%========================================================================
% APPENDIX CE: THE IMPROBABILITY OF THE PILE
%   Combined-evidence (no-coincidence) capstone. Wired into DFD_Main
%   2026-06-27. REWRITTEN 2026-07-02 after the
%   trials-factor audit and its threefold independent re-verification
%   (verify3x/, five audit scripts, all concordant). The former 5.5--7.1 sigma headline is
%   RETIRED; the corrected four-tier null ladder + the alpha-ladder
%   infeasibility theorem replace it (subsec:ce-trials-audit).
%   Every number below is grounded in the audit artifacts:
%     trials_factor_2026-07/ audit directory (E1..E6 scripts + outputs; shipped with the reproducibility package)
%   sigma sources: verify3x/synth_check.json (corrected_hostile,
%   conditional_base4, conditional_plus_a0_weak, alpha_ladder,
%   CE_as_written), E5_results.json (row prices), E6_synthesis.json
%   (space, selection_null).
%   Uses DFD_Main's theorem/proposition/remark/status environments
%   (do NOT redefine).
%========================================================================
\section{The Improbability of the Pile: A Combined-Evidence (No-Coincidence) Bound}
\label{app:CE:combined-evidence}

DFD derives, parameter-free, a set of fundamental numbers that the Standard Model
and $\Lambda$CDM treat as \emph{free} inputs. This capstone quantifies the combined
evidence \emph{after} a full trials-factor audit
(Sec.~\ref{subsec:ce-trials-audit}): every discrete menu the construction could
have scanned is enumerated and charged against the hits. Worst case first: under a
full across-observable selection charge (the most hostile admissible null) the
combined evidence is \textbf{not statistically significant}
($p\approx0.03$--$0.5$); under hostile menu accounting with no selection charge
the four surviving independent hits ($\sin^2\theta_W$, $\theta_{12}$-TM1, the
$H_0/\Lambda$ grammar block, $\Omega_\chi h^2$) give $\approx3.7$--$4.1\sigma$
(conservative windows) to $\approx5.0\sigma$ (moderate); \emph{conditional} on
DFD's forcing theorems as stated --- the corpus's pre-registered framing, on the
corrected row basis --- the floor is $\approx5.0$--$5.5\sigma$, a conditional
figure that is always to be read against the unconditional tiers just quoted.
Separately, and unconditionally, the $\alpha$-ladder infeasibility theorem
(Theorem~\ref{thm:ce-alpha-ladder}) stands: landscape scanning cannot produce the
$\alpha^{-1}=137.036$ match by chance. The former $5.5$--$7.1\sigma$ headline of
this appendix is \textbf{retired} (retraction paragraph,
Sec.~\ref{subsec:ce-trials-audit}). \textbf{We are explicit about what this is
and is not:} it is a \emph{preponderance / parameter-counting} argument --- the
same logic by which the Standard Model, BBN, and inflation were established ---
not a single falsification-test discriminator; each tier states exactly what it
grants; and it is \emph{leading-order} agreement ($0.2$--$4\%$), not precision.
The $\alpha$ match itself is \emph{excluded} from the pile rows ($\alpha$ is the
selector datum and must not be double-counted); its evidence is booked separately
in Theorem~\ref{thm:ce-alpha-ladder}.

\subsection{The surviving forced parameter-free rows (corrected basis)}
\label{subsec:ce-rows}
Each row is a quantity the SM/$\Lambda$CDM leave as a free parameter (or do not
predict at all), derived by DFD from $\{\alpha,\ \mathrm{M7}=\mathbb{CP}^2\times S^3\
\mathrm{topology}\}$ with \emph{no} fitted coefficient, and now \emph{priced}
against the menu of alternatives the construction could have scanned
(Sec.~\ref{subsec:ce-trials-audit}). Relative to the pre-2026-07 version of this
appendix, two rows are dropped and two are added (full retraction below): the CKM
$\sin2\beta$ row is dropped as \emph{zero-information} (Postulate~E.1 is an
independent normalization postulate and the CKM anti-theorem blocks apex forcing,
App.~\ref{app:ckm-update}; the companion Wolfenstein integers miss: $\lambda$ at
$\sim$$1.8\sigma$, $A$ at $\sim$$2.5$--$3.2\sigma$), and the fine $a_0$ row is
dropped because its former price ($p=0.005$) was $40$--$70\times$ finer than the
$\sim$$10\%$ empirical uncertainty on $a_0$ permits. Added: the $H_0/\Lambda$
grammar block ($3$ hits in the declared $135$-cell scan; look-elsewhere paragraph
of App.~\ref{app:de-status}) and the $\Omega_\chi h^2$ row (the finite
$SU(2)_{60}$ CS-vacuum Casimir abundance, App.~\ref{app:chi-matter},
Thm.~\ref{thm:chi-postinflation}). The PMNS $\theta_{12}$ row is \emph{restored}
at its pre-registered fair-menu price $1/14$: of the $14$ standard mixing
ans\"atze in the audit menu, only TM1 hits.

