Protective Switching and Sharp Observation Margins in a
Two-Node Ordinal-Sum System
Brian Theory
8 November 2026
Research preprint
Abstract
For fixed 0 < α < β <1 we analyze a two-node product-ordinal-sum network, with h = 1−β
independent of α. The range β/2 ≤ α < βis sufficient for global zero-basin identification, first-
use protection at every beta-critical point, and exact periodic worst schedules under a recurring
gap constraint. The sharp leading margins are (2 β/h)n−3 and (2 β/h)ζ(3)K −3. Parameter-
dependent energy estimates prove the coefficients. A certified critical-input counterexample
outside this range has zero margin even under alternating switches. The sufficient range is not
claimed maximal. Late forced first-use asymptotics remain valid for every fixed strict parameter
pair, despite possible failure of early protection and recurring robustness.
Keywords: ordinal sums; monotone maps; constrained switching; critical basins; observation
margins.
1 Introduction
Graded realizations of Boolean readouts distinguish a positive logical output from its distance to
the threshold. We study how switching changes that distance in a concrete two-node network.
The principal questions are global: which initial states admit zero margin, whether an eventual
protective mode is sufficient, and which admissible schedule gives the smallest margin. The an-
swers involve an exact basin boundary, a realizable greedy envelope, and an exchange inequality
identifying a periodic worst word.
This system supplies one explicit dynamical realization relevant to evolutionary correspondence-
matrix/logical-matrix (CM/LM) representations: the logical conjunction/disjunction readouts are
retained while a mode selects the graded operation. The representation alone does not determine
this dynamics; the state domain, ordinal sums, duality and switching contracts below are additional
declared choices. Finite compositions remain maps of the invariant state domain, but are not
asserted to remain in the two-element family of one-step maps. No closure under a new matrix
product or inverse is needed or claimed. The mathematical results concern this specified realization,
rather than every evolutionary CM/LM system.
The local critical geometry belongs to established nilpotent-parabolic theory [1]. Our proofs
are elementary and self-contained; the model-specific schedule comparison, terminal-margin transfer
and certified obstruction are the contribution examined here. Related constrained-switching and
monotone safety results [2, 3] are distinguished in the final discussion. Historical priority remains
unresolved.
1

2 Maps and contracts
Fix 0 < α < β <1, and set h = 1 − β >0. Throughout, h is not identified with α. On (0 , 1]2 define
Fq(r, s) = (Uq(r, s), V(r, s)), V (r, s) = r + s − rs, q ∈ {α, β},
Uq(r, s) =



rs/q, r, s ≤ q,
q + (r − q)(s − q)/(1 − q), r, s ≥ q,
min(r, s), otherwise.
For completeness, let bq = (1 +q)/2 and define Tq on [0, 1]2 as the product ordinal sum with blocks
[0, 1/2], [1/2, bq], [bq, 1]. Within a block [ l, u], its value is l + (x − l)(y − l)/(u − l); between blocks it
is min(x, y). Set Sq(x, y) = 1 − Tq(1 − x, 1 − y). The original update is ( x′, y′) = (Tq(x, y), Sq(x, y))
on (1/2, 1]2. The normalization r = 2x − 1, s = 2y − 1 gives the displayed maps: complementary
inputs lie in the common lowest block, hence V is independent of the mode. All finite states stay
strictly above the threshold 1 /2. An observation error strictly smaller than the margin in each
coordinate preserves this positive threshold readout. No error is injected into the state recurrence
in this definition. A schedule is an infinite word on {α, β}; adaptive choices also produce such a
word along each trajectory. It is called fair if each symbol occurs infinitely often, with no bound
on gaps. The full-trajectory observation margin is 1
2 inf t≥0 min(rt, st). All schedule infima fix the
initial state. Each mode is isotone, r decreases, s increases, and r1 ≤ min(r0, s0). Limits obey
R(1 − S) = 0.
