Results and open questions

Claims

Browse definitions, theorems, conjectures, counterexamples, and open problems from across FPRD Lab. Each record includes its exact statement, supporting evidence, dependencies, limitations, and links to related papers.

902 indexed claims and results

Starting points

A suggested reading order

01
Automata and formal languages · proof
A direct scaffold for even palindromes
Conditional on the complete Draft 5 Section 11.1 sequence contract, persistent receipts and simultaneous records give a direct finite-scaffold and PEG route with symbolic resource bounds.
FPRD-T128 Read the publication
02
Automata and formal languages · mechanically checked
Strict real-time multitape machines compile to scaffolds
A mechanically checked construction turns each fixed-tape real-time machine step into one bounded persistent scaffold update.
FPRD-T89 Read the publication
03
Automata and formal languages · informal proof
Reversed real-time multitape languages sit properly inside PEG
The transfer theorem and classical palindrome recognition together place even palindromes in PEG.
FPRD-T90 Read the publication
04
Automata and formal languages · mechanically checked
The scaffold-to-PEG direction is mechanically checked
The full sufficient direction of the Loff–Moreira–Reis correspondence is formalized for finite scaffolding automata.
FPRD-T110 Read the publication
05
Logic, semantics, and rewriting · proof
Local moves generate final behavioral equivalence
In the unary distance-one setting, detours, absorptions, garbage moves, and carrier slides connect exactly the histories with the same final behavior.
FPRD-T135 Read the publication
06
Logic, semantics, and rewriting · proof
Unary scaffold histories admit a finite coherent presentation
A finite family of cubical and bounded critical cells generates all parallel paths inside a final behavioral fibre.
FPRD-T137 Read the publication
07
Algebra and discrete mathematics · proof
Explicit metric-group products on the half-line
A nested readout transports bitwise XOR to group laws on the nonnegative reals whose product is also a metric.
FPRD-T115
08
Algorithms and complexity · mechanically checked
The direct GapCVP reduction reaches 1/30-hardness
Independently checked parameter choices preserve the direct reduction while exposing a retained-architecture frontier at 1/28.
FPRD-T114 Read the publication
09
Models of computation · informal proof
Compact finite feasibility sews into one presentation
Under the stated compactness hypotheses, feasibility through every observation horizon yields one uniform exact presentation.
FPRD-T60
10
Models of computation · informal proof
Escape modulus detects failure of exact sewing
Bounded escape identifies membership in one compact resource core; unbounded escape records finite feasibility without uniform realization.
FPRD-T61

Browse by research area

Automata and formal languagesAutomata, grammars, parsing, language recognition, residuals, pushdown structure, and scaffolding automata.203 claims Models of computationFinite persistent structures, abelian networks, schedule-sensitive computation, geometric machines, and exact interfaces between models.196 claims Algorithms and complexityReductions, lower bounds, automata resources, streaming memory, linear recurrences, matrix powers, and arithmetic circuits.260 claims Logic, semantics, and rewritingOperational semantics, causal equivalence, rewriting histories, confluence, cubical coherence, proof dynamics, and verification.50 claims Algebra and discrete mathematicsTensor categories, representation theory, elliptic integrable systems, mathematical physics, formal series, semirings, metric groups, linear invariants, and discrete topology.138 claims Number theory and discrete dynamicsInteger complexity, arithmetic iteration, guarded-affine traces, modular seams, recurrence, orbit structure, recovery budgets, and contraction.55 claims

All claims and evidence

902 records

Search by statement, research area, result type, or contribution assessment. Each row links to its proof or evidence, sources, dependencies, limitations, and open work.

SearchResearch areaResult type

55 of 902 entries

Stable ID and statement	Type and current status	Contribution assessment	Research area and topic	Proof or evidence	Dependencies	Source and review	Next action
FPRD-D38 Fix the accelerated map $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$ . A guarded-affine trace records a chronological word $w\in\{0,1\}^k$ , its affine action $T^k(n)=(3^s n+c_w)/2^k$ , its guard $r_w\equiv-c_w3^{-s}\pmod{2^k}$ , and—at a fixed layer $(K,S)$ — $D=2^K-3^S$ and $\Pi(w)=c_w-Dr_w$ .	DefinitionFixed and self-contained; no Collatz-conjecture implication	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Guarded-affine presentation and worked trace	No recorded dependencies	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Compare the presentation vocabulary with the closest parity-vector and 2-adic literature before any novelty language.
FPRD-T138 If a prefix $u$ has weight $p$ , endpoint $y_u$ , multiplier $a=3^p$ , and guard $r_u$ , then its exact horizon- $\ell$ action on a suffix guard is the affine permutation $F_u^{(\ell)}(q)=a^{-1}(q-y_u)\pmod{2^\ell}$ .	TheoremProved by direct modular substitution; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact affine sewing proof	FPRD-D38	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Compare the formulation with the closest parity-vector and 2-adic conjugacy literature before any novelty language.