% Corrected row prices and products (verify3x/synth_check.json +
% E5_results.json, 2026-07-02):
%  hostile-conservative: s2w 1/12, th12 1/14, H0/Lam 3/135, Ochi 0.136
%    product = (1/12)(1/14)(3/135)(0.136) = 1.799e-5  -> Z = 4.13 sigma
%    (x3 s13-menu surcharge: 5.40e-5 -> 3.87 sigma; lens-3 th12 price
%     -> 3.75 sigma; band 3.7--4.1)                [corrected_hostile]
%  hostile-moderate: s2w 1/56, th12 1/14, H0/Lam (3/135)*0.074, Ochi 0.136
%    product = 2.853e-7 -> Z = 5.00 sigma          [corrected_hostile.mod4]
%  conditional (forcing theorems granted): s2w 2d=0.003899,
%    th12 2d=0.074345, H0/Lam 3/135, Ochi 2d=1/30
%    product = 0.003899*0.074345*(3/135)*(1/30) = 2.147e-7 -> 5.06 sigma
%    (+ weak a0 x0.3 -> 6.44e-8 -> 5.28 sigma; two-stage H0 -> 5.53 sigma;
%     band 5.0--5.5)      [conditional_base4, conditional_plus_a0_weak]
\begin{center}\fittab{\begin{tabular}{llllll}\toprule
quantity (DFD parameter-free formula) & DFD & measured & agree & $p$ (hostile) & $p$ (cond.) \\\midrule
$\sin^2\theta_W=3/13$ (weak mixing, tree) & $0.2308$ & $0.2312$ & $0.19\%$ & $1/12$--$1/56$ & $0.0039$ \\
PMNS $\sin^2\theta_{12}$ (TM1; $1$ of $14$ ans\"atze) & $0.3184$ & $0.307$ & $3.7\%$ & $1/14$ & $0.074$ \\
$H_0/\Lambda$ grammar block ($\rho_\Lambda=\tfrac{3}{8\pi}\alpha^{57}\rho_P$) & $2.55$~meV & $2.24$~meV & in band & $3/135$ ($\times0.074$) & $3/135$ \\
$\Omega_\chi h^2$ ($SU(2)_{60}$ CS Casimir) & $0.118$ & $0.1200$ & $1.7\%$ & $0.136$ & $0.033$ \\\midrule
$n_s=1-2/60$ (spectral index)$^\dagger$ & $0.9667$ & $0.9649$ & $0.18\%$ & --- & --- \\
\bottomrule\end{tabular}}\end{center}

\noindent{\small $^\dagger$Kept for the record but excluded from every tier, as
before: the ``$1-2/k_{\max}$'' form is a numerical coincidence of the forced
$n_s=1-28.7\alpha/2\pi$ (App.~AH3a, Extended Derivations). The old
$\Omega_\chi/\Omega_b=16/3$ clue row (``a clue, not a derivation,'' App.~J,
App.~TP) is superseded by the direct $\Omega_\chi h^2$ Casimir row above; the
retired $\sin2\beta$ and fine-$a_0$ rows are documented in the retraction
paragraph of Sec.~\ref{subsec:ce-trials-audit}.}

\subsection{Trials-Factor Audit and Conditional No-Coincidence Bound}
\label{subsec:ce-trials-audit}