At a beta-critical point defined below, let zn = F n
β z0. Mn is the infimum of that margin over
words permitting exactly n beta updates, then one alpha, then an arbitrary tail; the prefix and
time zero are included. DL = min0≤n≤L Mn is the first-use deadline value. GK, for K ≥ 2, is the
infimum over words with no K consecutive betas, including in the initial block; thus the first alpha
index is at most K −1. Let PK denote the full-path margin of the particular word pK = (βK−1α)∞,
whether or not it is optimal.
3 The beta-critical boundary for arbitrary beta
Lemma 1. For every 0 < β <1, the constant-beta zero-margin basin is 0 < s < β, 0 < r≤ ϕβ(s),
where the boundary function is unique, continuous, strictly decreasing and involutive. Its endpoint
limits are ϕβ(0+) = β and ϕβ(β−) = 0. Boundary orbits tend to (0, β); strictly lower orbits tend
to (0, S), S < β. Write d = β − s. Then
Aβ(s) = d2
2β(h + d) < ϕβ(s) < Bβ(s) = d2
β(h + d) < d.
A beta-critical state means precisely a point on this boundary.
Proof. For the beta law, reaching s ≥ β preserves a positive floor min( r, β). Conversely R > 0
forces S = 1, hence a finite crossing. Also Uβ ≥ rs, strictly in the open square, so r + s strictly
increases there. Every zero orbit thus stays in r + s < βand satisfies
r′ = r(1 − d/β), d ′ = d − (h + d)r. (1)
The region r ≤ Aβ(s) is invariant: on its boundary, d′ = d − d2/(2β) > 0 and direct subtraction
yields
Aβ(s′) − r′ = d4(1 + β − d)
8β3(h + d)(h + d − d2/(2β)) > 0.
2

For smaller r, both successor coordinates decrease and Aβ decreases with s, proving invariance. If
S < β, then rt → 0 and Aβ(st) → Aβ(S) > 0, giving eventual strict entry.
The positive basin is open by finite crossing (an equality hit is followed by a strict crossing);
the zero basin is relatively closed and downward closed. Each vertical slice has an attained positive
zero endpoint below β − s. Its limit must have S = β, since otherwise strict certificate entry and
continuity would give larger zero points on that slice. Distinct ordered zero orbits have strictly
ordered terminal second coordinates: after at most one step their second-coordinate gap is positive
and is bounded below subsequently by multiplication by 1−¯rt. The product equals (1− ¯S)/(1−¯sm) >
0. Hence a smaller point has S < β, and every strict coordinatewise increase of a boundary point
has positive margin. This proves strict decrease. Points strictly below the graph lie in the open
zero interior by eventual strict certificate entry; points above lie in the open positive basin. These
two neighborhoods prove continuity of the graph. Input symmetry gives identical successors after
swapping coordinates, so the graph is symmetric and involutive. Its endpoint limits follow from
ϕβ(s) < β− s and involution.
The strict inner bound follows from strict certificate entry. For any zero orbit, set g(s) =
(β − s)/[β(1 − s)], which strictly decreases. Then rt − rt+1 = g(st)(st+1 − st). Summing and using
positive later increments gives r0 < g(s0)(β − s0) = Bβ(s0) < β− s0.
4 A sufficient parameter range and exact schedule ordering
Let H = (min( Uα, Uβ), V). Isotonicity and the shared second coordinate imply that one mode
realizes H at each step and H t(z) lies below every switched trajectory. Induction uses H(zH
t ) ≤
Fq(zH
t ) ≤ Fq(zq
t ). Thus the sharp arbitrary-word margin is the greedy terminal first coordinate
divided by two. Every finite greedy prefix has a fair extension, so arbitrary/fair uniform infima
agree; this does not assert fair attainment.
Theorem 2(Global sufficient range). If β/2 ≤ α < β, the switched zero basin equals the constant-
beta zero basin. Every first alpha use on a beta-critical orbit yields positive margin for all later
switching, including the full prefix. These statements include α = β/2. The range is sufficient;
maximality is not claimed.
