Obstruction, lifting, and sewing receipts

The reusable theorem pattern

A quotient is only half of the interface

The First Isomorphism Theorem explains why kernels govern homomorphic images. The Arsenal extends that discipline to presentations with local sections, admissible loci, dynamics, and resource constraints. For a declared quotient $q:X\to Q$, the operative questions are:

1. Which probes or future observations descend to (Q)?
2. What is the fibre, stabilizer, or null-probe system?
3. What cocycle records failure of chosen local lifts to agree?
4. When does that obstruction vanish, and what primitive repairs it?
5. Does the repaired lift lie in the intended admissible locus?

$$\text{quotient data} \;\; + \;\; \text{obstruction class} \;\; + \;\; \text{correction receipt} \;\; + \;\; \text{admissibility check}.$$

The same pattern appears as Jordan data inside an invariant fibre, a connecting class in a cochain quotient, monodromy in a graph cover, a growth bit between numeral charts, a product cokernel in a factorization lift, and a matching defect between anonymous graph cards.

FPRD-QA-MX-T01–T06 · matrix calibration

One matrix shows why a quotient needs a receipt

Fix $X=M_n(\mathbb C)$ and let$G=\mathrm{GL}_n(\mathbb C)$act by conjugation. Two matrices have the same polynomial invariant quotient exactly when they have the same characteristic polynomial. This records the eigenvalues with multiplicity, but not the sizes of the Jordan blocks attached to each eigenvalue. Procesi's invariant theory supplies the broader simultaneous- conjugation setting; the one-matrix statement here also follows directly from the coefficients of the characteristic polynomial.

Every invariant fibre contains one closed orbit, represented by the semisimple matrix with those eigenvalues. The balanced points in that orbit are the normal matrices, and they form one unitary conjugacy orbit. This is the one-matrix specialization of the Kempf–Ness minimum-norm picture. It does not select one canonical matrix: unitary symmetry remains.

The fibre receipt is a nullity tower

For an eigenvalue $\lambda$, put

The increment $d_{\lambda,k}-d_{\lambda,k-1}$is the number of Jordan blocks for$\lambda$ of size at least$k$. These increments are the conjugate Jordan partition, so the finite tower recovers the full similarity class once the eigenvalues are known. The same partition$\pi_\lambda$ gives the exact centralizer dimension

Shifted-power ranks can only decrease under orbit degeneration. Gerstenhaber's dominance theorem supplies the converse within a nilpotent fibre; the page does not claim that converse for arbitrary multi-eigenvalue degenerations without separating the primary parts.

Approaching the balanced core can be ill-conditioned

If the eigenvalues of $A$ are$\lambda_1,\ldots,\lambda_n$, then

Schur form proves the lower bound, and diagonal scaling of each Jordan block approaches it. The infimum is attained exactly when$A$ is semisimple. For one block$J_m(\lambda)$, conjugation by$\mathrm{diag}(t^{m-1},\ldots,t,1)$leaves excess Frobenius square $(m-1)t^2$ while that diagonal change of basis has condition number $t^{-(m-1)}$ for$0<t\le1$. This is an explicit tradeoff witness, not a global conditioning optimum.

Future observations define a different quotient

For a state $v\in V\cong\mathbb C^n$and declared outputs $CA^kv$, two states have the same complete future exactly when their difference lies in

Cayley–Hamilton shows that the first$n$ observations determine all later ones, that $N_C$ is$A$-invariant, and that$V/N_C$ is the coarsest linear quotient retaining this declared future. The horizon is sharp: for the nilpotent Jordan block with ones on the superdiagonal and$C=e_1^T$, the observability rank rises by one at every horizon through $n-1$. This is the classical observability interface associated with Kalman, specialized here to a quotient statement.

The dependency-free review audit reproduces all 44 partitions through dimension seven, 233 dominance comparisons, six exact scaling rows, the two-dimensional collision, and sharp observation chains through dimension seven. Audit program · canonical output · review certificate.

References for the matrix calibration

• George Kempf and Linda Ness, The Length of Vectors in Representation Spaces, in Algebraic Geometry, Lecture Notes in Mathematics 732, Springer, 1979, pp. 233–243, DOI 10.1007/BFb0066647. Used for the balanced/minimum-norm orbit framework.
• Claudio Procesi, The Invariant Theory of n × n Matrices, Advances in Mathematics 19 (1976), 306–381, DOI 10.1016/0001-8708(76)90027-X. Used as the broader simultaneous-conjugation source; the one-matrix quotient is the elementary specialization stated above.
• Murray Gerstenhaber, On Dominance and Varieties of Commuting Matrices, Annals of Mathematics 73 (1961), 324–348, DOI 10.2307/1970336. Used for the nilpotent orbit-closure dominance boundary.
• Rudolf E. Kalman, Mathematical Description of Linear Dynamical Systems, Journal of the Society for Industrial and Applied Mathematics, Series A: Control 1 (1963), 152–192, DOI 10.1137/0301010. Used for the classical observability viewpoint; the finite quotient proof above is self-contained.

FPRD-QA-T06 · FPRD-QA-T07 · FPRD-QA-T08

The cohomological sewing theorem, from first principles

A cochain complex is a sequence of abelian groups with maps$d:C^k\to C^{k+1}$ satisfying$d^2=0$. A cocycle is an element killed by $d$; a coboundary is an element of the form$da$. Cohomology identifies cocycles that differ by a coboundary. In the sewing problem, coboundaries are precisely the defects removable by changing a local choice.

A common source of this short exact sequence starts with an ambient presentation $P$, an admissible locus $A$, and a quotient$q:A\to Q$. An ambient cochain is descendable when its restriction to$A$ lies in the image of$q^*$; it is null when that restriction is zero. Pullback commutes with the differential, so both families are subcomplexes. Restriction then identifies their quotient with the descendable subcomplex inside$C^\bullet(A)$ by the First Isomorphism Theorem.

Let $N^\bullet\subseteq D^\bullet$be a subcomplex of null probes and let$Q^\bullet=D^\bullet/N^\bullet$be the surviving interface. For a quotient cocycle$q\in Z^k(Q)$, choose any lift $x\in D^k$. Because$q$ is closed, there is a unique cocycle $c\in N^{k+1}$with

Changing the lift changes $c$by a null coboundary, so its class$[c]\in H^{k+1}(N)$ is well-defined. This is the standard connecting homomorphism in the long exact cohomology sequence, used here as an explicit receipt rather than claimed as a new homological-algebra theorem.

$$q\text{ has a closed lift} \quad\Longleftrightarrow\quad [c]=0.$$

The correction is type-sensitive. If$c=d_Na$ for$a\in N^k$, then$x-i(a)$ is closed. The obstruction $c$ and its primitive $a$ live in adjacent degrees and must not be denoted by the same symbol.

Requiring $N$ to be a subcomplex is essential. In the finite control$D^0=D^1=\mathbb Z/4$ with identity differential, declaring$N^0=\{0,2\}$ and$N^1=\{0\}$ identifies 0 and 2 in degree zero but gives them different degree-one images. The proposed quotient differential is therefore not well-defined.

Local sections on a graph

For a finite oriented graph, let each vertex carry an abelian group and each edge $e:u\to v$carry an isomorphism $T_e$. Local lifts $x_v$ of a compatible quotient section have edge disagreements

These edge values are simultaneously the Čech overlap defect, the transported cycle receipt, and the cochain connecting class. A vertex correction kills every edge defect exactly when the class vanishes. On a connected graph this can be tested on any cycle-space basis: a spanning tree constructs the candidate correction, and each non-tree edge tests one circulation.

The unipotent three-cycle is the essential hostile example. The quotient transport is the identity, but closing the cycle sends$(a,b)\mapsto(a+b,b)$. The constant quotient section $b=1$returns displaced by one null-fibre unit, so no compatible lift exists upstairs.

Reduction in stages

For nested subcomplexes$N\subseteq M\subseteq D$, the first obstruction lies in$H^{k+1}(M/N)$. If it vanishes, an intermediate lift exists, but different intermediate lifts can change the next obstruction by the image of the connecting map for$0\to N\to M\to M/N\to0$. The canonical secondary receipt is therefore the coset

not a chosen raw representative. A class lifts all the way to$H^k(D)$ exactly when the primary class and this secondary coset both vanish. The$\mathbb Z/8$ control with$d(x)=4x$ has zero primary receipt and nonzero secondary receipt, proving that the second test is not redundant.

The calculus here is abelian. Nonabelian stabilizers require ordered products or pointed-set/groupoid data. Local quotient compatibility alone never licenses an admissible global lift.

The archived finite checks were replayed, and a separate implementation exhausted the same connecting and staged controls plus 2,401 lift/twist cases over$\mathbb Z/7$. See the independent certificate.

References for the sewing core

• Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002, especially Chapter 3 and the sections on Bockstein homomorphisms and local coefficients. Author-hosted PDF.

The long exact sequence, connecting homomorphism, Bockstein example, and cohomology with local coefficients are classical. The FPRD contribution in this unit is the common obstruction-and-receipt interface, the staged coset formulation, and the explicit finite calibrations under the stated hypotheses.

QA-CECH-A1 · QA-XR-T1 · QA-STAGE-T1

From compatible local data to one global object

Overlap agreement and global existence are different questions. A cover records where local presentations meet; an internal differential records whether each presentation satisfies the desired equation. A Čech double complex keeps both tests visible. For null cochains $N^q$ on a finite cover, write

Suppose $a$ is a collection of local lifts of a closed quotient cochain of internal degree$k$. Its internal defect is$b=da$, while its overlap defect is $c=\check\delta a$. The signed sum

is closed in the total complex. It changes by a total coboundary when the local lifts change. A strict global closed lift exists precisely when a null correction of Čech degree zero satisfies $O(a)=-D_{\mathrm{tot}}n$. Allowing a primitive of positive Čech degree instead produces higher coherence data; it does not by itself produce one global section. This distinction prevents an overlap-level solution from being silently promoted to strict descent.

Overlap compatibility without global descent

The extra descent hypothesis is necessary. Let$X=U\cup V$ and define a presheaf by$F(X)=0$ and$F(U)=F(V)=F(U\cap V)=\mathbb Z/2$. Both local restriction maps to the overlap are the identity; the restrictions from $X$ are zero. The pair $(1,1)$agrees on $U\cap V$, yet no global section restricts to it. Thus compatible local cocycles need not be globally extendable for an arbitrary presheaf.

Why the secondary obstruction is a quotient class

Let $\delta$ and$\partial$ be commuting differentials. If $\delta x=0$and the first defect is null in$\delta$-cohomology, choose$h$ with$\delta h=\partial x$. The corrected total cochain has next defect$\partial h$. But replacing$h$ by$h+z$, where$\delta z=0$, changes that defect by $\partial z$. Consequently the invariant secondary receipt is

equivalently the corresponding class on the next page of the associated spectral sequence. The raw representative can be useful in a calculation, but it is not canonical.

Split stages and the curvature receipt

For commuting Hamiltonian actions of a direct product$N\times H$ with split moment map, the exterior/Koszul data split, the extension curvature is zero, and direct reduction is compatible with reduction in two stages. At the simplicial level, the diagonal and iterated constructions are connected by the classical Alexander–Whitney and Eilenberg–Zilber chain-homotopy equivalences; they need not be literally the same chain complex.

The real Heisenberg group is the smallest hostile control. With

the central Maurer–Cartan form is$\theta_Z=dz-x\,dy$ and$d\theta_Z=-dx\wedge dy$. Any linear lifts of the two quotient basis vectors have bracket equal to the central basis vector. That nonzero curvature is the receipt missing from a naive split-stage model. This is a real Lie-group statement; no integral-lattice theorem is asserted.

The exact finite controls and the Heisenberg bracket check are in the independent audit certificate; the checker is published alongside it.

References for totalization and stages

• The Stacks Project, Double complexes, Tag 012X, and Čech cohomology of complexes, Tag 01FP: conventions and totalization for the two cochain directions. Double complexes; Čech complexes.
• Samuel Eilenberg and Joseph A. Zilber, Semi-Simplicial Complexes and Singular Homology, Annals of Mathematics 51 (1950), 499–513. doi:10.2307/1969364.
• Juan-Pablo Ortega, Optimal Reduction, 2002, arXiv:math/0206310. This is a broader classical comparator for reduction in stages, not the proof source for the direct-product calculation above. arXiv:math/0206310.

QA-BUCK-HUR-009–MOVING-017 · support, positivity, and lift boundary

Crossing budgets filter positive braid factorizations

Consider a positive factorization of the full-twist power $\Delta_n^{2k}$. Its factors may be half-twists about arcs between marked points or positive powers of Dehn twists about closed curves. Join the endpoints of every half-twist; the connected components $C_1,\ldots,C_r$ of this graph color the strands. If a Dehn-twist curve $c$ encloses $s_a(c)$ points of $C_a$, then every pair of distinct colors satisfies the exact crossing equation

Half-twists contribute zero between different components; a Dehn twist contributes two signed crossings for each enclosed cross-color pair. Thus a factorization made only of positive half-twists cannot have two endpoint components: its left side would vanish while the full twist has positive cross-component crossing. The endpoint transpositions therefore generate the full symmetric group. This explains why subgroup order cannot separate a positive central completion; it does not decide Hurwitz equivalence.

The pinned five-point boundary

In Buckman's five-point diagram the half-twist partition is $\{1,2,5\}\mid\{3\}\mid\{4\}$ and the three squared Dehn-twist supports are $\{1,2,3\},\{1,4,5\},\{3,4,5\}$ . Their cross-component contributions are respectively $(4,0,0),(0,4,0),(2,2,2)$ , exactly the required $(6,6,2)$. With this partition, target, proper-curve restriction, and squared-packet grammar fixed, three packets are minimal. Exhausting all 2,925 three-support multisets gives 55 solutions in three occupancy types: 1, 27, and 27.

The diagonal crossing equation restores the information omitted by those three off-diagonal coordinates:

For the three occupancy types it forces $h=-12,4,8$. Positivity eliminates the first type, and Buckman's four-half-twist inventory selects the second. The remaining eight-half-twist type belongs to a different factor-count chamber. After the two singleton strands are forgotten, it has the unique length-eight standard-positive shadow $x^4(yx^2y)^2=(xy)^6$ among all 256 words. That is a necessary three-strand shadow, not a five-strand lift.

Where the finite obstruction stops

In the canonical Birman–Ko–Lee section, a mod-three Burau vector computation rejects all 27 labelled eight-half-twist patterns: each singleton vector has 77 final images and is never fixed. Arbitrary winding changes the quantifier. The three atom types then collapse, support by support, to the same 40-element class, whose eighth product fills all 51,840 elements of $\operatorname{Sp}_4(\mathbb F_3)$. A Nielsen probe rejects one displayed finite-quotient lift but admits another lift for every support pattern. Therefore neither candidate rejection nor canonical-section rejection proves that the entire winding fibre is empty.

The crossing, diagonal, and positive-shadow checks were independently reproduced. The arbitrary-winding class closure and Nielsen full-fibre records remain at L3 because their archived replay scripts still require one missing helper; the stored certificates were inspected but not promoted. See the independent checker · canonical output · review certificate.

References for the positive-factorization boundary

• Richard E. Buckman, Positive Factorizations via Planar Mapping Classes and Braids, PhD dissertation, University of Massachusetts Amherst, 2023. Sections 3.1 and 3.3 fix the positive atom grammar; Figures 4.8–4.9 supply the five-point supports used above. doi:10.7275/36065157.
• Stepan Yu. Orevkov, On the Hurwitz action on quasipositive factorizations of 3-braids, Doklady Mathematics 91 (2015), 173–177. This supplies the stronger three-braid orbit context, not the mixed five-strand crossing equations. doi:10.1134/S1064562415020180 · arXiv:1409.4726.
• Norbert A'Campo, Tresses, monodromie et le groupe symplectique, Commentarii Mathematici Helvetici 54 (1979), 318–327; and Wade Bloomquist, Peter Patzt, and Nancy Scherich, Quotients of braid groups by their congruence subgroups, arXiv:2209.09889. These supply the ambient symplectic-image framework, not the support-tagged collapse calculation. A'Campo DOI · arXiv:2209.09889.
• Joan S. Birman, Ki Hyoung Ko, and Sang Jin Lee, A New Approach to the Word and Conjugacy Problems in the Braid Groups, Advances in Mathematics 139 (1998), 322–353. This is the source for the band-generator grammar used to define the canonical section. doi:10.1006/aima.1998.1761.

QA-HUR-FIB-T01–T08 · reviewed theorem core

Factor lifts interact through the ordered product

A Hurwitz move replaces adjacent factors $(a,b)$ by $(b,b^{-1}ab)$. It preserves their ordered product, but it does not make the choices of lifts independent. Suppose quotient factors $q_1,\ldots,q_m$ have chosen base lifts $g_i$, and the permitted corrections form affine subsets of an abelian kernel. With corrections written on the right, multiplying $g_i(1+\varepsilon x_i)$transports $x_i$ through the factors to its right. The attainable target defects therefore form one affine image, not a product of independent yes-or-no tests.

in a square-zero layer. Thus a target lifts exactly when its defect lies in the sum of the transported local direction spaces. The residual class in the cokernel is the obstruction. This formula uses right corrections; a left-correction convention gives the equivalent prefix-transported form. Stating the convention matters because the group need not commute.

The next congruence layer is ordered and quadratic

In the pinned mod-eight matrix model, corrections have the form $1+2x_i+4y_i$. Direct multiplication gives

Once the first-layer choices $x_i$are fixed, the remaining equation is affine in the$y_i$. Across the complete first-layer fibre it is quadratic, and the term $x_ix_j$ remembers order. For example, $e_{12}e_{21}=e_{11}$ while $e_{21}e_{12}=e_{22}$. An unordered collision count would therefore erase genuine lift information.

What is invariant under a Hurwitz path

If every local lift set is closed under the conjugations used by the Hurwitz action, a move bijects the complete lift fibres and preserves every total product. Consequently the attainable-product set—and, in a square-zero layer, its affine dimension, target membership, and cokernel class—is constant on the quotient Hurwitz orbit. The completeness quantifier is essential: rejecting one chosen section or one candidate lift says nothing about a different lift in the same fibre.

Appending the inverse target turns $g_1\cdots g_m=t$ into the product-one tuple $(g_1,\ldots,g_m,t^{-1})$. This is the relative Nielsen comparison used here. Classical Nielsen classes normally also impose generating and conjugacy-class conditions; those conditions are not silently added to this product-one slice.

The independent audit checks all 65,536 two-factor and 4,096 three-factor mod-eight matrix identities, the noncommutative order control, and 1,625 relative-Nielsen and split-central identities on the 125 length-three nonidentity tuples in $S_3$. Audit program · canonical output · review certificate.

This review covers the general affine, mod-eight, relative-Nielsen, and product-cokernel theorems. The later colored degree-four orbit counts remain at L3 because their archived generator still lacks one dependency; no count from that blocked family is upgraded here.

References for Hurwitz fibres

• Joel Brewster Lewis, A note on the Hurwitz action on reflection factorizations of Coxeter elements in complex reflection groups, The Electronic Journal of Combinatorics 27(2) (2020), article P2.54. The introduction states the Hurwitz action and its product-preservation property used here. doi:10.37236/9351.
• Michael D. Fried and Helmut Völklein, The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen 290(4) (1991), 771–800. This is a classical source for braid actions on product-one Nielsen classes; the affine lift and cokernel formulas above are self-contained FPRD deductions. doi:10.1007/BF01459271.
• Richard E. Buckman, Positive Factorizations via Planar Mapping Classes and Braids, PhD dissertation, University of Massachusetts Amherst, 2023. This supplies the planar-factorization setting, not the general extension theorems proved above. doi:10.7275/36065157.

FPRD-QA-MG-T01–T05 · local lifts, closed walks, and tours

Different graph questions require different quotients

Let a finite group $\Gamma$act on a finite graph represented by directed darts, with each undirected edge contributing two opposite darts. Keep vertex orbits and dart orbits as a multigraph; collapsing parallel edge orbits is a further quotient that may discard closed-walk data. For a dart $a:u\to v$, the lifts of its orbit from a fixed lift of $[u]$ are indexed by the coset set $\Gamma_u/\Gamma_a$. Their number is $[\Gamma_u:\Gamma_a]$, and local lifting is unique exactly when the two stabilizers agree. No quotient-group structure or torsor is asserted when $\Gamma_a$ is not normal.

Three levels now separate cleanly. The simple vertex-orbit graph preserves orbit reachability. The equitable matrix $B_{ij}=|N(x)\cap[v_j]|$, $x\in[v_i]$ preserves class-level walk counts, since $(B^k)_{ij}$ counts length-$k$ walks from a fixed vertex of class $i$ into class $j$. The dart-orbit multigraph together with a local coset label preserves the actual finite lift. None of these statements alone preserves a Hamiltonian cycle.

Voltage records how a closed walk sews

If the action is free on vertices and has no edge inversion, the orbit projection is a regular cover. Choosing one representative per vertex fibre assigns each quotient dart a voltage. The ordered voltage product of a closed walk is its monodromy: the lifted walk closes on a chosen sheet exactly when that sheet is fixed. For an $m$-sheet cover, the inverse image of a base Hamiltonian cycle splits according to the cycles of its monodromy permutation. It is one Hamiltonian cycle upstairs exactly when that permutation is an $m$-cycle. In a regular cover this requires the cycle voltage to generate the deck group; a noncyclic deck group can never satisfy this particular one-base-cycle lift criterion.

Conversely, an arbitrary Hamiltonian cycle upstairs usually projects to a closed walk rather than a simple base cycle. It must visit each quotient vertex exactly as many times as the size of that vertex fibre. This occupancy condition is necessary, not sufficient.

Small hostile examples

The half-turn quotient of $C_4$has two parallel edge orbits with voltages zero and one. Using the same orbit out and back lifts to two closed two-step walks; using the two different orbits sews into the four-cycle. Collapsing both edges to a simple $K_2$erases that distinction. On the $5\times5$ knight graph, the fixed center has four outgoing dart orbits and each has two lifts, so choosing a lift needs one bit for each selected orbit—not two bits. The independent audit also recovers the 4-by-4, 5-by-5, and 6-by-6 orbit counts, class-level walks through horizon four, and 6-by-6 quotient Hamiltonian cycles of both voltages: voltage zero splits into two 18-cycles, while voltage one lifts to a 36-vertex closed tour.

Independent checker · canonical output · review certificate.

References for graph lifting

• Jonathan L. Gross and Thomas W. Tucker, Generating All Graph Coverings by Permutation Voltage Assignments, Discrete Mathematics 18(3) (1977), 273–283. This is the primary source for the permutation-voltage representation of finite graph covers; the local coset and occupancy presentation above is self-contained. doi:10.1016/0012-365X(77)90131-5.
• Aleksander Malnič, Roman Nedela, and Martin Škoviera, Lifting Graph Automorphisms by Voltage Assignments, European Journal of Combinatorics 21(7) (2000), 927–947. Used as established voltage-lifting context, not as the source of the FPRD finite board calculations. doi:10.1006/eujc.2000.0390.
• Allen J. Schwenk, Which Rectangular Chessboards Have a Knight's Tour?, Mathematics Magazine 64(5) (1991), 325–332. Schwenk classifies existence of closed rectangular knight tours; the half-turn quotient and voltage examples above address a different lifting interface. doi:10.1080/0025570X.1991.11977627.

QA-GR-T02 · exact synchronization obstruction

Anonymous overlap counts need not share one naming

The deck of a graph is the multiset of its vertex-deleted cards. A type-slot synchronization asks a deliberately weaker question: assign an anonymous double-deletion isomorphism type to every pair of card rows so that each row sees exactly the multiset of types obtained by deleting one more vertex from that card. Passing every row test does not yet identify which vertex in one card is the same missing vertex seen from another card.

For $G=2P_3$, four rows come from deleting a leaf and two from deleting a path center. The 15 pair slots admit exactly 18 synchronizations. On the four leaf rows, one part of a synchronization determines a perfect matching $P$ from the leaf–leaf slot types; the center–leaf slots determine a second perfect matching $Q$. Swapping the two center rows and permuting the four leaf rows gives the full $S_2\times S_4$ row-profile symmetry. It splits the 18 states into two orbits:

Every row has four admissible vertex namings, hence every synchronization has $4^6=4096$ candidate systems of row names. All 4,096 are globally coherent in each of the six $P=Q$ synchronizations; none is coherent in any of the twelve $P\ne Q$ synchronizations. This is a counterexample to the anonymous type-slot receipt, not a counterexample to graph reconstruction: the genuine deck still comes from $2P_3$.

