IndisputableMonolith.Gravity.Analysis.ReggeTTCertificateScratch
Scratch module holding pre-cancellation algebraic certificate blocks for Regge TT gravity analysis. It packages the identity that four times the mass-like quantity M equals the literal per-tetrahedron sum, with edge weights s2 and s3 kept opaque. Offline lift cofactors vanish, so the difference is closed by ring (equivalently linear_combination with zero coefficients). Gravity analysts cite it when wiring certified per-tet sums into larger cancellation arguments.
claimPre-cancellation certificate: $4M$ equals the literal per-tetrahedron sum in the Regge TT block, with edge factors $s_2,s_3$ opaque. Offline lift cofactors are identically zero, so the polynomial identity $4M - \sum_{\mathrm{tet}}(\cdots)=0$ holds by ring arithmetic (equivalently $\mathtt{linear\_combination}$ with zero coefficients).
background
In the Gravity analysis layer, Regge-type discrete curvature is organized into tetrahedron (per-tet) sums whose algebraic cancellation must be certified before continuum or continuum-limit claims. The quantity $M$ is the mass-like aggregate appearing in the TT (transverse-traceless / certificate) bookkeeping; the factor four is the combinatorial prefactor that matches the expanded per-tet expression.
The module keeps $s_2$ and $s_3$ opaque so the certificate does not depend on specializing those edge weights. Offline lift cofactors are the coefficients that would appear if one transported the identity through a change-of-basis or embedding map; here they are zero, which collapses the certificate to a pure polynomial identity on the difference.
Imports are only Real.Basic, Ring, and LinearCombination: the content is elementary real algebra, not a full geometric development of Regge calculus.
proof idea
Two sibling certificate blocks (certificate_block_pertet, certificate_block_rational) state the pre-cancellation identity. Because lift cofactors vanish, each proof reduces to showing the difference is the zero polynomial: either ring1 on that difference, or linear_combination with all coefficients zero (same obligation). No geometric lemmas are invoked; opacity of $s_2,s_3$ is preserved throughout.
why it matters in Recognition Science
Supplies the certified per-tet building block that larger Regge TT cancellation pipelines in the Gravity domain can import without re-proving the $4M$ expansion. In the Recognition Science gravity stack, such certificates sit under discrete-curvature and mass-bookkeeping arguments that eventually feed continuum or ladder-side gravity claims. The scratch naming marks it as a local algebra dump rather than a public API theorem; downstream parents are expected to re-export a cleaned certificate once the full cancellation chain is assembled. No T0–T8 forcing step is discharged here; the module is pure algebraic support for gravity analysis.
scope and limits
- Does not prove geometric Regge curvature identities or continuum limits.
- Does not specialize or eliminate the opaque edge weights s2 and s3.
- Does not assert nonzero offline lift cofactors or nontrivial linear_combination witnesses.
- Does not export a stable public API; scratch module may be refactored or inlined.
- Does not connect directly to RS forcing chain steps T0–T8 or the mass ladder.