<PAGE>   1
                                                                   EXHIBIT 10.77






July 10, 2001



AppliedTheory Corporation
224 Harrison Street, 8th Floor
Syracuse, New York  13202


         I. Reference is made to the documents being entered into today by
AppliedTheory Corporation (the "Company"); its subsidiaries (the "Subsidiaries")
and the undersigned Lenders ("Lenders"), in connection with the "Current
Indebtedness" and "Receivables Financing" as such terms are defined in the
Security Agreement dated today among the Company, the Subsidiaries, the Lenders
and the Collateral Agent thereunder. Other terms used in this Part I and not
defined have the same meanings as in the Security Agreement.

         (A) Notwithstanding anything to the contrary in the documents
comprising the Current Indebtedness and Receivables Financing, the following two
actions (referred to below as the "Specified Actions") will not by themselves
constitute an Event of Default under any such documents; provided that (a) at
such time or times there is otherwise no Default or Event of Default existing
under the Security Agreement, (b) with respect to any claims, damages or
penalties of James G. Couch, a former shareholder of CRL Network Services, Inc.
("Couch") against the Company and the Subsidiaries under the Put Option
Agreement dated as of January 5, 2000 among the Company, Couch and other former
shareholders of CRL Network Services, Inc., such actions result in a full and
complete settlement of any and all claims as evidenced by a release, in form and
substance satisfactory to the Lenders, (c) with respect to any claims (including
governmental claims, fines and penalties) against the Company in connection with
services rendered to NYSERNet.net, such actions will result in a full and
complete settlement of all civil claims, and at the time of such actions there
shall not be pending or overtly threatened any criminal claims against the
Company or its subsidiaries arising from services rendered to NYSERNet.net
(collectively, the "NYSERNET Claims") as evidenced by a release and/or other
documentation, in form and substance satisfactory to the Lenders, (d) such
actions will not violate any applicable federal or state laws, (e) such actions
will not result in a diminution of Lenders' rights under the Current
Indebtedness or Receivables Financing pursuant to any laws relating to the
insolvency of debtors, fraudulent conveyance or transfer, or laws affecting the
rights of Creditors generally and (f) the Company shall deliver to the Lenders a
favorable
<PAGE>   2
opinion of outside counsel, reasonably satisfactory to the Lenders with respect
to the matters covered by (c), (d) and (e) above:

         The Specified Actions

         1.       The payment by the Company of not more than $1,000,000 in cash
                  collectively to both Couch and/or any claimants (including
                  governmental entities) in connection with the NYSERNET Claims;
                  and

         2.       The grant by the Company of a lien in certain Collateral under
                  the Security Agreement to Couch to the extent permitted under
                  Section 2.2(e)(ii) of the Security Agreement (as supplemented
                  by this letter) to secure payment of the balance of any amount
                  agreed to be paid to Couch in settlement of his claims and/or
                  as the price for the redemption of Couch's shares of the
                  capital stock of the Company; provided that any payments to
                  Couch with respect to such balance shall not be due or
                  actually paid until at least three months after the later of
                  (i) the final maturity of the Debentures and (ii) the
                  Revolving Credit Termination Date; and provided further that
                  Couch enters into an intercreditor agreement with the Lenders
                  in form and substance reasonably satisfactory to the Lenders.

         For the avoidance of doubt, it is hereby agreed that (i) the payment by
the Obligors of more than $1,000,000 in the aggregate (A) to Couch, and/or (B)
to any other person in respect of the NYSERNET Claims or (ii) the grant by the
Company of any security interest in any assets of the Company or any of its
subsidiaries in connection with the NYSERNET Claims, will constitute an Event of
Default under the Revolving Credit Agreement, the Security Agreement and the
Debentures. The parties also acknowledge and agree that (i) the reference to
stayed attachments or judgement Liens in Section 7.1(9) of the Revolving Credit
Agreement and (ii) the reference in Section 14(i) of the Amendment Agreement to
pending litigation disclosed in the Pre-Closing SEC Documents, shall be deemed
to refer only to the litigation, judgements and attachments arising out of the
claims made by Couch and the NYSERNET Claims.

         (B) Except as otherwise permitted under Section 2.2(e)(i) of the
Security Agreement or except for a Relevant Sale (as defined in the Amendment
Agreement), the Company and each of the Subsidiaries covenant that as long as
any Amended Debenture, Participation Certificate or Revolving Credit Note is
outstanding, neither the Company nor any Subsidiary shall enter into any
agreement outside its ordinary course of business (such ordinary course of
business being consistent with past business practices) without the express
prior written consent of the Lenders. The exclusion of a Relevant Sale from the
requirement of prior written consent is not intended to eliminate the existence
of an Event of Default that would otherwise occur under the Revolving Credit
Agreement as a result of the Relevant Sale.

         II. Reference is further made to the Amendment Agreement and the
Purchase Agreement referred to in the Security Agreement: for the purposes of
this Part II, capitalized terms which are not defined shall have the meanings
set forth in the Amendment Agreement or the Purchase Agreement.


                                       2
<PAGE>   3
         (A)(1) If and only if the SEC takes the position that, solely as a
result of the Company's execution and delivery of the July 2001 Transaction
Documents, the Company is required to file a post-effective amendment pursuant
to Item 512(a)(1)(iii) of Regulation S-K (the "SEC Position"), then the Company
shall deliver to the Investors and file with the SEC a post-effective amendment
to the Registration Statement in accordance with the rules and regulations under
the Security Act. Any such post-effective amendment (a) shall be filed with the
SEC within 15 days of the Company's first receipt of notice of the SEC's
position ("SEC Notice Date") and (b) shall be submitted to the Investors at
least two (2) Trading Days prior to its filing for review and (c) shall be
subject to the Investors' prior approval, which shall not be unreasonably
withheld or delayed. The Company shall use its best efforts (i) to cause any
such post-effective amendment, when filed, to become effective under the
Securities Act within 60 days of the SEC Notice Date and (ii) keep such
post-effective amendment effective during the period described in Section 5 of
the Registration Rights Agreement. All of the registration rights and remedies
under the Registration Rights Agreement shall apply to the post-effective
amendment and the Registrable Securities registered thereunder (including
without limitation the provisions providing for Default Payments and Premium
Redemption Price) except that references in the Registration Rights Agreement to
April 11, 2001 shall be deemed to refer to the 60th day following the SEC Notice
Date and references to June 11, 2001 shall be deemed to refer to the 75th day
date following the SEC Notice Date. The Company shall immediately notify in
writing (i) the Investors of the date on which the post-effective amendment has
been filed and the date on which the post-effective amendment has become
effective (notification of such filing and effectiveness shall be delivered to
the Investors by facsimile on the same day of such filing or effectiveness and
by overnight mail) and (ii) the Investors' Counsel of any request by the SEC for
additional information. Reference is also made to Section 2(b)(iii) of the
Registration Rights Agreement. Provided that a Blackout (as defined in the
Registration Rights Agreement) is triggered solely as a result of the SEC taking
the SEC Position, the Company shall be entitled to a Suspension Grace Period
that will commence on the SEC Notice Date (if any) and expire on the earlier of
(x) the sixtieth (60th) day thereafter and (y) the effective date of the
post-effective amendment. This Suspension Grace Period, if utilized, will
replace the Suspension Grace Period set forth in Section 2(b)(iii) of the
Registration Rights Agreement (which will no longer apply).

         (2) In the event that Section 3(b) of the Amendment Agreement is not
true solely as a result of the SEC Position, then no Event of Default shall have
occurred under such Section 3(b). Notwithstanding the preceding sentence, in the
event that Section 3(b) of the Amendment Agreement is not true the Company must
indemnify each of the Investors, each of its officers, directors and partners,
and each person controlling each Investor, within the meaning of Section 15 of
the Securities Act for any loss, expense or damage arising as a result of
Section 3(b) not being true, and shall reimburse each Investor, each of its
officers, directors and partners, and each person controlling such Investor for
any legal and any other expenses reasonably incurred in connection with
investigating and defending any such claim, loss, damage, expense, liability or
action.