Proof. For either mode, s ≥ β preserves r ≥ min(rcurrent, α) > 0. Therefore a zero greedy orbit has
terminal S ≤ β. Since Uq > rs in the open square, all its finite states satisfy r + s < β. In this
triangle min( r, s) < β/2 ≤ α and max(r, s) < β. Writing m = min(r, s), M = max(r, s), one has
Uα = m min(M/α, 1) > mM/β = Uβ. Thus every zero greedy orbit is a beta orbit. Conversely a
zero beta orbit bounds the greedy orbit above, forcing zero greedy margin. At any critical point,
the alpha successor has the same second coordinate and a strictly larger first coordinate than the
critical beta successor. The boundary separation proved above places it in the positive switched
basin. All prefix coordinates are positive, so the complete trajectory has positive margin. The
strict inequality r + s < βremoves any endpoint ambiguity when 2 α = β.
Proposition 3 (Fairness and uniform margin). For any strict parameter pair, the uniform infima
over arbitrary and fair words agree and equal RH /2, where RH is the terminal first coordinate of
the greedy orbit. This value is positive exactly when that orbit reaches s ≥ β at a finite time. If
β/2 ≤ α < β, every critical initial point has positive individual margin under each fair word, but
its uniform fair infimum is zero. Below the critical graph a fair zero-margin word exists. Above it
all words share a positive uniform margin.
3

Proof. The envelope comparison gives the lower bound and an arbitrary attaining word. Every
finite greedy prefix admits a fair extension; taking increasingly long prefixes proves equality of the
uniform infima. Since r1 ≤ s0, the full-path infimum is the terminal first coordinate divided by two.
After s ≥ β, either r ≤ α and both laws fix r, or r ≥ α and both preserve the floor α. Conversely
RH > 0 forces SH = 1 and therefore a finite crossing.
In the sufficient range, Theorem 2 applies at the first alpha of any fair word. Its finite preceding
prefix has positive coordinates. The constant-beta critical orbit gives zero arbitrary, hence zero
uniform fair margin. Below the graph the beta orbit tends to (0 , S) with S < β. Choose S < c < β
and a sufficiently late prefix so that r < c, s < c, and 2 r/(1 − c/β) < c− s. Alternate beta-alpha
from there. Before any putative first exit above c, a pair contracts r by at most c/β and raises s
by at most twice its starting r. Summing the geometric budget excludes both a whole-pair and a
half-pair exit. Thus s < cforever and r → 0, giving a fair zero word. The above-graph assertion
follows from global basin equality. No fair attainment of a positive worst margin is asserted by
equality of infima alone.
Lemma 4 (Exchange for every strict pair) . For any 0 < α < β <1, the region Ωα = {min(r, s) ≤
α} is invariant and
Fβ ≤ Fα, F α ◦ Fβ ≤ Fβ ◦ Fα on Ωα.
The word pK minimizes both coordinates at every finite horizon over words with no K consecutive
betas, for any start in this region.
Proof. Set fq(u) = min(u/q, 1). On this region Uq = mfq(M ); after one update r ≤ m ≤ α, proving
invariance. The ratio fβ/fα equals α/β up to α, then u/β up to β, then 1; it is nondecreasing.
With v = V (r, s) ≥ M , this orders mfβ(M )fα(v) ≤ mfα(M )fβ(v), the two first coordinates of the
compositions. Their second coordinates are ordered by fβ ≤ fα and isotonicity of V .
For a finite word, match the periodic prefix left to right. At the first mismatch the target
symbol must be beta: a target alpha is forced by the preceding K − 1 matched betas. Move the
first later beta left across the intervening alphas, or replace the current alpha if no beta remains.
Each swap alpha-beta to beta-alpha lowers the terminal state by the exchange and isotonic suffix
maps. It is legal: preceding beta-run length is zero except at the final swap, where the matched
prefix has length less than K − 1; the following beta-run shortens. Replacement with an all-alpha
suffix is also legal and lowers the state. Induction gives the periodic prefix. Every legal finite word
has an infinite legal extension, completing the comparison.