The 2026-07 trials-factor audit (run directory
\texttt{trials\_factor\_2026-07/} (shipped with the reproducibility package), artifacts E1--E6, independently
re-verified through five audit scripts, \texttt{verify3x/}) asked the
hostile question directly: \emph{if DFD were numerology, how many numbers could
it have scanned, and how surprising are the hits once every scan is charged?}

\paragraph{The address space.}
The first instrument is an explicit enumeration of every discrete choice
(``address coordinate'') the construction could have made. Fifteen dimensions
are charged: ten native coordinates from the corpus's own ledger (E1/E2
artifacts) plus five hostile enlargements added by the audit's skeptic pass
(SK1). Table~\ref{tab:ce-address} lists them. The core (numeric) address space
is $|\mathcal A|_{\rm core}=64$ (baseline reading) up to $8.01\times10^{9}$
(maximal-hostile: the E2 conservative core $185{,}472$ times the SK1 convention
multiplier $43{,}200$); with sign/axiom-adoption bits, $2.1\times10^{12}$.
Continuous inputs ($\alpha$ itself, $\hbar$, one scale, the fitted knobs of the
fitted-knob ledger) are excluded from $|\mathcal A|$ and booked as fits.

\begin{table*}[t]
\centering
\caption{The 15-dimension address space $\mathcal A$ charged by the trials
audit (artifacts \texttt{E1\_address\_space.json},
\texttt{E2\_enumeration.json}, \texttt{E6\_synthesis.json} in
\texttt{trials\_factor\_2026-07/}). ``Cons.''\ = maximal-hostile
count. The two SK1 factors marked $\hookrightarrow$ enlarge existing rows (CKM
Wolfenstein basis $\times3$; AU prefactor list $9\to30$, i.e.\ $\times10/3$)
and are folded there, so the SK1 total multiplies to
$43{,}200=10\cdot12\cdot3\cdot6\cdot2\cdot3\cdot(10/3)$. Core totals: $64$
(baseline) $\to185{,}472$ (E2 cons.) $\to8.01\times10^{9}$ (E2 cons.\ $\times$
SK1); with convention/axiom bits $2.1\times10^{12}$.}
\label{tab:ce-address}
\fittabFull{%
\begin{tabular}{llll}
\toprule
\# / dimension & baseline & cons. & provenance (one line) \\
\midrule
1. det-line charge $q_1$ & $\{3,6\}$ & $\{1,3,6\}$ & menu $3\mathbb{Z}$ proven (App.~F); $q_1\ge9$ tail $\alpha$-hit-dead (E2) \\
2. gauge partition & 2 (SM, PS) & 7 & anomaly audit; $\dim G\le28$ kills $SO(10)$, $E_6$; SM-embedding rivals $\alpha$-hit-degenerate \\
3. generation count $N_{\rm gen}$ & $\{1,\dots,6\}$ at $q_1{=}3$ & box to cap $k_{\max}{=}440$ & discrete input (App.~F); interval saturation proven; cap $=$ evaluated ladder \\
4. padding $n$ & $\{5,6\}$ & $\{5,6\}$ & $B{-}L$ spin-$\mathbb{Z}_4$ six-species list, exactly one $Y{=}0$ entry (App.~F) \\
5. assembly rule & 1 (det-line chain) & $+7$ alternates & dichotomy lemma; every alternate $\ge30\%$ off $\alpha$ (hit-dead) \\
6. CKM apex branch & 2 & 8 & anti-theorem names both branches (App.~\ref{app:ckm-update}); $\times$unit convention, $\times\bar\eta$; $\hookrightarrow\times3$ Wolfenstein basis (SK1) \\
7. closure grammar & 1 (forced) & 135 & declared $9\times15$ scan (App.~\ref{app:de-status}); $\hookrightarrow$ prefactors $9\to30$ (SK1) \\
8. sign bits & $2^2$ & $2^4$ & rank-2 GF(2) theorem; non-numeric, generate no rival hits \\
9. axiom-adoption bits & 1 & $2^4$ & audited 11-input list ($+$Y.10); adopt no value from a menu \\
10. $\chi$ primordial state & 1 & 4 & natural-preparation menu; $\Omega_\chi\le0.405$ hard cap for any state \\
11. surface/topology menu & 1 & 10 & SK1 hostile enlargement \\
12. lens-space quotients & 1 & 12 & SK1 hostile enlargement \\
13. $\dim X$ & 1 & 3 & SK1 hostile enlargement \\
14. formula grammar & 1 & 6 & SK1 hostile enlargement \\
15. $57$-subtraction convention & 1 & 2 & SK1 hostile enlargement \\
\bottomrule
\end{tabular}}
\end{table*}