         (B) In addition to the other criteria which define Marketable
Securities for purposes of determining whether a Sale of the Company is a
Relevant Sale under the terms of Section 2(c) of the Amendment Agreement, a
Relevant Sale must also satisfy the following (in addition to the criteria set
forth in said Section 2(c)): (i) the amount of the value of the Marketable
Securities received by the Investors that comprises the Relevant Sale
Consideration shall not exceed 30%


                                       3
<PAGE>   4
of the Total Dollar Volume (as defined below) of the Marketable Securities for
the 40 Trading Days prior to the date of the announcement of the Relevant Sale;
and (ii) the amount of the value of the Marketable Securities received by the
Investors that comprises the Relevant Sale Consideration shall not exceed 10% of
the Total Dollar Volume of the Marketable Securities during the six (6) month
period prior to the date of the announcement of the Relevant Sale. For purposes
of this Section II(B), "Total Dollar Volume" shall mean the sum of the product,
for each Trading Day during the period (40 Trading Days or 6 months, as
applicable) of the VWAP for such Trading Day multiplied by the total number of
shares of common stock traded on the principal market for such shares during
that Trading Day. For purposes of this Section II(B), trades of greater than
20,000 shares of common stock shall be excluded from the computation of Total
Dollar Volume. Marketable Securities shall be valued as set forth in Section
2(c) of the Amendment Agreement.


                                       4
<PAGE>   5
                                 LENDERS:

                                 HALIFAX FUND, L.P.
                                 By:      THE PALLADIN GROUP, L.P.
                                          Attorney-in-Fact

                                 By:       /s/ Maurice Hryshko
                                          --------------------------------------
                                          Name:  Maurice Hryshko
                                          Title:  Counsel

                                 PALLADIN PARTNERS I, L.P.
                                 By:      THE PALLADIN GROUP, L.P.
                                           Attorney-in-Fact

                                 By:       /s/ Maurice Hryshko
                                          --------------------------------------
                                          Name:  Maurice Hryshko
                                          Title:  Counsel

                                 PALLADIN OVERSEAS FUND LTD.
                                 By:        THE PALLADIN GROUP, L.P.
                                            Attorney-in-Fact

                                 By:       /s/ Maurice Hryshko
                                          --------------------------------------
                                          Name:  Maurice Hryshko
                                          Title:  Counsel

                                 DEAM CONVERTIBLE ARBITRAGE FUND, LTD.


                                 By:       /s/ Maurice Hryshko
                                          --------------------------------------
                                          Name:  Maurice Hryshko
                                          Title:  Counsel

                                 LANCER SECURITIES (CAYMAN) LTD.
                                 By:        THE PALLADIN GROUP, L.P.
                                            Attorney-in-Fact

                                 By:       /s/ Maurice Hryshko
                                          --------------------------------------
                                          Name:  Maurice Hryshko
                                          Title:  Counsel
<PAGE>   6
                                 ELLIOTT ASSOCIATES, L.P.


                                 By:       /s/ Paul E. Singer
                                          --------------------------------------
                                          Name:  Paul E. Singer
                                          Title:  General Partner

                                 ELLIOTT INTERNATIONAL, L.P.
                                 By:        ELLIOTT INTERNATIONAL CAPITAL
                                            ADVISORS INC.
                                            Attorney-in-Fact

                                 By:        s/ Paul E. Singer
                                          --------------------------------------
                                          Name: Paul E. Singer
                                          Title:  President
Agreed and Accepted

APPLIEDTHEORY CORPORATION


By:      /s/ Danny E. Stroud
   ----------------------------------------
         Danny E. Stroud, President


APPLIEDTHEORY GEORGIA CORPORATION


By:      /s/ Danny E. Stroud
   ----------------------------------------
         Danny E. Stroud, President

APPLIEDTHEORY SEATTLE CORPORATION


By:      /s/ Danny E. Stroud
   ----------------------------------------
         Danny E. Stroud, President

APPLIEDTHEORY CALIFORNIA CORPORATION


By:      /s/ Danny E. Stroud
   ----------------------------------------
         Danny E. Stroud, President
<PAGE>   7
APPLIEDTHEORY VIRGINIA CORPORATION


By:      /s/ Danny E. Stroud
   -----------------------------------------
         Danny E. Stroud, President

APPLIEDTHEORY AUSTIN CORPORATION


By:      /s/ Danny E. Stroud
   -----------------------------------------
         Danny E. Stroud, President

APPLIEDTHEORY COLORADO CORPORATION


By:      /s/ Danny E. Stroud
   -----------------------------------------
         Danny E. Stroud, President

                         [SIGNATURE PAGE TO SIDE LETTER]