壳符号之可比,量值之可比必待权之声明。

定义 2.8(旋向格)。 四名:$\mathsf A_1$,唯顺行切向搬运,禁径向禁逆行;$\mathsf A_2$,逐原子搬运 $\tau(m)\ge0$;$\mathsf A_3^\alpha$,逐转移净账 $\ell_\alpha(\theta)\ge0$;$\mathsf A_0^\alpha$,逐转移 $\ell_\alpha(\theta)=0$。$\mathsf A_0^\alpha$ 为 $\mathsf A_3^\alpha$ 之零纤维,列为中和扇区,非独立之第四钳。

定义 2.9(观测四层)。 四层确定压缩映射:毛搬运集 $\mathcal O_{\rm gross}(\theta)=M(\theta)$;切向净边流场 $J^\omega_\theta$,逐非退化壳之切向边依定向廓取正向,$J^\omega_\theta(e)=\#(e_+)-\#(e_-)$;壳净流 $\ell_j(\theta)=\sum_{e\subset\text{壳 }j}J^\omega_\theta(e)$;总账 $\ell_\alpha(\theta)$。零纤维之嵌套与严格性见第 6 节;核语义承〔P3〕§6。

定义 2.10(广义亏格与序贯三分)。 界面 $j$ 上,胞 $i$ 之任一内亲 $q$ 配偏移 $\delta_q(i)=(q-p_j(i))\bmod C_{j-1}$ 与广义亏格 $D(i,q)=i-\tilde p_j(i)-\delta_q(i)$;主亲处 $\delta=0$。序贯三分:主亲非常值者提升唯一,增量向量定;主亲常值者提升存在而不唯一,增量式不施,全胞同亲;周期和为 $C_{j-1}$ 之异倍者非序贯。

3 过程账、非正合性与有向词长

本节于合法转移上立账,证占据态不定瞬时流、累计账不正合,并以有向词长收束过程量对态量之辨。诸件之前件随件复录。

命题 3.1(三账相符)。 前件:同步复合之合法转移(定义 2.2 乙);示踪之乘者规则,逐搬运计入其被迁令牌;取向投影于退化壳两侧同零(定义 2.5)。则逐拍 $$\sum_{e\subset\text{壳 }j}J^\omega_{\theta_t}(e)=\ell_j(\theta_t)=\sum_a\bigl(w_{a,j}(t)-w_{a,j}(t-1)\bigr)\qquad(\forall j),$$ 即净边流账、壳账与诸令牌示踪解卷账逐拍同值;首一等式由定义 2.9 直得,次一等式为本命题。证。逐切向搬运恰一令牌被迁,其 $\tau$ 同时计入其边之净流、壳 $j$ 之净流与该令牌之解卷账;三和皆同一有限和之异序,交换即得。径向搬运于三账俱零,故标签之跨壳不生边界项。$\blacksquare$〔T1〕

引理 3.2(裸占据不定瞬时流)。 存在位形 $x$ 与两合法转移 $\theta,\theta'$ 同自 $x$ 出而 $\boldsymbol\ell(\theta)\neq\boldsymbol\ell(\theta')$。证。取一非退化饱和壳:顺行整壳刚转与逆行整壳刚转皆为合法之占据恒等转移(闭链,定义 2.2),账分别 $+C_j$ 与 $-C_j$。故单拍净流非占据态之函数。$\blacksquare$〔T1〕

定理 3.3(非正合性)。 前件:库为占据恒等类之饱和转移库(定义 2.2 丙),至少含一非退化饱和壳之顺行刚转可入调度;权 $\alpha$ 任意正。其一,占据层:不存在占据态之实值函数 $\Psi$ 使 $\ell_\alpha(\theta)=\Psi(x_{\rm 终})-\Psi(x_{\rm 始})$ 对库中一切转移成立。盖占据恒等而账 $\alpha_jC_j\neq0$ 之转移在库(刚转),等式左非零而右恒零。其二,带标签层:标签态之实值函数亦无。盖 $C_j$ 次顺行刚转组成一闭历史,标签复位而累计账 $C_j\cdot\alpha_jC_j\neq0$;任何标签态函数沿闭历史之全差为零。其三,有限扩展层:凡有限态扩展 $E$ 承认该刚转史之无穷提升者,$E$ 上亦无正合原函数。盖提升轨道于有限 $E$ 内终归周期,一周期之累计账为 $\alpha_jC_j$ 之正倍数而非零,任何 $E$ 上函数沿周期之全差为零。$\blacksquare$〔T1〕

推论 3.4(收窄)。 有限状态可承载旋向、相位之模类与离散流档;上定理所排除者,唯实值正合原函数之全域存在;凡欲精确积分无界实值累计账之扩展,必须无限。〔T1〕

命题 3.5(预算之别)。 径向侧:内向原子步之预算有限,恰为势差 $\Delta\Phi_r$,承〔P3〕推论 3.1a 之账本恒等式与其势定界诸定理,此处不复证。切向侧,诸支各以可重复转移之存在为条件:无钳之占据恒等库含某非退化饱和壳之双向刚转者,累计账可向 $\pm\infty$;$\mathsf A_3^\alpha$ 库含可重复 $\theta_+$ 使 $\ell_\alpha(\theta_+)>0$ 者,自零起单向无界;$\mathsf A_0^\alpha$ 库存在可重复之 $v\in\mathrm{Spec}(\boldsymbol\ell)\setminus\{0\}$ 且 $\alpha\cdot v=0$ 者,逐壳账无界往复;无该向量则不作此支之结论。主句:径向下降有有限之倒计时,角向累计无 $\Phi_r$ 所供之倒计时。〔T1;"洞能永转而不能永落"之语,唯于诸支前件成立处为条件拼装之读法,层级 T2。〕