\paragraph{The $\alpha$-ladder infeasibility theorem.}
Within this space, the number of \emph{hit-capable} $\alpha$ ladders ---
distinct (ladder, convention) assignments that could have produced an
$\alpha^{-1}$ match at all --- is $10{,}800$.

\begin{theorem}[$\alpha$-ladder infeasibility of landscape scanning]
\label{thm:ce-alpha-ladder}
Let a false theory scan the fully enlarged address space of
Table~\ref{tab:ce-address}, containing $n=10{,}800$ hit-capable ladders. The
observed match is $\alpha^{-1}(k_{\max}{=}60)=137.03599985$ against CODATA
$137.035999084$, a residual of $5.6\times10^{-9}$ fractional ($0.0056$~ppm).
The per-ladder probability of a false match \emph{at least as good as the
observed $5.6\times10^{-9}$ fractional match} is $p=6.6\times10^{-7}$ (rung
spacing $\simeq1.7\times10^{4}$~ppm of $\alpha$ at $k=60$), so the expected
number of false matches at least as good as observed, across the whole space, is
\[
\fiteq{\mathbb{E}[\text{false matches at least as good as observed}]
 = n\,p \approx 7.2\times10^{-3}}
\]
($0.007$--$0.05$ under honest family inflation; $\le0.26$ under maximal
hostile family and convention inflation --- never $\ge1$). The infeasibility
margin is $\times140$ (headline), $\ge\times4$ (maximal hostile), and
$\times7.8$ under the pre-registered $\pm0.1$~ppm falsifier band. Landscape
scanning cannot produce the $\alpha^{-1}=137.036$ match by chance.
\end{theorem}

\noindent\emph{Proof sketch.} (1)~$\alpha^{-1}(k)$ is strictly monotone on
$k=1,\dots,2000$ (computed), so any sub-percent window contains at most one
integer rung: enlarging \emph{any} address cap adds trials but no new
hit-capable addresses beyond those counted (hit-capability lemma, E2).
(2)~Enumerating the space of Table~\ref{tab:ce-address} yields $n=10{,}800$
hit-capable ladders. (3)~The per-ladder false-hit probability is the two-sided
window-to-step ratio, $p=2\times137.036\times5.6\times10^{-9}/2.318
=6.6\times10^{-7}$. (4)~Linearity of expectation gives
$\mathbb{E}=np\approx7.2\times10^{-3}$; hostile family/bit inflation multiplies
$\mathbb{E}$ by at most $\sim$$36$, topping out at $0.26<1$. The wording of the
match criterion is load-bearing: under the weaker criterion ``within $1$~ppm''
the expectation is $1.28\ge1$ and no theorem survives --- the theorem is true
only for matches \emph{at least as good as the observed $5.6\times10^{-9}$
fractional match}, and it is stated only in that form. (Artifacts:
\texttt{verify3x/synth\_check.json}, \texttt{partI\_verify.json}.)\hfill$\square$

\paragraph{The four-tier null ladder (worst case first).}
% Tier sources (all 2026-07-02):
%  tier 1: E6_synthesis.json selection_null grid (cons_M12 0.227,
%          cons_M20 0.456, mod_M12 0.016, mod_M20 0.049);
%          adopted band p ~ 0.03--0.5 (verify3x synthesis);
%          median-product check: synth_check.json median_prod3of20
%          = 3.2e-4 > 2.5e-4 flagship-conservative product.
%  tier 2: synth_check.json corrected_hostile.cons4 = 1.80e-5 (4.13s),
%          cons4_x3_s13menu = 5.40e-5 (3.87s), lens-3 price 3.75s
%          (REPORT Sec. 2) -> band 3.7--4.1 sigma.
%  tier 3: synth_check.json corrected_hostile.mod4 = 2.85e-7 (5.00s).
%  tier 4: synth_check.json conditional_base4 = 2.15e-7 (5.06s),
%          conditional_plus_a0_weak = 6.44e-8 (5.28s), two-stage-H0
%          5.53s (REPORT Sec. 2) -> band 5.0--5.5 sigma.
Each tier is a complete, self-consistent null; each grants strictly more than
the one above it. Quote the tier that matches what your audience grants.
\begin{center}\fittab{\begin{tabular}{llll}\toprule
null (most hostile first) & what it grants & $p$ & $Z$ \\\midrule
full selection charge ($M{=}12$--$20$ booked comparisons, best-$k$ pick) & nothing & $\approx0.03$--$0.5$ & n.s. \\
hostile menus, conservative windows, no selection charge & row independence & $(1.8$--$5.4)\times10^{-5}$ & $3.7$--$4.1\sigma$ \\
hostile menus, moderate windows, no selection charge & row independence & $2.9\times10^{-7}$ & $5.0\sigma$ \\
\emph{conditional}: forcing theorems granted (pre-registered framing, corrected rows) & the derivations & $(0.6$--$2.1)\times10^{-7}$ & $5.0$--$5.5\sigma$ \\
\bottomrule\end{tabular}}\end{center}
Under the full selection charge the conservative flagship product
($2.5\times10^{-4}$) is genuinely unremarkable: the \emph{median} product of the
three smallest of $20$ uniform draws is $3.2\times10^{-4}$, i.e.\ a hostile
skeptic picking the three best of twenty booked comparisons expects to do this
well half the time.

\begin{tcolorbox}[colback=yellow!5!white, colframe=orange!70!black,
 title={Adopted combined-evidence claim (independently re-verified, 2026-07-02; worst case first)}]
Under a full across-observable selection charge (the most hostile admissible
null), the combined evidence is \textbf{not statistically significant}
($p\approx0.03$--$0.5$). Under hostile menu accounting with no selection charge,
the four surviving independent hits ($\sin^2\theta_W$, $\theta_{12}$-TM1, the
$H_0/\Lambda$ grammar block, $\Omega_\chi h^2$) give $\approx3.7$--$4.1\sigma$
(conservative windows) to $\approx5.0\sigma$ (moderate). Conditional on DFD's
forcing theorems as stated --- the corpus's pre-registered framing, on the
corrected row basis --- the floor is $\approx5.0$--$5.5\sigma$. The former
$5.5$--$7.1\sigma$ headline is retired: its $\sin2\beta$ row is zero-information
(Postulate~E.1 is an independent normalization postulate and the CKM
anti-theorem blocks apex forcing; $\lambda$ and $A$ miss at $\sim$$1.8\sigma$
and $\sim$$2.5\sigma$), and its $a_0$ row was priced $40$--$70\times$ finer than
the empirical uncertainty permits. Separately, and \emph{unconditionally}: the
$\alpha$-ladder infeasibility theorem stands --- across the fully enlarged
address space ($\sim$$10{,}800$ hit-capable ladders), the expected number of
false $\alpha$ matches \emph{at least as good as the observed $0.0056$~ppm} is
$\sim$$0.007$ ($\lesssim0.05$--$0.26$ under maximal hostile family and
convention inflation; margin $\times140$ headline, $\ge\times4$ hostile,
$\times7.8$ under the $\pm0.1$~ppm falsifier band) --- landscape scanning cannot
produce the $\alpha^{-1}=137.036$ match by chance. Strong evidence, not proof.
\end{tcolorbox}

