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R0047-24411 · q-20261015-advance-20261113-r06-gradient-vector-0086
Use mode=partials_exist_not_differentiable, positive scale=11, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
G001 — Use mode=partials_exist_not_differentiable, positive scale=11, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [gradient-boundary-counterexample]
After
derived = [[mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary]]
Conditions: none
List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Gradient diagonal exact scale half := div(11, 2)
After
Gradient diagonal exact scale half = 11/2
Conditions: none
Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [partials_exist_not_differentiable, 11]
After
derived = [partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm]
Conditions: none
Axis quotients give[s,0],so the candidate linear map is sh.On(h,h),f=sh/2 and the remainder divided by sqrt(2)|h| is−s*sign(h)/(2sqrt(2)),not0.Partial existence does not prove total differentiability.
Rule: Gradient_mode_specific_counterexample_with_actual_hypotheses · Uses: G001, S002
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute Gradient diagonal exact scale half with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ed0a73461716b7697c07d3fd29e1cf500985cf026dcb094d036ea3b6c04bd765candidate record SHA-256: 6ec4d42e06b7f9224fe1d9e71fb68816e361e84ce3a70b914c2c52e12e295a83semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED