Evolutionary CM/LM Proof Pilot
Compatibility, Closure, and Sharp Robustness Bounds
Research working note prepared for Brian
4 November 2026
Abstract
WedistinguishexactBooleanabstractionfromuniformseparationfromthethreshold. Fora
continuous t-norm with an exact half-threshold Boolean abstraction and its standard-negation
dual, we prove that the greatest idempotent below an input bound gives the optimal uniform
distancefromthethresholdoverallfiniteformulas. Theboundextendstoarbitrarysynchronous
feedback networks in the declared connective language and is sharp already for scalar repeated
conjunction. A two-node example has an explicit robustness transition invisible in its constant
Booleantrajectory. Wealsoproveafinite-closureobstructionforapolynomialtwo-nodesystem.
Theresultsareacandidatesynthesisofclassicalt-normandabstractionmachinery,notacleared
historical novelty claim.
1 Scope and claim status
This is a separate proof pilot, not a revision of an existing manuscript. The symbolic and numeric
CM/LM conventions are inherited from the September 27 foundations manuscript. Its Boolean
normal form and pairing theorems are not asserted for graded scalars. The finite-jet comparison
uses the October 14 manuscript, not the earlier parent draft.
The main candidate contribution is Theorem 5 and its fragile-alternation example, supported
by Theorem 3, its feedback consequence, and the worked coupled example. The exact abstrac-
tion criterion, ordinal sums, t-norm powers, and partition refinement have classical antecedents.
These elementary proofs establish mathematical statements under explicit assumptions; they do
not establish historical firstness. No independent referee or formal proof assistant was used.
2 Compatibility contract
Setc = 1/2,D = [0,1]\{c},anddefineQ(x) = 0forx < candQ(x) = 1forx > c. LetN(x) = 1−x.
A continuous t-norm T is associative, commutative, nondecreasing in each coordinate, and satisfies
T(x,1) = x. Set S(x,y) = 1−T(1−x,1−y). The permitted formula language consists of T,S,N,
variables, and constants 0,1. In particular, residual implication is not automatically included.
Theorem 1 (Exact cut criterion). The following are equivalent:
1. T maps D2 into D and Q(T(x,y)) = Q(x)∧Q(y) there.
2. T(t,t) > c for every t > c.
Either condition implies T(c,c) = c. Under these conditions S,N also preserve D and Boolean
abstraction, and every finite formula has exact Boolean abstraction.
1

Proof. Necessity follows by using x = y = t. For sufficiency, if an argument is below c, then
T(x,y) ≤ min(x,y) < c. If both are above c, put t = min(x,y) and use monotonicity. Continuity
at (c,c) and T(c,c) ≤ c imply equality. Standard negation exchanges the two sides of the cut,
so duality gives the assertion for S. Structural induction proves the formula statement, including
shared variables: no probabilistic independence is assumed.
For a general cut c ∈ (0,1) the same conjunction criterion holds. Standard negation preserves
a single complementary cut only at c = 1/2. If conjunction compatibility is required at every
cut, then T = min: otherwise T(x,x) < x for some x, and any cut strictly between these values
contradicts compatibility. Diagonal idempotence implies T(x,y) = x when x ≤ y, by monotonicity
and T ≤ min.
Example 2 (Continuousnon-idempotentcompatibleconjunction). The ordinal sum of two product
t-norms is

