slug: nilforoushan-parvaresh-conjecture-8-1 bibkey: nilforoushanparvaresh2026kdecks doi: 10.48550/arXiv.2609.23106 url: https://arxiv.org/html/2609.23106v1 triage: theorem motivation_gids:
- D5/S1/Words/Complexity/ExactDecks/Asymptotics/PositivePairExactDeckAsymptotics.actual_exactKDeckImage_weightedLyndon_isTheta
Nilforoushan-Parvaresh Conjecture 8.1
Problem
Conjecture 8.1 of Arman Nilforoushan and Farzad Parvaresh, Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters, arXiv:2609.23106v1, states:
D_{q,k}(n) = Theta(n^{E_q(k)}) for every fixed q >= 2, k >= 1.
The paper defines D_{q,k}(n) as the number of distinct exact k-decks of
length-n words over a q-letter alphabet and defines
E_q(k) = sum_{j=1}^k j L_q(j) - 1, where L_q(j) is the number of actual
length-j Lyndon words over that alphabet. Preregistration issue #9512 records
this complete target, not one of the cases already proved in the source.
The formal target quantifies over an arbitrary finite linearly ordered type
A, assumes exactly 2 <= Fintype.card A, takes every natural k with
1 <= k, and proves Real.IsTheta at Filter.atTop for the real-valued
cardinality of exactKDeckImage A k n. Its exponent is
weightedLyndonExponent A k - 1, with
weightedLyndonExponent A k = sum_{i=0}^k i * actualLyndonCount A i and
actualLyndonCount A i the cardinality of the subtype of actual Lyndon words
of length i. The zero term vanishes, so this is the source exponent. There is
no free proxy parameter and no extra hypothesis on n or on k.
Motivation
The source proves the general upper bound but leaves the matching lower bound
open outside k=2 and the binary k=3 case. A complete resolution must
therefore build many decks realized by actual positive words, preserve exact
source length, and prove that the actual Lyndon coordinates determine the
complete deck. Dimension counting or a free family of abstract parameters is
not sufficient.
Gap
The paper’s affine-dimension calculation concerns relations generated by the subsequence-count product rule and explicitly distinguishes that calculation from realizability by actual decks. The formal gap is the general lower bound with the source exponent, together with an unconditional passage from Lyndon coordinates to all scattered counts. The existing frozen Vivion theorem is a different binomial-complexity refutation and supplies only reusable definitions and basic count identities.
The exact arXiv HTML, OpenAlex, and Crossref were checked on September 24,
2026. The arXiv source still labels the statement Conjecture 8.1, OpenAlex
reported zero citing works, and Crossref’s first three exact-title results did
not match a later proof. Semantic Scholar was rate-limited with HTTP 429 and is
ASSUMED-UNVERIFIED. This bounded search does not support a universal absence
or worldwide-priority claim.
Route
The route has 32 connected owners in nine natural subdomains. Recursive standard bracketing gives each actual Lyndon word an integral homogeneous polynomial with that word as leading monomial. Binary and indexed fixed-source schedule theorems retain occurrence identity, including duplicate factor occurrences, and show that an interleaving of nonincreasing Lyndon factors cannot exceed their concatenation.
The actual Magnus polynomial has scattered-count coefficients. Recursively generated positive pairs agree below their level, and their first possible difference is the corresponding standard-bracket direction. The full indexed family spans every standard bracket. Literal power substitution then scales the first unequal degree while preserving all lower degrees.
One selects as many independent actual positive-pair directions as there are
actual Lyndon words. Direction-major, scale-minor digit blocks give equal-length
positive words whose lower coefficients are fixed and whose degree-r
coefficients encode a complete base-2^r digit array injectively. Combining
these blocks over all degrees yields the weighted-Lyndon lower growth of the
actual positive-word cutoff-Magnus ball.
Padding a ball representative to a fixed length loses only the one
source-length degree of freedom and injects the construction into actual exact
k-decks. Conversely, the overlap-infiltration identity and the indexed
shuffle leading order prove unconditionally that agreement on actual Lyndon
coordinates through k implies agreement on every scattered count through
k. For a fixed deck length, erase one singleton Lyndon coordinate and recover
it from total length; the remaining coordinates inject exact decks into the
expected box. The lower and upper bounds are then the two directions of
Real.IsTheta at atTop.
Falsifier
The resolution would fail if any formal object counted proxy directions rather
than the actual Lyndon-word subtype, if duplicate indexed shuffles were
identified and changed product multiplicities, if the positive-pair words did
not share the asserted lengths and lower coefficients, if the multi-scale map
were not injective, if padding changed the cutoff data, if exact-deck equality
did not imply all shorter counts at a common source length, if Lyndon recovery
required the conclusion as a guard, or if the erased singleton coordinate
could not be recovered from the common total length. A proof only for a fixed
alphabet, fixed k, sufficiently large k, or an abstract free parameter
would not settle the registered statement.
Evidence
The formal settlement is
D5/S1/Words/Complexity/ExactDecks/Asymptotics/PositivePairExactDeckAsymptotics.actual_exactKDeckImage_weightedLyndon_isTheta.
It has exactly the finite-alphabet, linear-order, q >= 2, and k >= 1
hypotheses described above and no source-length hypothesis. Its conclusion is
Real.IsTheta at Filter.atTop for the cardinality of the actual
exactKDeckImage and the power with exponent
weightedLyndonExponent k - 1.
The 31 preceding owners expose the standard bracket, occurrence-indexed shuffle order, actual Magnus coefficients, full-family span, central digits, positive-word-ball lower bound, exact-deck lower bound, and unconditional Lyndon-coordinate recovery used by the final proof. They are prerequisites of the single full-target settlement, not separate open-problem resolutions.
Triage
theorem. The result is a symbolic theorem for arbitrary finite linearly
ordered alphabets and all positive deck lengths. It is not bounded enumeration,
a numeric reduction, a certified finite instance, or a checker result.
ASSUMED-UNVERIFIED
Semantic Scholar was unavailable because of HTTP 429. The completed literature checks were bounded, and no claim is made that every later publication or private manuscript was searched. The typed resolution records this repository’s exact formal settlement of the cited statement; it does not by itself certify historical novelty or first-publication priority.