FPRD-T139 For every $\ell\ge3$ , the prefixes $u_s=1^s$ induce $2^{\ell-2}$ distinct exact horizon- $\ell$ seam maps. Thus an interface naming the complete map on every suffix guard needs at least $\ell-2$ bits, while slope and intercept use $O(\ell)$ bits; exact seam-profile demand is $\Theta(\ell)$ .	Theorem · exact resource boundProved for the explicitly defined exact-profile interface; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact seam-profile demand proof	FPRD-D38; FPRD-T138	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Keep this exact-interface bound separate from recognition lower bounds; seek a closer comparator for the profile-count formulation.
FPRD-T140 For a contracting layer $(K,S)$ and split $w=uz$ after $j$ symbols, $\Pi(uz)=2^j(B_u-\delta_z-D\mu_z)$ , where $\delta_z\ge0$ is loss from the best fixed-weight suffix and $\mu_z\ge0$ is the modular seam lift.	Theorem · exact decompositionProved algebraically and finitely corroborated; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Prefix recovery calculus and proof	FPRD-D38; FPRD-T138	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Use the exact decomposition only with joint guard–disorder information; marginal counts have already failed to give contraction.
FPRD-T141 Prefix budgets satisfy $2B_{ub}=B_u-\varepsilon_bD-bG_{L,q}$ . Sending a child slot $\nu$ to the parent slot $\varepsilon_b+2\nu$ gives disjoint parity images, so $N^{\mathrm{slot}}_{u0}+N^{\mathrm{slot}}_{u1}\le N^{\mathrm{slot}}_u$ and the level potential is nonincreasing.	Theorem · conservation lawProved; equality cases refute uniform one-step contraction; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	One-bit recurrence and slot proof	FPRD-D38; FPRD-T140	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Do not seek a universal one-step factor loss; any improvement must use block structure or a stronger joint statistic.
FPRD-T142 If $t_1\ge\cdots\ge t_L\ge0$ are decreasing guard thresholds and $A_i\ge0$ are unconditional bounds for increasing disorder sublevels, then $M_L=t_L+\sum_{i=2}^L\min(t_{i-1}-t_i,A_i)+A_1 \ge\min_i(t_i+A_i).$	Theorem · no-gain obstructionProved by an elementary induction and internally audited; no external review	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Nested-threshold theorem and proof	Nonincreasing thresholds; Nonnegative unconditional sublevel bounds	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Do not add more unconditional Ferrers bands; seek a statistic that retains the joint guard–disorder correlation.
FPRD-T143 Let $N_u^{\mathrm{cap}}$ bound admissible guard extensions, $W_u=\binom{L}{q}$ count fixed-weight suffixes, $m_u=\min(N_u^{\mathrm{cap}},W_u)$ , and $P_h=\sum_{\|u\|=h}m_u$ . Then the accepted count is at most $P_{h+1}\le P_h$ , with equality at full depth, and the local loss obeys the exact bottleneck identity recorded below.	Theorem · capped potentialWritten all-length proof; internally audited; no independent checker for this theorem	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Lowering-capped potential proof	FPRD-D38; FPRD-T140	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Seek joint guard–disorder loss; the capped potential alone gives no uniform factor contraction.
FPRD-C03 Determine whether the forced continuation admits an all-length joint estimate for $r_w\le R,\qquad \delta_w\le B-Dr_w,$ strong enough to improve the leading candidate exponent, or instead construct an exact accepted-trace counterfamily showing that actual forced-suffix disorder can preserve exponential mass.	Open problemOpen; marginal routes are closed and no exponent improvement is claimed	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Surviving frontier and exact alternatives	FPRD-T140; FPRD-T142; FPRD-T143; FPRD-FAIL-COLLATZ-LF5	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Analyze the actual uniquely forced suffix, not another marginal capacity envelope or a larger finite enumeration.
FPRD-FAIL-COLLATZ-LF3 The exact monotonicity $P_{h+1}\le P_h$ and its additive bottleneck-loss identity do not imply a uniform bound $P_{h+1}\le\rho P_h$ with $\rho<1$ ; capacity-one paths can preserve their only slot.	Failed approach · negative boundaryClosed as a universal contraction route	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Why monotonicity supplies no universal rate	FPRD-T143; FPRD-FAIL-COLLATZ-LF5	Collatz post-freeze failure ledger · LF3Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Use the monotone potential for pruning and exact accounting only; require a stronger joint statistic for rate loss.
FPRD-FAIL-COLLATZ-LF4 At layer $(K,S)=(27,17)$ and depth $h=14$ , exact reconstruction gives $2{,}321$ live prefixes, all with capped multiplicity one, so $P_{14}=2{,}321$ and $F_{14}=0$ ; forced continuation therefore does not by itself mean that few traces survive.	Computational finding · failed scarcity proxyIndependently reproduced with an exact-integer checker	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact LF4 reconstruction and evidence boundary	FPRD-T143	Collatz post-freeze failure ledger · LF4Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Preserve the executable regression and avoid treating this finite row as an asymptotic contraction result.
FPRD-FAIL-COLLATZ-LF5 For every fixed $C>0$ , there are $2^{\gamma K-O(\log K)}$ genuine fixed-weight-feasible capacity-one prefixes, with $\gamma=(1-\log_3 2)(1-\log_2(3/2))=0.1531779921\ldots,$ that cross a $C\log_2K$ block with zero attrition under maximal-completion capacity.	Theorem · counterfamilyWritten all-length proof; critical-layer estimate has an external dependency	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Forced-family obstruction and proof boundary	Critical-layer estimate h=Z+O(log K); FPRD-T143	Collatz lowering-and-contraction supplement · Section 6Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Move to actual forced-suffix disorder; do not seek a universal capacity-only contraction over logarithmic blocks.
FPRD-FAIL-COLLATZ-LF7 The marginal cap $m_u=\min(N_u^{\mathrm{cap}},W_u)$ records how many guard slots and fixed-weight tails exist, but not which tail occupies which slot or its actual disorder; it is therefore insufficient by itself to resolve the forced-continuation frontier.	Failed abstraction · limitationSubstantiated as a limitation; the replacement joint statistic remains open	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Why the marginal cap loses the matching	FPRD-T143; FPRD-FAIL-COLLATZ-LF5; FPRD-C03	Collatz post-freeze failure ledger · LF7Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Track the joint assignment of exact guards to actual forced suffix deficits.
FPRD-FAIL-COLLATZ-LF8 The strongest preserved candidate estimate remains exponential, $2^{0.34416K+o(K)}$ ; none of the guarded-affine, marginal-capacity, or forced-family results proves finiteness of paradoxical sequences, Terras's conjecture, or the Collatz conjecture.	Nonclaim · scope boundaryExplicitly established as the program boundary	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Explicit no-Collatz boundary	FPRD-T142; FPRD-T143; FPRD-FAIL-COLLATZ-LF5; FPRD-C03	Collatz post-freeze failure ledger · LF8Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Preserve dormancy unless a concrete joint guard–disorder lemma, exact counterfamily, or external audit reopens the route.
FPRD-IC-O01 Selfridge asked whether $\lVert 2^n\rVert=2n$ for every $n$ , where $\lVert m\rVert$ is the least number of ones needed to build $m$ using addition, multiplication, and parentheses.	Recognized open problemOpen; the FPRD track proves only restricted expression-family results	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Problem, scope, and current frontier	No recorded dependencies	Selfridge's integer-complexity question; current context in Konyagin--Oganesyan (2026)Reviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep the restricted track paused pending a new normal-form or effective-height theorem; more residue tables are not the current frontier, and unrestricted Selfridge remains open.
FPRD-IC-D01 The active restricted grammar has value $M[A(B+C)(D+E)+F]$ , where displayed operands are positive $(2,3)$ -smooth constructions and all three displayed additions are paid. After cost-preserving normalization, $\gcd(B,C)=\gcd(D,E)=1$ , the inner equation is primitive, and a fork is proper when both inner sums are nonsmooth.	Definition and cost-preserving normal formPrecisely stated and proved within the restricted grammar	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Grammar and normalization	No recorded dependencies	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Preserve the exact paid-atom cost and unit charges in every subsequent reduction.
FPRD-IC-T01 Every normalized primitive proper fork $3^a(B+C)(D+E)+2^f3^g=2^N$ satisfies $f\le2$ . Moreover the odd parts of the two nonsmooth inner sums have opposite values of the quadratic character $\chi(x)=+1$ for $x\equiv1,3\pmod8$ and $-1$ for $x\equiv5,7\pmod8$ .	Computer-assisted restricted theoremProved and independently reconstructed within FPRD Lab	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Thin-branch theorem	FPRD-IC-D01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the restriction to keep all later attacks inside the surviving f=0,1,2 branches.
FPRD-IC-T02 In the normalized $g=0$ , positive-type-I branch, the ladder $3^aP=(2^{2rx}-1)/(2^x+1)$ has no surviving canonical negative-character solution with $r\ge2$ . The remaining $r=1$ controls cost at least $2N+2$ .	Exact computer-assisted restricted theoremProved with complete periodic certificates and an independent evaluator	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Cyclotomic ladder closure	FPRD-IC-D01; Michael Bennett, Theorem 1.6, for inherited boundary equations	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep this branch closed unless a defect is found in a displayed periodic certificate.
FPRD-IC-T03 The complete odd- $N$ , $g=0$ , positive-type-I branch of the normalized proper fork has no displayed expression of cost at most $2N-1$ . Its only solution in the two nonsmooth canonical shapes is $7\cdot73+1=2^9$ , whose two presentations cost $2N+2$ and $2N+3$ .	Exact restricted branch theoremProved with congruence and primitive-divisor arguments; independently audited	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Odd-exponent branch	FPRD-IC-D01; FPRD-IC-T02; Zsigmondy's theorem on one boundary	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not reopen the odd-N branch; concentrate on the surviving even-N families.
