Research · Papers · Adders, counters and the heap law · MF-119
Formula for canonical constant-heap complexity H_n(K)
H_n(K) = 4n - 6 - λ_n(K) for the canonical constant-heap class
Published 2026-09-04
For everyone
Plain summary
In digital logic design, counting expensive multiplication gates helps engineers evaluate circuit efficiency. This entry proves an exact formula, H_n(K) = 4n - 6 - λ_n(K), for the multiplication count within a specialized family of binary circuits called the canonical constant-heap class. Across a benchmark set of 32 instances (the 32-J distribution), this formula accounts for 3,801 total multiplication products.
This exact count holds only within the canonical constant-heap architecture. It is not an unrestricted lower bound or general multiplicative complexity result for arbitrary logic circuits, and it does not set a record gate reduction compared to alternative interface designs.
Result
For the canonical constant-heap class under the GF(2) XOR-AND graph (XAG) cost model, the multiplicative complexity is:
H_n(K) = 4n - 6 - λ_n(K)
For the 32-J distribution, the evaluation polynomial across instances is:
15u^120 + 5u^119 + 7u^118 + 2u^117 + 2u^116 + u^114
This distribution totals 3,801 products across the 32 instances.
Setting and definitions
The setting is multiplicative complexity over GF(2) XAG networks: linear operations (XOR) carry zero cost, while nonlinear AND gates carry unit cost.
- H_n(K): multiplicative complexity for parameter n and configuration K within the canonical constant-heap class.
- λ_n(K): structural parameter specific to configuration K of size n.
- 32-J distribution: polynomial in indeterminate u recording the multiplicity of product counts across 32 structural instances.
Method
The formula and distribution were derived by formal construction and structural verification (evidence tier P + FR). Verification artifacts were produced and validated across AX162 execution runs:
- Class theory and width-32 analysis:
heap-and-width32.receipt.json. - Structural heap verification:
heap-verification-receipt.json. - Independent module replay:
independent-module-replay.json.
Derivations and analyses appear in reports CONST-THEORY.md, CONST-BUILD.md, and STATE-OF-PROGRAM-V2.md.
Discussion
The formula H_n(K) = 4n - 6 - λ_n(K) applies strictly within the canonical constant-heap class. It is not an unrestricted multiplicative complexity equality across general Boolean circuit spaces, nor does it establish an empirical record saving over alternative architectures.
Evaluating the 32-J distribution gives 3,801 products, leaving a conditional 857-row gap against the 92-product interface scenario. The result characterizes exact cost behavior within the canonical constant-heap family without claiming optimality outside this envelope.
For everyone — the takeaway
What this means
This result gives an exact formula for counting multiplication gates in canonical constant-heap circuits. While it doesn't solve general circuit complexity or beat every alternative design, it gives engineers a precise tool to calculate implementation costs across this specific family of circuits.
Register references
- Register entry: MF-119
zkgolf-decomp/reports/CONST-THEORY.mdzkgolf-decomp/reports/CONST-BUILD.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.mdzkgolf-decomp/const-theory-scratch/heap-and-width32.receipt.json(SHA-25697495ea7cb13222225e45598ceef9956d61e9684d0c2001e96613a41f83a4da0)zkgolf-decomp/const-build-scratch/heap-verification-receipt.json(SHA-2569488889c6b66095cb02dc5ed84f3c3482ba28d9f114842f34edfe5e6cade5fa9)zkgolf-decomp/const-build-scratch/independent-module-replay.json(SHA-256831b91df8cc5d025471d1dc841120e54c2b633f442b43197f356273ee6287d62)
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 3 of 6 receipt files bundled (36 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.