Exact carry and Majority program families
The R5 Σ0/carry and Majority low-bit wall can be replaced by finite exact program families, and any candidate simplification that survives into R6 can be detected without approximation.
Diagnostic work, exact algebra and local capabilities are not treated as end-to-end mining advantage.
What was tested?
The R5 Σ0/carry and Majority low-bit wall can be replaced by finite exact program families, and any candidate simplification that survives into R6 can be detected without approximation.
Why the test is meaningful
For fixed context, each low-bit Boolean output is a finite truth table. Canonicalization groups identical tables into exact families; this is lossless compression of semantics, not statistical fitting. Majority is evaluated as Maj(x,y,z)=(x∧y)⊕(x∧z)⊕(y∧z), while modular additions are represented by boundary carry functions.
Maj(x,y,z) = (x ∧ y) ⊕ (x ∧ z) ⊕ (y ∧ z)family(f) = canonical truth table of f on the registered source cubeH₀: no exact R6 family candidate survives the registered context searchHow it was tested
Enumerate the registered source cubes, compile exact carry families at each boundary and exact Majority programs for bits 0–3, join them into an R5 program library, then search the connected R6 contexts. Replay every compiled family in an independent test engine and separately validate the exploratory E5 side channel.
What happened
The exact family library was reproduced with zero semantic disagreement. The atlas contained 14, 76, 382 and 520 unique Majority programs at bits 0–3, respectively. The connected R6 search returned zero exact family candidates, and the fresh E5 side validation promoted none.
Exactness and statistical controls
The result separates deterministic and empirical claims. Family equivalence was checked exactly over each complete registered cube; the fresh test engine passed. Candidate counts were then reported as zero rather than treating absence of a selector as a positive result. No economic verdict was issued.
What the result means
V21 successfully resolves an opaque operation into auditable exact objects, but the hoped-for connected R6 simplification was not found. The value is structural: it identifies the next black box and prevents a local representation gain from being mislabeled as mining advantage.
Limitations
- Exactness is conditional on the registered source cubes and inherited context.
- The number of canonical families is not a runtime-cost measurement.
- No connected reject/survivor flow or ASIC benchmark was established.
Evidence trail
Re-enumerate every source cube, canonicalize truth tables, replay the compiled carry/Majority programs, and repeat the R6 candidate search on fresh contexts while preserving a zero-error semantic oracle.
Canonical variants
SUBENGINE-V21ASUBENGINE-V21BSUBENGINE-V21CSUBENGINE-V21DSUBENGINE-V21ESUBENGINE-V21FSUBENGINE-V21GSUBENGINE-V21HSource: internally audited canonical reports. Local filesystem structure, private headers and operational identifiers are excluded from publication.