FPRD-IC-T04 At the first even layer $a=1$ , all but two large-x residue families and one x=4 control lift are excluded. In the x=4 family, every positive solution of $3(4+3^v)(1+16\cdot3^y)+1=2^N$ must satisfy $(v,y,N)\equiv(0,0,8)\pmod{(576,144,360)}$ , $v_3(2^N-13)=\min(v,y)+1$ , and $v+y>10^{70}$ .	Exact restricted reduction and lower boundProved and independently audited as an intermediate reduction; the lift is closed by FPRD-IC-T05	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Even-layer reduction	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain the Hensel class as the input to FPRD-IC-T05; do not treat the x=4 lift as an active survivor.
FPRD-IC-T05 The inherited $a=1$ , II2, $x=4$ equation $3(4+3^v)(1+16\cdot3^y)+1=2^N$ has no positive solution. Its first 3-adic Hensel digit forces $N\equiv3248\pmod{9720}$ ; modulo $19441$ , the right side is then $15812$ , while the nine possible left-side residues omit $15812$ .	Exact computer-assisted restricted theoremProved by a finite nine-residue certificate and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Exceptional-lift closure	FPRD-IC-T04	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T07 for the type-II2 continuation and concentrate on the remaining large-x type-II1 family.
FPRD-IC-T06 Every remaining $a=1$ large-x solution would already have displayed cost at most $2N-1$ . The 2-adic gates force $v\ge55313581$ in II1 and $v\ge60254788$ in II2. In the II2 subbranch $y<v$ , $y\ge5$ , an exact modulus- $19441$ quotient eliminates 17 of the 28 inherited classes for $x\pmod{540}$ , leaving 11 necessary classes.	Exact restricted reduction and partial branch theoremProved and independently audited as an intermediate reduction; II2 is closed by FPRD-IC-T07	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Large-x frontier	FPRD-IC-D01; FPRD-IC-T05	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain the 17/11 split as the input to FPRD-IC-T07; do not treat the eleven classes as active survivors.
FPRD-IC-T07 The large-x equation $3(4+3^v)(1+2^x3^y)+1=2^N$ has no solution under the inherited $a=1$ , type-II2 conditions. Modulo 7 forces $3\mid y$ ; the cases $y=2,4$ fall immediately and $y=3$ falls modulo 73. For $y\ge5$ , exact quotients modulo $19441$ , $6481$ , and $13$ eliminate all 28 inherited classes for $x\pmod{540}$ .	Exact computer-assisted restricted branch theoremProved with displayed finite certificates and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete type-II2 closure	FPRD-IC-D01; FPRD-IC-T05; FPRD-IC-T06	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T08 for the type-II1 continuation; do not treat this family as an active survivor.
FPRD-IC-T08 The last $a=1$ type-II1 family $3(2+3^v)(1+2^x3^y)+1=2^N$ , with $v$ odd, $N\equiv16\pmod{18}$ , and $x$ in the sixteen inherited classes modulo $270$ , has no solution. Modulo $7$ forces $(v,y)\equiv(3,0)\pmod6$ ; modulo $13$ , the left residues are then $\{0,7\}$ and the right residues are $\{3,10\}$ . Together with the preceding canonical closures, this completes the full $a=1$ layer of the active normalized branch.	Exact elementary restricted layer theoremProved by two displayed residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete first-layer closure	FPRD-IC-D01; FPRD-IC-T05; FPRD-IC-T07	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-B01 and FPRD-IC-T09 for the first a=2 family; do not reopen a=1.
FPRD-IC-B01 For the $a=2$ type-I2 equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ , the coarse necessary classes $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ cannot be excluded by any finite collection of moduli coprime to $6$ : the formal boundary $(v,y,N)=(-1,0,6)$ supplies compatible residues. The sharper 3-adic condition $N\equiv42$ or $78\pmod{108}$ breaks this obstruction and enables FPRD-IC-T09.	Exact coarse-sieve method-boundary lemmaProved algebraically and checked on independent finite quotients	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Next-layer boundary	FPRD-IC-D01; FPRD-IC-T08	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Apply the 3-adic refinement before coprime order moduli; FPRD-IC-T09 shows that this closes the family.
FPRD-IC-T09 The equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ has no positive solution in the inherited $a=2$ type-I2 branch. Modulo $7$ and $13$ force $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ . An exact 3-adic comparison at $2^6$ sharpens this to $N\equiv42$ or $78\pmod{108}$ . Modulo $271$ , the resulting 25 left residues and 10 right residues are disjoint.	Exact computer-assisted restricted branch theoremProved by displayed valuation and finite residue certificates; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	First a=2 family closure	FPRD-IC-D01; FPRD-IC-T08; FPRD-IC-B01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T10 for the final a=2 family; do not treat a=2 as an active survivor.
FPRD-IC-T10 The remaining odd-exponent type-II2 equation $9(4+3^v)(1+2^x3^y)+1=2^N$ , with $v,y\ge1$ and $x=v_2(37+3^{v+2})$ , has no solution. Modulo $73$ and the primary 2-adic gate force $v\equiv5\pmod{12}$ and $(x,y)\equiv(1,3),(4,11),(7,7)\pmod{(9,12)}$ . Modulo $577$ , where $3=2^{105}$ and $\operatorname{ord}_{577}(2)=144$ , the required logarithm classes $\{1,4,7\}\pmod9$ are disjoint from the available classes $\{2,3,5,6\}\pmod9$ . Together with the preceding canonical closures, this completes the $a=2$ layer of the active normalized branch.	Exact computer-assisted restricted layer theoremProved by displayed finite quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete second-layer closure	FPRD-IC-D01; FPRD-IC-T09	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T11 for the a=3 entry table; do not reopen any entry with x at most 20.