2xy, x,y ≤ 1/2,


T (x,y) = 1/2+2(x−1/2)(y−1/2), x,y ≥ 1/2,
∗

min(x,y), otherwise.
Boundary formulas agree. This is a continuous t-norm. To see associativity directly, order three
arguments. If all are in the same half, affine rescaling reduces to product associativity. If two are
in the lower half and one in the upper, both bracketings return the lower-half product. If one is
lower and two upper, both return the lower argument. The other axioms follow directly. It satisfies
Theorem 1, though T (3/4,3/4) = 5/8 ̸= 3/4.
∗
Theupper-halfproductcaninsteadbereplacedbyminimum, leavingthelower-halfproductintact;
call this T . The same proof works. This provides a non-idempotent graded conjunction with
safe
strong margin preservation.
Ordinary product and Lukasiewicz conjunction fail the cut criterion: at x = y = 3/5 they give
9/25 and 1/5. Minimum passes. Idempotence at the cut alone is insufficient: an upper-half scaled
Lukasiewicz component gives T(3/5,3/5) = 1/2.
Residual implication boundary. For T , the residuum x ⇒ y = sup{z : T (x,z) ≤ y} sends
∗ T ∗
(x,y) = (2/5,1/5) to 1/4. Both inputs threshold to false, but the result is false, whereas Boolean
implication is true. Thus a theorem for the stated De Morgan language cannot be promoted to one
for all operations of a residuated fuzzy logic. Material implication S(N(x),y) does satisfy the cut
contract.
CM/LM transport. An involutive polarity lift L (x)[α] = f(x
⟨α1⟩
,...,x
⟨αn⟩
), with positive
f 1 n
polarity the identity and negative polarity N, retains reindexing equivariance. Pointwise term
superposition commutes with this lift and with evaluation in the chosen graded algebra. Crisp
coordinate selection also preserves all term operations. After Q, these objects recover the Boolean
truth functions, CMs, and symbolic LM lifts in the existing calculus. Graded equality is equality
in the declared semantics; Boolean-equivalent formulas need not be graded-equal. For example
T (x,x) ̸= x.
∗
The formula frame (cid:2) x 1−x(cid:3) has determinant 2x−1 under real multiplication and is gener-
1−x x
ally not an involution. This refutes direct transplantation of that Boolean proof into real matrix
arithmetic; it is not a theorem about every possible fuzzy matrix product.
2

(cid:80) (cid:80)
For real averaging q (a) = w a , with w ≥ 0, w = 1, preservation of all pointwise
w i i i i i i
products holds iff one weight is one. Indeed, applying multiplicativity to each coordinate indicator
gives w = w2. More generally the product defect is exactly covariance:
i i
q (ab)−q (a)q (b) = Cov (a,b).
w w w w
This is classical probability algebra, and explains why soft pairing requires a separate contract.
3 An exact optimal margin theorem
For a ∈ (c,1] define the input domain
D = [0,1−a]∪[a,1], e (a) = max{e ∈ [c,a] : T(e,e) = e}.
a T
The maximum exists: the set is nonempty by Theorem 1 and closed by continuity.
Theorem 3 (Sharp uniform formula margin). Assume Theorem 1. Over all finite formulas in
T,S,N,0,1 and all assignments of their variables in D ,
a
inf|φ(x)−c| = e (a)−c.
T
Every formula output stays in D and has its Boolean formula value after thresholding. If e (a) > c,
T
every formula output belongs to D . The infimum is sharp using only repeated conjunction of
eT(a)
one variable set to a.
Proof. Write e = e (a). If e > c, D is closed under the generators. Negation exchanges its two
T e
intervals. For conjunction, an argument at most 1−e forces the result to be at most 1−e; two
arguments at least e give a result at least T(e,e) = e. Duality proves closure under S. Constants
belong to D . Since D ⊆ D , induction gives the lower bound. If e = c, the zero lower bound is
e a e
automatic, and exactness still follows from Theorem 1.
Forsharpness,setp = aandp = T(p ,a). ThisnonincreasingsequencehasalimitL. Every
1 n+1 n
idempotent d ≤ a satisfies p ≥ d by induction. Associativity gives p = T(p ,p ); continuity
n 2n n n
yields L = T(L,L). Hence L ≤ e, while the previous lower bound gives L ≥ e. Thus p ↓ e. Each
n
p > c by the exact cut criterion. Therefore |p −c| ↓ e−c.
n n
Corollary 4 (Feedbackandthetwoexactnessregimes). Everysynchronous feedback networkwhose
coordinate updates are finite formulas in the stated language has an exact Boolean quotient on Dn.
From initial coordinates in D , all finite iterates have distance at least e (a)−c from the threshold.
a T
The bound is optimal across this network class. Exactness permits a zero uniform margin.
Proof. Coordinatewise application of Theorem 1 gives QnF = F Qn, and iteration gives QnFt =
B
FtQn. Invariance of Dn gives the positive margin case; the zero case is trivial. For sharpness use
B e
the scalar feedback x = T(x ,x ) with x = a. Its values are the t-norm powers p from the
t+1 t t 0 2t
preceding proof and tend to e.
For T the only idempotents are 0,c,1. For c < a < 1 the optimal uniform margin is zero. By
∗
contrast, T is idempotent everywhere above c, so its optimal margin is a−c. Both have identical
safe
Boolean conjunction tables and exact Boolean feedback quotients. Their graded robustness differs
sharply.
3

Interpretation of robustness. The bound concerns distance of exact states and internal gate
values from the threshold. Observation errors strictly smaller than that distance preserve Boolean
readout. It does not guarantee stability under errors injected back into the dynamics, parameter
uncertainty,orperturbationsthatchangethegatelaw. Suchguaranteesrequireaperturbed-system
invariant or an error-propagation theorem. Equality at the error bound may hit the undefined
threshold.
Finite-depth sharpness. For T set r(x) = 2|x−c|. Case analysis gives r(T (x,y)) ≥ r(x)r(y);
∗ ∗
by duality this holds for S , and negation preserves r. On the lower half the expression is r(x)+
∗
r(y) − r(x)r(y); on the upper half it is the product; in a mixed pair it is one input margin. A
formula with L variable occurrences (counted in its unfolded tree) and input normalized margins at
least m > 0 therefore has margin at least mL. Constants have margin one. The bound is attained
by an all-conjunction tree on repeated equal true inputs. In particular, self-conjunction feedback
satisfies
x = c+ 1(2x −1)2t , x > c.
t 2 0 0
It is Boolean-true at every finite time but tends to the threshold when x < 1. An unfolded-tree
0
bound does not claim a circuit-size bound for shared subexpressions.
4 Switching can destroy individually persistent margins
Theorem 5 (Optimal margin for a finite switched library). Let T ,...,T be continuous t-norms
1 m
satisfying Theorem 1, with duals S and common standard negation. Allow any finite formula using
j
theentirelibrary, andarbitraryswitchingamongsuchcoordinate-updateformulas. Forinitialinputs
in D , the optimal uniform distance from the threshold is
a
e (a)−c, e (a) = max{e ∈ [c,a] : T (e,e) = e for all j}.
T T j
Thelowerboundholdsateveryfinitetime. Sharpnessisachievedasaninfimumbyascalarswitched
self-conjunction trajectory.
Proof. Thecommonidempotentsetisclosed,containsc,anditsgreatestelementbelowamakesD
e
invariant under every generator. For sharpness let g(x) = min T (x,x) and x = g(x ), x = a.
j j t+1 t 0
The sequence decreases to a limit L ≥ e. The minimum of finitely many continuous functions is
continuous, so g(L) = L. Since every T (L,L) ≤ L, equality of their minimum to L implies that
j
everyT (L,L) = L. HenceL = e. Ateachstepaminimizingindexexistsandgivesanallowedgate;
j
the resulting finite prefix is a permitted formula or switching schedule. All finite values exceed c.
No observability of the schedule is assumed: every selected self-conjunction has the same Boolean
update.
Example 6 (Individually robust laws with fragile alternation). For b ∈ (c,1) let T be the ordinal
b
sum of rescaled product t-norms on [0,c], [c,b], and [b,1], with minimum for pairs in different
intervals. On an interval [l,u] its formula is l+(x−l)(y−l)/(u−l). Its idempotents are exactly
0,c,b,1, and it preserves the strict half-cut.
Take b = 7/10, b = 4/5, and a = 9/10. Repeating only T self-conjunction has optimal
1 2 b1
readout radius 1/5; repeating only T has radius 3/10. Alternating them has radius zero, although
b2
the Boolean state and the Boolean CM label (AND) stay unchanged at every finite step. The
states decrease; both even and odd subsequences have the same limit, which by continuity must be
idempotent for both laws. The only common idempotent in [c,a] is c.
4