\begin{remark}[Scope: what the audit kills, and what it cannot]
\label{rem:ce-audit-scope}
The audit closes the \emph{landscape-scanning} charge against the $\alpha$
match: no admissible enlargement of the address space makes a false
$\alpha^{-1}$ hit likely (Theorem~\ref{thm:ce-alpha-ladder}), and every
surviving pile row now carries its menu price. What the audit does \emph{not}
--- and cannot --- adjudicate is the \emph{grammar/framing choice} itself: a
skeptic who rejects DFD's forcing theorems is charged no trials and simply
reads the unconditional tiers, down to non-significance. What kills or confirms
the framing is the pre-registered falsifier set (App.~\ref{app:RM-kills}):
Belle-II $|V_{cb}|$ at $0.5\%$ (a confirmed inclusive $42.2\times10^{-3}$ kills
$A=108\alpha$ outright), FLAG $\xi$ firming at $1.206$ (kills the locked-CKM
$\Delta M_s/\Delta M_d$), the Simons Observatory $\sigma_8$ determination, the
$a_0(z)$ sign, and $m_\beta=9.16$~meV.
\end{remark}

\paragraph{Retraction of the former headline.}
The pre-2026-07 version of this appendix claimed $5.5$--$7.1\sigma$ (it printed
$P_6=1.4\times10^{-12}\Rightarrow7.1\sigma$ and a ``referee-proof floor''
$3\times10^{-8}\Rightarrow5.5\sigma$; exact recompute
$1.65\times10^{-12}\Rightarrow6.96\sigma$ and
$4.59\times10^{-8}\Rightarrow5.34\sigma$). Both rested on two invalid rows and
are \textbf{retired}, together with the related E3 central/maximal
$6.5/6.8\sigma$ figures (which retain the invalid $a_0$ window):
(a)~the $\sin2\beta$ row carries zero information --- Postulate~E.1
(App.~\ref{app:ckm-update}) is an independent normalization postulate, the CKM
anti-theorem blocks apex forcing, and the companion Wolfenstein integers miss
($\lambda$ at $\sim$$1.8\sigma$, $A$ at $\sim$$2.5$--$3.2\sigma$);
(b)~the $a_0$ row was priced at $p=0.005$, $40$--$70\times$ finer than the
$\sim$$10\%$ empirical uncertainty on $a_0$ permits (and it silently used DFD's
own $H_0$).
A post-apex-fix re-audit (2026-07, 3/3 independent confirmations;
\texttt{downgrade\_reaudit\_2026-07/}) \emph{strengthens} the zeroing rather
than reopening it: under the corpus's own PDG~2024 centrals
(App.~\ref{app:microsector}) \emph{neither} apex integer is even the nearest
point of its own grid --- $\bar\rho$ measures $21.80\alpha$ (nearest
half-integer $22$, not $\tfrac{43}{2}$) and $\bar\eta$ measures $48.28\alpha$
(nearest integer $48$, not $49$) --- so the apex pair carries menu price
$p_{\rm menu}=1.0$ outright; under the earlier PDG-generation snapshot
($\bar\rho=21.67\alpha$, $\bar\eta=48.62\alpha$) the per-channel prices
$0.66\times0.76$ with the $\times2$ half-integer charge again give a pair
price of $1.0$. The four-Wolfenstein-integer input block scores
$\chi^2/4\,\mathrm{dof}=9.7$ ($p=0.046$) resp.\ $9.4$ ($p=0.053$)
\emph{identically} before and after the 2026-07 apex-slot fix, which improves
five downstream observables but touches none of the zeroing legs; the
corrected four-tier ladder above is therefore unchanged by the tension-hunt
batch.
The $5.5\sigma$ floor \emph{value} survives, but \emph{only} on the
corrected row basis of Sec.~\ref{subsec:ce-rows} and only as the
\emph{conditional} tier of the ladder above --- this is a rewrite of the claim,
not a relabel. All audit artifacts and every number in this subsection are
reproducible from \texttt{trials\_factor\_2026-07/} (E1--E6 scripts
and JSON, \texttt{verify3x/} audit scripts and synthesis recompute, and the
post-apex-fix re-audit in \texttt{downgrade\_reaudit\_2026-07/}).