FPRD-IC-T11 In the active normalized branch $27P(1+2^x3^y)+1=2^N$ , where $P$ is one of the four negative-character canonical factors, the two $x=1$ families and every entry with $3\le x\le20$ have no solution. Modulo $81$ gives $N\equiv18$ or $36\pmod{54}$ . The exact gate $27P+1\equiv2^x3^y\pmod{2^{2x}}$ , followed by a complete quotient modulo $73$ , forces $(x,y)\equiv(0,6),(3,2),(6,10)\pmod{(9,12)}$ . Exact finite covers eliminate $x=3,6,9,12,15,18$ . Hence every survivor has $x\ge21$ and $3\mid x$ ; FPRD-IC-T12 closes this remote tail.	Exact computer-assisted entry theorem and finite range closureProved by exact residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Third-layer entry theorem	FPRD-IC-D01; FPRD-IC-T10	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T12 for the remote tail; do not reopen the a=3 positive-type-I entry table.
FPRD-IC-T12 The remote branch $27P(1+2^x3^y)+1=2^N$ , with $x\ge21$ , $3\mid x$ , and $P$ one of the three surviving negative-character shapes, has no solution. Modulo $262657$ , where $\operatorname{ord}(2)=27$ and $\operatorname{ord}(3)=14592$ , the complete quotient leaves eight residue rows. Six fail modulo $7$ and two fail modulo $13$ . Together with FPRD-IC-T11, the complete $a=3$ positive-type-I layer is empty.	Exact computer-assisted restricted layer theoremProved by complete residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Remote third-layer closure	FPRD-IC-D01; FPRD-IC-T11	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Treat positive type I as closed and move to the three low positive-type-II layers.
FPRD-IC-T13 For every normalized primitive proper equation $3^aPQ+1=2^N$ with $f=g=0$ and $a\ge4$ , both canonical odd types $Q=1+2^x3^y$ and $Q=2^x+3^y$ are impossible. The valuation identity gives $27\mid N$ , so modulo $73$ and $262657$ both equations force $Q=0$ . Every canonical negative factor $P$ is nonzero at both primes, while the complete exponent classes making $Q=0$ at the two primes have no simultaneous lift.	Uniform exact computer-assisted restricted theoremProved by displayed subgroup and incompatible-lift certificates	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform high-layer obstruction	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not resume exterior-layer induction above a=3; only the low positive-type-II layers remain in the g=0 branch.
FPRD-IC-T14 Every normalized primitive proper $f=g=0$ fork whose positive-character factor is $1+2^x3^y$ has displayed cost at least $2N$ . The cyclotomic and odd- $N$ theorems close $a=0$ ; FPRD-IC-T08 and T10 close $a=1,2$ ; FPRD-IC-T11 and T12 close $a=3$ ; and FPRD-IC-T13 excludes every $a\ge4$ .	Restricted branch-closure corollaryProved by the cited exhaustive branch decomposition	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete positive-type-I branch	FPRD-IC-T02; FPRD-IC-T03; FPRD-IC-T08; FPRD-IC-T10; FPRD-IC-T12; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Classify positive type II at a=1,2,3; the uniform theorem already supplies its high-layer boundary.
FPRD-IC-T15 The normalized complementary family $27P(v)(2^x+3^y)+1=2^N$ , with $P$ any canonical negative-character factor, has no solution. Its exact gate quotient contains 524288 states; relaxed but solution-preserving filters modulo $6481,257,73,7,193$ leave respectively $16512,1268,390,124,0$ states.	Exact computer-assisted restricted layer theoremProved by a complete finite quotient and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary third-layer closure	FPRD-IC-D01; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep a=3 closed and concentrate on the exact residual quotient at a=1,2.
FPRD-IC-T16 Every canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ with $a\in\{1,2,3\}$ satisfies $x\le10$ . A coupled quotient at the aligned primes $17,193,257,12289$ leaves two high-gate rows modulo $(512,512,6144,6144)$ ; all 162 simultaneous lifts to periods $(1536,1536,18432,18432)$ fail modulo $7,13,73,97,577$ .	Uniform exact computer-assisted restricted theoremProved by coupled exponent quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform high-x obstruction	FPRD-IC-D01; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not resume a high-x search; the low-x quotient is now classified by FPRD-IC-T21.
FPRD-IC-T17 Every remaining canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ , $a\in\{1,2,3\}$ , lies in one of 21 displayed periodic rows modulo $(v,y,N)=(1536,1536,18432)$ , all at $a=1,2$ , with $x\in\{1,3,4,5,6,8\}$ . These are necessary lift classes, not a classification.	Exact residual-quotient theoremProved as a complete necessary quotient; every row is now classified by FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Twenty-one residual lift classes	FPRD-IC-T15; FPRD-IC-T16	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use this finite quotient only as the input to the completed second-gate classification FPRD-IC-T21.
