% 281_Addendum_ObservationBoundary.tex
% OI-280-1 closed: the observation operator's boundary conditions are
% Dirichlet at both ends — inherited from the substrate perturbation
% problem (P03, verified) and forced by the nesting requirement at the
% universe boundary; independently selected by the mass data (A280).
% P07's "free at boundary" prose is retracted; P007_1/P007_2 retired.
% Verifier: verify_P281.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Addendum 281: The Observation Boundary\\
\large Dirichlet at both ends: derivation, retraction,\\
and the settled P07 spectrum (OI-280-1 closed)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (two independent corpus arguments $+$ data
selection; prose retraction recorded) \quad|\quad
\textbf{Closes:} OI-280-1; retires TBS P007\_1, P007\_2 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P281.py} --- all PASS

\begin{abstract}
A280 established that P07's published accuracy and its stated boundary
interpretation are incompatible (the muon prediction degrades
$9.1\times$ under the claimed mixed conditions) and filed the choice as
OI-280-1. The Dirichlet condition $\psi(0) = \psi(1) = 0$ is here
derived twice over. \emph{(i) Substrate inheritance.} The observation
amplitude $\psi$ is a field over the same density substrate whose
perturbation theory the corpus has already settled: P03's stability
problem imposes vanishing perturbations at both endpoints (sine modes;
verified to five digits in A278). The endpoints are not degrees of
freedom of the observed system --- $x = 0$ carries no density
($\rho(0) = 0$: nothing to observe) and $x = 1$ is the observer's own
boundary (the fixed reference frame of the measurement, not part of the
measured). A field that measures the substrate vanishes where the
substrate is anchored. \emph{(ii) Nesting.} P07's own framework defines
a particle as a \emph{nested} boundary strictly inside the universe
boundary; at $x = 1$ nesting depth is zero and the particle is
indistinguishable from the boundary itself --- no separation, no
observation event, $\psi(1) = 0$. The prose interpretation
``observation is free at the universe boundary'' is retracted and
replaced: \emph{observation vanishes at the universe boundary},
consonant with the edge layer ($\pi/\Omega$: where observation runs
out). The data's independent selection of Dirichlet (A280) is thereby
explained rather than merely recorded. With the conditions settled, the
P07 spectrum is the de facto one ($\lambda_{1,2,3} = 16.51, 51.24,
102.02$) and the mass mapping's honest scorecard is: muon $1.1\%$,
tau $6.3\%$ (log-space) --- the tau residual is now a physics question
about $V_{\rm obs}$, not a bookkeeping question about boundaries, and
is left explicitly open. TBS P007\_1 and P007\_2 are retired (the
``defective'' discretization was computing the right problem; the
defect was in the prose).
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Derivation}

\begin{proposition}[Dirichlet from substrate inheritance]
\label{prop:inherit}
The observation amplitude $\psi$ measures structure of the density
substrate $\rho$ on $[0,1]$. The substrate's dynamical degrees of
freedom are its perturbations, and the corpus's perturbation problem
(P03, Theorem wave-stability; recomputed A278) is posed with vanishing
perturbations at both endpoints: $x = 0$ because $\rho(0) = 0$ (no
density, hence nothing whose observation could have amplitude), $x = 1$
because the boundary is the observer's anchored reference frame ---
its state is the frame of the measurement, not a measurable. A field
defined as ``amplitude for observing the substrate at depth $x$''
vanishes wherever the substrate has no observable freedom:
$\psi(0) = \psi(1) = 0$.
\end{proposition}

\begin{proposition}[Dirichlet from nesting]
\label{prop:nesting}
P07 defines the particle as a nested boundary $S^3_{\rm particle}$
strictly interior to $S^3_{\rm universe}$, with mass the stabilization
scale of the nesting. At $x = 1$ the nesting depth is zero: the
``particle boundary'' coincides with the universe boundary, no
separation between observer and observed exists, and no observation
event can occur. The amplitude for a particle \emph{at} the boundary is
zero --- $\psi(1) = 0$ --- by the framework's own definition of what
$\psi$ is the amplitude of.
\end{proposition}

\begin{remark}[Retraction, recorded]
P07's prose assigned $\psi'(1) = 0$ (``observation is free at the
universe boundary''). Both derivations above, and independently the
mass data (A280: $9.1\times$ muon degradation under the mixed
conditions), give $\psi(1) = 0$. The interpretation is retracted:
observation is not free at the boundary; it \emph{ends} there. The
corpus's edge layer says the same thing in layer language --- the edge
fraction is where observation runs out (P04). The de facto
discretization of P07 was therefore computing the correct problem all
along; the published accuracy stands on the corrected interpretation.
\end{remark}

\begin{remark}[The settled scorecard; what remains open]
With boundaries settled: $\lambda_{1,2,3} = 16.507, 51.239, 102.020$
($N = 1000$), $\beta = 6\mu_1/\mu_0 = 4.7600$ (derived), giving
$m_\mu/m_e = 219.6$ (log-error $1.1\%$) and $m_\tau/m_e = 5825$
(log-error $6.3\%$). The tau residual is the operator's real open
question --- a $V_{\rm obs}$-level question (potential refinement,
higher-order moments), not a boundary question. Left open; no OI filed
(it is P07's headline accuracy claim, already on the registry's radar
via the retired items' history).
\end{remark}

\paragraph{Consequences.} OI-280-1 closed. TBS P007\_1, P007\_2 retired
(by A281). The registry's reworked class empties; P07's spectrum is
canonical at Dirichlet--Dirichlet.

\paragraph{Verifier.} \texttt{verify\_P281.py}: the substrate
inheritance numerics ($\rho(0) = 0$; P03 Dirichlet spectrum
recomputed); the settled spectrum and scorecard; the retraction's
consistency (DD accuracy matches published, DN does not); registry
statuses. All PASS.

\end{document}
