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R0019-10126 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0051
On all R, continuous f has ascending coefficients [0], constant term first. For G(x)=∫_-1^x f(t)dt, give (G(-1),G′(-1)) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, continuous f has ascending coefficients [0], constant term first. For G(x)=∫_-1^x f(t)dt, give (G(-1),G′(-1)) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.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
raw integrand f=[0];ascending powers on all R
After
normalized coefficients=[0/1];declared continuous real polynomial
Conditions: none
The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof.
Rule: declared_FTC_continuity_hypothesis · Uses: G001
Before
zero primitive coefficient1 := div(0, 1)
After
zero primitive coefficient1 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
integrand power=0;coefficient=0/1
After
primitive power=1;coefficient=0/1
Conditions: none
For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive.
Rule: polynomial_primitive_coefficient_rule · Uses: G001, S002
Before
original integrand coefficients=[0/1]
After
zero-constant primitive coefficients=[0/1]
Conditions: none
This is one convenient representative,not a claim that the primitive is unique without its constant family.
Rule: zero_constant_polynomial_primitive · Uses: G001, S003
Before
constructed P0:constant coefficient=0/1
After
derivative coefficients=[0/1]
Conditions: none
A constant differentiates to the zero polynomial,represented as[0/1],not a fabricated negative power.
Rule: constant_polynomial_derivative_zero · Uses: G001, S004
Before
P0 derivative coefficients=[0/1];f=[0/1]
After
P0'=f on all R in this declared polynomial model
Conditions: none
The full coefficient identity establishes the primitive relation in the declared model.With the supplied continuity premise use the FTC endpoint difference,not a function guessed from sample points.
Rule: constructed_primitive_coefficient_identity · Uses: G001, S005
Before
P0 at anchor Horner multiply0 := mul(0, -1)
After
P0 at anchor Horner multiply0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
P0 at anchor Horner add0 := sum(0, 0)
After
P0 at anchor Horner add0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
anchor a=-1;P0(a)=0/1
After
G(t)=P0(t)-(0/1);G'(t)=f(t)
Conditions: none
The anchor contributes the t-independent constant P0(a),which vanishes under differentiation.Do not use G's zero anchor value as its derivative.
Rule: FTC_accumulation_anchor_constant · Uses: G001, S008
Before
f at derivative point Horner multiply0 := mul(0, -1)
After
f at derivative point Horner multiply0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
f at derivative point Horner add0 := sum(0, 0)
After
f at derivative point Horner add0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
anchor integral same-endpoint difference := sub(0, 0)
After
anchor integral same-endpoint difference = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
G(a)=0/1;f(a)=0/1
After
ordered value-derivative pair=(0/1, 0/1)
Conditions: none
Equal endpoints give integral value0.The derivative comes separately from f(a) and may be nonzero.Retain the requested field order.
Rule: anchor_value_not_derivative · Uses: G001, S012
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute anchor integral same-endpoint difference with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute f at derivative point Horner add0 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute f at derivative point Horner multiply0 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute P0 at anchor Horner add0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute P0 at anchor Horner multiply0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute zero primitive coefficient1 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.(0/1, 0/1)
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: d579807c5ed4161ba75673c43537670c805a574c37a7c08b9a094327a1a8db1dcandidate record SHA-256: d083d1db15cdb82263fa375303b678d258ddb5b293b95f56a6278135d9f55309semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED