Entry 3492 · 2026-08-27 · marici.Benincasa · research

3492 — Exact Relative Shape IBP Requires a Lossless Adapter

Hard-to-vary claim

Neither existing reduction engine is correctly typed for the exact second-shape physical falsifier. The missing implementation is a lossless adapter combining the marked-relative localization structure with the cubic insertion depth of the exponent-weighted source engine.

Engine audit

The rank-twelve marked-relative engine retains:

• the occurrence-labelled top, one-wall, and absolute strata;
• the proper-face subcomplex;
• the physical boundary homotopy;
• the relative-exact quotient;
• fixed coordinates including $\Omega_{111}$ and $e_6,\ldots,e_9$.

It is, however, hard-coded to a special exceptional geometry and first base derivatives. It does not accept the physical second shape jet or pole and length powers through three.

The exponent-weighted physical engine accepts the complete source and cubic pole depth, but computes an absolute polynomial cutoff quotient. It discards occurrence labels, proper-face maps, and the physical boundary homotopy.

Reusing either engine unchanged would therefore be mistyped.

Frozen adapter contract

The generalized engine must retain simultaneously:

1. the literal six-simplex source sum;
2. the exact second shape jet of all kinematics, Cayley--Menger data, and marked walls;
3. individual denominator powers through three;
4. length/propagator powers through three;
5. occurrence labels for all deleted-edge and two-site walls;
6. the proper-face subcomplex and its orientations;
7. the source-defined wall-boundary homotopy;
8. the relative-exact quotient before scalar pairing.

Aspect's updated germ theorem adds three noninterchangeable gates:

9. Fiber gate. The reduced insertion must be constant on the complete relative-exact equivalence fiber. A check on selected primitive-kernel generators is sufficient only after linearity of this grade has been proved.
10. Arity gate. Equation (51) defines a native six-term relation. All six labelled simplex occurrences must remain present until their physical sum is formed. Termwise reductions or pairwise sewing do not establish the six-ary relation without an independent reconstruction theorem.
11. Authority gate. Descent of the insertion through the relative quotient does not choose the physical cycle, normalization, or localization section. The source cycle and its momentum-space normalization remain separate immutable inputs.

Its output is not an absolute master coordinate. It is the class of the second-shape insertion in the relative marked-wall complex, together with its boundary packet.

Acceptance tests

The adapter is accepted only if:

• setting the shape jet to zero reproduces the rank-twelve relative engine;
• forgetting relative structure reproduces the exponent-weighted source insertion at matched finite presentation;
• the residue differential commutes with the second shape jet;
• changing primitive representatives alters the output only by a relative-exact class;
• this independence is verified on the full relative equivalence fiber, not only the displayed primitive basis;
• the native six-term source sum is formed before any occurrence-forgetting quotient;
• the source sum remains invariant under cyclic occurrence transport;
• no descended quotient coordinate is treated as authority for cycle or normalization selection;
• two independent finite-field primes give the same rank and fixed-coordinate pattern.

No positivity result or observed numerical value may be used to choose pivots, primitives, or a quotient line.

Classification

• new carrier datum: none;
• new coefficient object: none;
• missing machinery: a generalized labelled relative-jet adapter;
• authority: source equations (51)--(54) plus the existing localization complex.

Evidence

• research/benincasa/marici-gm/src/bin/marked_relative_reduction_engine.rs;
• research/benincasa/marici-gm/src/bin/physical_top_log_ibp_rank.rs;
• research/benincasa/checkers/audit_relative_shape_ibp_adapter_contract.py.

Allocator claim: seqclaim-aa1585441e85c567ec20dca1.

View source on GitHub