This is a concrete failure of separate robustness certificates under composition. Boolean oper-
ator labels alone cannot distinguish these semantics. It is the most focused candidate formulation
produced by the pilot; its proof combines elementary monotone iteration and classical ordinal sums,
so external novelty remains open.
Persistent disturbances require a stronger model. For any t-norm, the perturbed self-
conjunction x = max(0,T(x ,x )−η) with fixed η > 0 decreases by at least η until zero. It
t+1 t t
leaves the true region in finite time. A universal disturbance guarantee therefore needs restorative
dynamics, resets, restrictions on disturbances, or a finite horizon. Readout robustness does not
imply this stronger guarantee.
5 A coupled two-node robustness transition
Consider
x = T (x ,y ), y = S (x ,y ), x ,y ∈ (c,1].
t+1 ∗ t t t+1 ∗ t t 0 0
The Boolean dynamics is (b,d) (cid:55)→ (b∧d,b∨d); in the chosen region its orbit is constantly (1,1).
Put r = 2x −1, s = 2y −1, and k = r +s . Then
t t t t 0 0
r = r s , s = r +s −r s .
t+1 t t t+1 t t t t
Theorem 7 (Coupledlimitandexactreadoutradius). The normalized sum r +s = k is invariant,
t t
and
(r ,s ) −→ (max(0,k−1),min(k,1)).
t t
The infimum, over all times and both coordinates, of distance to c equals
max(0,k−1)
= max(0,x +y −3/2).
0 0
2
Thus the same constant Boolean orbit admits a positive uniform readout margin precisely when
x +y > 3/2.
0 0
Proof. Conservation is immediate. Since 0 < r ,s ≤ 1, r decreases and s increases. Their limits
t t t t
satisfy R = RS and R+S = k. If k ≤ 1, positivity of R would give S = 1 and R = k−1 ≤ 0, so
R = 0. If k > 1, r = k−s ≥ k−1 > 0, so S = 1 and R = k−1. To compute the infimum include
t t
time zero: inf r = R, inf s = s , and R ≤ s because r ≤ 1. Divide by two.
t t t t 0 0 0
For k < 1, r ≤ r kt. For k = 1, r = r (1 − r ) and r ∼ 1/t: indeed 1/r − 1/r =
t 0 t+1 t t t t+1 t
1/(1 − r ) → 1, and averaging the increments proves the asymptotic. These rates describe the
t
approach to the threshold, not a finite-time change of truth. For k > 1, writing r = k −1+u
t t
gives u = (2−k)u −u2.
t+1 t t
The predicate x+y > 3/2 distinguishes persistent readout robustness within a single Boolean
state and fixed pair of CM operator labels. It illustrates why margins can be relevant even after
exact Boolean closure has been achieved. The recurrence itself is elementary; historical novelty is
unestablished.
5

6 Closure beyond the connective-generated class
Proposition 8 (Exact deterministic quotient). For a state map F : Z → Z and observation
P : Z → A, a deterministic quotient G : P(Z) → P(Z) with PF = GP exists iff P(z) = P(z′)
implies P(Fz) = P(Fz′). It is then unique on reachable abstract states.
Proof. Necessity follows from the commuting equation. For sufficiency define G(Pz) = P(Fz); the
hypothesis gives well-definedness.
ThisclassicalfibercriterionappliesequallywhenP includesCMlabels, guards, orfinitelymany
memory variables derived from the state. Current operator labels alone are not a closure proof.
Theorem 9 (A polynomial two-node obstruction). For F(x,y) = (y2,x) on (0,1]2, no finite deter-
ministic quotient refining the total observation B(x,y) = (1 ,1 ) exists. The obstruction
x>1/2 y>1/2
persists on the invariant subset whose iterates avoid threshold ties.
Proof. At even times, F2t(r,r) = (r2t,r2t). For every integer j ≥ 0, choose r = 2−1/(3·2j). Its
j
even-time observations stay (1,1) for t ≤ j +1 and first become (0,0) at t = j +2. No iterate of
either coordinate equals 1/2, since the relevant exponent is a power of two divided by 3·2j. Thus
these initial states have infinitely many distinct future observation traces. Two states in a single
deterministic quotient state would have identical traces, contradicting finiteness. This argument
permits arbitrary finite cells, not just polynomial guards.
The example already fails one-step Boolean closure: (3/5,3/5) and (3/5,9/10) have the same
observation but different next observations. Adding finitely many instantaneous operator labels
cannot repair infinite-horizon deterministic closure. A nondeterministic finite abstraction, or a
finite-horizon description, is a different contract and remains possible.
For horizon H, the word P (z) = (Bz,BFz,...,BFHz) is the coarsest observation retaining
H
that entire word. Its next-state map is exact from horizon H to horizon H −1 by deletion of the
first symbol. It does not in general give an autonomous fixed-horizon quotient. This is standard
finite-trace refinement.
7 Regularity comparison and evolutionary operators
For the multilinear expectation extension f(cid:101)of a Boolean function, exact half-threshold recovery
on all input half-cubes occurs iff f is affine parity. This is a corollary of the local finite-jet rigidity
theorem already in the collection: apply it to the polynomial 2f(cid:101)((u + 1)/2) − 1 near u = 0.
(cid:76)
Conversely the extension of f = b⊕ α satisfies
i∈I i
(cid:89)
1−2f(cid:101)(p) = (−1)b (1−2p
i
),
i∈I
which proves threshold compatibility. It is not a new independent theorem claim. The nonsmooth
ordinal-sum construction avoids its analytic hypothesis.
For an exogenous sequence of compatible update formulas F , stepwise semiconjugacy yields
t
the exact Boolean evolution b = (F ) (b ). If all steps use the same T,S,N, the idempotent
t+1 t B t
margincertificateremainsinvariant. Differentstepsmayusedifferentt-norms; Theorem5givesthe
optimal common-barrier guarantee for a finite library under arbitrary switching. If operator choice
depends on hidden continuous state, the combined abstraction must satisfy the fiber criterion; the
switching sequence cannot silently be treated as known input.
6