命题 3.6(有向词长,群同而账异)。 前件:饱和核,$\mathsf A_1$ 库;此库之非空转移恰为诸非退化壳顺行刚转之不交并。以 $\mathbf k=(k_j)\in\prod_j\mathbb Z_{C_j}$,$0\le k_j相同可达群不推相同过程成本:过程量对态量之辨,于此闭合。〔T1〕

4 刚转谱、幅值界与反号对称

本节立单壳之谱刚性、任意转移之幅值上界,并以反号对合立谱之零点对称。诸件前件随件复录。

定理 4.1(单壳刚转量子化)。 前件:壳 $j$ 非退化($C_j\ge3$)且饱和;转移 $\theta$ 为同步复合之合法转移(定义 2.2 乙),其搬运尽居壳 $j$ 之切向边,且占据恒等。则 $\theta$ 之搬运子图之每一非空连通分量为环上有向圈;无弦简单环之有向圈唯三类:单边二圈(邻换)、整壳顺行圈、整壳逆行圈。故 $$\ell_j(\theta)\in\{0,\pm C_j\},$$ 且 $\ell_j(\theta)=\pm C_j$ 当且仅当 $\theta$ 于壳 $j$ 含同向之整壳刚转。证。饱和加占据恒等使 $\theta$ 于壳 $j$ 之限制为该环胞集上之置换,partial permutation 三款使其分解为有向圈;环无弦,故圈之支撑或为一边(邻换,账 $+1-1=0$)或为全环(刚转,账 $\pm C_j$)。$\blacksquare$〔T1〕

命题 4.2(径向格之单拍谱)。 前件:诸壳饱和;库为诸壳内切向占据恒等转移之不交并(无径向搬运)。则向量谱为直积 $$\mathrm{Spec}(\boldsymbol\ell)=\prod_{C_j\ge3}\{0,\pm C_j\},$$ 退化壳分量恒零;加权总谱为带号子集和 $\{\sum_j\varepsilon_j\alpha_jC_j:\varepsilon_j\in\{-1,0,1\}\}$。证。诸壳互不相扰,逐壳独立施定理 4.1,直积即得。$\blacksquare$ 注:总谱之洞依 $\{C_j\}$ 与 $\alpha$ 而定;边步规范下容量 $(3,4)$ 之双壳谱缺 $\pm2,\pm5,\pm6$,精确枚举(C1-R)之全表归附录 C。有洞之谱为组合谱,非物理能谱。〔T1〕

命题 4.3(普遍幅值上界与等号)。 前件:任一同步复合之合法转移,占据任意,含空位之开链在内。边步规范下 $$|\ell_{\mathbf 1}(\theta)|\le\sum_{C_j\ge3}C_j=:M.$$ 等号当且仅当 $\theta$ 于每一非退化壳上之限制为同一方向之整壳刚转;退化扇区可另含与之不交之零账活动。证。partial permutation 使逐胞至多一出移;计账者唯非退化壳之切向步,逐步 $\pm1$,故上界为非退化壳胞数。等号须每一非退化壳胞出一切向步且全图同号;占据约束下逐壳"每胞迁于其顺邻"即整壳刚转,反之显然。退化壳之搬运计账皆零,故其活动不碍等号。$\blacksquare$〔T1〕

定义 4.4(反号对合)。 于占据恒等转移(闭链之并,定义 2.2 丙)上,置 $\rho(\theta)=\{b\to a:a\to b\in\theta\}$,即逐搬运反向。$\rho(\theta)$ 仍为合法占据恒等转移,且 $\boldsymbol\ell(\rho\theta)=-\boldsymbol\ell(\theta)$。〔D〕

引理 4.5(对合计数与不动点)。 占据恒等库上 $\rho$ 为对合,故库之枚举满足 $$N_{\rm all}=N_0+2N_+,$$ $N_0$ 为账向量零者之数,$N_+$ 为正账者之数,正负对偶。不动点刻画:$\mathrm{Fix}(\rho)$ 恰为对换之并;无径向搬运之库中 $\mathrm{Fix}(\rho)=\{\boldsymbol\ell=\mathbf 0\}$(定理 4.1);一般库中普遍唯 $\mathrm{Fix}(\rho)\subseteq\{\boldsymbol\ell=\mathbf 0\}$,严格包含条件于库含 $\rho(\theta)\neq\theta$ 而 $\boldsymbol\ell(\theta)=\mathbf 0$ 之补偿转移。证。对合性与账反号由逐搬运反向直得;计数恒等由轨道之长唯一与二。不动点即逐搬运自反者,恰对换。$\blacksquare$ 注:容量 $(1,3,4)$ 之洋葱,全库 $1995=521+2\times737$、无径向库 $53=27+2\times13$,精确枚举(C1-R)在附录 C。〔T1〕

引理 4.6(含空位单壳之幅值界)。 前件:壳 $j$ 非退化,占据含 $v\ge1$ 空位;同步复合之合法转移,搬运尽居壳 $j$ 切向。则 $|\ell_j(\theta)|\le C_j-v$,等号于同步平移全部 $C_j-v$ 令牌时取得。证。逐源至多一出而源须占据,故切向步数不逾令牌数 $C_j-v$,逐步 $\pm1$ 即得上界;达界者,诸令牌各迁其顺邻,逐靶至多一入且终占据仍属 $\{0,1\}$(平移之像),合法,账 $C_j-v$。$\blacksquare$〔T1〕