FPRD-IC-T18 For the five residual rows of $9P(v)(64+3^y)+1=2^N$ inherited from FPRD-IC-T17, every solution satisfies $v\equiv v_0\pmod{36288000}$ , $y\equiv y_0\pmod{36288000}$ , and $N\equiv12\pmod{48384000}$ , where $(v_0,y_0)$ is its displayed boundary residue. Twelve simultaneous multiplicative-order quotients preserve exactly one lift of each row.	Exact computer-assisted lift theoremProved and independently reconstructed; superseded as a frontier by FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Shared five-row boundary lock	FPRD-IC-T17	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain as an exact intermediate theorem; do not extend this lock after FPRD-IC-T21 closes every nonboundary lift.
FPRD-IC-T19 Every non-boundary solution in the five-row cluster of FPRD-IC-T18 satisfies $v+y>26774635116096000$ and $N\ge42436792629504012$ . Equivalently, after writing $v=v_0+36288000r$ , $y=y_0+36288000s$ , every solution has $r+s>737837167$ .	Certified computer-assisted height theoremProved, but superseded by the complete nonboundary exclusion FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	One-sided approximation barrier	FPRD-IC-T18	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not extend the height sweep; the full second gate now excludes every lift to which this conditional bound applied.
FPRD-IC-T20 Let $F(v,y)=3^aP(v)(2^x+3^y)+1$ . Eleven rows of FPRD-IC-T17 have fixed valuation $v_2(F)=N_0$ . In each of the other ten rows, the solutions of $F(v,y)\equiv0\pmod{2^e}$ form a one-dimensional Hensel graph: modulo $L_e=3\cdot2^{e-2}$ , there are exactly $2^{e-11}$ states and projection to the $v$ -coordinate is bijective.	Exact 2-adic lifting theoremProved by LTE and complete finite base checks; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Full valuation gate and Hensel graph	FPRD-IC-T17	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Reuse the full-side valuation gate before applying odd-prime quotients in the remaining fork orientations.
FPRD-IC-T21 The normalized positive-type-II family $3^aP(v)(2^x+3^y)+1=2^N$ with $a\ge1$ has exactly ten canonical proper solutions, representing $3\cdot5\cdot17+1=2^8$ , $3\cdot31\cdot11+1=2^{10}$ , $9\cdot5\cdot91+1=2^{12}$ , and $9\cdot13\cdot35+1=2^{12}$ . Every presentation costs at least $2N+4$ .	Exact computer-assisted restricted branch classificationProved from the complete residual quotient, full 2-adic gate, and one aligned prime; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Positive-type-II classification	FPRD-IC-T13; FPRD-IC-T15; FPRD-IC-T16; FPRD-IC-T17; FPRD-IC-T20	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Move to the complementary a=0<g orientation and final 2-adic exponents f=1,2; do not reopen the classified tower.
FPRD-IC-T22 Let $RS+3^g=2^K$ be the reduced $a=0<g$ equation, with $R$ the negative-character canonical odd factor and $S$ the positive-character factor. Of the twelve possible canonical pairs modulo 24, the equation and the two Jacobi identities leave exactly $(R,S;K,g)\equiv(5,1;1,1),(5,11;0,0),(7,1;0,0),(7,17;1,0),(13,1;0,1),(13,19;0,0)\pmod{(24,24;2,2)}$ .	Exact quadratic-residue restrictionProved elementarily and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary quadratic gate	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the six-row gate before any 3-adic lifting or odd-prime quotient in the complementary orientation.
FPRD-IC-T23 No normalized primitive proper solution with $a=0<g$ and $f=2$ has $v_2(B+C)=v_2(D+E)=1$ . Indeed, the two odd parts must have opposite character and hence residues $5$ and $17\pmod{24}$ , a pair excluded by FPRD-IC-T22.	Exact restricted branch exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Double-even branch exclusion	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Classify the three surviving f=1 residue rows, beginning with their unique-minimum and tied 3-adic strata.
FPRD-IC-T24 In every normalized primitive proper solution in the three complementary $f=1$ residue rows, let $(1+3^b)/2$ be the odd part of the even factor and write the other factor as $W=2^h+2^q3^y$ . Then $\min(b,y,g)=1+v_3(K+1-h)$ . All six tied-minimum strata whose leading 3-adic digit could cancel are impossible modulo 7 and 73.	Exact valuation theorem and tied-stratum exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=1 valuation law	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Combine this law with the f=2 valuation theorem and classify the first two minimum layers m=1,2.
