Cofinal Invariance

Canonicity for the live growth rate: is the exponent a property of the system, or of the tower?

Standing. Separate from the frozen quotient–completion–observer triple. This note attacks the canonicity question left open by Regular-Growth Identification. Nothing in Resolution STOP is modified.

The problem#

Define the live growth exponent of a tower $P$

\delta_P \;:=\; \limsup_{j\to\infty}\frac{\log L_j(P)}{j\log 2},

where $L_j$ is the number of live level- $j$ vertices — those with a non-empty boundary below them. By Regular-Growth Identification, $\delta_P = \overline{\dim}_B(\partial T_P)$ under the visual metric. The question is whether

\delta_P \;=\; \delta_{P'}

for two admissible refining towers representing the same behavioural system. Three outcomes are possible: exact invariance, invariance after normalisation, or no invariance at all.

The answer is the third, for the raw quantity — and the second, once the right normalisation is used. We do them in that order.

Outcome 3: the raw exponent is not canonical#

Let $P$ have resolution $\varepsilon_j = 2^{-j}$ and live counts $L_j = 2^j$ , so $\delta_P = 1$ . Define a slower tower by using every second refinement:

P'_j \;:=\; P_{2j}.

The boundary set is unchanged: $\varprojlim_j P'_{j} = \varprojlim_j P_{2j} = \varprojlim_j P_j$ , since a cofinal restriction of a cofinal inverse system has the same limit. But the visual metric changes. Depth in $P'$ is $r' = r/2$ , so

d_{P'} \;=\; 2^{-r'} \;=\; 2^{-r/2} \;=\; d_P^{\,1/2}.

The gauge law#

If two towers have depths related by $r' \sim \alpha\, r$ , then $d_{P'} \asymp d_P^{\alpha}$ , and because balls of $d$ -radius $\varepsilon$ are $d^{\beta}$ -balls of radius $\varepsilon^{1/\beta}$ ,

\boxed{\;\dim_{d^{\beta}}(X) \;=\; \frac{1}{\beta}\,\dim_{d}(X).\;}

So with $\beta = \tfrac12$ ,

\delta_{P'} \;=\; 2\,\delta_P \;=\; 2 \qquad\text{while}\qquad \delta_P = 1 .

Measured:

quantity	$P$	$P'=P_{2j}$
raw $\delta$ (level-index visual metric, base 2)	1.0000	2.0000
gauge check $\dim_{d^{\beta}}$ for $\beta=\tfrac12,1,2$	—	$2,\;1,\;\tfrac12$

The raw exponent is gauge-dependent. The visual metric has two gauge freedoms — the chosen base, and the depth parametrisation — and both move the dimension. Same for $q_c$ , which is read off the level index and inherits the gauge. Outcome 3 is confirmed for the raw quantity, exactly as suspected.

The set did not change. The metric did.

The fix: intrinsic resolution#

The gauge freedom disappears if the metric is built from physical resolution rather than level number. Let $\varepsilon_j$ be the mesh — the actual diameter of the partition cells at level $j$ — and define the physical metric on the boundary by first-disagreement depth:

d_{\mathrm{phys}}(\xi,\eta) \;:=\; \varepsilon_{\,r(\xi,\eta)} .

Then

\boxed{\;D_P \;:=\; \limsup_{j\to\infty}\frac{\log L_j}{-\log \varepsilon_j}.\;}

This is reindexing-invariant for a trivial reason: reindexing moves $\varepsilon$ and $L$ together. For $P'_j = P_{2j}$ we get $\varepsilon'_j = \varepsilon_{2j}$ and $L'_j = L_{2j}$ , so the physical metric is literally the same function:

d_{\mathrm{phys}}'(\xi,\eta) = \varepsilon'_{r'(\xi,\eta)} = \varepsilon_{2r'} = \varepsilon_{r} = d_{\mathrm{phys}}(\xi,\eta).

Verified numerically at several depths: $\varepsilon_r(P)$ and $\varepsilon'_{r/2}(P')$ agree to machine precision, and $D_P = D_{P'} = 1.000000$ . The exponent is now a gauge-invariant, because it is the upper box dimension of the boundary in a metric that depends only on the base geometry and the physical scales.

Cofinal Tower Invariance#

Theorem. Let $P,P'$ be admissible refining towers on the same compact metric space $(K,d)$ whose mesh sequences satisfy $\varepsilon_j\to0$ , and suppose

cofinality — for every $j$ there is $k(j)\to\infty$ with $\varepsilon_{k(j)}(P')\le \varepsilon_j(P)$ , and symmetrically; and
common boundary — the inverse limits are canonically identified.

Then

\boxed{\;D_P \;=\; D_{P'}.\;}

Proof sketch#

$D_P$ is the upper box dimension of $\partial T$ in $d_{\mathrm{phys}}$ , and $d_{\mathrm{phys}}$ is determined by the mesh sequence of the base space, which the two towers share. Level- $j$ cylinders of a tower with mesh $\varepsilon_j$ cover the boundary and are $\varepsilon_{j-1}$ -separated, so the covering function satisfies $N(\varepsilon) = L_j$ for $\varepsilon\in[\varepsilon_j,\varepsilon_{j-1})$ . Cofinality says the two towers supply the same physical scales up to constants; equal boundary means the same covering problem is being solved at each scale. Hence $N_P(\varepsilon) \asymp N_{P'}(\varepsilon)$ for all $\varepsilon\to0$ , and the limsup defining $D$ agrees. $\blacksquare$

What is not claimed: that the towers’ level indices correspond, or that the visual metrics agree. Only the physical scale functions need to line up.

The canonical STOP exponent#

The same normalisation fixes STOP. Replace the level-indexed

O_q \;=\; \sum_j n_j (1-q)^j

(which is gauge-dependent, since $j$ is arbitrary) by a resolution-weighted operator with the physical scales as weights:

O_s \;=\; \sum_j n_j\,\varepsilon_j^{\,s}, \qquad \boxed{\;s_c \;:=\; \inf\Bigl\{s:\textstyle\sum_j n_j\varepsilon_j^{s} < \infty\Bigr\}.\;}

This is the abscissa of convergence of a generalised Dirichlet series, and it is invariant under reindexing.

Cauchy–Hadamard, generalised. Since $n_j\varepsilon_j^s \ge 0$ ,

\boxed{\;s_c \;=\; \limsup_{j\to\infty}\frac{\log n_j}{-\log \varepsilon_j}.\;}

Caution: this is a root test, not a ratio test. The ratio $\bigl(n_{j+1}/n_j\bigr)\bigl(\varepsilon_{j+1}/\varepsilon_j\bigr)^s$ overestimates the abscissa for oscillating level counts. The half-live tower $L_j = 2^{\lfloor j/2\rfloor}$ with $\varepsilon_j=2^{-j}$ gives ratio-test value $1.000$ but true value $0.500$ . Always use the root form.

Relation to q_c. In the regular case $\varepsilon_j = 2^{-j}$ the series is $C(2^{-s})$ , convergent iff $2^{-s}<R$ , so

s_c \;=\; \log_2 \frac{1}{R}, \qquad q_c \;=\; 1-2^{-s_c}.

$s_c$ is the gauge-free replacement for $\log_2(1/R)$ ; the frozen $q_c$ is its exponential re-encoding.

The full/live gap survives normalisation#

Write $s_c^{\text{full}}$ for the exponent built from $n_j=|X_j|$ and $s_c^{\text{live}}$ for the one built from $L_j$ . Since $L_j\le n_j$ ,

\boxed{\;s_c^{\text{live}} \;\le\; s_c^{\text{full}},\;}

with equality iff dead ends do not asymptotically dominate. Measured:

tower	`n_j`	`L_j`	$s_c^{\text{full}}$	$s_c^{\text{live}}$	gap
regular	$2^j$	$2^j$	1.0000	1.0000	0.0000
reindexed	$4^j$	$4^j$	1.0000	1.0000	0.0000
spine + flourish	$2^j$	$1$	1.0000	0.0000	1.0000
half-live	$2^j$	$2^{\lfloor j/2\rfloor}$	1.0000	0.5000	0.5000

Note the reindexed row: the raw $q_c$ moved (R changed with the level index) but $s_c$ did not. The gap is now a gauge-free quantity.

The canonical triple#

\boxed{\;\text{quotient growth } \{n_j\} \;\longrightarrow\; \text{resolution-normalised dimension } D \;\longrightarrow\; \text{critical STOP exponent } s_c\;}

with

D = s_c^{\text{live}} = \limsup_j \frac{\log L_j}{-\log\varepsilon_j}, \qquad s_c^{\text{full}} \ge D .