节末三明书。其一,定理 4.1 之量子化条件于饱和、搬运局壳与无弦环,三者缺一即坏,破坏之道即下节之题。其二,总谱依权:边步与转数两规范下同一库之谱形不同,量值之较必先声明权。其三,一切谱皆离散组合谱;自组合谱至任何物理谱之桥,俱属接口,本文不立。

5 界面子群、退化捷径与破刚

界面闭链账受一整数子群之钳,其代数格距由容量、增量与亲偏移立法。 此为本节冠句,亦为本文第二主结果。单壳之谱既刚(定理 4.1),跨壳搬运一旦入库,新账值落于何格、格距几何,答案不在库开多宽,而在界面结构生成何数。本节先立三类原始圈为生成元,继立局部子群定理与全局格距,终以谱格点推论与格距之梯收束。

引理 5.1(原始圈之构造)。 设界面 $j$,记号从定义 2.3 与 2.10;库为占据恒等之合法转移(定义 2.2 丙),诸壳饱和。其一,共亲三圈:环邻胞 $i,i{+}1$ 共内亲 $q$(经任一亲)者,闭链 $(j,i)\to(j,i{+}1)\to(j{-}1,q)\to(j,i)$ 合法,账 $+1$;反向账 $-1$。其二,增量圈:界面序贯且主亲非常值、二壳非退化时,逐列 $i$ 之闭链 $\gamma_i$ 为:列 $i{+}1$ 降至 $p(i)+g_i$,壳下逆行 $g_i$ 步至 $p(i)$,升至列 $i$,壳上顺行一步;边数 $g_i+3$,账 $1-g_i$;$g_i=0$ 予三圈账 $+1$(即其一之主亲形),$g_i=2$ 予五边圈账 $-1$。其三,副亲圈:胞 $i$ 有副亲 $q$、偏移 $\delta=\delta_q(i)\ge1$ 者,闭链自 $i$ 降经 $q$,壳下逆行 $\delta$ 步至 $p(i)$,升返 $i$;账 $-\delta$;$\delta=1$ 予三圈。其四,退化款:下壳 $C_{j-1}\le2$ 而上壳非退化者,取向投影置零下壳,构造圈账恒 $+1$,不依 $g_i$。证。诸构造皆简单圈,饱和下为胞集之置换,同步三款俱备故合法;账由取向投影逐步计得。$\blacksquare$〔T1〕

定理 5.2(界面子群定理)。 前件:二壳非退化;主亲序贯且非常值;$\theta$ 为占据恒等之合法转移,支撑于壳对 $\{j{-}1,j\}$。则其边步账落于 $$G_j\mathbb Z,\qquad G_j=\gcd\bigl(C_j,\;C_{j-1},\;\{|1-g_i|\}_i,\;\{\delta_q(i)\}_{i,q}\bigr),$$ 且 $G_j$ 恰为该壳对原始简单圈账集之最大公因。证。设 $d$ 整除定理所列全部生成元。广义亏格满足二同余式 $$D(i{+}1,p(i{+}1))-D(i,p(i))=1-g_i\equiv0,\qquad D(i,q)-D(i,p(i))=-\delta_q(i)\equiv0\pmod d,$$ 又 $D$ 之周期漂移为 $C_j-C_{j-1}\equiv0$,故一切径向端点之 $D$ 同余于一常数 $c_j$。立 $d$ 值位势 $\psi(j{-}1,x)=x$、$\psi(j,i)=i-c_j$,俱模 $d$,列坐标依定义 2.5 之共定向取。则逐搬运:切向步之账 $\tau=\pm1$ 恰为 $\psi$ 之增量,绕行处以 $d\mid C$ 仍合;径向步账零而 $\psi$ 增量 $\equiv q-(i-c_j)\equiv c_j-D(i,q)\equiv0\pmod d$。$\theta$ 分解为点不交之圈(定义 2.2),逐圈 $\psi$ 归位,故总账 $\equiv0\pmod d$。生成侧:$C_j$ 与 $C_{j-1}$ 由整壳刚转实现,$|1-g_i|$ 由增量圈,$\delta_q(i)$ 由副亲圈,皆原始简单圈,故圈账之最大公因整除诸生成元而不粗于 $d$;合上含得等。$\blacksquare$〔T1〕

推论 5.3(全局格距与退化捷径)。 尾段款:$\theta$ 支撑于非退化尾段、逐界面满足定理 5.2 前件者,置 $d=\gcd_jG_j$,各界面径向同余常数为 $c_j$,递归取 $s_{j}-s_{j-1}=c_j\pmod d$、起始壳 $s_{j_*}=0$,立 $\psi(j,i)=i-s_j$ 模 $d$;则一切径向边之 $\psi$ 增量为零、切向边为 $\tau$,任一闭链账 $\equiv0\pmod d$。全局款:全图支撑之占据恒等合法转移,其账落于 $G^{\deg}\mathbb Z$, $$G^{\deg}=\gcd\bigl(d,\;\{1:\text{凡下壳退化而上壳非退化之界面}\}\bigr),$$ 退化界面由引理 5.1 其四得生成元 $1$,含法平凡而生成实;不含退化界面者归尾段款。退化之芯为单位生成捷径:于至少含一非退化壳之全饱和占据恒等闭链库,含单胞或双胞内壳之洋葱全局代数格距恒为一,账之量子化唯于远离退化枢纽之尾段存续。容量 $(2,12,4)$ 之例,尾段格距二而全局一,精确枚举(C1-R)在附录 D。$\blacksquare$〔T1〕

推论 5.4(谱格点)。 前件全列:全饱和核;占据恒等;同步 partial permutation;完整占据恒等库;诸非退化界面满足定理 5.2 前件。则 $$\mathrm{Spec}(\ell_{\mathbf 1})\subseteq G^{\deg}\mathbb Z\cap[-M,M].$$ 域外为伪之登记:含一空位之同壳单跳账为 $\pm1$,于全局格距为四之塔即出格;故格点句唯施闭链库,不施含空位之谱。证由推论 5.3 与命题 4.3。$\blacksquare$〔T1〕

格距之梯与破刚判据。 定理 5.2 使格局三分。格距 $G_j=C$(二壳等容、单亲、增量全一):一切界面闭链账为绕数乘 $C$,定理 4.1 之刚性于跨壳库存续。格距 $1单位格距不推单拍 $\pm1$,更不推谱无洞:存在单亲界面例,$G=1$ 而原始圈账集不含 $\pm1$,其全谱大破(C1-R,附录 D);自子群至装填之桥为开放问题,归第 9 节。

表 5.1(界面四例)。 逐例之证明与精确枚举俱在附录 D。

界面结构	生成元特征	$G_j$	谱之钳
等容单亲均匀塔	增量全一,无偏移	$C$	谱 $\subseteq C\mathbb Z$,刚性存续
单胞中心	退化捷径	$1$	中心处处共亲,单位档
同余单亲塔	增量恒同余	中间值	谱 $\subseteq d\mathbb Z$,偶格例在册
缺省双亲	偏移恒一	$1$	处处破刚,与共亲语言合流

6 观测压缩与零纤维

本节短而立一事:角向信息沿四层确定压缩逐层损耗,其损耗之形恰为零纤维之嵌套;净流为最浅可见层,非唯一可见层。四层映射已由定义 2.9 立。

命题 6.1(零纤维之嵌套与严格性)。 前件:同步复合之合法转移,占据任意,权 $\alpha$ 任意正。普遍包含 $$\ker\mathcal O_{\rm gross}\subseteq\ker J^\omega\subseteq\ker\boldsymbol\ell\subseteq\ker\ell_\alpha,$$ 其中 $\ker$ 谓该层取零之转移类,$\ker\mathcal O_{\rm gross}$ 即空转移。三包含之严格性各以见证条件化。其一,毛搬运可见而净边流零:同边邻换(一边二向相消)与纯径向对换,俱非空而 $J^\omega\equiv0$。其二,净边流可见而壳账零:同一非退化壳上,二不交边各承一顺一逆之合法单跳并拍,如容量五、令牌居列零与三、空位居一二四,搬运 $0\to1$ 与 $3\to2$;二边净流分别 $+1$ 与 $-1$ 而壳账零。其三,壳账可见而总账零:中和补偿,条件于库与权,转数规范下任二非退化壳之对转即是,边步规范则须等容二壳之对转;库无该向量者,此包含不严。证。包含由逐层为前层之函数直得;诸见证之各层值逐一计得。$\blacksquare$〔T1〕

命题 6.2(盲区三限定)。 角向账之不可见,唯限于下列三观测,逐条具名而有限。其一,占据投影盲于账:饱和核上顺行刚转之调度与逆行刚转之调度,占据轨迹逐拍全同(恒为饱和常态)而累计账反号悬殊;精确枚举之见证(C1-R)在附录 C。其二,占据商之熵账盲于账:前篇〔P3〕之熵计数定义于占据商,其值于任一占据恒等转移下不变。此二者皆因其定义域为占据或其商之函数,而占据恒等而账非零之转移在库(定义 2.2 丙、定理 3.3)。其三,整壳同步剥蚀之跨壳释放统计盲于账,另证:该统计非纯占据之函数,其读取标签之出生壳与释放轮;然纯切向调度不改任一标签之壳归属,整壳同步剥蚀唯按壳次序释放,故跨壳逆序统计于任何切向调度之插入下不变;旋转盲之逆序见证(C1-R)在附录 C。$\blacksquare$〔T1〕

合而言之:粗压缩层非零,必蕴含其上游细层非零;上游细层非零,可于继续压缩时归零。 故完整搬运纪录可见,不保净边流可见;净边流可见,不保壳账可见;壳账可见,不保加权总账可见。净流为最浅可见层而非唯一可见层,其义即此:凡在净流层留痕者,其上游诸层俱留痕;上游之痕,未必达净流层。第 1 节判否其三之正读亦即此:本文不称一切观测皆盲于角向,所立之盲唯命题 6.2 之三名,而上游净边流层所见更细。

7 角向账不改核形,且不提供径向支撑

本节立本文之负结果:角向活动既不改变光滑核之形,亦不在本模型中提供径向支撑。前件随件复录;本节兼用二时序语义,逐件明书。

引理 7.1(单向滑动之互达)。 前件:非退化壳 $C\ge3$,占据数 $0

定理 7.2(核形不变)。 前件:固定令牌数 $N$;逐次单跳语义;库自由单跳钳至顺行($\mathsf A_2$ 于单跳层面,此层面 $\mathsf A_3$ 与之合一,盖单搬运之净即其步)。则前篇〔P3〕命题 3.2 所立固定 $N$ 之唯一光滑 sink 强连通类,其三性原封:其一,任一非光滑态仍有有限步严格降 $\Phi_r$ 之路径,内落之合法唯依占据,不问取向;其二,光滑扇区封闭如故;其三,表皮壳于顺行单跳下仍强连通,由引理 7.1;$C\le2$ 之壳引理 7.1 不适用,改引〔P3〕命题 3.2 原证之绕环推空位,$C\ge2$ 皆行。故切向旋向之钳不改核之形:唯一光滑 sink 之存在、唯一与封闭俱不动。$\blacksquare$〔T1〕

命题 7.3(环带位形无径向支撑)。 称位形 $x$ 为环带位形,若 $N>0$,壳 $0$ 至 $h{-}1$ 全空($h\ge1$)而占据尽居壳 $h$ 至 $H$。前件:同步复合语义;库为无钳之合法转移;环带含至少一合法切向搬运(饱和壳 $C\ge3$ 之刚转,或非满壳之邻空单跳)。三款如下。

其一,通道与预算:内向通道恒开,盖最低占据壳之内亲居空壳;达光滑扇区所需之内向原子步数恰为 $\Delta\Phi_r$(承〔P3〕推论 3.1a 账本恒等式),不因任何切向穿插增减;总调度拍数则可因穿插任意延长。所不变者步数,非拍数。

其二,存续与账:凡含合法切向搬运之环带,存在保腔之无穷纯切向调度,逐拍取一合法切向转移而永不排径向,故腔之持续可由调度拒径向步而得。零账支为附加条件:库另含可重复之非平凡零账转移者(饱和壳之邻换、二占据邻胞之对换,或他种 $\ker\boldsymbol\ell$ 之非空元),同一腔形亦可于逐拍零账下持续。此非普遍结论,其确条件为:环带若含他壳,而他壳具可重复之非零反号切向步,或自身已有零账转移,则逐拍零账调度存在;反例在册:环带唯此一壳,仅一令牌二空位($C=3$),每一非空切向步 $|\ell|=1$,无对换可用,逐拍零账调度不存在。故账值非径向通道之开关;旋转纯属并演。

其三,瞬态:任一非空环带因内层为空而非光滑,于定理 7.2 之动力学中属瞬态,不入封闭常返类。

合而正句:形可由调度长期维持,而模型无任何项使角向账进入径向合法性或径向预算;环带之寿命是调度之节制或立法之钳,非自旋之功。名之无离心者,确义唯此句,不作普适物理否定。$\blacksquare$〔T1〕

8 条件测度下的差旋统计

本节以声明测度下之抽样统计佐前文诸定理。凡结论俱相对测度而立,测度非自然,无一为自然律之称;本节数值皆 C1-R(§1),协议全文、种子、精确矩与逐项全表在附录 E,复算材料与哈希在附录 F。

测度与协议之声明。组件宇宙取诸壳双向刚转、全体对换与限长六之简单圈逐向,按支撑与账向量商化,逐枚过定义 2.2 之合法判据;三族为容量 $(1,3,4,5,6)$ 之缺省双亲洋葱、容量 $(12,4)$ 之同余单亲塔与容量 $(4,4,4)$ 之等容单亲均匀塔,坐标为其全局代数格距(推论 5.3)。自由段之测度为宇宙上均匀,拍独立;钳段之 $\mathsf A_2$ 依定义 2.8 逐搬运;补偿段以声明混率抽补偿对;中和段之测度为半数零账单组件、半数不交零和组件对;表皮段为单环之随机胞位序贯 sweep,令牌同拍可再动。凡有精确矩者,以宇宙直和之精确值为准,经验唯对拍。

表 8.1(差旋统计总表;纯组合测度抽样,非时序演化)。

对象	设定	要目
归一化壳漂移之等与不等	$\mathsf A_2$ 均匀组件测度,三族	同余单亲塔相等($0.164$ 对 $0.166$);均匀塔中壳倍速($0.200,0.372,0.197$);缺省双亲族内快外慢($0.071,0.152,0.116,0.036$)
邻壳协方差之号	自由均匀组件测度	同余塔强反相 $-0.750$;均匀塔强同相 $+0.351$;缺省族混号($-0.004,+0.023,-0.067$)
补偿两形之率	$\mathsf A_3$ 混合测度,混率 $0,\tfrac14,\tfrac12$	件内补偿底率 $0.440$;件间补偿随混率升至 $0.968$;总账恒升
中和内部流强	零账混合测度	均值 $2.608$(区间 $2.582$ 至 $2.635$),总账逐拍恒零
逐拍增量方差率	同上	沿壳外增,$0.661$ 至 $1.259$
迁移曲线峰	单环 $C=12$,随机胞位 sweep	峰于半充($v=6$,迁移率 $0.356$),两端俱衰(附录 E)

四句读法,逐句冠其测度,层级俱 T2。

其一。在限长六、按支撑与账向量商化、均匀组件抽样且 $\alpha=\mathbf 1$ 之测度下,邻壳协方差之号为该截断圈枚举构成之秤:补偿型圈主导者反相,同向长圈主导者同相,两型相持者混号;表 8.1 之三族分列其三读。不作一般图族自然耦合常数之称。

其二。同测度而钳依定义 2.8 之 $\mathsf A_2$ 下,正账组件唯诸壳刚转者,归一化壳漂移相等;正向混圈一旦入库,廓线一般差旋。该相等性来自转移库与抽样测度之组合对称,非本文另立之动力方程。

其三。于 $\mathsf A_3$ 之声明混合测度且 $\alpha=\mathbf 1$ 下,补偿两形分立:单件非负而壳分量可负者为件内补偿,不交正负件同拍者为件间补偿;局部逆行由全局进账埋单,钳不禁差旋,唯禁总退。此为命题 3.5 单向支之统计定量;换权则足额之判据随变。

其四。于半数零账单组件、半数不交零和组件对之测度下,总账静止不蕴含逐壳过程静止:总账逐拍恒零而内部流强逐拍逾二,逐壳账以线性率往复扩散。此为命题 3.5 中和支之统计定量;不外推于一切 $\mathsf A_0$ 调度。

9 开放数学问题与物理接口债

问题 9.1(装填)。 一个自然之猜测谓:破刚之后,占据恒等谱必普遍填满区间 $[-M,M]$。此猜测为伪:单位格距不推谱无洞,第 5 节之界桩在册,故撤之。可辩护之形收窄为猜想:缺省双亲之增长洋葱与单亲 floor 之增长洋葱,其占据恒等谱恒为 $[-M,M]$ 全段。二族之全局格距恒一:前者由副亲偏移(表 5.1);后者亦然,盖 floor 增长使容量逐界面严格增,非降提升之增量周期和 $C_{j-1}$ 小于项数 $C_j$,必有零增量,环邻共亲而 $\pm1$ 圈在(引理 5.1 其一)。三族之穷举与一大族之构造证书支持之(C1-R,附录 D),一般装填证明未立。层级 Open,不入承重。

问题 9.2($\pm1$ 原始圈之全刻画)。 引理 5.1 之四路构造为充分,定理 5.2 之子群为阻断;中间带实存:有单亲界面例,格距为一而原始圈账集不含 $\pm1$(附录 D)。$\pm1$ 原始圈存在之充要刻画,开。

问题 9.3(非序贯账理论)。 主亲交叉者提升不存,增量语言失效;小例之圈账反丰。非序贯内亲结构上之账理论,开。

三纪之角向声部(T2)。 前篇之三纪以径向叙事立;其角向声部可自本文诸定理逐句拼得,兹录其读法,每句注其零件。下降纪:落料之切向偏置逐拍写账,径向步于角账取零,二账并行不悖(定义 2.5,命题 3.1)。饱和之拍:壳一旦充满,单跳语义即死,闭链成为唯一载体(定义 2.2);含 $v\ge1$ 空位之单壳单拍账界为 $\pm(C-v)$(引理 4.6);饱和之拍谱之腹亡而谱之极生,内部诸值尽灭,$\pm C$ 初次可达(定理 4.1)。示踪之壳坐标自此永冻(承〔P3〕引理 3.3 之标签径向锁),占据全史与账全史彻底脱钩(命题 6.2 其一)。冻结循环纪:刚转可由调度常驻,单壳谱 $\{0,\pm C_j\}$,跨壳格距由界面立法(定理 4.1,定理 5.2)。蒸发纪:整壳同步剥蚀之跨壳释放统计承〔P3〕之序而对纯切向账盲(命题 6.2 其三);单环之迁移曲线示,满占据端与空竭端为同一组合迁移曲线之两个低活动翼(表 8.1 末行,C1-R 与 T2)。以塌缩与蒸发之物理名命此两翼者,唯 toy 层之接口称谓,层级 T3,不承重。

接口猜想(T3,整段不承重)。 本模型之动量层为体例所截,归接口,承〔P2〕之划分。由此可猜:所谓无拖曳、无能层来历、无继承律与无离心,四缺席同根于此一截。逐条明书本文实证之界:无离心有 T1 之确义,即命题 7.3 之合句,模型无任何项使角向账进入径向合法性或径向预算;无拖曳之模型内形同出 7.3,而拖曳之物理名未立;能层之来历与角动量之继承,本文全未立,唯接口层之观察,盖 $J_{\rm phys}$ 本身未立,塌缩之 $J$ 于本模型无处交割。四者之同根为猜想,非本文之定理。

10 结论与防火墙

本文之结论三句。其一,角向净流居转移层,其累计账一般非正合。其二,刚转谱及其界面格距由局部图结构和径向条款共同立法。其三,角向活动既不改变光滑核形,也不在本模型中提供径向支撑。

接口之债四项,单列于此。壳权 $\alpha^{\rm phys}$:自组合账至物理量之逐壳权重,本文唯声明边步与转数二组合规范,物理规范未立。时间尺度:拍与物理时之桥未立,一切"率"皆逐拍之率。能量接口:账与能量之任何联系未立,有洞之谱为组合谱,非能谱。$J_{\rm phys}$ 之交割:物理角动量非本文之量,其态性、守恒与继承俱不属本文辖区。