FPRD-IC-T25 In every normalized primitive proper solution in the complementary $f=2$ , split- $(2,0)$ branch, let $V_b=(1+3^b)/4$ , where $b\ge3$ is odd, and write the other odd factor as $W=2^h+2^q3^y$ . Then $\min(b,y,g)=1+v_3(K+2-h)$ . All ten tied-minimum strata whose leading 3-adic digit could cancel are impossible: nine by complete periods modulo 7 and 73, and the sole residual class by incompatible complete periods modulo 487 and 2593.	Exact valuation theorem and tied-stratum exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=2 valuation law	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Combine the f=1 and f=2 laws at minimum layers m=1,2, seeking a finite order-modulus cover before opening the f=0 rows.
FPRD-IC-T26 In the normalized complementary $f=1$ and split- $(2,0)$ $f=2$ branches, impose $m=\min(b,y,g)\in\{1,2\}$ . Of the 32 canonical minimum strata, nine are empty. Every state in the other 23 lies on one of 40 exact periodic lift graphs with common coordinate period $P=20{,}652{,}025{,}680$ . Twenty-nine graphs are rooted at seventeen proper harmless presentations and eleven at nine improper rational boundary identities. Every noncontrol proper solution therefore has output exponent $N\ge P$ .	Exact modular lift classification and height barrierProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary minimum-layer lift lock	FPRD-IC-T22; FPRD-IC-T24; FPRD-IC-T25	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the next 3-adic digit and an archimedean descent on the eleven boundary-rooted lift graphs; do not merely enlarge the order-modulus period.
FPRD-IC-T27 In the normalized complementary $f=1$ and split- $(2,0)$ $f=2$ branches, every solution with $m=\min(b,y,g)\in\{1,2\}$ is one of exactly seventeen explicit proper presentations. All seventeen cost at least $2N+6$ . The eleven lift graphs rooted at improper rational identities are empty modulo $3^6$ ; 26 control-rooted graphs freeze completely; and the remaining three reduce to $2^{K+1}=11\cdot3^b+173$ or $2^K=3^g+295$ , whose nonzero lifts are excluded by Matveev's explicit lower bound and exact finite modular descent.	Exact computer-assisted classificationProved with one classical linear-forms dependency; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary minimum-layer classification	FPRD-IC-T26; FPRD-IC-T24; FPRD-IC-T25	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Test whether the valuation period and an explicit linear-forms cutoff cross uniformly in m; otherwise open the complementary f=0 quotient.
FPRD-IC-T28 The normalized complementary $f=0,1,2$ family has only finitely many presentations. More precisely, its eight fixed factor-shape patterns contain at most $8\exp(7\cdot30^{15})$ presentations in total. This is a direct corollary of the Evertse--Schlickewei--Schmidt bound for nondegenerate S-unit equations: after expansion and division by $2^{K+f}$ , five positive $\{2,3\}$ -units sum to one in a multiplicative group of rank at most six.	Classical S-unit corollaryProved from an established theorem; not an effective height bound	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform complementary finiteness	FPRD-IC-D01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not treat the solution-count bound as a height cutoff; obtain finite congruence quotients inside the remaining f=0 strata.
FPRD-IC-T29 In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the quadratic gate leaves forty canonical factor placements and exactly twenty possible tied-minimum 3-adic cancellations. Only two can have $v=y=g$ . One has no solutions; the other has the unique solution $(2+3)(1+8\cdot3)+3=2^7$ , whose displayed construction costs $18=2K+4$ ones.	Exact elementary classification with finite residue certificateProved; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=0 triple-minimum classification	FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T30 for the pair-minimum continuation; do not treat the eighteen strata as active survivors.
FPRD-IC-T30 In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the eighteen pair-minimum strata whose first nonzero 3-adic digit can cancel contain exactly one solution, $(4+3)(1+8\cdot3)+3^4=2^8$ , of displayed cost $29=2K+13$ . Together with FPRD-IC-T29, the entire exceptional cancellation locus consists of this control and $(2+3)(1+8\cdot3)+3=2^7$ . Every other solution satisfies $\min(v,y,g)=1+v_3(K-r-u)$ .	Exact computer-assisted restricted theoremProved with complete finite quotients and independent reconstruction	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=0 cancellation closure	FPRD-IC-T22; FPRD-IC-T29; LTE	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Pause layer-by-layer enumeration; reopen this branch only with an effective six-term reduction, a uniform freezing theorem, or another structural bridge.
FPRD-D39 Fix $b\ge2$ , $A_b=\{0,\ldots,b-1\}$ , and $n\in\mathbb N_0$ . Write the canonical expansion $n=\sum_{i=0}^{L-1}a_i b^i$ least-significant first, with $L=1,a_0=0$ for zero and $a_{L-1}\ne0$ otherwise. Define $\operatorname{rev}_b(n)=\sum_i a_i b^{L-1-i}$ , $T_b(n)=n+\operatorname{rev}_b(n)$ , $c_i=a_i+a_{L-1-i}$ , $\gamma_0=0$ , $s_i=(c_i+\gamma_i)\bmod b\in A_b$ , and $\gamma_{i+1}=\lfloor(c_i+\gamma_i)/b\rfloor$ , retaining $\gamma_L$ . Reversal-created leading zeroes do not change the integer. For canonical digits $d_i$ of $m$ , $D_b(m)=\#\{i:d_i\ne d_{M-1-i}\}$ counts ordered positions.	DefinitionExact conventions reviewed; no theorem is embedded in this record	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Definitions and conventions	No recorded dependencies	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Find the smallest update-closed receipt that carries this one-step presentation across iteration.
FPRD-T144 For each fixed $b\ge2$ and $n\in\mathbb N_0$ , the quantities in FPRD-D39 satisfy $\gamma_i\in\{0,1\}$ and $T_b(n)=\sum_{i=0}^{L-1}s_i b^i+\gamma_L b^L$ . After the symmetric fold supplies $c_0,\ldots,c_{L-1}$ in increasing digit index, the carry stage is a deterministic subsequential transducer over $C_b=\{0,\ldots,2b-2\}$ , with states $\{0,1\}$ , initial state 0, transition $\lfloor(c+\gamma)/b\rfloor$ , output $(c+\gamma)\bmod b$ , and terminal output 1 exactly when the final carry is 1. Within this folded one-step presentation, carry is the only orientation-dependent sequential state.	Written theorem · fixed-base transducer familyProved on the public page; finite replay is corroboration, not proof	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Folded carry theorem and proof	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare this elementary decomposition with prior reverse-and-add and finite-transducer literature before making any novelty claim.
FPRD-T145 Under FPRD-T144, if $\gamma_L=0$ (equivalently, the step does not increase digit length), then for $j=L-1-i$ , $s_i-s_j\equiv\gamma_i-\gamma_j\pmod b$ and the integer $\gamma_i-\gamma_j$ lies in $\{-1,0,1\}$ . This is a residue statement, not an absolute-distance bound; its residue-set corollary is nonrestrictive in bases 2 and 3.	Written corollary · modular digit structureProved on the public page; finite replay is corroboration, not proof	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Mirrored carry law and proof	FPRD-D39; FPRD-T144	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit prior art, especially the documented observation that a palindromic output requires a palindromic carry pattern.
FPRD-T146 A positive decimal input has a palindromic one-step reverse-and-add output exactly in one of two cases: every symmetric column sum is at most $9$ , or every sum lies in $\{0,11\}$ and the outer sum is $11$ . On $1\le n\le1{,}000{,}000$ , the two classes contain $151{,}250$ and $808$ inputs.	Digit-structure theorem · reproduced finite censusGeneral classification proved; complete bounded census independently reproduced	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Classification proof and bounded census	FPRD-D39; FPRD-T144	Vaughn Suite, reverse-and-add carry note (16 September 2003); FPRD independent proof and censusReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Translate the decimal carry proof to general bases, treating the exceptional base-two growth pattern separately.
FPRD-T147 For the outputs $T^k(196)$ , $1\le k\le4{,}000$ , the reflection distance has minimum $2$ , its minimum normalized value is $4/33$ at step $68$ , and its mean normalized value over the $3{,}771$ outputs longer than $100$ digits is $0.659803985542$ .	Reproduced bounded computational findingExact finite window independently replayed by two implementations	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Reflection-distance observation	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the same pinned statistic with convergent controls; do not extrapolate the finite lower bound to the infinite orbit.
FPRD-T148 For the first $3{,}000$ transitions from $196$ , the mean normalized shifted reflection distance is $0.496558797244$ at shift $0$ on the $1{,}735$ non-growing outputs and $0.497066641824$ at shift $+1$ on the $1{,}265$ growing outputs; the mismatched shifts tested from $-2$ through $+2$ lie between $0.884$ and $0.897$ .	Reproduced bounded computational findingExact finite table independently replayed; universal shifted-axis identity unproved	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Axis-shift correction	FPRD-D39; FPRD-T145	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Derive or refute an exact shifted-index identity; retain the present result as a finite table until then.
FPRD-T149 Across the states $T^k(196)$ , $0\le k<4{,}000$ , every within-state decimal digit block through length $4$ and every incoming-carry block through length $12$ occurs. The pinned replay gives carry-in density $0.499366103113$ , mirrored-digit mutual information $0.563591475633$ bits, and mirrored carry-in mutual information $0.000013795714$ bits.	Reproduced bounded computational findingExact finite window and estimators independently replayed	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Raw-complexity observation	FPRD-D39; FPRD-T144	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add explicit control sequences under the same estimators before making comparative complexity claims.
FPRD-FAIL-LYCHREL-01 The aperiodic-monotile substitution analogy was stopped because the sampled raw digit and carry streams showed none of the low-complexity hierarchy that route intended to exploit. This failure does not rule out conditional, finite-state, backward-preimage, or anti-concentration approaches.	Important failed approachScope-corrected negative research record	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Failed-route boundary	FPRD-T149	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Preserve the fold-and-carry representation while avoiding claims that raw block complexity excludes richer invariants.
FPRD-C04 Does there exist a positive decimal integer whose reverse-and-add orbit never reaches a palindrome? In particular, does the orbit of $196$ avoid palindromes forever?	Open problemOpen; no FPRD resolution	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Open boundary	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Pursue finite structural results and exact interfaces without treating bounded orbit searches or random-like statistics as an infinite proof.