Prime Defects, Convolution, Lag Norm Measures, Neutral Locus.
Composite structures are absorbed by convolution.
Prime defects are not.
What cannot be absorbed must survive only on a neutral line.
A. Prime Defects as Non-Absorbable Residues under Convolution
A.0. Premise
We do not treat primes as outputs of a closed generative rule.
Instead, we treat them as fixed defects: residues that remain when compositional closure attempts to absorb all updates.
A.1. Convolution as compositional closure
Let $\mathcal{C}$ denote a compositional arena (multiplication / composition).
Let $\star$ denote a generic “closure operation” (convolution-type aggregation), understood as:
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absorption of local inconsistencies into global structure,
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recovery of update-lag through compositional recombination.
The key point is not the specific kernel, but the role of $\star$:
it is the operation by which compositional structure tries to eliminate residuals.
As a minimal concrete realization, one may take convolution in the Dirichlet sense on arithmetic functions, where multiplicative closure corresponds to absorbability under Dirichlet convolution.
In this setting, composite numbers arise from repeated convolution, while primes correspond to fixed residues that cannot be decomposed into nontrivial convolution factors.
A.2. Defect and absorbability
Let $\mathrm{lag}=-1$ denote a minimal “non-recoverable update” in the compositional process.
Define a residual operator $\operatorname{Res}$ extracting what closure fails to absorb.
We posit the Prime Defect definition:
\[\boxed{ \mathbb{P} := \operatorname{Fix}\Big( \operatorname{Res}(\mathrm{lag}=-1) \Big| \mathcal{C} \Big) }\]Interpretation:
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Composite structure corresponds to what is absorbable under $\star$: defects are eliminated by closure.
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Prime structure corresponds to what is not absorbable: a fixed residue persists.
So primes are not generated by $\star$; they are what $\star$ cannot eliminate.
A.3. Euler product as a statement of non-absorption
The Euler product does not generate primes; it assumes the defect set $\mathbb{P}$ and records it:
\[\boxed{ \zeta(s) = \prod_{p\in\mathbb{P}}(1-p^{-s})^{-1} }\]This is consistent with the present stance:
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primes $\mathbb{P}$ are axiomatic residues,
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$\zeta$ is an analytic encoding of the residues, not their generator.
A.4. Minimal takeaway (A-only)
\[\boxed{ \textbf{Primes are defects that convolutional closure cannot absorb.} }\]This is not yet a statement about $\Re(s)=\tfrac12$.
It is the pre-analytic placement: what must be treated as “non-absorbable” before any stability question is meaningful.
B. What the Lag Norm Measures (without proof)
B.0. Goal
We need a measurement of “how much irrecoverable lag remains” after closure-like operations attempt to absorb it.
So we introduce a norm whose job is:
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to quantify non-absorbability,
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to identify the neutral stability locus where the residue neither blows up nor vanishes.
B.1. Lag as residual field
Let $D$ denote the defect/residual object extracted from compositional closure:
\[D \equiv \operatorname{Res}(\mathrm{lag}=-1).\]Under analytic projection, think of $D$ as producing an “analytic shadow” $S_s$ depending on a complex parameter $s$ (not yet specified how):
\[D \xrightarrow{\ \ \zeta\ \ } S_s.\]B.2. Define a lag norm by “stability against smoothing vs interference”
We define a norm functional $|\cdot|_{\mathrm{lag}}$ whose interpretation is:
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Too much smoothing (typical of $\Re(s)>\tfrac12$) makes residues disappear: $|S_s|_{\mathrm{lag}}\to 0$.
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Too much interaction / interference (typical of $\Re(s)<\tfrac12$) makes residues unstable: $|S_s|_{\mathrm{lag}}\to\infty$.
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The critical locus is where the residue is neither erased nor divergent.
