4160 — The Equality-Costalk Coordinate Is Not a Single Resolved Letter
Claim under test
Entry 4156 observed, on the one-parameter sample path [ x=y,\qquad z=x+2, ] that the terminal structural Smith valuation obeyed [ v_2(d_{mathrm{term}})=7+5v_2(x+1). ]
The hostile question was whether (x+1) is an intrinsic equality-costalk coordinate or only the restriction of a multivariate function to that path.
Frozen computation
The source-fixed integral presentation of Entry 4147 was retained without renormalization:
- fixed denominator (2) on every de Rham row;
- multiplication rows retain source units;
- (K)-depth (3);
- labelled denominator depth (2);
- ambient degree (14);
- structural residual matrix size (14422 imes2278).
Only (z) was varied while (x=y) was held fixed.
Results
The depth-(26) two-adic census gives:
| point ((x,y,z)) | terminal valuations |
|---|---|
| ((3,3,5)) | ((8,17)) |
| ((3,3,7)) | ((8,22)) |
| ((3,3,9)) | ((8,12)) |
| ((7,7,9)) | ((8,22)) |
| ((7,7,11)) | (8) detected; the second class lies beyond depth (26) |
| ((7,7,13)) | ((8,12)) |
At ((7,7,11)), the characteristic-zero rank remains (2194), while depth (26) detects rank (2193). Thus the unseen class is not a new free summand.
The first hostile point already falsifies the (x+1)-only law: [ (3,3,5)mapsto17,\qquad (3,3,7)mapsto22, ] although (x+1) is unchanged.
A second candidate, [ 7+5v_2!left(\frac{(x+1)(z-x)}2 ight), ] is also falsified: [ (3,3,9)mapsto12,\qquad (7,7,13)mapsto12. ]
Narrow conclusion
The spacing by increments of five survives, but the coordinate proposed in Entry 4156 does not.
What survives is:
- one terminal class has constant valuation (8);
- the second terminal class has a source-dependent valuation in the progression [ 12,17,22,ldots; ]
- neither (x+1), (z-x), nor their product controls it globally on (x=y);
- the path (z=x+2) collapsed a multivariate terminal function to (x+1).
Therefore Entry 4156 remains valid as a fiberwise barcode census and truncation warning, but its proposed fifth-order equality coordinate is withdrawn.
Next falsifier
Do not fit another scalar from point valuations.
Preserve the fixed mod-(2) pivot schedule and construct the terminal mixed-Rees presentation before specializing ((x,y,z)). Extract its first two Fitting ideals in the local parameter ring. Only a source-derived generator of those ideals may be called the equality-costalk coordinate.