What the logarithmic derivative can and cannot extract

What the Logarithmic Derivative Can and Cannot Extract

Working notes with verified computations. Not a research announcement.

What this is. A record of three connected investigations, each built and checked numerically from first principles rather than cited. Essentially all individual results are classical. The value, if any, is in the juxtaposition, the explicit verification, and the failure modes — several of which are more informative than the successes.

What this is not. A unified theory. An earlier draft claimed "localization" as an organizing thesis; that framing covers the residue theorem, stationary phase, spectral theory, and compressed sensing equally well, which means it predicts nothing. It is dropped. The honest organizing observation is narrower and stated in §1.

---

1. The organizing observation

For a product $F = \prod_i (z - r_i)$, the logarithmic derivative $$\frac{F'(z)}{F(z)} = \sum_i \frac{1}{z - r_i}$$ converts a product into a sum over distinguished points, each weighted by inverse distance. This one identity, in various guises, drives:

• Newton's method. The step $z - F/F'$ is $z - 1/(F'/F)$; the nearest root's term dominates the sum, which is why the iteration converges locally — and why its complex basin boundaries are fractal when the "vote" is nearly tied.
• The argument principle. $\oint z^k F'/F$ returns power sums over enclosed roots, feeding Newton's identities.
• Euler's product for $\zeta$. $\zeta'/\zeta$ yields the von Mangoldt function; the explicit formula follows.
• Euler's pentagonal product. $x F'/F$ yields the divisor sum $\sigma(n)$ (§3.3).
• Spiral-mode transforms. $G'/G$ locates complex growth-plus-rotation rates (§2.4).

The three parts below examine where this works, where it breaks, and where there is nothing for it to find. The failures are the point: each is a specific, identifiable obstruction rather than a limitation of effort.

---

2. Part I: Mellin transform on the universal cover

2.1 The topological constraint

A kernel $z^s = r^s e^{is\theta}$ is single-valued on $\mathbb{C}^*$ only for $s \in \mathbb{Z}$: traversing $\theta \to \theta + 2\pi$ multiplies it by $e^{2\pi i s}$. Verified at $r=2$, $s = 0.4+0.7i$: ratio $-0.00995 + 0.00723i$, matching $e^{2\pi i s}$, nowhere near 1.

This is $\pi_1(S^1) = \mathbb{Z}$, and it is the same constraint that makes Taylor coefficients integer-indexed — a Taylor coefficient is a Fourier coefficient, since $f(re^{i\theta}) = \sum_n a_n r^n e^{in\theta}$ and Cauchy's formula is coefficient extraction divided by $r^n$ (verified recovering $a_4 = 3^{-4}$ to 15 digits).

The standard consequence is the Fourier–Mellin transform, with $s$ continuous and $n$ discrete, used in image processing for scale- and rotation-invariant matching. Its measure requires no reconciliation: the polar element $r\,dr\,d\theta$ factors as $(r\,dr)(d\theta)$, and $r\,dr$ is Mellin's multiplicative Haar measure. Verified by splitting a joint integral into independent 1D Mellin and Fourier-series computations, agreeing to 15 digits.

2.2 Removing the constraint

On a logarithmic spiral $z(t) = e^{(a+ib)t}$, one full turn returns to a different radius; the curve never closes. Equivalently, work on the universal cover of $\mathbb{C}^*$, where $w = \log z$ is single-valued.

This is not a new transform. Mellin transform is Laplace transform after the substitution $x = e^t$; that is its standard construction. What the spiral picture supplies is the domain on which a fully complex exponent is legitimate, and the geometric reading of the result: for $g(w) = e^{pw}$ with $p = a+ib$, $$G(s) = \frac{1}{s-p},$$ a simple pole whose real part is radial growth rate and imaginary part is rotation rate — both read from one point. Verified: $|G|$ tracks $1/\mathrm{dist}$ exactly ($2.0, 10.0, 100.0, 1000.0$ at distances $0.5, 0.1, 0.01, 0.001$).

The price is explicit: one describes functions on the cover, not on $\mathbb{C}^*$, and "once around" becomes a trackable event rather than a return.

2.3 Derivative property

On the spiral $z\,d/dz = d/dw$, so

• bilateral (no endpoint): $\mathcal{T}[z f'](s) = s\,\mathcal{T}[f](s)$;
• one-sided (cut at $z_0$): $\mathcal{T}[z f'](s) = s\,\mathcal{T}[f](s) - f(z_0)$,

verified to 20 digits. This is not a discovery but it is a useful clarification: the three classical transforms' apparently different derivative rules differ only in whether the domain has an endpoint. Laplace's $f(0)$ is not an anomaly; it is the transform reporting an initial condition. Fourier has no endpoint. Mellin's term vanishes for any $f$ whose transform converges.

