Euclid Collaboration, R. Ingrao, S. Borgani, M. Costanzi, A. Saro, T. Castro, L. Baumont, M. Aguena, S. Grandis, C. Murray, S. Bhargava, E. Munari, B. Altieri, S. Andreon, N. Auricchio, C. Baccigalupi, M. Baldi, S. Bardelli, P. Battaglia, A. Biviano, E. Branchini, M. Brescia, S. Camera, V. Capobianco, C. Carbone, J. Carretero, M. Castellano, G. Castignani, S. Cavuoti, A. Cimatti, C. Colodro-Conde, G. Congedo, L. Conversi, Y. Copin, F. Courbin, H. M. Courtois, H. Degaudenzi, G. De Lucia, F. Dubath, X. Dupac, S. Escoffier, M. Farina, R. Farinelli, S. Farrens, S. Ferriol, F. Finelli, P. Fosalba, M. Frailis, E. Franceschi, M. Fumana, K. George, B. Gillis, C. Giocoli, J. Gracia-Carpio, A. Grazian, F. Grupp, S. V. H. Haugan, H. Hoekstra, W. Holmes, F. Hormuth, A. Hornstrup, K. Jahnke, M. Jhabvala, B. Joachimi, S. Kermiche, A. Kiessling, B. Kubik, H. Kurki-Suonio, A. M. C. Le Brun, S. Ligori, P. B. Lilje, V. Lindholm, I. Lloro, G. Mainetti, E. Maiorano, O. Mansutti, O. Marggraf, M. Martinelli, N. Martinet, F. Marulli, R. J. Massey, S. Maurogordato, E. Medinaceli, S. Mei, M. Meneghetti, E. Merlin, G. Meylan, A. Mora, M. Moresco, L. Moscardini, C. Neissner, S. -M. Niemi, C. Padilla, S. Paltani, F. Pasian, K. Pedersen, V. Pettorino, S. Pires, G. Polenta, M. Poncet, L. A. Popa, F. Raison, A. Renzi, J. Rhodes, G. Riccio, E. Romelli, M. Roncarelli, R. Saglia, Z. Sakr, A. G. Sánchez, D. Sapone, B. Sartoris, P. Schneider, A. Secroun, E. Sefusatti, P. Simon, C. Sirignano, G. Sirri, L. Stanco, P. Tallada-Crespí, A. N. Taylor, I. Tereno, N. Tessore, S. Toft, R. Toledo-Moreo, F. Torradeflot, I. Tutusaus, J. Valiviita, T. Vassallo, Y. Wang, J. Weller, G. Zamorani, F. M. Zerbi, E. Zucca, M. Ballardini, A. Boucaud, E. Bozzo, C. Burigana, R. Cabanac, M. Calabrese, A. Cappi, J. A. Escartin Vigo, J. García-Bellido, T. Gasparetto, S. Hemmati, L. Ingoglia, J. Macias-Perez, M. Magliocchetti, R. Maoli, J. Martín-Fleitas, N. Mauri, P. Monaco, A. Montoro, C. Moretti, A. Pezzotta, M. Pöntinen, I. Risso, E. Sarpa, V. Scottez, M. Sereno, M. Tenti, M. Tucci, M. Viel, M. Wiesmann, J. A. Acevedo Barroso, Y. Akrami, G. Alguero, I. T. Andika, S. Anselmi, M. Archidiacono, G. Arico, F. Atrio-Barandela, M. Baes, L. Bazzanini, P. Bergamini, D. Bertacca, M. Bethermin, F. Beutler, L. Blot, M. Bonici, M. L. Brown, S. Bruton, A. Calabro, B. Camacho Quevedo, F. Caro, C. S. Carvalho, F. Cogato, A. R. Cooray, P. Corcho-Caballero, B. Csizi, O. Cucciati, H. Dannerbauer, T. de Boer, F. De Paolis, G. Desprez, A. Díaz-Sánchez, S. Di Domizio, J. M. Diego, V. Duret, M. Y. Elkhashab, Y. Fang, A. Farina, A. Finoguenov, A. Franco, K. Ganga, R. Gavazzi, E. Gaztanaga, Z. Ghaffari, F. Giacomini, F. Gianotti, E. J. Gonzalez, G. Gozaliasl, A. Gruppuso, M. Guidi, C. M. Gutierrez, A. Hall, N. A. Hatch, C. Hernández-Monteagudo, H. Hildebrandt, J. J. E. Kajava, Y. Kang, V. Kansal, D. Karagiannis, K. Kiiveri, J. Kim, C. C. Kirkpatrick, A. Kovács, I. Kova{č}ić, K. Koyama, S. Kruk, M. C. Lam, F. Lepori, G. Leroy, G. F. Lesci, J. Lesgourgues, T. I. Liaudat, L. Linke, S. J. Liu, F. Mannucci, C. J. A. P. Martins, M. Migliaccio, M. Miluzio, G. Morgante, S. Nadathur, K. Naidoo, A. Navarro-Alsina, S. Nesseris, F. Oppizzi, F. Pace, D. Paoletti, F. Passalacqua, K. Paterson, C. Pattison, R. Paviot, D. Potter, G. W. Pratt, S. Quai, M. Radovich, W. Roster, S. Sacquegna, M. Sahlén, D. B. Sanders, A. Schneider, D. Sciotti, E. Sellentin, S. Serjeant, I. Szapudi, K. Tanidis, F. Tarsitano, G. Testera, R. Teyssier, S. Tosi, A. Troja, C. Uhlemann, C. Valieri, A. Venhola, D. Vergani, G. Verza, S. Vinciguerra, M. von Wietersheim-Kramsta, N. A. Walton, L. Wang, A. H. Wright

We present \texttt{CosmoPostProcess}, a simulation-based forward-modelling algorithm calibrated to reproduce Euclid optical cluster observables. Its main deliverable is a correction for stacked surface-density profiles, binned in richness and redshift, accounting for selection systematics in richness-selected samples relative to unbiased references. We focus on the Euclid richness definition foreseen for cosmological analyses, which does not apply a colour selection; red-sequence richness is not considered. The algorithm processes $N$-body simulations by painting galaxies with a halo-occupation model and emulating survey detection and richness assignment. We also implement a novel estimate of optical cluster centres from projected galaxy densities, validated against Euclid pipelines. Baryonic effects are included through a correction calibrated on hydrodynamical simulations; the baryon-corrected excess surface density agrees within \(2\,\%\) over \(r\in[0.1,\,5]\,h^{-1}\,\mathrm{Mpc}\). Selection-bias contributions are assessed by varying cosmology and the mass--richness relation. Projection-induced selection bias follows a robust pattern: correlated large-scale structure projected along the line of sight enhances the stacked profile near the one-halo to two-halo transition, peaking at about \(1\,h^{-1}\,\mathrm{Mpc}\) with an amplitude of \(20\!-\!40\,\%\), depending on richness and redshift. The effect is mild at low and intermediate redshift ($z\lesssim0.7$), at the few-percent level, but becomes more relevant at higher redshift ($z\gtrsim0.7$). Baryonic modifications remain sub-dominant outside the core, at about \(2\,\%\) beyond \(r\gtrsim0.3\,h^{-1}\,\mathrm{Mpc}\). The framework delivers radial profile corrections with uncertainties, combining projection-induced selection bias, baryonic physics, and miscentring, to control systematics in Euclid DR1 cluster cosmology. (abridged)