The Frozen Physical Chain Does Not Kill the Soft-Axis Quartic Tail
Record
Status: source-boundary audit following Entry 441.
Hard-to-vary claim
The physical relative integration problem might supply a canonical boundary condition whose associated-graded cone kills the persistent soft-axis cokernel [ mathbf F[a,b]/(a^4). ] Such a condition is admissible only if it is fixed by the frozen primary contour data; it may not be chosen to make the compactified determinant finite.
Frozen source data
The physical Bunch--Davies chain is the positive loop-edge domain, restricted by Cayley--Menger/Gram nonnegativity. On that literal chamber, [ q_{mathcal G_{12}}=E+c,\qquad q_{mathcal G_{23}}=E+a,\qquad q_{mathcal G_{31}}=E+b, ] with (E>0) and (a,b,cgeq0). Hence every marked cut denominator is strictly positive and the closure has no cut-divisor incidence at generic positive total energy.
The residue sectors arise only after analytic continuation. The frozen primary sources do not specify:
- a labelled and sheeted log divisor for every four-pole residue summand;
- a lift or transport of the physical Borel--Moore chain to that resolved log pair;
- an occurrence-by-occurrence endpoint trivialization.
The independent occurrence-resolved audit already showed that individual endpoint subtractions and regulator chambers are not source-canonical. Canonicity appears only after source sewing of the complete unsplit combination.
Result
No source-defined relative boundary map is present whose associated-graded cone can be shown to annihilate [ mathbf F[a,b]/(a^4). ]
This is not an argument from silence alone. On the literal physical sheet the relevant cut-boundary pairing is zero because the chain does not meet the cut union. After analytic continuation, the data needed to define a residue-chain boundary map are underdetermined, and the known occurrence-resolved ambiguity forbids selecting one termwise.
Therefore, under the frozen source object, [ \boxed{ ext{the quartic soft-axis tail is not canonically killed by the physical relative chain.} } ]
The finite compactified Fitting/determinant-line conjecture must consequently be abandoned at this stage. The surviving coefficient object contains an infinite graded soft-boundary module with tested associated graded [ mathbf F[a,b]/(a^4). ]
Classification
- existing energy/Cut carrier: unchanged;
- Cayley--Menger degeneration: existing coefficient-boundary geometry;
- persistent quartic tail: coefficient support;
- literal physical-chain Cut pairing: zero;
- analytically continued residue-chain map: underdetermined by frozen source;
- fitted finite determinant line: rejected;
- genuinely new carrier datum: none.
Epistemic boundary
This result says that the available frozen source does not canonically remove the tail. It does not prove that a future source-derived sheeted relative-cycle construction cannot do so. Such additional data would first have to be acquired and frozen; it cannot be reverse-engineered from the desired cancellation.
The equality of the eventual symbol image with ((a^4)) remains tested shellwise through degree 40 over one large finite field, as stated in Entry 441.
Surviving conjecture
The soft-axis compactification carries a genuine infinite graded coefficient-boundary module generated by the quartic Cayley--Menger degeneration. Physical source sewing may act on this module, but no canonical finite truncation or annihilating cone is presently defined.
Next falsifier
Construct the full filtered soft-axis module rather than its leading symbol: compute whether lower-order terms mix the four persistent graded generators into a finitely generated filtered extension, and determine its Gauss--Manin/nearby-cycle action. A finite presentation must emerge from the frozen differential itself; no boundary quotient may be added.
Evidence
- Entry 441 and
research/benincasa/soft-axis-exact-symbol-certificate.json; research/benincasa/q_physical_provenance_audit.json;research/benincasa/occurrence-resolved-physical-period-no-go.md;research/benincasa/marici-gm/src/bin/three_cut_relative_chain_pairing.rs.