Correction to an earlier draft. A "damped Chebyshev identity" was claimed as new: $$T_n(\cosh a\cos b + i\sinh a\sin b) = \cosh(na)\cos(nb) + i\sinh(na)\sin(nb).$$ It is true, but it is the classical $T_n(\cos z) = \cos(nz)$ at complex argument: $\cosh a\cos b + i \sinh a\sin b = \cos(b - ia)$ exactly (verified to 20 digits), so both sides are $\cos(n(b-ia))$. The correct statement of the observation is a logarithmic spiral is a complex angle, which is the same content as §2.2 and carries no independent novelty.

2.4 Extraction, and an obstruction

Near a pole, $G'/G$ is dominated by that pole: at $s = p_1 + (0.05+0.02i)$ with a second mode present, $G'/G = -16.98+7.21i$ against the isolated value $-17.24+6.90i$.

Two readings. First, $G'/G = -\mathbb{E}_s[w]$, an exponentially tilted (Esscher) mean of $w = \log z$ — real part a typical log-scale, imaginary part a typical phase. Second, poles are repelling under Newton's method on $G$; one iterates on $1/G$ instead, giving $s \mapsto s + G/G'$, which converges to $p_1$ in four steps to 17 digits. The argument principle confirms simplicity: winding number $-1$ around one pole.

The obstruction. A sum of $m$ simple poles has numerator degree $m-1$ over a common denominator, hence $m-1$ extraneous zeros sitting between the poles. Any contour enclosing all modes also encloses these. With poles at $0.3+1.5i$ and $-0.5+2.7i$ (residues $1, 0.6$), the induced zero is at $-0.2+2.25i$; the $k=0$ integral returns $Z - P = -1$ rather than $-2$, and Newton's identities then recover wrong modes. Small isolating contours recover both exactly.

This is a real difference from the polynomial case, where $F'/F$ has poles only, and it is why practical mode extraction builds a pole-only characteristic polynomial directly from samples (Prony, matrix pencil, ESPRIT) rather than contour-integrating an observed transform.

2.5 Estimation limits (standard, included for the failure taxonomy)

For a two-mode signal ($p_1 = 0.3+1.5i$, $p_2 = -0.5+2.7i$, $\Delta = 0.05$, noise $\sigma = 0.05$), median error in the weaker decaying mode:

Three distinct effects, which is the reason to include an otherwise textbook exercise:

1. SVD rank truncation beats polynomial root-finding by $\sim10\times$ at low noise, because root-finding amplifies coefficient noise.
2. ML does not attain the CRB at $N{=}40$ — started at the true parameters, the optimizer moves away to lower cost. Standard threshold effect; the CRB is asymptotic.
3. The $N{=}160 \to 320$ plateau is a genuine information wall. Fisher information for a decaying mode's rate accumulates as $\sum_n t_n^2 e^{-2\gamma t_n}$, a convergent series: totals $4.56, 10.92, 14.20, 14.40$ for $N = 40, 80, 160, 320$. A decaying signal carries a finite total budget of information about its own decay rate.

A fourth hypothesis — that the $N{=}40$ gap was an initialization problem — was tested with multi-start refinement across four pencil parameters and produced byte-identical results. Rejected.

General criterion. Total Fisher information about a parameter is finite iff the sensitivity function $\partial s/\partial\theta$ is square-summable. For a mode $s = e^{pt}$ and the parameter $p$, $\partial s/\partial p = te^{pt}$, so the criterion is exactly $\mathrm{Re}(p) < 0$. Verified: totals saturate for $\mathrm{Re}(p) = -1, -0.5, -0.1$ and diverge for $\mathrm{Re}(p) = 0, +0.3$.

The consequence is stronger than a slow rate. Finite total Fisher information means the CRB does not tend to 0 as $N \to \infty$, so no consistent estimator exists — a decaying mode's own decay rate cannot be determined exactly even with unlimited data. The plateau in the table above is that fact, not a numerical artifact.

2.6 Application: LPPL

$\ln p(t) = A + \mathrm{Re}[K(t_c-t)^{m+i\omega}]$ is a spiral mode in $w = \log(t_c - t)$. On synthetic data ($t_c=100$, $m=0.4$, $\omega=8.0$) the fit recovers $100.03, 0.397, 8.010$.

Correction. An earlier draft reported that the fit "correctly returns $\omega \approx 0.03$ on a random walk rather than hallucinating structure." That was one realization, and it was lucky. Repeating on 30 independent random walks: 22 converge, median $|\omega| = 4.05$, maximum $172$, and 86% return $|\omega| > 2$ — values that would read as genuine log-periodicity. The seven-parameter fit hallucinates structure on null data most of the time. This is a second, independent strike against predictive use, alongside lead-time collapse.

Lead time collapses sharply:

The failure is not graceful — sharp, wrong, confident answers. This is the numerical shape of the standing objection to LPPL crash prediction. The mathematics is sound; the predictive claim is contested, and lead-time collapse is why. The idea stands on firmer ground in materials failure, where $t_c$ is externally verifiable.

---

3. Part II: Euler's pentagonal theorem

Independent of Part I. Included because §3.3 is the same log-derivative identity in a combinatorial setting, and §3.4 is the one observation in these notes I have not seen stated explicitly.

3.1 Setup

$\prod_{n\ge1}(1-x^n) = \sum_k (-1)^k x^{k(3k-1)/2}$, verified to order 60. Nonzero coefficients occur only at generalized pentagonal numbers $0,1,2,5,7,12,15,22,26,35,40,51,57$. Since every finite subset $S\subset\mathbb{N}$ contributes $(-1)^{|S|}x^{\sum S}$, almost all subsets must cancel in parity-matched pairs.

3.2 Franklin's involution

For a partition into distinct parts, let $s$ = smallest part, $m$ = length of the top diagonal, $r$ = number of parts. If $s\le m$ and $s \ne r$: delete the smallest row, add 1 to each of the top $s$ parts. If $m<s$ and not ($m=r$ and $s=m+1$): the inverse. The two exclusions are the fixed points.

Verified exhaustively for $n \le 16$: a valid free $\mathbb{Z}/2$ action with

Exactly one at each generalized pentagonal number, none elsewhere; the two staircase families are the branches $k(3k\mp1)/2$.

The signed sum then collapses to fixed points alone — the mechanism of the Lefschetz fixed point theorem and of Duistermaat–Heckman / Atiyah–Bott localization. The partition recurrence follows and reproduces $p(0..20)$ exactly.

3.3 The log-derivative, again

$x F'(x)/F(x) = -\sum_N \sigma(N)x^N$. Matching coefficients gives a recurrence for $\sigma$ with the same pentagonal memory positions, but the term whose lookback lands at index 0 supplies $\sigma(n)$ itself rather than a boundary constant. Solving for it amounts to substituting the current index $n$ wherever $\sigma(0)$ would sit. Verified against direct divisor sums to $n=30$.

A distinct species of self-reference: the recurrence's index leaks into its value table at exactly one position per step. The partition recurrence has the constant $p(0)=1$ in the same slot.

3.4 Stabilizer size, stability, and selection are independent

An earlier draft claimed a clean two-way distinction: Franklin's fixed points have full stabilizer (a generic partition moves; the staircases cannot, being too rigid for the move to be defined), while spontaneously broken vacua have reduced stabilizer (the Higgs Lagrangian is $SU(2)_L\times U(1)_Y$ symmetric; the chosen vacuum is invariant only under $Q = T^3+Y/2$). Both statements are true. But the framing implied that stabilizer size is what marks a point as the one that matters, and testing that against further cases breaks it.

The test uses a control parameter to move a single point across categories. In mean-field Landau theory, $F(m) = a(T-T_c)m^2 + bm^4$, the point $m=0$ has full $\mathbb{Z}_2$ stabilizer at every temperature. Its curvature is $F''(0) = 2a(T-T_c)$: above $T_c$ it is a stable minimum and is the realized state; below $T_c$ it is an unstable maximum and is not. Stabilizer unchanged, status inverted.

Four cases, three properties:

Full stabilizer co-occurs with all three selection outcomes. And Feigenbaum's universal function is unstable — that instability is precisely what $\delta$ measures — yet it governs the asymptotics of the entire period-doubling route. So neither stabilizer size nor dynamical stability determines whether a distinguished point carries the information.

A further correction: two of the mechanisms merge. I claimed Franklin's selection was purely algebraic — cancellation with no variational character — and that this made it independent of energetic minimization. That is wrong.

Subbarao observed that Franklin's involution preserves not only size but perimeter ($\lambda_1 + \ell(\lambda) - 1$), so it acts within each perimeter class. Testing the consequence: among partitions of $n$ into distinct parts, the staircase is the unique minimizer of perimeter, verified at every generalized pentagonal $n \le 35$. And the minimum-perimeter class is a singleton exactly at generalized pentagonal numbers — verified for all $n \le 30$:

This is a complete explanation rather than a coincidence. Franklin's involution partitions each perimeter class into pairs; a class of odd size must leave one survivor; the minimum-perimeter class has size 1 precisely at pentagonal $n$; and the staircase — the most tightly packed distinct-part partition, with no room for the involution's move — is that survivor.

So Franklin's fixed points do have a variational characterization.

But this does not generalize — a correction to the correction. Chen and Liu (arXiv:0911.5019) give a Franklin-type involution on partitions into distinct parts with smallest part odd, whose fixed points sit at $n = k^2$. If "involution fixed points extremize a preserved functional" were a general mechanism, some natural functional should single out the squares. Tested: perimeter, largest part, number of parts, $\lambda_1 - \ell$, and smallest part. None reproduces the correspondence — perimeter already fails at $n=9$ (min class has size 2) and at $n=6,7,11,12$ (odd class, non-square).

So the collapse is specific to the classical Franklin involution, not a principle. Two selection mechanisms coincide in one case without being the same mechanism.

What survives. The three mechanisms stand as distinct — algebraic cancellation, variational minimization, asymptotic dominance — with the caveat that they can coincide on particular objects, and that such coincidence should not be read as identity. Franklin is a case where cancellation and minimization happen to select the same set; Chen–Liu is a case where they do not.

The defensible residue is a warning rather than a thesis: when two settings both involve a group action with a distinguished invariant locus, that structural similarity does not license transferring conclusions between them, because what makes the locus matter can differ completely. §3.5 is an instance — the analogy to RG fixed points is real at the level of orbit structure and fails at the level of local spectral theory.

3.5 Where the RG analogy stops

Franklin's fixed points resemble RG fixed points: generic objects cancel or flow away, an invariant locus carries the information, and Feigenbaum's universal function is literally a fixed point of $T[g](x) = \alpha g(g(x/\alpha))$.

The analogy has a sharp edge. Feigenbaum's $\delta$ is an eigenvalue of the linearization of $T$ — a statement about the neighborhood. Franklin's setting is discrete: no tangent space at a staircase partition, no nearby non-fixed objects, hence no $\delta$ analogue at all. What transfers is only the narrow claim that rare invariant points carry disproportionate information.

---

4. Part III: Non-archimedean dynamics

The most self-contained part. §4.4 is the clearest single finding in these notes.

4.1 Ostrowski and the ultrametric

Every nontrivial absolute value on $\mathbb{Q}$ is the ordinary one or a $p$-adic one; the list is complete, and the places are tied by $\prod_v|x|_v = 1$. The $p$-adic ones satisfy $|x+y|_p \le \max(|x|_p,|y|_p)$, which the ordinary one provably fails ($|50+3| = 53 \not\le 50$). Everything below follows from this.

4.2 No fractal basin boundary

Over $\mathbb{C}$, Newton's method for $z^3-1$ has a Julia set: points $2\times10^{-19}$ apart converge to different roots, and one unit-length line crosses the boundary 73 times. The mechanism is the "vote" of $F'/F$ being nearly tied and amplified.

Ultrametric ties are not infinitely finely balanced. Verified for $x^2-1$ mod $7^{20}$: ten starting points with wildly different high-order digits but the same residue class mod 7 all converge to the same root, with correct digits doubling each step ($1,2,4,8,16$) — Hensel's lemma, quadratic convergence.

4.3 Good reduction forbids chaos; bad reduction permits it

This section previously claimed "chaos is archimedean." That claim is false and is retracted. The correction is recorded rather than deleted because the error was instructive.

What is true. For $f(x) = 4x(1-x)$, the exact identity $f(x)-f(x') = 4(x-x')(1-x-x')$ makes the multiplier a $p$-adic unit away from the critical point, so valuations are preserved exactly. Verified at $p=5$ mod $5^{25}$: points agreeing to $k$ digits still agree to exactly $k$ digits after 15 iterations, for $k=3,6,10$. At $p=2$, where $4=2^2$ degenerates the map, the valuation grows by 2 per step — super-contraction, not chaos. This is the "good reduction implies empty Julia set" theorem (Rivera-Letelier, Benedetto).

What is false. Generalizing this to "no $p$-adic map is chaotic" ignores bad reduction. Take $f(z) = z^2 + c$ with $|c|_p > 1$. On the Julia set, which sits at $|z|_p = |c|_p^{1/2} = r$, the identity $f(z)-f(w) = (z-w)(z+w)$ gives $|f(z)-f(w)| = |z-w|\cdot r$ — genuine expansion by $r > 1$ per step.

Verified concretely at $p=5$, $c = 1/25$ (so $r=5$). The fixed points solve $z^2 - z + c = 0$, giving $z^* = (1+\sqrt{21}/5)/2$ with $\sqrt{21}$ obtained by Hensel lifting in $\mathbb{Z}_5$. The multiplier is $|2z^*|_5 = 5 > 1$: a repelling fixed point, hence on the Julia set. Perturbing it at digit 8 and iterating mod $5^{30}$:

The gap multiplies by exactly 5 each iteration. Positive Lyapunov exponent $\log 5$, in $\mathbb{Q}_5$. The dynamics on this Julia set is conjugate to the full 2-shift.

The error, and why it was easy to make. A first attempt tested "sensitivity" on orbits that were simply escaping to infinity, where gaps grow trivially in any setting; that measures the basin of infinity, not the Julia set. Finding a repelling periodic point is what forces the test onto the Julia set itself.

Corrected statement. Chaos is not archimedean. It requires bad reduction, which is a condition on the map's coefficients relative to $p$ — not a property of the absolute value alone. The logistic map at $r=4$ is tame at every prime because it happens to have good reduction at odd $p$ and to degenerate rather than expand at $p=2$; that is a fact about that map, not about $\mathbb{Q}_p$.

Is there a $p$-adic Feigenbaum cascade? No — but for a different reason than the retracted one. For $z^2+c$ the period-2 cycle has multiplier $4(c+1)$, so the parameter set with an attracting 2-cycle is $\{c : |4(c+1)|_p < 1\}$ — an ultrametric ball, hence clopen, with empty boundary. Over $\mathbb{R}$ the analogous set is the interval $(-5/4, -3/4)$, whose boundary points are the bifurcations, and the cascade is an accumulation of such boundary points. A totally disconnected parameter space has no connected path along which a multiplier can move continuously through $-1$, so there are no bifurcation points to accumulate. Chaos is present or absent according to whether $|c|_p > 1$, with no route between the two.

The still-standing negative result. The product formula constrains a fixed number's norms across places and supplies no mechanism relating a map's Lyapunov exponent at one place to another. There is no adelic compensation principle. The adelic view supplies Call–Silverman height theory, $\hat h = \sum_v \hat h_v$, in which almost all local terms are trivial.

4.4 Roughness splits in two: a place-dependence test

The real setting conflates two sources of fractal dimension that the $p$-adic setting separates. The following is a controlled comparison — same kind of object (a Cantor set), measured at two places, differing only in whether its dimension is generated geometrically or dynamically.

Instrument check. For digit-restriction sets $S = \{x \in \mathbb{Z}_p : \text{all base-}p\text{ digits lie in } D\}$, $|D| = d$, the sum $\sum_n d^n p^{-ns}$ has a pole at $\log d/\log p$. Verified finite just above and divergent just below the predicted value for $(p,d) \in \{(3,2),(5,2),(5,3),(5,4),(7,3),(7,5),(11,4)\}$ — seven cases, all exact.

Geometric case: dimension is place-independent.

Both equal $\log 2/\log 3 = 0.63093$. The construction is digit restriction in both cases; changing the absolute value does not change the answer.

Dynamical case: dimension is place-dependent.

Take $f(z) = z^2 - 3$. Since $|{-}3| > 2$, the Julia set over $\mathbb{R}$ is a Cantor set. Generated by inverse iteration ($10^6$ points, depth 20) and box-counted:

The $p$-adic side follows from $f(z)-f(z') = (z-z')(z+z')$: the gap's valuation is preserved unless $z+z'$ is a non-unit, i.e. near the critical point. Verified at $p=5$ mod $5^{25}$ — starting points agreeing to $2$, $5$, or $9$ digits still agree to exactly that many digits after 20 iterations. Zero expansion, hence empty Julia set (good reduction).

