# Question 11638: A Symbolic Heat Residual Identity, Not a Point Sample

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0022-11638 · q-20261013-advance-20261024-r06-partial-differential-equation-0088



## Original question and givens

Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].


- G001 — Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].





## Definition and scope

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.


- [mko-partial-differential-equation](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/)





## Step-by-step derivation

S001 · impliesBefore

original coefficients=[0, 0, 2, 1, 1, 0];basis=[1,x,y,x^2,xy,y^2]

After

u(x,y)=(0)+(0)*x+(2)*y+(1)*x^2+(1)*x*y+(0)*y^2

Conditions: none

Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic.

Step identifiers and dependenciesRule: bivariate_scalar_polynomial_basis_roles · Uses: G001





S002 · calculationBefore

u_xx constant := mul(2, 1)

After

u_xx constant = 2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

u_yy constant := mul(2, 0)

After

u_yy constant = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

u coefficients=[0, 0, 2, 1, 1, 0];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

After

u_xx=2;u_xy=1;u_yy=0

Conditions: none

Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem.

Step identifiers and dependenciesRule: polynomial_second_partials_explicit_monomials · Uses: G001, S003





S005 · calculationBefore

heat diffusivity times u_xx := mul(1, 2)

After

heat diffusivity times u_xx = 2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

heat constant residual := sub(2, 2)

After

heat constant residual = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

u_y=(2)+(1)*x+(0)*y;diffusivity*u_xx=2

After

residual coefficients in [1,x,y]=[0, 1, 0];identically zero=No

Conditions: none

Here y is time and x is space.Collect the whole residual u_y−d*u_xx in[1,x,y].Linear independence of monomials makes all-zero coefficients equivalent to the polynomial identity.A nonzero coefficient cannot be repaired by cancellation at a sampled point.

Step identifiers and dependenciesRule: heat_residual_coefficient_identity_not_samples · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute heat constant residual with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute heat diffusivity times u_xx with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute u_yy constant with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute u_xx constant 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.





## Result

([0, 0, 2, 1, 1, 0], 1, [0, 1, 0], No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 80b46b7a9444d0c04e0cc0a99aa9a9f7554253d555a0923138225de689a78437
- candidate record SHA-256: 39430d2cb8b79b86d1e9e6b889c3fe3400e72fd3a3a6541bd8dd67f3e5806369
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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