正名句复诵:本篇先立组合角向输运账,自该账至物理角动量之壳权、时间尺度与能量接口均未立。第 1 节之四句判否,于全文恪守如初。

附录 A 原始圈构造与合法性

以下以坐标列式给出构造。诸构造之合法性俱依定义 2.2 乙之三款逐款验:逐源至多一出、逐靶至多一入、终占据不变;饱和下诸构造为其胞集之置换,三款自动俱备。

甲,共亲三圈(引理 5.1 其一):环邻 $i,i{+}1$ 共内亲 $q$,搬运列 $(j,i)\to(j,i{+}1)$、$(j,i{+}1)\to(j{-}1,q)$、$(j{-}1,q)\to(j,i)$;取向投影逐步 $+1,0,0$,账 $+1$;逐搬运反向即账 $-1$。乙,增量圈 $\gamma_i$(其二):$(j,i{+}1)\to(j{-}1,p(i){+}g_i)$;壳下逆行 $g_i$ 步至 $(j{-}1,p(i))$;$(j{-}1,p(i))\to(j,i)$;$(j,i)\to(j,i{+}1)$;边数 $g_i{+}3$,投影 $0,(-1)^{\times g_i},0,+1$,账 $1-g_i$;$g_i=0$ 退为甲之主亲形,$g_i=2$ 为五边圈账 $-1$。丙,副亲圈(其三):$(j,i)\to(j{-}1,q)$;壳下逆行 $\delta_q(i)$ 步至 $(j{-}1,p(i))$;$(j{-}1,p(i))\to(j,i)$;账 $-\delta_q(i)$;$\delta=1$ 为三圈。丁,退化款(其四):下壳 $C_{j-1}\le2$ 时投影置零下壳,乙构造之账恒 $+1$,不依 $g_i$。诸圈皆简单,支撑胞两两异,故为原始圈。

附录 B 全局递归位势之长证

推论 5.3 尾段款之全证。设尾段诸界面俱满足定理 5.2 前件,$d=\gcd_jG_j$。其一,逐界面常数之在:由定理 5.2 证中二同余式,界面 $j$ 之一切径向端点广义亏格 $D(i,q)$ 模 $d$ 同余,记其值 $c_j$;此值不依提升代表之选,盖 $D$ 之周期漂移 $C_j-C_{j-1}\equiv0\pmod d$。其二,壳常数之递归:取尾段起始壳 $s_{j_*}=0$,逐界面 $s_j\equiv s_{j-1}+c_j\pmod d$;良定,盖每界面唯一常数。其三,位势:$\psi(j,i)=i-s_j\bmod d$,列坐标依定义 2.5 之共定向廓取;$d\mid C_j$ 使环坐标之绕行于模 $d$ 下无碍。其四,逐搬运增量:壳 $j$ 之切向步,$\psi$ 增量为 $\pm1$ 恰等于取向投影 $\tau$,即其账;界面 $j$ 之径向步 $(j,i)\to(j{-}1,q)$,账零而 $\psi$ 增量为 $q-s_{j-1}-(i-s_j)\equiv q-i+c_j\equiv c_j-D(i,q)\equiv0\pmod d$,升步同理反号。其五,归位:占据恒等之合法转移分解为点不交之闭链(定义 2.2),逐链逐胞归位,$\psi$ 之总增量为零,故账之总和 $\equiv0\pmod d$。$\blacksquare$ 全局款由此加退化界面之生成元 $1$(附录 A 丁)即得,含法平凡而生成实。

附录 C 解析定理之精确枚举对拍(C1-R)

诸数俱出附录 F 所列脚本之一次标准运行,并经独立复算(C1-R 之义见第 1 节)。容量 $(1,3,4)$ 之洋葱:占据恒等库单成分 $642$(对换 $18$,有向圈 $624$),合法复合 $1995$;反号对合之计数 $1995=521+2\times737$,无径向库 $53=27+2\times13$(引理 4.5 之见证);全谱为 $[-7,7]$ 满段;有向词长之回恒等最小正价 $9$(命题 3.6 之实例)。边步规范下容量 $(3,4)$ 双壳之径向格总谱缺 $\pm2,\pm5,\pm6$(命题 4.2 注)。容量 $(1,3,4,5)$:全谱 $[-12,12]$ 满段,谱点 $25$。容量 $(1,3,4,5,6)$:幅 $M=18$,$0$ 至 $18$ 逐值构造证书全得。容量 $(1,2,8)$:退化壳投影置零验证,全谱满段,幅 $8$。界面子群双算:标准增长族六界面之圈账最大公因与定理 5.2 解析式逐一相等,皆 $1$;缺省双亲等容塔 $C=3$ 至 $6$ 亦然。盲区见证:占据轨迹逐拍全同而累计账悬殊之双调度,与旋转盲之剥蚀逆序,俱在册(命题 6.2 之见证)。

附录 D 边界例与反例

其一,等容单亲均匀塔(表 5.1 行一):容量 $(4,4,4)$,主亲恒等,增量全一无偏移,定理 5.2 予 $G=\gcd(4,4,0)=4$;枚举证一切圈账属 $4\mathbb Z$,全谱亦然。刚性于跨壳库存续,账即绕数乘容量。其二,单胞中心(行二):$C_0=1$ 使壳 $1$ 全胞共中心亲,共亲三圈予 $\pm1$,全局格距恒一。其三,同余单亲塔(行三):容量 $(12,4)$,主亲 $p(i)=3i$,增量恒 $3$,$G=\gcd(4,12,2)=2$,枚举证全谱属 $2\mathbb Z$;双界面塔 $(36,12,4)$ 同法逐界面 $2$,全图限长圈账皆偶。其四,缺省双亲(行四):偏移恒一入生成元,$G=1$;容量 $(4,4)$ 之副亲三圈账 $\mp1$ 为其见证,与共亲语言合流。其五,中间带例(问题 9.2):容量 $(7,5)$,单亲,增量 $(3,1,1,1,1)$,既无环邻共亲,增量亦不属 $\{0,2\}$;$G=1$ 而原始圈账集为 $\{0,\pm2,\pm5,\pm7\}$,不含 $\pm1$;全谱为 $\{0,\pm2,\pm5,\pm7,\pm12\}$,幅 $12$ 而洞十六处。单位格距不推谱无洞之界桩(问题 9.1)。其六,退化捷径之在类见证(推论 5.3):容量 $(2,12,4)$,尾段界面格距 $2$ 且尾段圈账最大公因实测 $2$,而捷径三圈账 $+1$ 合法、含奇账 $+5$ 之合法转移在、全图圈账最大公因 $1$。其七,序贯三分(定义 2.10):主亲常值之 $(3,3)$ 例,提升存在而不唯一,全胞同亲径得 $G=1$;交叉之 $(3,3)$ 例($p=(0,2,1)$),提升不存,圈账集 $\{\pm1,\pm2,\pm3,\pm4\}$,登记为问题 9.3 之素材。

附录 E 差旋统计之协议与全表

协议。组件宇宙:诸壳双向刚转、全体对换、限长六之简单圈逐向,按支撑与账向量商化,逐枚过定义 2.2 之合法判据。三族与坐标如第 8 节。抽样:逐段局部随机数发生器,种子在册:自由段 $501$,钳段 $503$,补偿段 $507$,中和段 $509$,表皮段 $511+v$;拍独立,$T=20000$(表皮段逐 $v$ 四千)。环境:Python 3.12.3,NumPy 2.4.4;确定段逐位复现,随机段同版环境位同。bootstrap:独立重抽,$B=400$,分位 $2.5$ 与 $97.5$。凡有精确矩者以宇宙直和之精确值为准,经验唯对拍。

全表。宇宙:组件 $1398/32/114$,其中刚转 $8/4/6$;全局格距 $1/2/4$,宇宙账逐族整除断言过。自由段:逐壳漂移精确为零,经验相符;邻壳协方差(精确)缺省双亲族 $(1,2)\,{-}0.004$、$(2,3)\,{+}0.023$、$(3,4)\,{-}0.067$,同余塔 $(0,1)\,{-}0.750$,均匀塔 $(0,1)$ 与 $(1,2)$ 皆 $+0.351$;刚转占比精确 $0.006/0.125/0.053$。钳段:$\mathsf A_2$ 组件 $537/6/35$,零账占比 $0.201/0.670/0.231$;转数率缺省族 $(0.0709,0.1519,0.1155,0.0363)$、同余塔 $(0.1641,0.1655)$、均匀塔 $(0.1997,0.3720,0.1972)$。补偿段(混率 $0/0.25/0.5$):补偿率 $0.000/0.247/0.500$,逐拍逆向流强 $0.440/0.698/0.968$,逐拍总升 $0.988/0.947/0.892$。中和段:内部流强均值 $2.608$,中位 $2.0$,区间 $(2.582,2.635)$;逐拍增量方差率 $(0.661,1.143,1.236,1.259)$。表皮段($C=12$):

$v$	1	2	3	4	5	6	7	8	9	10	11
迁移率	.096	.179	.246	.297	.331	.356	.350	.331	.295	.226	.133
增量方差	1.26	2.39	3.34	4.15	4.48	4.69	4.62	4.28	3.76	2.75	1.60

附录 F 复算材料与哈希

本文数值出自四脚本与一前篇依赖之一次标准运行,逐件 SHA-256 如下;发表包之包哈希依系列双锚体例登于出版记录元数据。

spin.py:7a60653de726d98b6ddd00f0152f9922bcad051e3528316666f45f7d6b96278a。angular.py:b200a17d7497cbf352aad8d2082ab5673ecacf5251b6f313be3dfe86d8bd7faa。families.py:fb4bd9611d68177fdafb0621b6b8f3f5b7ac50ef31a984140f415a292efc2509。diffrot.py:b605ccbe7f89250a27d56c5f275550ca34b6274378d7efddbae709adbc3aaf64。依赖 arealaw.py 属前篇〔P3〕发布包,SHA-256:39ca2ab687935caa69dfa1c831db59f05899fa064cd40b954968e7f128347c07。脚本包 SHA-256:24312a41889bae7fbd455e8127c81ad69f42fec9eec354cb65d08eb35b95cae6。标准运行:于同一目录依次执行 python3 spin.py、python3 families.py、python3 diffrot.py(后二者依赖 angular.py 与 arealaw.py),即复现附录 C 与 E 之全数;环境 Python 3.12.3 与 NumPy 2.4.4;确定段逐位相同,随机段于声明环境位同;诸数俱经独立复算通过,故记 C1-R。

附录 G 完整状态表

D(定义与模型输入):定义 2.1、2.2、2.3、2.5、2.6、2.7、2.8、2.9、2.10、4.4;组件宇宙与诸抽样测度;三规范。T1(条件式定理,前件正文逐件复录):引理 2.4;命题 3.1;引理 3.2;定理 3.3;推论 3.4;命题 3.5 主句;命题 3.6;定理 4.1;命题 4.2;命题 4.3;引理 4.5;引理 4.6;引理 5.1;定理 5.2;推论 5.3;推论 5.4;命题 6.1;命题 6.2;引理 7.1;定理 7.2;命题 7.3。T2(条件拼装读法):命题 3.5 之"洞能永转而不能永落";第 8 节四句;第 9 节三纪之角向声部与两翼之组合读法。T3 与 Open:问题 9.1(Open)、9.2、9.3;接口猜想段;两翼之物理命名;壳权、时间尺度、能量接口与 $J_{\rm phys}$ 之交割。C1-R:附录 C 与 E 之全数。

鸣谢

作者感谢多个大型语言模型系统于推演校核、对抗性审读与数值独立复算中之辅助;文中一切主张与错误,文责俱归作者。

资助与利益冲突:本研究未受外部资助;作者声明无利益冲突。

参考文献

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