Extension: dimension is a place-indexed family. Given §4.3, a sharper case is available: a map whose Julia set is nonempty at both places, with different dimensions. For $f(z)=z^2+c$ with $|c|_p>1$, the $p$-adic Julia set is a Cantor set with two inverse branches each contracting by $r=|c|_p^{1/2}$, so $\dim = \log 2/\log r$. For $c = 5^{-2k}$ as a real number, $c$ is tiny and inside the main cardioid, so the complex Julia set is a quasicircle with $\dim \approx 1$ (Ruelle).

The $p$-adic dimension was verified by direct box count rather than assumed: enumerating residues that survive $n$ iterations without escaping gives $N(5^{-n}) = 2^n$ exactly for $n = 1,\dots,7$, hence $\dim = \log2/\log5$.

Both positive, both nontrivial, and they move in opposite directions as $k$ grows. Neither the sum nor the product is constant, so there is no product-formula analogue for dimension — as expected, since dimension is not a norm, but worth ruling out explicitly.

Reading. Two Cantor sets, both with dimension strictly between 0 and 1 over $\mathbb{R}$, indistinguishable by that measurement alone. Changing the place separates them completely: the digit-restriction set keeps its dimension, the Julia set loses it entirely. Fractal dimension in the real setting is therefore not a single quantity — it can have a geometric component that is intrinsic to the construction and a dynamical component that exists only because the archimedean absolute value permits expansion. The two are not distinguishable from within $\mathbb{R}$, where they routinely co-occur.

4.5 Computability

Hensel's condition $|f(x_0)|_p < |f'(x_0)|_p^2$ is a finite, checkable certificate guaranteeing convergence to a unique nearby root — stronger than anything real Newton's method offers, where rigorous isolation needs separate machinery (Sturm sequences, interval arithmetic).

This does not generalize. Equality of computable $p$-adic numbers is undecidable for the same halting-flavored reason as in $\mathbb{R}$. Only the algebraic case improves.

4.6 What survives of Mellin

Tate's local zeta integral is the $p$-adic Mellin transform, $Z(f,s) = \int_{\mathbb{Q}_p^*} f|x|_p^s\,d^*x$. For $f = \mathbf{1}_{\mathbb{Z}_p}$ it sums to $(1-p^{-1})/(1-p^{-(s+1)})$, the local Euler factor (verified to 15 digits). Multiplicative convolution becomes a product — verified on shell indicators — because $v_p(xy) = v_p(x)+v_p(y)$.

---

5. Status of claims

Classical, cited not derived: Euler's pentagonal theorem; Franklin's involution (1881); Lefschetz and Atiyah–Bott localization; Ostrowski; Hensel; Tate's thesis; good-reduction results in non-archimedean dynamics; Prony, matrix pencil, ESPRIT; Cramér–Rao and threshold effects; LPPL and its critiques; Call–Silverman heights; the Fourier–Mellin transform.

Corrected from an earlier draft: the "damped Chebyshev identity" is the classical identity at complex argument (§2.3); the "unified spiral transform" is Mellin transform on the universal cover, not a new object (§2.2); "localization" as a unifying thesis was too general to be load-bearing and has been dropped.

Retracted after testing: "chaos is archimedean" (§4.3) — bad-reduction maps have genuinely chaotic $p$-adic Julia sets, demonstrated at a repelling fixed point; and the independence of algebraic cancellation from variational selection (§3.4) — Franklin's fixed points are the unique perimeter-minimizing distinct-part partitions.

Offered, modestly: the perimeter-minimization characterization of the pentagonal numbers and the odd-class-size explanation (§3.4) — the perimeter invariance is Subbarao's, the minimization consequence I have not seen stated; the place-indexed dimension family with no product-formula analogue (§4.4); the finite-Fisher-information criterion and its non-consistency consequence (§2.5), corroborated by the CRB itself plateauing at $0.0132$ for $N\ge160$; the LPPL false-positive rate on null data (§2.6); the failure taxonomy throughout — extraneous-zero obstruction (§2.4), threshold effect and information wall (§2.5), lead-time collapse (§2.6), no adelic compensation (§4.3), no $\delta$ analogue (§3.5).

Verification scope. Illustrations, not proofs. Franklin exhaustive only to $n \le 16$; $p$-adic dynamics at $p \in \{2,5\}$ at fixed modulus; the estimation study is one signal at one parameter set. Broader sweeps would strengthen but not alter claims that are either classical or explicitly framing.

Parts are independent. Nothing in Part I is used to derive anything in Part II or III. They share a tool (§1) and a habit of attending to failure modes. That is the honest extent of the connection.