\subsection{The conditional no-coincidence bound (corrected rows)}
\begin{proposition}[Conditional combined-evidence bound]
Grant DFD's forcing theorems as stated (the corpus's pre-registered framing;
this is the \emph{conditional} tier of Sec.~\ref{subsec:ce-trials-audit}, and it
is capped by the unconditional tiers there). Treat each surviving row as a blind
parameter-free formula that had to land within its achieved fractional agreement
$\delta_i$ of the measured value, in an $O(1)$ prior range
($p_i\simeq\min(1,2\delta_i)$), except the $H_0/\Lambda$ block, which is priced
by its declared-grammar count:
\[
\fiteq{p\big(\sin^2\theta_W\big)\!\approx\!0.0039,\ \
p(\theta_{12})\!\approx\!0.074,\ \
p(H_0/\Lambda)\!=\!3/135,\ \
p(\Omega_\chi h^2)\!\approx\!0.033 .}
\]
The four being independent (distinct sectors: electroweak, neutrino mixing,
cosmological clock/vacuum, dark sector), the combined conditional probability is
\[
\fiteq{\boxed{\,P_{\rm cond}=\prod_i p_i\approx2.1\times10^{-7}
\ \Longrightarrow\ Z\approx5.06\sigma\,}}
\]
rising to $5.28\sigma$ if a deliberately weak, measurement-limited $a_0$ row
($p=0.3$) is appended, and to $5.53\sigma$ under the two-stage $H_0$ booking:
the conditional band is $5.0$--$5.5\sigma$, and it must always be quoted
together with its unconditional cap ($3.7$--$4.1\sigma$ conservative-hostile,
$5.0\sigma$ moderate-hostile, not significant under the full selection charge).
\end{proposition}
\noindent\textbf{Why a $\chi^2$ ``consistency'' statistic is the wrong tool here.} The tree-level $\sin^2\theta_W
=3/13$ matches to $0.19\%$, but the measurement is precise to $0.02\%$, so a naive pull is $\sim\!-11\sigma$. That
is an \emph{artifact} of comparing a leading-order parameter-free value to a precision measurement: DFD predicts
the leading number; radiative/higher-order corrections supply the remaining digits. The figure of merit is
the \emph{fractional agreement} a blind formula achieved (the table), not the measurement error.

\subsection{What this is, and is not}
\begin{remark}
\begin{itemize}\itemsep2pt
\item \textbf{It IS} a preponderance / no-coincidence argument, now stated as a
four-tier null ladder with the worst case first
(Sec.~\ref{subsec:ce-trials-audit}): not significant under the full selection
charge $\to$ $3.7$--$4.1\sigma$ (hostile-conservative) $\to$ $5.0\sigma$
(hostile-moderate) $\to$ $5.0$--$5.5\sigma$ (conditional on the forcing
theorems). The same kind of evidence that established the Standard Model and
BBN --- a wall of independent numbers --- but with every menu priced.
\item \textbf{It is NOT} statistically significant under the most hostile
admissible null (full across-observable selection), and it is NOT a single
falsification-test discriminator (DFD$\neq$standard, confirmed in isolation).
Those (the SN--lensing sign $\sim$1.5$\sigma$, App.~SL; the clock LPI, under
systematic audit) await LSST/Roman or a clean verdict.
\item \textbf{Conditional means conditional:} the $5.0$--$5.5\sigma$ tier holds
only if each forcing theorem is granted as stated; a skeptic who rejects the
derivations reads the unconditional tiers, down to non-significance.
\item \textbf{Leading-order:} $0.2$--$4\%$ agreement, not precision; the last
digits need higher-order terms.
\item \textbf{The $\alpha$ match is booked separately} (it is the selector
datum, never multiplied into the pile): unconditionally, landscape scanning
cannot fake it (Theorem~\ref{thm:ce-alpha-ladder}). The full CKM matrix
contributes zero information here (Postulate~E.1) and is excluded.
\end{itemize}
\end{remark}
\noindent\textbf{One-line claim:}
\begin{equation*}
\fiteq{\boxed{\begin{aligned}
&\text{Fully hostile: not significant. Hostile menus: }3.7\text{--}4.1\sigma
\text{ to }5.0\sigma.\\
&\text{Conditional on the forcing theorems (and capped by those tiers): }
5.0\text{--}5.5\sigma.\\
&\text{Unconditionally: landscape scanning cannot fake }\alpha^{-1}=137.036.\\
&\text{Strong evidence, not proof.}
\end{aligned}}}
\end{equation*}