For a critical state already in Ω α, the beta law is the greedy law for all future time, even when
α < β/2. A first alpha strictly raises the state above the critical beta successor, so the constant-beta
basin theorem and the same envelope comparison give positive tail margin. Hence
GK = PK > 0 for every K ≥ 2 (2)
for these state-specific starts. In the sufficient parameter range every critical point belongs to Ω α,
since its minimum is below β/2. Positive margin forces a finite s ≥ β crossing; after the first
update r ≤ α and both modes freeze r at crossing. The terminal R controls the whole path because
R ≤ r1 ≤ s0, so its original margin is R/2. Requiring both symbols in every same-length K block,
or adding ordinary fairness, leaves this sharp value unchanged because pK lies in each class.
4

5 Parameter-dependent almost-invariant bounds
Use the following constants; none assumes α = h:
C = 1
2βh , A = 1
2β , B = 2β + h
6β2h2 , L = 1
βh , D = 3β + h
2β2h .
Choose
δ = min
(
β
4 , h
2 , β − α
2 ,
r
αβh
2
)
> 0, J = (h + δ)L, S = 2/h.
Then Lδ2 ≤ α/2, J δ≤ 3/8, Aδ ≤ 1/8. Define E(r, d) = r − Cd2 − Ard + Bd3. Direct expansion of
(1) gives
∆βE = −3Brd 3 + Ar2d + Dr2d2 + 3Br 2d3 − B(h + d)3r3. (3)
Set
c1 = 3B, c 2 = A + Dδ + 3Bδ 2, c 3 = B(h + δ)3, Q = S(c1 + c2L + c3L2δ),
X = CJ 2 + ALJ + BJ 3δ2, H 1 = 2Q + X + AL/β, H ∗ = 2Q + X + 3AL/β.
These are finite positive fixed-parameter constants, not uniform constants over all parameter pairs.
Lemma 5. On any segment whose prestates satisfy 0 ≤ dt ≤ ρ ≤ δ, rt ≤ Lρ2 and P rt ≤ Sρ, the
sum of absolute hypothetical beta energy drifts is at most Qρ4. At a critical point with d ≤ δ,
|E(r, d)| ≤Qd4, r = Cd2 + (AC − B)d3 + O(d4), AC − B = h − 4β
12β2h2 .
Consequently ndn → 2β and n2rn → 2β/h on every fixed critical orbit. The sign of the cubic
coefficient is not assumed negative.
Proof. Equation (3) bounds the absolute drift by c1rd3 +c2r2d+c3r3. Summing and using r ≤ Lρ2,
d ≤ ρ ≤ δ gives the stated Qρ4. On a critical tail, telescoping (1) gives
r =
X
t≥0
dt
β(h + dt) (dt − dt+1) ≤ d2
β(h + d) ≤ Ld2,
X
rt ≤ d/h ≤ Sd.
Since E tends to zero, the drift estimate proves the critical energy bound. Solving r(1 − Ad) =
Cd2 − Bd3 + E gives the expansion, with remainder bounded by [ Q + A|AC − B|]d4/(1 − Aδ). In
particular rn ∼ Cd2
n and
1
dn+1
− 1
dn
= (h + dn)rn
dndn+1
→ hC = 1/(2β).
Cesaro summation gives the two orbit limits.
6 One-use and recurring transfer
Take a critical pre-alpha state with d = ρ ≤ δ. Its coordinates obey r ≤ Lρ2 < αand s ≥ β −δ > α.
Before escape alpha leaves r fixed; beta decreases it by ϵ = rd/β. Both modes have d′ = d−(h+d)r.
Alpha energy change is exactly
∆αE = ∆βE + ϵ(1 − Ad′). (4)
5

The critical first beta successor has d′ > 0 since d′ ≥ ρ − ρ2/β > 0. The alpha successor strictly
exceeds it in r and belongs to Ω α, so its arbitrary tail has positive uniform margin by Lemma 1
and the envelope argument. Thus a crossing occurs in finite time under the beta-greedy or periodic
tail.
At crossing, −J ρ2 ≤ dτ ≤ 0, R ≤ Lρ2, and P
t<τ rt ≤ (ρ + J ρ2)/h < Sρ. Both laws freeze r
thereafter. The exit error is |E(R, dτ ) − R| ≤Xρ 4. Initially |E| ≤Qρ4; all hypothetical beta drifts
together cost at most Qρ4, including the last update whose new d′ can be negative. Also |d′| ≤ρ.
For one alpha followed by beta-greedy switching, (4) therefore gives
|2Mn − rndn/β| ≤H1d4
n (dn ≤ δ). (5)
Here the entire initial beta prefix is included: terminal R ≤ rn ≤ s0 for n ≥ 1; for n = 0, the alpha
conjunction itself is at most s0. All earlier first coordinates are at least R and second coordinates
at least s0. Hence the sharp full-path margin equals R/2.
For the periodic word set tj = jK − 1 and take ρ = dK−1 before its first alpha, and set
r∗ = r(t1) = rK−1. Write ϵK,j = r(tj)d(tj)/β before crossing and zero after. Let SK = P
j ϵK,j.
Each nonzero kick after the J0th is preceded by K − 1 beta losses at least as large as that kick,
since positive r, ddecrease. The blocks are disjoint, so
X
j>J0
ϵK,j ≤ r(tJ0)
K − 1 (6)
when tJ0 precedes crossing, and the tail is zero otherwise. In particular SK ≤ r∗ρ/β + r∗/(K − 1).
For Kρ ≥ β, 1 /(K − 1) ≤ 2ρ/β, giving SK ≤ 3Lρ3/β. The total error in dropping the factors
1 − Ad′ is at most AρSK. Thus the periodic terminal coordinate satisfies
|2PK − SK| ≤H∗ρ4 (ρ ≤ δ, Kρ ≥ β). (7)
Both conditions hold eventually for every fixed parameter pair and critical start sinceKdK−1 → 2β.
Theorem 6 (Leading laws and their scope) . For every fixed 0 < α < β <1 and fixed beta-critical
initial point,
Mn ∼ 2β
h n−3, P K ∼ 2β
h ζ(3)K−3.
If the critical start lies in Ωα, then GK = PK for all K ≥ 2, so the latter law holds for GK. In
particular it holds for every critical start throughout the sufficient range β/2 ≤ α < β. In that
range DK−1 ∼ (2β/h)K−3 and GK/DK−1 → ζ(3). Early Mn or DL outside that range need not be
positive; no recurring-law assertion is made there without a state condition.
Proof. From (5) and the critical limits, n3Mn → 1
2β (2β/h)(2β) = 2 β/h. This is local along the
late beta orbit and does not require the sufficient global range. For each fixed integer j, scaled
periodic pre-kick coordinates obey K2r(tj) → (2β/h)j−2 and Kd(tj) → 2β/j . To justify this, start
at t1 = K − 1, set u = K2r, v = Kd, θ = 1 + (t − t1)/K. Beta updates are u+ = u − uv/(βK ),
v+ = v − hu/K − uv/K 2; alpha changes only the first formula to u+ = u. On 1 ≤ θ ≤ j
before and including a possible first crossing, scaled prestates are bounded. The limiting equations
˙u = −uv/β, ˙v = −hu have solution ( u, v) = ((2 β/h)θ−2, 2β/θ). On a compact rectangle the
vector field is Lipschitz. Taylor comparison and the at most j alpha omissions yield em+1 ≤
(1 +Lj/K)em + Cj/K2 + (Cj/K)1α. Iteration gives max em ≤ eLj j(e0 + C′
j/K) = o(1). A possible
first crossing would contradict the strictly positive limiting v ≥ 2β/j , closing the stopped estimate.
Constants here fix j.
6

Thus K3ϵK,j → (4β/h)j−3 for each fixed j. By (6), the scaled tail beyond J0 has limsup at most
(2β/h)J −2
0 . Truncate, take K → ∞, then J0 → ∞to obtain K3SK → (4β/h)ζ(3). The error in
(7) vanishes after multiplication by K3, proving the law for PK. Equation (2) gives the GK claim.
