Context-relative crossing compilation and explicit integer pieces

Context-relative crossing compilation and explicit integer pieces

The target-summary construction optimizes a finite response summary and a specified weighted dynamic programme. It does not itself implement a component of Anthropic’s crossing recursion. Here we supply two checked components: an exact context-quotient network compiler, including its effect on the actual eight-square gadget, and explicit integer factorizations completing the single-bit rank observation in Appendix H of the cited preprint [3]. The quotient principle, Bell basis and single-bit identity are established mathematics. The contribution is their precise executable interface, coefficient bounds, proof of admissible contexts and demonstrated component costs. It is not a new crossing identity or an improvement of the paper’s strongest general exponent.

The full context contract and exact scalar recovery

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Consider a closed integer tensor network on a graph G = (V, E), with one string ze ∈Fλe
2
per edge.
The tensor at a vertex reads all its incident edge strings. On each edge supply a full-row-rank binary
map He : Fλe
2 →Fre
2 and supply the vertex tensor in the form

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Wv((ze)e∋v) = W v((Heze)e∋v).

This is a representation promise, not a claim inferred from a few tensor queries. The admitted surrounding contexts are all integer or rational vertex functions of these quotient values, with arbitrary dependencies among the incident quotient coordinates.

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Theorem C.1 (Context-certified wire quotient). For the declared context class, put D = ∑︁
e(λe −re).
Then
Z =
∑︂
∏︂
v
Wv(z) = 2D
∑︂
∏︂
W v(y) = 2DZ.

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z
e Fre
2
v
y∈∏︁

The replacement is valid uniformly over every admitted surrounding context, including signed ones. It requires no enumeration of the discarded fibres. The factor must be restored for the full value; it is not a discarded normalization convention.

Proof. Each is surjective and has 2 preimages of each output. For each full quotient assignment , every summand in its fibre has the same vertex product. Edge fibres are independent Cartesian factors, so their common cardinality is 2. Sum over

A binary right inverse is found by Gaussian elimination in polynomial work in the supplied matrices. The code checks rank and constructs lifted basis vectors; it does not enumerate 2 candidates to find a lift. If dense original tables, rather than quotient tables or a proved factorized evaluator, are supplied, establishing the representation can require scanning those tables. That cost cannot be omitted. Computing 2 writes up to additional output bits; denominators and output lengths are charged. The context promise is necessary: on one two-bit edge let retain only the low bit and let both endpoint tensors be [ = 0]. The original contraction is one, whereas replacing each by its values at representatives 0 1 and restoring the two-element fibre factor gives two. The tensors do not factor through; representative agreement alone was insufficient. This theorem concerns a computational representation of a nominated boundary response. It does not remove an accessible physical archive or shrink a causal quantum memory. Exact symmetry-aware tensor representations are well established[15]; their resource minima can depend on topology and admitted symmetry[14]. No unrestricted minimum-bond-dimension assertion follows here.

Target projection can spoil an inherited decomposition

In the paper’s Eq. (27), for three-bit strings, write

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σ(p, q) = (−1)
∑︁
i≤j piqj+q1p2p3+p1q2p3+p1p2q3.

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Its Eq. (30) defines

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R = [x ⊕y ⊕u ⊕v = 0]σ(x, y)σ(v, u),
C = [x ⊕y ⊕u ⊕v = 0]σ(x, u)σ(v, y),

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with R + C = 2[x = v][y = u]. Row indices are (x, y) and column indices (u, v); the alternative split
uses (x, u) and (y, v). The ranks of R in those two directions are (8, 64), and those of C are (64, 8).
Now admit every surrounding context that depends on each three-bit boundary only through its
low bit, H(x) = x mod 2. Let  ̃︁R and  ̃︁C be the sums of the original pieces over each four-boundary
fibre. In row order 00, 01, 10, 11, exact enumeration gives

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⎛
2
0
0
1
0
−1
2
0
0
2
−1
0
1
0
0
2
⎞
⎛
2
0
0
−1
0
1
2
0
0
2
1
0
−1
0
0
2
⎞
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎠,
 ̃︁C = 8
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎠.
 ̃︁R = 8

Both pieces have rank four in both directions. Their sum is 32, where single bits. The factor 32 is 2 from the paper’s identity times 4 original crossed-string lifts. It is not 4, because the crossing constrains two pairs. Put1, and 1) = 0. The identity already noted in the paper’s Appendix H is

Its integer factors are explicit: is one product at the column split and a four-term coordinate-selector sum at the row split; is one product at the row split and a four-term coordinate-selector sum at the column split. Thus recomposing the same fibre-summed target as 32 gives directional ranks (4 1) and (1 4). Each piece’s directional bit sum is two, compared with four for either projected octonion piece. There are still two terms. The new scalar 32 costs one more magnitude bit than the largest projected entry 16; this growth is accounted for rather than interpreted as a probability.

