Prove exact Cayley path denominators (no new κ)

research/cayley-path-exact-denominator/PROOF.md

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+# Exact denominators of skew-path Cayley blocks
+
+Produced by Nick Petalas. Apache-2.0.
+
+## Claim and source boundary
+
+This is a scoped written algebraic proof with finite exact corroboration.
+It sharpens the arithmetic separation step in Ali Romman's
+[PR #5339](https://github.com/CrocSwap/integer-mult-bounds/pull/5339), source
+head `26fe40cbca6972a0e392e9248a9cb8a84368530b`. The predecessor supplies
+the complete rational register grammar and the funded error implication in
+its proof, sections 1 and 2. Those results are premises here. Its scientific
+replay is not the deliverable and was not rerun.
+
+The new result is the exact common denominator of each labelled path matrix,
+including its necessary cancellation, and the resulting smaller sufficient
+precision. It does not settle the predecessor's native-compatible execution
+question or discharge a selected multiplication-framework assumption.
+
+## 1. One connected block
+
+Let K be the skew adjacency matrix of an n-vertex path. Its upper
+off-diagonal entries are +1 and its lower ones are -1. Put
+
+    O = (b I - a K)^(-1) (b I + a K),
+
+where a,b are positive coprime integers. Let den(O) be the least positive
+integer h for which hO is an integer matrix. Define
+
+    D_0 = 1,  D_1 = b,
+    D_n = b D_(n-1) + a^2 D_(n-2).
+
+Then, for every n >= 1,
+
+    den(O) = D_n / gcd(D_n, 2b).                         (1)
+
+The same formula holds for O inverse. Signs on individual path edges do not
+change it. One optional closed form is
+
+    D_n = sum_(0 <= k <= floor(n/2)) binom(n-k,k) a^(2k) b^(n-2k).
+
+These continuant and path-matching identities are classical. The exact
+cancellation and its application to this witness interface are the step
+established here; global literature priority is unclaimed.
+
+### Proof of the denominator formula
+
+Write M=bI-aK. Expansion along the last row gives det(M)=D_n. The recurrence
+and positive a,b show D_n>0. Modulo a it gives D_n congruent to b^n.
+Consequently gcd(D_n,a)=1.
+
+Since bI+aK=2bI-M,
+
+    O = 2b adj(M)/D_n - I.
+
+Every numerator is divisible by h=gcd(D_n,2b). Hence multiplying O by
+D_n/h gives an integer matrix. This proves that den(O) divides D_n/h.
+
+For n>=2, the far-corner cofactor has absolute value a^(n-1). This follows
+by successively expanding the minor obtained by deleting the last row and
+first column. It is triangular along the unique path. Thus the off-diagonal
+corner entry of O has absolute value
+
+    2b a^(n-1) / D_n.
+
+Its reduced denominator is exactly D_n/gcd(D_n,2b), because a and D_n are
+coprime. Any common denominator must clear that one entry. This proves (1).
+For n=1, K=0 and O=I; (1) gives b/gcd(b,2b)=1 as required.
+
+Real skew symmetry gives O inverse = O transpose, so the denominator is
+unchanged. For arbitrary signed unit path edges there is a diagonal matrix
+S with entries in {+1,-1} such that K_signed=S K S. Conjugation by S
+changes only entry signs, preserving the least common denominator. This
+completes the proof.
+
+## 2. Label runs and primitive scaling
+
+The source uses t=A/B in reduced form and
+
+    O_label(t) = (I-t K_label/2)^(-1) (I+t K_label/2).
+
+Set g=gcd(A,2B), a=A/g and b=2B/g. Apply (1) to these coprime parameters.
+Applying it directly to A,2B without cancelling g can give a false answer.
+For example t=2/9 has a=1,b=9. A two-vertex block has denominator 41,
+whereas the unscaled expression incorrectly gives 82.
+
+Edges not carrying the selected label are absent in K_label. Its connected
+components are the consecutive runs of selected edges, together with
+isolated vertices. A run of ell edges has n=ell+1 vertices. The matrix and
+its Cayley transform are block diagonal in these components. Therefore
+
+    Q_label = lcm_(components c) [D_(n_c)/gcd(D_(n_c),2b)]       (2)
+
+is the exact denominator of the entire label matrix and its inverse.
+An isolated vertex contributes 1. Repeated component sizes contribute once
+to the least common multiple. Distinct component denominators need not
+divide one another, so replacing the lcm by a maximum is unsound.
+
+At t=1/4, runs of two and one edges have denominators 33 and 65. Their
+joint denominator is 2145. Two separate two-edge runs have denominator 33,
+rather than 33 squared. These are denominator statements, not physical
+component-sharing or free native-call claims.
+
+## 3. The complete register lattice and separation
+
+Retain the source grammar over Q. Every role is returned. Assignments read
+old source values. Instructions are rational linear assignments, applications
+of named O_label or its inverse, and
+
+    dst += c O_label^(+/-1) src,
+