In the sufficient range all finitely many early Mn are positive. The asymptotic bounds on later Mn
and that positive finite minimum imply DL ∼ (2β/h)L−3, without assuming monotonicity.
At β = 3/5 the coefficients reduce to 3 and 3 ζ(3).
7 A fixed-parameter logarithmic refinement
Proposition 7. Only in this proposition, fix (α, β) = (2 /5, 3/5) and a fixed beta-critical initial
point. Then
Mn = 3
n3 − 21 logn
n4 + O(n−4).
The remainder constant and onset may depend on the initial point.
Proof. Lemma 5 gives r = (25/12)d2 − (625/216)d3 + O(d4). Substitution into d′ = d − (2/5 + d)r
yields d′ = d − (5/6)d2 − (25/27)d3 + O(d4). Consequently, for wn = 1/dn,
wn+1 − wn = a + b
wn
+ O(w−2
n ), a = 5/6, b = 175/108.
Already wn ∼ an by Lemma 5. First summation gives wn = an + O(log n). Hence 1 /wn − 1/(an) =
O(log n/n2) is absolutely summable, as are the O(w−2
n ) terms. Summing once more gives
wn = 5
6 n + 35
18 log n + O(1), d n = 6
5n − 14 logn
5n2 + O(n−2).
The one-use transfer bound (5) implies Mn = (125 /72)d3
n + O(d4
n). Expanding its cubic proves
the stated coefficients 3 and −21. All estimates use fixed parameters and a fixed critical orbit; no
parameter-uniform logarithmic law is asserted.
8 A certified failure outside the sufficient range
Take β = 3/5, α = 1/100, s0 = 1/5, and the exact implicit critical coordinate r0 = ϕβ(1/5). The
rational certificates detailed below establish
11/50 < r0 < 6/25. (8)
The lower endpoint enters the strict beta zero certificate after three exact updates, with slack
666485001263/272735592187500 > 0. The upper endpoint crosses s = β by update 23, certified by
downward rounding its orbit, so is beta-positive. Lemma 1 yields the strict bracket, rather than
assigning exact criticality to a rounded point.
Throughout this initial interval both inputs exceed alpha, and
Uβ(r, 1/5) − Uα(r, 1/5) = 2(35r − 2)
495 > 0.
Thus the first alpha successor lies strictly BELOW the critical beta successor with the same second
coordinate. Lemma 1 implies its constant-beta tail has zero margin; hence M0 = 0. This is a
long-time basin conclusion, not merely the one-step comparison.
7

More strongly, ( αβ)∞ has zero margin from every initial point in the box in (8) at s0 = 1/5.
After 15 pairs, upward rounding of the upper corner supplies
¯r = 364294600485702452729
302231454903657293676544 , ¯s = 629770882596016946421965
1208925819614629174706176 .
With c = 11/20, the exact strict cap slack is
(c − ¯s) − 2¯r
1 − c/β = 830182727008321022239
6044629098073145873530880 > 0.
To see why this certifies the infinite tail, start each alpha-beta pair below the cap withr ≤ ¯r < c < β.
Alpha cannot increase r; the subsequent beta multiplies it by at most c/β < 1 as long as the
intermediate second coordinate stays below c. Each pair increases s by at most twice its starting r.
The geometric total is bounded by 2¯r/(1 − c/β) < c− ¯s. A first-exit argument bounds a putative
half-pair exit as well and closes the cap induction. Therefore s < cforever and r → 0.
This alternating word satisfies both-symbols-in-every-two-updates, hence both one-sided and
two-sided block contracts for every K ≥ 2. Consequently GK = 0 for all K ≥ 2 at this critical
state. The rational upper start (6 /25, 1/5) is beta-positive but has the same alternating zero
tail; therefore global switched/constant-beta zero-basin identification also genuinely fails. The
bracket, finite iterations and cap are reproduced by the companion certificate code. The finite
dyadic computations use the full piecewise maps and round both coordinates outward after each
update; isotonicity propagates the resulting enclosures. This isolated certified obstruction does not
characterize the maximal good parameter range or the complete behavior below α = β/2.