The matching defect

The three perfect matchings on four points can be identified with the three nonzero vectors of $\mathbb F_2^2$. Define $d(P,Q)=v(P)+v(Q)$. Then $d=0$ exactly when $P=Q$, so vanishing is exactly the sewing criterion. Permuting the leaves permutes the three nonzero vectors through $\mathrm{GL}_2(\mathbb F_2)\cong S_3$; therefore zero versus nonzero is independent of the chosen vector labels. The 18 states distribute as six with zero defect and four for each nonzero defect.

The new dependency-free audit reconstructs the 18 synchronizations, both symmetry orbits, all 73,728 candidate row namings, and all 864 actions of the full row-profile symmetry group. It also confirms on all 52 unlabeled graphs through order five that vertex-deletion-type classes equal marked-vertex automorphism orbits, explaining why the proposed one-card marking adds no information to this smallest obstruction. Audit program · canonical output · review certificate.

References for graph decks

• J. A. Bondy and R. L. Hemminger, Graph reconstruction—a survey, Journal of Graph Theory 1(3) (1977), 227–268. This supplies the classical deck and reconstruction-conjecture setting; the type-slot quotient is an FPRD-defined auxiliary object. doi:10.1002/jgt.3190010306.
• Brendan D. McKay, Reconstruction of small graphs and digraphs, Australasian Journal of Combinatorics 83(3) (2022), 448–457, arXiv:2102.01942. McKay proves reconstruction for graphs through 13 vertices by a different exhaustive method; that result is context, not evidence for the synchronization theorem. arXiv:2102.01942.
• Carla Groenland, Tom Johnston, Alex Scott, and Jane Tan, Reconstruction from smaller cards, Israel Journal of Mathematics 273(2) (2025), 823–860. This supplies the modern smaller-card comparison; it does not assert the type-slot obstruction. doi:10.1007/s11856-025-2858-3.

FPRD-QA-COL-D01–FAIL02 · finite traces, completions, and obstructions

Parity words are exact charts, but their completions need not be integers

Use the accelerated Collatz map $T(n)=n/2$for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. For a chronological parity word $w\in\{0,1\}^k$, let $s(w)$ count its ones. A uniquely determined integer $a_w$ gives

$$T^k(n)=\frac{3^{s(w)}n+a_w}{2^k},\qquad n\equiv-3^{-s(w)}a_w\pmod{2^k}.$$

Terras proves the underlying parity-residue correspondence. The FPRD guard atlas fixes a chronology and tracks how two blocks sew: for $w=uv$, $a_{uv}=3^{s(v)}a_u+2^{|u|}a_v$. It also gives the sharp response cost: knowing $n\bmod 2^{h+k}$ determines $T^k(n)\bmod 2^h$, while one fewer bit does not. The all-even inputs $2^{h+k-1}$ and $2^{h+k}$ are the hostile pair.

Integral, positive, and 2-adic lifts are different claims

A periodic realization of $w$ must solve $(2^k-3^{s(w)})n=a_w$. Thus an integer lift exists exactly when the coefficient divides $a_w$; positivity needs additional sign conditions. The coefficient is odd and nonzero for every nonempty word, so it is invertible in $\mathbb Z_2$ and every word has one 2-adic fixed lift. For $w=001$, that lift is $4/5$, not an integer. The coherent residues $2^k-1$ for $1^k$ likewise converge to $-1\in\mathbb Z_2$, outside the nonnegative integers. A coherent residue tower represents a nonnegative integer exactly when its least residues are bounded, equivalently eventually constant.

The selector theorem is finite-depth: choose one outgoing edge at each vertex of the depth-$h$ binary de Bruijn chamber graph. Every such functional graph has a unique equivariant 2-adic section. An integral section also requires integral values on every directed cycle and integral inverse-branch values on the trees feeding those cycles. This is not a classification of coherent selectors at unbounded depth.

The independent audit checks 131,070 parity words, 98,304 block seams, 98,305 adjacent swaps, all 8,800 primitive parity necklaces through period sixteen, and all 65,812 selectors through depth four. Its positive-cycle control means precisely that the induced graph on ${1,\ldots,100000}$ has only the cycle $1\leftrightarrow2$; it does not exclude larger cycles. Independent checker · canonical output · review certificate.

References for Collatz guard chambers

• Riho Terras, A stopping time problem on the positive integers, Acta Arithmetica 30 (1976), 241–252. Theorems 1.1–1.2 supply the affine parity-prefix and residue-class foundations. doi:10.4064/aa-30-3-241-252.
• Corrado Böhm and Giovanna Sontacchi, On the existence of cycles of given length in integer sequences like x(n+1)=x(n)/2 if x(n) is even, and x(n+1)=3x(n)+1 otherwise, Rendiconti Lincei 64 (1978), 260–264. This is a classical source for the periodic-word divisibility calculation. primary scan · EuDML record.
• Jeffrey C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arithmetica 56 (1990), 33–53. This supplies the rational-cycle setting used to distinguish integral from rational lifts. doi:10.4064/aa-56-1-33-53.
• Daniel J. Bernstein and Jeffrey C. Lagarias, The 3x+1 Conjugacy Map, Canadian Journal of Mathematics 48(6) (1996), 1154–1169. This is the primary 2-adic conjugacy reference; the finite selector census above is an FPRD computation. doi:10.4153/CJM-1996-060-x.

FPRD-QA-RA-D01–FAIL02 · adjacent numeral-length charts

One bit is the complete one-step length-seam receipt

Fix a base $b\ge2$. In a canonical $m$-digit input, pair digits symmetrically. Their admissible pair sums form a mirror profile $q\in Q_{b,m}$. The weighted symmetric sum $V_{b,m}(q)$ is exactly the first reverse-and-add output, and it is injective on each fixed- length profile chart. The global atlas is the disjoint union of these charts before equal output values are identified.

An $m$-digit source has an output of length $m$ or $m+1$. Therefore an output of length $M$ can receive at most one profile from chart $M$ and at most one from chart $M-1$. Every global profile fibre has size at most two, and the overlap graph is a matching. The bit $\varepsilon=M-m\in\{0,1\}$recovers the chart; fixed-chart injectivity recovers the profile.

The bit is necessary in every base. For $b\ge3$, the two- and three-digit sources $(b-1)2_b$ and $110_b$ both map to $(b+1)^2$. At the hostile base-two boundary, $110_2$ and $1000_2$ both map to $1001_2$.

Let $f_m(y)$ be the number of raw $m$-digit sources mapping to $y$. The complete raw fibre is the disjoint union of the two possible strata, so its cardinality is $f_{M-1}(y)+f_M(y)$. Crossing the seam adds at most one bit beyond the larger within-chart source receipt. This is an information budget for the declared quotient, not a lower bound on arbitrary algorithms.

A separate implementation exhausts 1,208,130 admissible profiles for bases two through twelve and lengths one through eight, finding 310 two-chart overlap outputs and reproducing a sharp collision in every base. The proof itself is all-base and uses only the output-length bound and fixed-chart injectivity. The matching has no one-step seam monodromy; any unbounded behavior must occur in repeated-response towers, causal implementations, or raw-source fibres. Independent checker · canonical output.

Eight later Stage 6 archival records remain at L1 because their proof payloads are unavailable. They are not evidence for this Stage 5 theorem family and were not promoted by this review.

Provenance for the cross-length atlas

The fixed-length mirror-profile construction and all three cross-length theorems are FPRD deductions from elementary base-$b$ digit arithmetic. No external theorem is used in the proof, and no literature-priority claim is made. The earlier carry-structure page gives the underlying folded-digit definitions and keeps its classical reverse-and-add comparisons separate.

Result families

Evidence and maturity policy

What “complete import” means here

Superseded revisions and provenance aliases were reconciled to one public record rather than duplicated. Exact claims with a written proof and successful hostile replay enter at L4 or L5. Eight reverse-and-add records whose proof payloads never completed durable transfer enter at L1 only. Several later Hurwitz computations enter at L3 because the stored certificates and audits exist, but the imported replay scripts depend on a missing helper module. Their limitation is visible in each record.

A localized source-audit dispute in a recent paper remains outside the public mathematical claim set. The usable cross-axis theorem is stated here with an internally consistent sign convention, without publishing an erratum claim about another author's work. Earlier Buckman theorem QA-BUCK-HUR-008 is an alias of the already published QA-BK-T05, so it is preserved as provenance rather than duplicated.