This is the canonical form of the bridge. The level-indexed $q_c = 1-R$ is the same content written in coordinates that a reindexing of the tower can move.

Transient complexity is not cofinal-invariant#

Write the transient complexity — the gap between the observer’s exponent and the geometry’s — as

\Delta s \;=\; s_c^{\text{full}} - s_c^{\text{live}} \;\ge\; 0 .

Answer: no. $\Delta s$ is not invariant under admissible cofinal presentations.

Structural constraints#

Any realizable discovery tree satisfies:

$(T_1)$ $L_j \le n_j$ ;
$(T_2)$ $L_j$ is non-decreasing — every live vertex has at least one live child, and distinct live vertices have distinct parents, so $v\mapsto$ (a live child of $v$ ) injects $\mathrm{live}(j)\hookrightarrow\mathrm{live}(j+1)$ ;
$(T_3)$ $n_{j+1}\ge L_j$ .

$(T_2)$ is easy to forget and it rules out the obvious construction (an alternating live count is impossible).

Counterexample#

Take

L_j = 2^{j}\ \ \forall j, \qquad n_j = \begin{cases}4^{j}, & j \text{ odd}\\ 2^{j}, & j \text{ even}\end{cases}

This satisfies $(T_1)$ – $(T_3)$ (checked explicitly), so it is a realizable tree. With $\varepsilon_j = 2^{-j}$ :

tower	$s_c^{\text{full}}$	$s_c^{\text{live}}$	$\Delta s$
$P$	2.000000	1.000000	1.000000
$P' = P_{2j}$	1.000000	1.000000	0.000000

Same boundary, same live exponent — $s_c^{\text{live}}$ is unchanged, exactly as Cofinal Tower Invariance requires — but $\Delta s$ moves from $1$ to $0$ .

Why#

$s_c^{\text{live}}$ is cofinal-invariant because $L_j$ is a covering count: it answers a question about the boundary, and covering numbers are functions of physical scale. $s_c^{\text{full}}$ is not, because $n_j$ is not a covering count — dead-end vertices cover nothing. So $n_j$ has no intrinsic scale-function interpretation; it is only a shell-indexed sequence, and a cofinal restriction is free to sample a different subshell of it. $\Delta s$ is a difference of a scale function and a shell sequence, so it inherits the shell sequence’s presentation-dependence.

When it is invariant#

If both

a_j := \frac{\log n_j}{-\log\varepsilon_j}, \qquad b_j := \frac{\log L_j}{-\log\varepsilon_j}

converge, then every cofinal subsequence has the same limits, so $\Delta s = \lim a_j - \lim b_j$ is invariant. Verified: $n_j = 3^j, L_j = 2^j$ gives $\Delta s = 0.584963$ both before and after reindexing. The counterexample above works precisely because both quantities are $\limsup$ s attained on non-equivalent subshells — the full rate on odd shells, the live rate on even ones.

Consequence. Transient complexity is not an invariant of a behavioural system from counts alone. It becomes one under a regular-variation hypothesis on the admissible presentations. This is a strictly weaker situation than the geometry: $D = s_c^{\text{live}}$ is presentation-independent under cofinality, while $\Delta s$ is not.

Status#

Claim	Status
Raw $\delta_P$ and raw $q_c$ are gauge-dependent (reindexing changes them)	proved — explicit counterexample, $1\to2$
Gauge law $\dim_{d^\beta} = \beta^{-1}\dim_d$	proved (standard metric scaling)
The physical metric $d_{\mathrm{phys}} = \varepsilon_{r}$ is reindexing-invariant	proved — immediate, verified numerically
Cofinal Tower Invariance: $D_P = D_{P'}$	proved under cofinality + common boundary
Canonical STOP exponent $s_c$ , invariant under reindexing	proved (generalised Cauchy–Hadamard, root form)
$s_c^{\text{live}}\le s_c^{\text{full}}$ , equality iff no dominant dead ends	proved (immediate from $L_j\le n_j$ )
Canonicity of the gap $\Delta s = s_c^{\text{full}}-s_c^{\text{live}}$	refuted — explicit realizable counterexample: $\Delta s$ is $1$ for $P$ and $0$ for $P'=P_{2j}$
$\Delta s$ invariant when the two exponents converge (regular variation)	proved — a limit is subsequence-invariant

Machinery used, not claimed. Metric scaling of box dimension; dyadic cylinder covering; abscissa of convergence of non-negative Dirichlet-type series; Cauchy–Hadamard.