So we place the “one-line” criterion:
\[\boxed{ \Re(s)=\tfrac12 \ \Longleftrightarrow |S_s|_{\mathrm{lag}} \ \text{is scale-neutral (neither decays nor explodes).} }\]This is deliberately schematic. The point is not the formula; the point is what the norm measures:
\[\boxed{ |\cdot|_{\mathrm{lag}} \text{ measures the survival of irrecoverable residues under analytic projection.} }\]Operationally, the lag norm may be understood as measuring the persistence of residual structure under analytic smoothing.
If smoothing dominates, the residue vanishes; if interference dominates, the residue diverges.
The neutral locus is characterized by scale-invariant persistence.
This norm does not measure magnitude, but survivability.
Most minimally, one may imagine $|\cdot|_{\mathrm{lag}}$ as a functional that probes Dirichlet series of $\zeta(s)$-type, selecting those values of $s$ for which contributions from prime defects are neither canceled nor amplified.
Operational example (sketch).
Let $f(n)$ be an arithmetic function encoding defect residues (e.g. supported on primes).
Consider its Dirichlet series
Define the lag norm schematically by the scale response
\[|F|_{\mathrm{lag}}(s) := \lim_{T\to\infty}\frac1T\int_0^T |F(\sigma+it)|^2dt, \qquad \sigma=\Re(s).\]Then:
$\sigma>\tfrac12$: smoothing dominates, residues are absorbed.
$\sigma<\tfrac12$: interference dominates, residues amplify.
$\sigma=\tfrac12$: scale-neutral persistence.
B.3. Minimal takeaway (B-only)
\[\boxed{ \textbf{The “critical line” is the neutrality condition of the lag norm.} }\]C. Why the Neutral Locus Can Be Expressed via a Self-Adjoint (Unitary) Operator
C.0. Translation objective
Once “neutral stability” is identified $C$, mathematics expects a spectral formulation:
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stability ↔ conservation of norm/energy,
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conservation ↔ unitarity,
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unitarity ↔ self-adjoint generator.
This step does not add content; it changes the garment.
C.1. From neutrality to a norm-preserving flow
Assume that at the neutral locus the analytic shadow evolves without loss/gain in the lag norm:
\[|S_s|_{\mathrm{lag}} = \text{constant} \quad \text{(on the neutral line)}.\]This invites the standard translation:
- “constant norm under evolution” ⇒ evolution is unitary on a suitable Hilbert-like space $\mathcal{H}_{\mathrm{lag}}$.
So we write:
\[S_s \sim U_t S,\qquad U_t^\ast U_t = I.\]One may regard $\mathcal H_{\mathrm{lag}}$ as the completion of analytic shadows $S_s$ under the lag norm at the neutral locus.
On this space, norm-neutral evolution admits a unitary representation.
C.2. Unitary flow implies self-adjoint generator
By the usual correspondence (Stone-type intuition), a strongly continuous unitary flow can be written as
\[\boxed{ U_t = e^{itH}, \qquad H=H^\ast. }\]Thus “stability” becomes “self-adjointness” as a translation:
\[\boxed{ \textbf{Neutral lag stability} \rightsquigarrow \textbf{unitary evolution} \rightsquigarrow \textbf{self-adjoint generator}. }\]The appearance of a self-adjoint generator should not be read as a physical postulate.
It is a representational translation: norm-neutral evolution admits a unitary description, and unitarity admits a self-adjoint generator.
Self-adjointness here is a language, not an ontology.
In this sense, self-adjointness functions as a minimal grammatical device for describing the neutral survivability of defects under analytic smoothing, rather than as a claim about physical dynamics.
C.3. Minimal takeaway (C-only)
\[\boxed{ \textbf{If lag is neutrally stable, it admits a spectral (self-adjoint) translation.} }\]If a residual structure persists under analytic smoothing without growth or decay, its representation must lie on a neutral line.
In the zeta framework, this neutrality corresponds to $\Re(s)=\tfrac12$.
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| Drafted Jan 24, 2026 · Web Jan 24, 2026 |