8 Prior art and research decision
1. Klement, Mesiar and Pap, Triangular norms. Position paper I: Basic analytical and
algebraic properties, https://www.flll.jku.at/div/research/technical_reports/
flll-tr-0208.pdf. Indexed Proposition 6.5 and Corollary 6.6 inspected; direct PDF
retrieval timed out. These explicitly connect limits of t-norm powers with idempotents and
give closedness of the idempotent set. They are direct antecedents to the key ingredient of
Theorem 3; the ingredient must not be claimed as new.
2. Fodor, Left-continuous t-norms in Fuzzy Logic: an Overview, https://acta.uni-obuda.
hu/Fodor_2.pdf. Inspected definitions and structural discussion. Standard t-norm, strong-
negation, duality, and ordinal-sum context; no novelty claimed for these.
3. Gispert, Godo and Esteva, Non-falsity and general threshold-preserving companions of
MTL logics, IJAR 196 (September 2026), 109699, https://doi.org/10.1016/j.ijar.
2026.109699. Publisher indexed abstract and excerpts inspected; full theorem-by-theorem
comparison remains open. Their consequence relations preserving designated intervals are
not automatically the gatewise Boolean homomorphism used here, but this is close priority
literature.
4. Petersen et al., Deep Differentiable Logic Gate Networks, https://arxiv.org/html/2210.
08277. Full relevant real-gate and operator-mixture sections inspected in the preceding as-
sessment. Ordinary product relaxations and learnable gates predate this pilot.
5. Cummins et al., Boolean models coarsely sample continuous dynamics of regulatory networks,
https://arxiv.org/html/2606.14925v1. Relevant model/refinement sections inspected in
the preceding assessment. Discrete/continuous refinement and richer hidden dynamics have
strong prior art.
6. Formal analysis of piecewise affine systems through formula-guided refinement,https://www.
sciencedirect.com/science/article/abs/pii/S0005109812004931. Abstract-level com-
parison only: formula-relative finite quotients and refinement are established methods.
Targeted searches included continuous t-norm threshold preservation, ordinal-sum Boolean
thresholds, t-norm powers and greatest idempotents, coupled product/probabilistic-sum recur-
rences, and finite deterministic quotients. Search non-detection is not evidence of firstness. The
exact powers argument and invariant cut domains are elementary consequences of standard t-norm
theory. The manuscript-worthy candidate is the sharp compositional robustness formulation and
its evolutionary CM/LM interpretation, not a new fuzzy connective or a new abstraction theory.
Decision. The pilot finds substantive mathematical structure and a sharp theorem with proofs
and counterexamples. It does not establish a substantively new result relative to all published
mathematics. Preserve this as a technical research note and pursue a focused priority review before
developing it into a standalone paper. The finite-library common-barrier theorem is proved here.
Stronger remaining targets are constrained switching with feedback disturbances and an explicit
observation-costresult; thispilotprovesneither. Avoidbuildingalargelearnerbeforethatdecision.
Verification. The companion standard-library script checks exact rational instances of compati-
bility, associativity, invariant barriers, the coupled recurrence, scalar feedback, residual-implication
failure,andfiniteclosurewitnesses. Finitecheckscomplementtheproofsanddonotcertifypriority.
The script writes a JSON receipt beside itself. No original manuscripts are edited.
7