9 Singular limits, positioning and scope
Every limit above fixes both parameters and the critical initial point. If α = β, the modes coincide
and critical margins are zero under all words; the positive coefficients for strict pairs do not extend
by taking that parameter limit first. As α ↑ β, entry into the mixed strip requires d < β− α and
can be arbitrarily late. As β → 0 or β → 1, the local radius, constants and admissible initial-state
domains change; the coefficient 2β/(1 − β) is not a uniform theorem in those limits. No interchange
of limits or parameter-uniform remainder is claimed.
The primary antecedents have distinct scopes. In the coordinates u = d, v = −(h + d)r/h,
the beta map is u′ = u + hv, v′ = v(1 − u/β)(1 + hv/(h + u)). It satisfies analytic case 3 of
Cuf ´ ı-Cabr´ e–Fontich Theorem 2.1 (the 2020 preprint version linked in [1]): c = h > 0, p = 0,
k = 4, l = 2, bl = −1/β < 0. Their local leading coefficients match this independent special-
case derivation. That antecedent does not supply the global schedule ordering or obstruction.
Automaton-constrained switching [2], Definition 1 and Theorem 2.3, concerns linear multinorm
stability; monotone repeated-control safety [3], Theorems 1–2, concerns selected safe policies. Nei-
ther result establishes this nonlinear worst-word ordering or these margin coefficients. Historical
priority remains unresolved. The result is a focused theorem for the declared model; its scope does
not extend automatically to general fuzzy or matrix-algebra systems. Observation margin concerns
exact trajectories; no process disturbance or stochastic robustness guarantee is inferred.
8

10 Reproducible certificates and limitations
The companion audit uses exact symbolic arithmetic and directed rational state bounds. In par-
ticular, at β = 1/10, α = 1/20, a positive critical-graph cubic coefficient is allowed; at α = 1/4,
β = 1 /2 the sufficient-range endpoint is tested. At these and other stated parameter pairs, the
exact initial point is specified implicitly as (ϕβ(β/3), β/3). Rational inner/outer certificates bracket
its first coordinate, and outward full-map propagation certifies positive period-two margins. The
obstruction uses the different exact implicit state specified in its section. These computations
corroborate proofs; finite tests do not establish the continuum theorems or historical novelty.
The core proofs above are self-contained. The certificate programs are supplied with the source
package; they require Python and SymPy and use exact rational or symbolic arithmetic for cer-
tification. The remaining open classification is the maximal good parameter/state region below
α = β/2; no further result on that region is claimed. Observation-margin robustness does not
imply robustness to process disturbances.
Research status and AI assistance
This research preprint was developed with generative AI assistance in drafting, proof development,
computation, and internal adversarial checking. The internal checks do not constitute external
human peer review or formal proof verification. Historical priority remains unverified. The com-
panion source package identifies the reviewed version, executable certificates, and the scope of each
check.
References
[1] C. Cuf ´ ı-Cabr´ e and E. Fontich,Differentiable invariant manifolds of nilpotent parabolic points ,
Discrete and Continuous Dynamical Systems 41 (2021), no. 10, 4667–4704. https://doi.org/
10.3934/dcds.2021053. The theorem numbering used here is from the 2020 preprint: https:
//mat.uab.cat/dpt/Publ/prep/p04_20.pdf.
[2] M. Philippe, R. Essick, G. E. Dullerud and R. M. Jungers, Stability of discrete-time switching
systems with constrained switching sequences, Automatica 72 (2016), 242–250. https://arxiv.
org/abs/1503.06984.
[3] S. Sadraddini and C. Belta, Safety Control of Monotone Systems with Bounded Uncertainties ,
CDC 2016. https://sadraddini.com/wp-content/uploads/2018/04/cdc16monotone.pdf.
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