The equality is proved on all 16 basis contexts of the four retained bits, which by linearity is every rational context in the declared class. It is not just agreement for one selected contraction. This supplies a strict improvement over projecting the inherited pieces separately and using their full-rank interfaces. It is not a gain over a comparator that already recomposes the quotient crossing directly. The example explains why target projection should be followed by decomposition selection, not by assuming that an inherited low-rank advantage automatically survives.

Polynomially evaluable integer realization of Appendix H's rank hint

The paper’s Appendix H points out the rank saving from the single-bit identity. To use a rank statement in Definitions 1 and 7, integer factors and their evaluation costs are needed. The following explicit construction supplies them without rational Gaussian elimination. For integer1 and strings of bits define by

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min(t,k)
∑︂
t∏︂
t
∑︂
(︄
t
z
)︄
i=0
BiT i,
F(t, k) =
3z,
j=1
(Ij −TAj) =
z=0

with) = 0 for 0. Use the four integer Bell vectors

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v0 = (1, 0, 0, 1),
v1 = (1, 0, 0, −1),
v2 = (0, 1, 1, 0),
v3 = (0, 1, −1, 0).

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They are orthogonal and all have squared norm two. A Bell pattern d of length t gives vd = ⨂︁
j vdj, an
integer vector with entries 0, ±1 and squared norm 2t.

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Theorem C.2 (Explicit integer crossing pieces). The integer pieces A′
i = 2tBi, 0 ≤i ≤t, sum to 2tSt,
where St = [x = v][y = u] on t-bit strings. Their exact row and column ranks are

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rankrow Bi = F(t, t −i),
rankcol Bi = F(t, i).

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In each direction A′
i has an explicit integer factorization through the displayed number of terms, with
factors of magnitude at most 22t. An individual factor, the piece value and its index are evaluable in
polynomial bit work in t. Thus these are pieces in the paper’s Definition 7 with

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ρ = 1
4 log2 12 ≃0.8962406252,
crow
i
+ ccol
i
≤t log2 12 + 2.

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Proof. At T = 1 each factor is the single-bit crossing, proving the sum identity. At the row split, I is
the identity matrix and A = v3v⊤
3 has eigenvalue two on v3, zero on the other three Bell vectors. For a
pattern with z occurrences of 3, the eigenvalue of Bi is (−2)i(︁z
i
)︁. It is nonzero exactly when z ≥i. The
number of such patterns is F(t, t −i).
At the column split, I = v0v⊤
0 and A = J ⊗J, where J =
(︁0 1
−1 0
)︁. On (v0, v1, v2, v3) the polynomial
I −TA has eigenvalues (2 −T, T, T, −T). Let z count the nonspecial entries dj ̸= 0, and let n3 count
entries equal to 3. The coefficient eigenvalue is

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cd = (−1)n3+i−z2t−i
(︄
t −z
i −z
)︄
(z ≤i),

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and zero otherwise. This gives rank F(t, i).
Since ∑︁
d vdv⊤
d = 2t Id, at either split

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2tBi =
∑︂
d:cd̸=0
(cdvd)v⊤
d .

Both factors are integer; zero-padding gives an index oflogbits. The displayed coefficients are bounded by 2, and a Bell vector entry is a product of entries in. Patterns with at most nonspecial entries are indexed lexicographically by suffix counts), so unranking one pattern uses polynomially many integer operations on)-bit counts. Binomial coefficients, eigenvalues and the polynomial coefficient defining are likewise polynomially evaluable. No list of all 4 patterns is constructed by a factor query. Finally, so the product of the two ranks is at most 12. Taking two ceilings proves the directional bit bound. The piece count is + 1, the common scalar is 2, and coefficient sizes and evaluators meet Definition 7 after increasing its absolute constant

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The rank calculation is already motivated in Appendix H; the result above is its executable integer-
factor completion, not a new rank discovery. Its ρ is better than the main text’s simple 11/12 estimate
but worse than Appendix B’s 1/
√

2. Consequently it supplies a transparent alternative component, with polynomial-time entry evaluators in place of that construction’s stated 2 bound, not a better general ETH exponent. As always, actual evaluation costs and term enumeration must be compared jointly; a cheaper evaluator alone does not settle the tradeoff. The code implements these pieces, not the entire recursive ETH algorithm.

Rational adapter, network costs and fair comparisons

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For supplied rational quotient tables choose the positive least common denominator qv at each vertex,
set W int
v
= qvW v, and let Q = ∏︁
v qv. Then

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Z = 2DZint/Q.

The same is used once; no scalar or discarded-bit factor is restored twice. Signed tables remain algebraic weights rather than probabilities. Their exact likelihood interpretation requires a separately admitted nonnegative normalized source and readout model. The integer network fits Anthropic’s Definition 2 using equality edge factors, with

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e∋v
re},
B = max{1, max
v,y bitlength |W int
v (y)|}.
U = 2
∑︂
e
re,
P = max{1, max
v
∑︂

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Thus |W int
v | < 2B. Exact table/evaluator cost is τ. For an admitted fixed K satisfying |V | ≤g = 3K8,
pad with isolated unit vertices to g. Conditional on an achievable pair (K, e), Anthropic’s Lemma 1
gives the following bound in its stated word-RAM operation model:

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C1 2e(U+gP/K)+2gP/K(P + B)C1τ,

plus denominator clearing, output recovery and padding-description work. If, this padding argument does not apply at that; one must choose a suitable larger achievable pair and charge its changed constants, or use a different reduction. No uniform bound is obtained by treating those new constants as fixed while the network size grows. This is an explicit direct integer-network bridge. It avoids modular reconstruction but does not prove a uniformly better complexity bound than the rational OpenAI adapter. When grows, Appendix A(b) has the factor [. Here is the common clique-edge string width and the table-entry magnitude parameter in that external theorem, not an individual original edge width. Precision dependence cannot then be described as a fixed polynomial independent of; its operation model must also be distinguished from the bit-complexity statement of the OpenAI adapter. The implemented ordinary enumeration backend evaluates the quotient network exactly and restores its scalar. For a complete four-vertex network with three-bit edge strings, one retained parity per

edge and positive supplied vertex tables, both independent calculations give = 45 621 248. Full enumeration uses 262 144 assignments and 1 048 576 vertex queries; quotient enumeration uses 64 assignments and 256 vertex queries, plus polynomial representation checks and a 12-bit recovery shift. The full comparator additionally builds 48 local edge-readout entries. This is an actual executed finite comparison. At six vertices, quotient enumeration uses 32 768 assignments and 196 608 vertex queries, returning 215 011 853 527 416 832. The 2 full assignment count is theoretical and was not executed. A fair specialized comparator can already apply exact bond quotienting or exploit another low-width structure. It has the same compressed task; no advantage over that comparator is claimed. General contraction-tree costs and rank decompositions are also different computation models [1]. The restricted reciprocal improvement here is relative to the explicitly inherited component pipeline, not to all algorithms or to the paper’s worst-case theorem.

Coverage, verification and retained limits

The implementation checks all 4,096 octonion identities, all 16 quotient basis contexts and 40 additional signed contexts. It verifies the two projected rank matrices, explicit integer Bell factors in both directions, exact ranks through three retained bits, 1,306 pattern-unranking cases, 2,145 right-inverse queries, 30 signed rational networks and independent full small-network contractions. Tests and exact parameters are supplied. Appendix D’s sparse checks were examined but not modified into a claimed faster algorithm. A target that annihilates some unwanted assignments can restrict a coverage obligation only after that vanishing is proved and its representation cost charged. A target with full support need not remove any obligation. Signed cancellation of the final answer does not justify dropping assignments from the paper’s nonnegative stopping certificate. No general reduction in check count or load bound is proved. Likewise, the graded operators in Appendix B supply an algebraic factorization; they are not a derived physical interaction, detector, Born selector or consciousness mechanism. This component extension preserves the source–readout distinction between information required for a target and a uniquely identified physical source. It changes no delayed-trine memory or complete-history resource bound.