+including src=dst. Across both programs, let C be the product of the positive
+reduced denominators of every scalar-coefficient occurrence, exactly as in
+the predecessor. Let the q named-call occurrences have labels a_1,...,a_q.
+Define
+
+    H = C product_(j=1,...,q) Q_(a_j).                      (3)
+
+For every integral input bank, every returned difference coordinate has
+denominator dividing H. Equivalently, the full rational linear bank map
+has entries in H^(-1) Z. No role is assumed to be clean or discarded.
+For inputs with common denominator U, replace H by UH.
+
+To prove this, maintain a common denominator on each program side. At a
+linear instruction multiply by the product of that instruction's scalar
+denominators. At an application multiply by Q_label. At a query multiply
+by Q_label and the coefficient denominator. The old destination and source,
+including an alias, use the denominator before this multiplication. Unchanged
+registers can be represented with the enlarged denominator too. Induction
+proves the claim for each side. The product of the two side denominators
+clears their difference and is (3).
+
+In particular the predecessor's selected integral basis input satisfies
+
+    f != 0  implies  |f| >= eta_new = 1/H.                  (4)
+
+The integer H is a valid common denominator, not necessarily the least
+denominator of a complete program. Compositions, cancellations and unused
+directions can reduce it. Matrix-level sharpness in (1) does not imply
+that (3), (4), or the final numerical precision is sharp.
+
+## 4. Smaller sufficient grid with the same error contract
+
+Use the predecessor's exact bank gains beta_i>=1 and entry-error gains
+gamma_i. Write G=product beta_i, Gamma=max gamma_i, and m for the combined
+instruction count. An empty product has G=1, and Gamma=0 when there are no
+instructions. The cited error implication is
+
+    |fhat-f| <= E 2^(-p),   E=m G (Gamma G+1),
+
+for entrywise truncation of forward tables toward zero, transposed tables
+for inverse calls, and one toward-zero data truncation per new coordinate.
+All old-value and scalar-gain hypotheses remain in force. The only change
+required in that implication is the sufficient choice
+
+    2^p >= 4 max(1,E) H.                                  (5)
+
+Equations (4) and (5) give error at most eta_new/4. Threshold eta_new/2
+therefore distinguishes a zero selected endpoint from a nonzero one.
+A zero at this input/output/parameter tuple is not a universal identity test.
+
+For the original reduced A/B, each exact Q_label divides
+det(2B I-A K_label). The predecessor bounds that determinant by (3B)^d.
+Thus (3) never exceeds its coarse C(3B)^(dq). No claim asserts that H
+divides that coarse numeric upper bound.
+
+### A strict, fully specified fixture
+
+Take t=1/4 and a path of nine vertices. Its eight edge labels, in spatial
+order, are a,b,a,b,a,b,a,b. Each label has four two-vertex components, so
+Q_a=Q_b=65. On one role apply b,a,b,a,b,a,b,a in chronological order.
+Compare with the empty program. On input e_8 the coordinate-zero difference
+is (16/65)^8. Reaching vertex zero in eight calls requires taking the
+leftward edge at every call, so there is exactly one contributing route.
+
+Here C=1, q=m=8, G=9^8, Gamma=9, and
+
+    E=8*9^8*(9*9^8+1)=133417453941706320.
+
+The old H bound is 12^72 and the new one is 65^8. The least integer p
+satisfying each version of the same sufficient inequality is respectively
+318 and 108. The reduction is exactly 210 fractional bits. The verifier
+executes both dyadic grids and checks every returned coordinate against the
+cited error budget. This is an explicit local guard improvement, not a
+least-precision result or a performance measurement.
+
+## 5. Scope and credit
+
+The written proof covers every finite signed path and the stated rational
+register grammar. Finite corroboration covers only the enumerated domains.
+No source scientific census is rerun, and no predecessor theorem is audited.
+No native Clifford binding, production precision policy, complete physical
+cost, stock, tape, recursive compiler admission, new kappa, selected-frontier
+change or complete multiplication theorem is established. Independent human
+and formal proof-kernel review are unclaimed.
+
+Credit Ali Romman #5339 for the separation/error consumer and #5315 for the
+rational orthogonal witness, Krzysztof Wojt #5262 for the complete inverse
+compiler and shortest-word theorem, zxv #5235 for the retained certificate
+lineage, and all predecessor notices for dirty-echo/tensor/phase and
+rational-series work. Cayley transforms, continuants, adjugates and integer
+lattice separation are classical. The exact-denominator specialization,
+precision refinement, proof and verifier were produced by Nick Petalas.

research/cayley-path-exact-denominator/README.md

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+# Exact Cayley path denominators and a smaller sufficient grid
+
+This scoped algebraic proof determines the exact least common denominator of
+every signed path Cayley block. It improves one arithmetic step in Ali
+Romman's [PR #5339](https://github.com/CrocSwap/integer-mult-bounds/pull/5339).
+For coprime positive a,b and an n-vertex connected path,
+
+```text
+D_0 = 1, D_1 = b, D_n = b D_(n-1) + a^2 D_(n-2)
+den((bI-aK)^(-1)(bI+aK)) = D_n/gcd(D_n,2b).
+```
+
+The inverse has the same denominator. Disconnected label runs combine by
+least common multiple. The far-corner matrix entry proves minimality.
+[PROOF.md](PROOF.md) proves the formula for every finite signed path and
+substitutes it into the predecessor's complete rational-register separation
+and error implication. No predecessor scientific replay or audit is performed.
+
+In a specified nine-dimensional alternating-label example with eight calls,
+the same sufficient precision inequality drops from **318 to 108 fractional
+bits**, a reduction of 210. This is a guard improvement, not least precision,
+a runtime measurement, native admission or a new multiplication exponent.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and only its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cayley-path-exact-denominator
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cayley-path-exact-denominator/verify.py
+```
+
+A different CPython 3.12 path can replace this machine's interpreter path.
+The registered commands run the warning-strict verifier in normal and
+optimized modes. Expected scientific marker:
+`PASS_CAYLEY_PATH_EXACT_DENOMINATOR`.
+
+The verifier compares the formula with dense rational inversion, enumerates
+signed supports, checks complete dirty-input bank maps, executes both dyadic
+grids in the strict example, and exercises rejection controls. It emits a
+generated stdout receipt bound to all seven package file hashes. It writes
+no files. [VALIDATION.md](VALIDATION.md) records domains and execution limits.
+
+## Limitations
+
+The exact per-matrix denominator need not be the least common denominator
+of a composed program. The new sufficient grid need not be optimal. The
+funded detector's error theorem is credited to #5339. Its general arithmetic
+kernel, native compatibility, physical costs and upstream coefficient
+producer are not audited here. The written theorem is unbounded in path
+size; the finite checker limits matrices to dimension 32 and tests the
+explicit smaller domains in VALIDATION.md.
+
+No selected caller, production precision policy, physical supplier, new
+kappa, selected-frontier change or complete multiplication theorem is
+claimed. Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+- Can program-level cancellation reduce the lattice H below the product of
+  the exact per-call denominators? Start with one literal inverse-pair word
+  and retain all scalar and dirty-return effects. Matrix-level minimality
+  does not settle this question.
+- Can this arithmetic refinement lower a bound for a native-compatible
+  witness with a complete paid error contract? First pin its actual operator
+  relations and caller. That binding and its physical cost remain open.
+
+## Related open work
+
+The live publication search found no open PR studying these skew-path Cayley
+denominators or this orthogonal-witness precision bound. Historical #5315,
+#5339 and #5352 are credited predecessors, not open work. Search coverage
+and its sparse-local limitation are recorded in TASK.json.
+
+## Credit
+
+This contribution was produced by Nick Petalas. Credit Ali Romman #5339 for
+the separation/error consumer and its open optimization question, and #5315
+for the rational orthogonal realization; Krzysztof Wojt #5262 for the inverse
+compiler and shortest-word theorem; zxv #5235 and the retained
+dirty-echo/tensor/phase and rational-series lineage. Cayley transforms,
+continuants, adjugates and integer lattice separation are classical. No
+global literature-priority claim is made. Apache-2.0.

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